STRUCTURAL OPTIMISATION
IN BUILDING DESIGN PRACTICE :
CASE -STUDIES IN TOPOLOGY OPTIMISATION
OF BRACING SYSTEMS
Robert Baldock Corpus Christi College June 2007
A dissertation submitted for the Degree of Doctor of Philosophy Cambridge University Engineering Department Declaration
Except where otherwise stated, this thesis is the result of my own research and does not include the outcome of work done in collaboration. This thesis has not been submitted in whole or in part for consideration for any other degree of qualification at the University or any other institute of learning. The thesis contains 49 figure, 14 tables and less than 42,000 words. Robert Baldock Corpus Christi College Cambridge June 2007 Abstract
Keywords: structural topology optimisation, structural design practice, bracing design, Evolutionary Structural Optimisation, Pattern Search, Optimality Criteria, Genetic Programming, computer-aided design, large-scale structural size optimisation
This thesis aims to contribute to the reduction of the significant gap between the state of the art of structural design optimisation in research and its practical application in the building industry. The research has focused on structural topology optimisation, investigating three distinct methods through the common example of bracing design for lateral stability of steel building frameworks. The research objective has been aided by collaboration with structural designers at Arup. It is shown how Evolutionary Structural Optimisation can be adapted to improve applicability to practical bracing design problems by considering symmetry constraints, rules for element removal and addition, as well as the definition of element groups to enable inclusion of aesthetic requirements. Size optimisation is added in the optimisation method to improve global optimality of solutions. A modified Pattern Search algorithm is developed, suitable for the parameterised, grid based, topological design problem of a live, freeform tower design project. The alternative objectives of minimising bracing member piece count or bracing volume are considered alongside an efficient simultaneous size and topology optimisation approach, through integration of an Optimality Criteria method. A range of alternative optimised designs, suitable for assessment according to unmodelled criteria, are generated by stochastic search, parametric studies and changes in the initial design. This study is significant in highlighting practical issues in the application of structural optimisation in the building industry. A Genetic Programming formulation is presented, using design modification operators as modular "programmes", and shown to be capable of synthesising a range of novel, optimally directed designs. The method developed consistently finds the global optimum for a small 2D planar test problem, generates high performance designs for larger scale tasks and shows the potential to generate designs meeting user defined aesthetic requirements. The research and results presented contribute to establishing a structural optimisation toolbox for design practice, demonstrating necessary method extensions and considerations and practical results that are directly applicable to building projects. Acknowledgements
I wish to thank my academic supervisor, Kristina Shea, for her dedicated support, guidance and encouragement throughout the course of this project. I am also greatly indebted to Geoff Parks for his efficiency and advice in the role of advisor and subsequently as administrative supervisor. Thanks are due to Marina Gourtovaia and Andrew Flintham for their valuable assistance in computing matters and to all my friends and colleagues in the Engineering Design Centre, Cambridge for many stimulating discussions. The collaboration with Arup has been fundamental to this research. I therefore wish to express my sincere gratitude to Ed Clark and Alvise Simondetti, as industrial supervisors, as well as Damian Eley, Chris Neighbour, Steve McKechnie, Martin Holt, Colin Jackson, Jan Peter Koppitz, Chris Carroll, Pat Dallard and Peter Young, all of whom generously gave time to aid me in various aspects of this project. Additionally, the support of Chris Kaethner and Stephen Hendry, in relation to Oasys GSA, has been very beneficial. I could not have completed this thesis without the fantastic friends who have inspired, distracted and kept me sane. I have been blessed with loving and loyal parents who have supported me from my first steps to the conclusion of this thesis. I owe them the greatest thanks of all. This research has been made possible through funding by the Engineering and Physical Sciences Research Council and an Industrial CASE studentship from Arup. Additional financial support from Cambridge University Engineering Department, Corpus Christi College, Cambridge and the Royal Commission for the Exhibition of 1851 is also gratefully acknowledged. Contents
1. INTRODUCTION ………………………………………………………….. 1
1.1. The nature of design optimisation ……………………………………. 1 1.2. Optimisation of structures ……………………………………………. 3 1.3. The design process for building structures ………………………… ... 5 1.4. Drivers and barriers for structural optimisation in the building industry 7 1.5. Summary of research contributions .………………………….……… 10 1.6. Thesis structure …………………………………………….………… 11 2. STATE OF THE ART: RESEARCH AND PRACTICE OF DESIGN OPTIMISATION IN STRUCTURAL ENGINEERING …………………... 12 2.1. Structural design optimisation research ……………………………… 12 2.1.1. Section size optimisation ………………………………………... 14 Optimality Criteria ………………………………………………… 14 Mathematical Programming ………………………………………. 14 Fully Stressed Design ……………………………………………… 15 Additional considerations ………………………………………….. 15 2.1.2. Discrete topology optimisation methods ………………………… 16 Ground structure approach ………………………………………… 16 Ruled based approaches …………………………………………… 18 2.1.3. Evolutionary Algorithms in topology optimisation ……………… 19 Genetic Algorithms …………………………………………...….... 19 Genetic Programming ……………………………………………… 20 Evolutionary Strategies …………………………………………….. 21 Evolutionary Programming ……………………………………….... 21 2.1.4. Continuum based optimisation methods …………………………. 21 Homogenisation …………………………………………………….. 22 Bubble method ……………………………………………………… 22 Evolutionary Structural Optimisation ……………………………… 22 2.1.5. Computer based conceptual design methods …………………….. 24 2.2. Optimisation in building engineering design practice ………………… 25 2.2.1. Comparison of structural design in the automotive and aeronautical industries versus the building industry …………………………… 25 2.2.2. Commercial optimisation software ……………………………….. 27 2.2.3. Published literature on industrial applications ……………………. 29 Section size optimisation …………………………………………… 29 Evolutionary Structural Optimisation (ESO) ……………………….. 30 Parametric optimisation …………………………………………….. 31 Non parametric optimisation ……………………………………….. 31
2.2.4. Facilitating structural optimisation ……………………………….. 32 Software …………………………………………………………….. 32 Parametric optimisation case studies ……………………………….. 33 Non parametric discrete optimisation and design generation case studies ………………………………………………………………………. 33
2.3. Conclusions …………………………………………………………….. 34 2.4. Justification of case study ……………………………………………… 35 2.5. Context of research contributions ……………………………………... 37 Evolutionary Structural Optimisation ………………………………. 37 Pattern Search and Optimality Criteria ……………………………... 37 Genetic Programming using design modification operators ………... 38 3. CONTINUUM TOPOLOGY OPTIMISATION OF BRACED STEEL FRAMES ……………………………………………………………………….. 40
3.1. Introduction …………………………………………………………….. 40 3.2. Background …………………………………………………………….. 40 4.7.1 Method overview ………………………………………………….. 40 3.2.1. Addition considerations and extensions …………………………... 41 3.2.2. Evolutionary Structural Optimisation (ESO) for stiffness and displacement constraints ………………………………………….. 43 3.2.3. Bi directional Evolutionary Structural Optimisation (BESO) …… 45 3.3. Benchmark problem: structural model specifications ………………… 45 3.4. Optimisation for minimal mean compliance ………………………….. 46 3.5. Optimisation for displacement constraint …… ……………………….. 47 3.6. Including optimisation of domain thickness …………………………... 53 3.7. Including architectural requirements and pattern definition …………... 58 3.8. Discrete interpretation of continuum topologies ………………………. 60 3.9. Conclusions ……………………………………………………………. 64 3.10. Guidelines for practical use …………………………………………… 64 4. BRACING TOPOLOGY AND SECTION SIZE OPTIMISATION BY A HYBRID ALGORITHM: AN INDUSTRIAL CASE STUDY ………………………... 67 4.1. Introduction …………………………………………………………… 67 4.2. Background ………………………………………… ………………… 68 4.4.1. Overview of studies ……………………………………………… 69 4.3. Design task definition …………………………………………………. 69 4.3.1. Structural models ………………………………………………… 69 4.3.2. Topology optimisation models …………………………………… 72 Optimisation model A ……………………………………………… 72 Optimisation model B ……………………………………………… 72 4.4. Pattern Search method ………………………………………………… 73 4.5. Live project optimisation ……………………………………………… 75 4.5.1. Topology optimisation by Modified Pattern Search ……………... 75 4.5.2. Parametric studies ………………………………………………… 76 4.5.3. Outline proposals …………………………………………………. 77 4.6. Characterisation of design space ………………………………………. 78 4.7. Topology optimisation method development …………………………. 80 4.7.1. Objective function formulation …………………………………... 83 Formulation 1 ………………… ……………………………………. 83 Formulation 2 ………………… ……………………………………. 83 4.7.2. Comparative investigation ………………………………………... 84 Evolving designs from fully braced initial configuration ………….. 85 Alternative objective function formulations ……………………….. 86 Scheduling of exploratory moves ………………………………….. 86 Performance of designs evolved from randomly generated initial configurations …………………………………………………… 86 Use of pattern moves ………………………………………………. 87 4.8. Topology optimisation: structural model B …………………………… 87 4.8.1. Results ……………………………………………………………. 89 4.8.2. Observations ……………………………………………………… 91 4.8.3. Diversity ………………………………………………………….. 91 4.9. Size optimisation ………………… …………………………………… 92 4.9.1 Overview ………………… ………………………………………. 92 4.9.2. Derivation of iterative approach from Optimality Criteria ……….. 92 4.9.3. Pitfalls ……………………………………………………………. 97
Complex values of Ai. ………………………………………………. 97 Convergence failure ………………… ……………………………... 97
* Negative values of Cj and Cj. ………………… ……………………. 97
4.9.4. Assignment of discrete sections .…………………………………. 98
4.9.5. Size optimisation of fully braced configuration .…………………. 99 4.9.6. Size optimisation by Optimality Criteria with bending moments ... 102
4.10. Integration of topology and size optimisation .………………………... 102 4.10.1. Results ……………………………………………………………. 106 4.10.2. Observations ……………………………………………………… 107 4.11. Summary of results from optimisation model B ……………………… 108
4.12. Conclusions .…………………………………………………………… 112 5. STRUCTURAL TOPOLOGY OPTIMISATION OF BRACED STEEL FRAMEWORKS USING GENETIC PROGRAMMING ………………….. 114 5.1. Introduction …………………………………………………………… 114 5.2. Background …………………………………………………………… 115 5.3. Genetic Programming method ………………………………………... 115 5.3.1. Introduction ……………………………………………………… 115 5.3.2. GP for bracing design ……………………………………………. 116 Creating initial designs …………………………………………….. 118 Analysis and fitness ………………………………………………... 120 Generating subsequent populations ………………………………... 120 Handling geometrically infeasible designs ………………………… 121 5.4. Bracing design for a 2x6 framework ………………………………….. 125 5.5. Bracing design for a 6x30 framework ………………………………… 130 5.6. Defining aesthetic style ……………………………………………….. 138 5.7. Further work …………… …………………………………………….. 139 5.8. Conclusions …………………………………………………………… 139 6. CONCLUDING REMARKS ……………………………………………….. 141 6.1. Review of contributions ………………………………………………. 141 6.2. Recommendations for future work ……………………………………. 145 Evolutionary Structural Optimisation …………………………….... 145 Pattern Search Optimality Criteria ……………………………….. 145 Genetic Programming ……………………………………………… 146 6.3. Application of structural optimisation in practice …………………….. 147 6.4. Projected trends in structural design automation and optimisation in practice ……………………………………………………………………… 148 6.5. Closing notes ………………………………………………………….. 149 APPENDIX 1. STRUCTURAL ANALYSIS …………………………………. 151 APPENDIX 2. SOFTWARE DEVELOPMENT AND PROTOTYPING ……. 152 REFERENCES …………………………………………………………………. 153 List of Figures
Figure 1.1: Structural optimisation tasks illustrated through the example of the design of a simply supported, centrally point loaded structure .………………. 5
Figure 2.1: Michell truss subjected to load F at point A and fixed at a circular support at point B, after Michell (1904) .……………………………………….. 13
Figure 2.2: Optimal self adjoint cantilever trusses with six and eleven joints, subjected to load F at point A and fixed at support points B (Prager 1977) ……… 13
Figure 2.3: Fully connected ground structure for a relatively simple (3x6) grid . 17
Figure 2.4: Concept sketches for bracing design of 122 Leadenhall St. Building (reproduced by kind permission of Chris Neighbour, Arup) …………… 34
Figure 3.1: Weighting factors used for averaging sensitivity numbers across elements to avoid checkerboarding ……………………………………………….. 42
Figure 3.2: Real loads (left), including member groupings and geometric specifications, and virtual load (right). ASCE standard section specifications (below) ………………………………………………………………….. 46
Figure 3.3: Design topology of Liang et al. (2000): δ=0.024, element retention = 22% (left) Comparative result to Liang et al. (2000) topology: δ=0.024, element retention = 23% (right) …………………………………………………. 47
Figure 3.4: Elements removed in the top left bay unit in the first iteration, based on maximum cross strain energy (left) and sum of strain energies (right) in pairs of elements grouped by the horizontal symmetry condition ………………. 48
Figure 3.5: ESO results with element removal determined by cross strain energy. 25.4mm designable domain, 8 elements removed per iteration (left: maximum of sensitivity number in pairs of elements, right: sum of sensitivity number in pairs of elements) ……………………………………………………….. 49
Figure 3.6: Minimum volume designs satisfying the displacement constraint derived by BESO for varying domain thickness and starting configuration ……….. 51
Figure 3.7: Process flowchart for ESO with domain thickness optimisation …... 54
Figure 3.8: Flowchart for domain thickness optimisation loop ………………… 55
Figure 3.9: Process history for simultaneous topology and domain thickness optimisation with a single thickness group …………………………….. 57
Figure 3.10: Best designs derived by simultaneous thickness and topology optimisation, with one, three and six thickness groups ………………… 58
Figure 3.11: Evolving topologies with prescribed symmetry, using simultaneous thickness optimisation of appropriate groups …………………………... 60 Figure 3.12: Discrete bracing topologies (with circular solid sections) optimised for minimum mass satisfaction of displacement constraint ………………… 63
Figure 4.1: Fully braced analysis model (left to right): plan view; side elevation; isometric view (shown with two spirals highlighted); isometric split sections …………………………………………………………………………… 70
Figure 4.2: Split elevation view of the upper section of structural model 1, with spiral numbering and bracing members at the tip of each element highlighted . 75
Figure 4.3: Parametric Studies ………………………………………………….. 77
Figure 4.4: Designs generated for consideration for outline proposal ………….. 78
Figure 4.5: 2D simplified representation of design domain, model A ………….. 80
Figure 4.6. A sample exploratory move ………………………………………… 81
Figure 4.7: Pattern Search topology optimisation flowchart …………………… 82
Figure 4.8: Design solutions from topology optimisation of structural model 2 .. 89
Figure 4.9: Size optimisation flowchart ………………………………………… 99
Figure 4.10: Convergence of size optimisation algorithm from maximum section sizes in fully braced design …………………………………………………… 101
Figure 4.11: Convergence of size optimisation algorithm from minimum section sizes in fully braced design …………………………………………………… 101
Figure 4.12: Flowchart for combined size and topology optimisation algorithm . 105
Figure 4.13: Arup design proposal, without requirement for bracing members to be grouped in continuous spirals …………………………………………… 109
Figure 4.14: Volume reduction by simultaneous versus sequential topology and size optimisation routines ……………………………………………………. 111
Figure 5.1: Tree representation of mathematical equation: y = 4/(X*X) + 5*(7 X) 116
Figure 5.2: Function set for GP trees representing bracing designs ……………. 117
Figure 5.3: Seeded framework ………………………………………………….. 118
Figure 5.4: Development of an initial design by application of design modification operators ………………………………………………………………… 119
Figure 5.5: Linear rank based weighting system for parent selection ………….. 120
Figure 5.6: Genetic programming evolutionary process flowchart …………….. 121
Figure 5.7: Example of geometric infeasibility, with units overlapping and extending beyond the orthogonal framework ……………………………………… 122 Figure 5.8: Repair algorithm in the context of mutation or crossover ………….. 124
Figure 5.9: Example of initial population, penalised designs shown in grey (Population size = 30, Crossover ratio = 0.9, run number 22) ………………………. 129
Figure 5.10: Example of final population, penalised designs shown in grey (Population size = 30, Crossover ratio = 0.9, run number 22), best of run design top left …………………………………………………………………………... 129
Figure 5.11: Example of evolution history (Population size = 30, Crossover ratio = 0.9, Run number 22) ………………………………………………………… 130
Figure 5.12: Example of most efficient tree representation of the optimal double echelon design (left), with the actual representation found in Run number 22, Population size = 30, Crossover ratio = 0.9 (right) …………………….. 130
Figure 5.13: Geometry and cross section groupings for 6x30 framework ……... 132
Figure 5.14: Initial population of randomly generated designs ……………….... 134
Figure 5.15: Final generation of designs (run 1), including best of run design (top left) …………………………………………………………………………... 135
Figure 5.16: Evolution history of run 1 ………………………………………… 136
Figure 5.17: Best of run designs and performance (* indicates displacement constraint violation) ………………………………………………………………... 136
Figure A1.1: Structural analysis flowchart ……………………………………... 151 List of Tables
Table 2.1 Design and production of a typical automotive component versus a steel building structure ……………………………………………………….. 27
Table 4.1 Comparison of structural models ……………………………………. 71
Table 4.2: Statistical analysis of 10000 randomly generated designs ………….. 81
Table 4.3: Statistical summary of 20 runs per case …………………………….. 85
Table 4.4: Performance of designs derived from fully braced initial configuration 90
Table 4.5: Performance of randomly generated initial designs and solutions derived from them through bi directional topology optimisation ………………. 90
Table 4.6: Catalogue of circular hollow sections and corresponding areas available for bracing members ……………………………………………………….. 93
Table 4.7: Size optimisation of fully braced design from different initial distributions …………………………………………………………………………… 100
Table 4.8: Performance of optimised designs derived from fully braced initial configuration ……………………………………………………………. 106
Table 4.9: Performance of initial and optimised designs derived from random initial configurations …………………………………………………………... 107
Table 4.10: Summary of best designs from structural model 2 ……………….... 109
Table 5.1: Batch characteristics in parametric study …………………………… 127
Table 5.2: Cross sections made available and selected in fully stressed design .. 133
Table 6.1: Summary of methods used in this thesis ……………………………. 143
Structural Optimisation in Building Design Practice: Case-studies in topology optimisation of bracing systems ROBERT BALDOCK
Summary
Keywords: structural topology optimisation, structural design practice, bracing design, Evolutionary Structural Optimisation, Pattern Search, Optimality Criteria, Genetic Programming, computer-aided design, large-scale structural size optimisation
This thesis aims to contribute to the reduction of the significant gap between the state-of-the- art of structural design optimisation in research and its practical application in the building industry. The hypothesis that optimisation can be successfully and appropriately applied in practice through consideration of industry specific issues is explored. The research has focused on structural topology optimisation, investigating three distinct methods through the common example of bracing design for lateral stability of steel building frameworks. The research has been aided by collaboration with structural designers at Arup. It is shown how Evolutionary Structural Optimisation can be adapted to improve applicability to practical bracing design problems by considering symmetry constraints, rules for element removal and addition, as well as the definition of element groups to enable inclusion of aesthetic requirements. Size optimisation is added in the optimisation method to improve global optimality of solutions. A modified Pattern Search algorithm is developed, suitable for the parameterised, grid-based, topological design problem of a live, freeform tower design project. The alternative objectives of minimising bracing member piece count or bracing volume are considered alongside an efficient simultaneous size and topology optimisation approach, through integration of an Optimality Criteria method. A range of alternative optimised designs, suitable for assessment according to unmodelled criteria, are generated by stochastic search, parametric studies and changes in the initial design. This study is significant in highlighting practical issues in the application of structural optimisation in the building industry. A Genetic Programming formulation is presented, using design modification operators as modular "programmes", and shown to be capable of synthesising a range of novel, optimally- directed designs. The method developed consistently finds the global optimum for a small 2D planar test problem, generates high-performance designs for larger scale tasks and shows the potential to generate designs meeting user-defined aesthetic requirements. The research and results presented help to establish a structural optimisation toolbox for design practice, demonstrating necessary method extensions and considerations and practical results that are directly applicable to building projects. The research hypothesis is hence strongly supported.
1. Introduction
The research presented in this thesis is motivated by the disparity between the vast volume of academic literature in the field of structural optimisation and the very modest uptake of these methods in building design practice. The core research objective is therefore to contribute towards reducing the gap between research and industry. The accompanying central hypothesis is that optimisation can be successfully and appropriately applied in practice through consideration of industry specific issues. Collaboration with the structural engineering consultants, Arup, has allowed observation of and involvement in live projects, providing useful insights into the pertinent issues to be addressed in furthering the application of optimisation in the building industry. The research objective is achieved through the investigation of three optimisation methods: Evolutionary Structural Optimisation, Pattern Search with Optimality Criteria for simultaneous section-size optimisation and Genetic Programming using design modification operators, all applied to test problems in the field of topological bracing design for lateral stability of steel building frameworks. At the start of each chapter, research questions are posed, with corresponding proposals stated and subsequently developed in detail. Significant research contributions are made in each of these studies, as stated in section 2.5, following a discussion of the state-of-the-art of structural optimisation in research and practice. Major themes in this work are generating a range or selection of high performance designs for assessment according to unmodelled criteria, such as aesthetics and the integration of size and topology optimisation. This introductory chapter begins to explore the issues associated with the research objective, considering the nature of optimisation in general, specific characteristics of optimisation of structures and the design process in the building industry. Benefits and barriers to optimisation are discussed, aided by the opinions of practising structural designers. The structure of the thesis is then presented with an overview of each subsequent chapter.
1 1.1. THE NATURE OF DESIGN OPTIMISATION
Design optimisation is loosely defined by Papalambros and Wilde (2000) as the selection of the "best" design within the available means. When stated so simply, optimisation seems an obvious objective of any design task. Yet when the problem is ill-structured (defined by Simon (1973) as lacking definition in some respect), including a possible absence of appropriate tools and knowledge, or if the expenditure in finding an optimal solution places a high premium on the design cost, a good design that meets a defined tolerance on all requirements is generally accepted. The American political scientist and pioneer of Artificial Intelligence, Herbert Simon, coined the term "satisficing" to describe the process of finding such designs (Simon 1955) . Indeed, this is the standard approach adopted in manual design. Automated and directed search are often considered under the name of optimisation. In this case, referred to as design exploration, the process is more focused on examining a broad range of the design space, in search of diverse and novel high- performance designs, without emphasis on strict global optimality. Here the search may be seeking satisficing designs where none was previously known. Papalambros and Wilde (2000) observe that design optimisation involves: "1. The selection of a set of variables to describe the design alternatives. 2. The selection of an objective (criterion), expressed in terms of the design variables, which we seek to minimise or maximise. 3. The determination of a set of constraints, expressed in terms of the design variables, which must be satisfied by any acceptable design. 4. The determination of a set of values for the design variables, which minimise (or maximise) the objective, while satisfying all the constraints." A corresponding mathematical definition of a classical optimisation model with equality and inequality constraints and mixed discrete-continuous design variables is as follows:
2 T Minimise: f( x), x = (x 1, x 2,…, x n) objective function Eq 1.1
subject to: gj(x) ≤ 0, j = 1,…,p inequality constraints Eq 1.2
hj(x) = 0, j = 1,…,m equality constraints Eq 1.3
xi∈Di, D i = (d i1 ,d i2 ,…,d iq i); i = 1,…n d discrete variable set
Eq 1.4
where f is the objective function of x, a set of n design variables, nd of which are
discrete, the remainder being continuous. qi is the number of available discrete
values within Di for each xi, gj is the set of p inequality constraints (including bounds
on continuous variables), hj is the set of m equality constraints.
Deviations from this form of definition are frequently observed, for example the number of variables in the problem may not be fixed or multiple conflicting objectives may exist. This latter class of problem is known as multi-objective optimisation and has been extensively researched (Collette and Siarry 2003). It should also be noted that objectives and constraints may not always be readily mathematically defined nor their values quantifiable. An obvious example with respect to the subject matter of this thesis is aesthetic appeal. Complications arise in practical design problems on account of multi-modal design spaces with numerous local optima, from which a small deviation of any combination of variables will increase the objective value, despite the existence of a better solution elsewhere in the design space. The design space may also be fragmented, with several disconnected feasible regions, surrounded by infeasible space.
1.2. OPTIMISATION OF STRUCTURES
Prior to considering existing optimisation methods, it is useful to define the framework associated with structural design optimisation. This section presents a classification of the design tasks themselves and is followed by a discussion of the typical phases of the structural design process. In increasing order of complexity, structural design optimisation tasks are generally considered to be:
3 - Optimisation of size (and shape) of cross-section for discrete structural members, such as beams and columns, or thickness of continuous material, such as panels or floor slabs. This is often referred to as size optimisation. - Shape Optimisation, varying positioning of nodes or connections and definition of lines, curves and surfaces that describe structural form. - Topology Optimisation, varying the configuration and connectivity of members or material. These tasks are illustrated in figure 1.1, noting the trend in the stage of the design process at which the tasks are addressed. Whilst it is possible to assign a fixed set of variables in defining an optimisation model for size and shape optimisation, this is generally not the case for topological optimisation, hence an infinite number of solutions may exist. The requirement for modelling member connectivity in topology design is a significant barrier to application of many classical optimisation methods, as noted by Deb (2001). Shape optimisation is often considered to include cross-sectional size optimisation; in turn topology optimisation may include both shape and cross-sectional size optimisation. It is possible to define shape and topology optimisation tasks parametrically, for example by defining control points on a curve or varying the number of columns on the perimeter of a building, although this obviously places restrictions on the search space. Additionally, it is possible to consider optimisation of plan layout, for example for maximising potential letting revenue, type of structural system or material selection. The field of structural design optimisation includes a number of unique characteristics and corresponding methods. Many structural design tasks are ill-structured, especially those in the earlier stages of the design process, where decisions carry the greatest influence on final efficiency. A crucial part of a potential optimisation process in the building industry is the evaluation of structural designs, generally by finite element analysis, which often carries a significant time cost.
4 Initial Design
Size
Optimisation Increased generality of design task
Shape (+ Size) Optimisation
Progression of design process of design Progression Topology Increased ease of task formulation of ease Increased (+ Shape + Size) Optimisation
Figure 1.1: Structural optimisation tasks illustrated through the example of the design of a simply-supported, centrally point-loaded structure.
1.3. THE DESIGN PROCESS FOR BUILDING STRUCTURES
It is vital to the successful implementation of optimisation in structural design that the optimisation tasks detailed above are linked to the appropriate phase of the design process. The structural design process essentially follows the same progression as any other design task. However, the interdisciplinary nature of building design, with input from clients, architects and structural and building services engineers, serves to complicate the process and may lead to a large number of iterations and revisions, even revisiting earlier design phases. With reference to design of topology and form and section allocation, it is useful to consider the corresponding stage in the design process for each of these tasks. Structural systems and topologies are developed earlier in the design process, with the optimisation problem less well-defined, the design space larger and hence a greater range of possible solutions. Section sizes are not finalised until the latter stages of the
5 design process. Although section-size optimisation is a much more straightforward task, a strong driver for optimisation prior to this stage is provided by estimates suggesting that up to four-fifths of the total resources in an engineering project are committed in the early design stages (Deiman 1993). The classical design process follows the following stages: In the conceptual design stage, a set of initial concepts is generated in an attempt to satisfy the broad design requirements prescribed, in the case of design of buildings, by the architect or client. The preliminary design stage further develops one (or more) conceptual design(s). At this point, the general building system functionalities that were determined previously will be subject to further refinement in order to furnish a more accurate cost estimate for the project. The detailed design stage finalises all information required for construction. In these latter stages, member-sizing, joint-detailing and similar well-defined tasks are undertaken in structural design. Whilst these design stage definitions are widely used throughout the design community, the Plan of Work Stages 1999 as described by the Royal Institute of British Architects (RIBA 1999), (Phillips 2000) is recognised and implemented throughout the construction industry. Stages A to L include tasks undertaken both before and after the design stages described above, e.g. tendering, construction and completion. However, the following stages roughly correspond to those detailed above: "B: Strategic Briefing Preparation of Strategic Brief by, or on behalf of, the client confirming key requirements and constraints. Identification of procedures, organisational structure and range of consultants and others to be engaged for the project. [Identifies the strategic brief (as CIB Guide) which becomes the clear responsibility of the client.] C: Outline proposals . Commence development of strategic brief into full project brief. Preparation of outline proposals and estimate of cost. Review of procurement route. D: Detailed proposals . Complete development of the project brief. Preparation of detailed proposals. Application for full development control approval. E: Final proposals . Preparation of final proposals for the Project sufficient for co- ordination of all components and elements of the Project."
6 1.4. DRIVERS AND BARRIERS FOR STRUCTURAL OPTIMISATION IN THE
BUILDING INDUSTRY
It is necessary to validate the overriding objective of aiding the adoption of structural design optimisation in the building industry, demonstrating it to be a worthwhile endeavour. This can be achieved by highlighting the drivers for computational design optimisation and search in the building industry. In the validation process it is also necessary to establish that it is possible to overcome the common barriers to the use of optimisation in structural design practice. The subsequent discussion is augmented by the opinions of Arup engineers relating to application of structural engineering in the building industry, as canvassed by Shea (2003) and the author.
Drivers
- Rapid generation and evaluation of a large number of design alternatives. This is desirable in the early stages of a project, permitting a wider and more thorough exploration of the design space than could be achieved manually. - Discovering previously unknown feasible solutions. In the case of some highly complex structures, there may be uncertainty as to the existence of a feasible design within the defined constraints. Starting from an initial infeasible design, optimisation or heuristic search methods have the potential to find feasible solutions where none was previously known, as seen in section 2.2.3. - Cost benefits such as reduced material or construction cost and increased potential revenue. Financial savings are an obvious potential driver for use of optimisation methods. Cost is often simplistically equated to structural weight, especially in research investigations, whereas piece-count and connection detailing are also important in construction costing. Further, maximisation of floor space and quality will control the potential letting revenue in office and residential buildings, influencing the financial feasibility of a project as a whole. - Time savings through computer assisted search . A well-executed optimisation process has the potential to save design time, avoiding the need to iterate a design by hand to find structurally or financially feasible solutions, as well as reducing the tedium of routine tasks. - Marketability of optimisation capabilities . Substantial interest has been reported in using computational optimisation for gaining a market-edge by offering clients,
7 for example, maximisation of net lettable floor space or minimisation of steel tonnage. - Decision support in the design process . Decisions in the earlier stages of design are frequently made based on previous experience or intuition of the engineer, but without rigorous justification. Increased rigour provided by optimisation or search could provide leverage for the structural designer in multidisciplinary design decisions.
Barriers
- Problems may be ill structured. Topology design problems, for example, do not present a fixed set of variables, which creates difficulties in implementing numerical optimisation techniques. However, various methods are capable of handling such complications, an example being domain-specific methods, such as those discussed in Chapter 2. - Architectural constraints often heavily influence the structural design of buildings. These are difficult to incorporate into an optimisation model and often not conducive to using gradient-based optimisation methods. However, stochastic methods, with a random component to the search method, can be used to present a range of high-performance options, for assessment according to unmodelled criteria. - Time required for modelling, method customisation and running optimisation processes on a specific project may be prohibitive. Since design time is often at a premium, optimisation cannot be allowed to become a critical path with large amounts of time devoted to method development or parameter tuning. Each structural analysis can also be computationally expensive, although analysis time can be reduced by appropriate approximations and simplifications to improve efficiency. Repeated use of appropriate optimisation and analysis techniques for routine tasks within a company will develop a skills base, with increasing efficiency of implementation. It is highly desirable that in-house tools, techniques or expertise developed on one project should be reusable. Thus initial investments may need to be made to develop capabilities, before cost-benefits are achieved. - Issues of scale in extending small research case study problems to real world design scenarios. A number of optimisation methods detailed in research literature could not feasibly handle large-scale industrial problems due to the large
8 numbers of design variables, especially discrete variables, possible lack of a feasible initial solution and the number of function evaluations required. Adopting simplified models for the design task may ease these issues, otherwise alternative methods may be required. - Specifications for a project may change rapidly at certain points in the design process. This possibility presents the danger that the results of an optimisation process may be obsolete before they are even generated, due to the rapid information exchange between architects and engineers. Optimisation tools must therefore be versatile and adaptable to facilitate rapid incorporation of specification changes. - Designers lack tools, experience and knowledge required to implement optimisation methods. Commercial optimisation programmes require technical and theoretical expertise and hence can be inaccessible to structural designers who do not use them on a regular basis. As will be discussed in chapter 2, many commercial tools are targeted at the aerospace and automotive industries. Practical concerns also include the cost of software licenses, which are a substantial expenditure, especially for a single project. - User scepticism. 20 years after Berke commented on the doubts held by designers (Sobieski et al. 1986) regarding effective benefits, reliability and appropriate methodologies of structural optimisation, these reservations still persist. Cohn (1994) notes that "structural engineers willing to optimise their designs have no alternative but to learn many optimisation procedures and then decide which of these fit their real problems". However, an increased volume of successful case studies on "live" building projects is likely to reduce scepticism and accelerate uptake. - Optimisation results can be hard to verify. Although it may be hard to demonstrate that a design is globally optimal with respect to the defined optimisation problem, an improvement on a manual design may provide sufficient endorsement for optimisation results to be accepted in practice. In summary, the obstacles highlighted present important issues to be addressed, without negating the potential for structural optimisation in the building industry.
9 1.5. SUMMARY OF RESEARCH CONTRIBUTIONS
A thorough discussion of the research contributions of this thesis, in the context of previous work, is presented in section 2.5 following a review of the state-of-the-art in academic research and industry. This section summarises them as follows: - The methods discussed in research chapters 3 to 5 contribute to the development of a structural optimisation toolbox appropriate for use in the building industry. - The practical consideration of constraints, repetition and symmetry given precedence in consideration of Evolutionary Structural Optimisation. - A unique example of "live" structural topology optimisation is presented on a full- scale building project, with associated method development. A modified Pattern Search algorithm is used for this purpose. Constraint handling methods are developed to suit the problem formulation. - Means of efficiently integrating size optimisation into topology optimisation processes are presented for the methods relating to the previous two points, with modest increases in structural analysis requirements. - Genetic Programming is used to rapidly generate novel and diverse design concepts. Through the use of design modification operators in the development of design blueprints, a new tool is developed for optimally directed and controlled pattern generation.
1.6. THESIS STRUCTURE
Chapter 2 presents a review and comparison of the state-of-the-art in academic research and building engineering practice, to explore the reasons for the low transfer of techniques and tools. This section also compares structural optimisation in the context of the building industry with that in the automotive and aerospace industries, where the state-of-the-art is significantly more developed. The chapter closes with a clear statement of the research contributions to be subsequently presented and proven. In research chapters 3 to 5, three distinct optimisation and search methods are investigated, developed and results presented. Their present or future potential impact on structural design of buildings will be assessed. These methods are all applied to the common topological design problem of bracing configuration in steel building frameworks to meet lateral stability requirements, the choice of which is justified in Chapter 2. This area of application is not intended to be exclusive, since generality is
10 desirable in any method, but rather to provide a unifying theme to the distinct elements of this thesis. Further threads are provided through investigating the integration of size and topology optimisation in two of these cases and considering control over aesthetic form of solutions. Chapter 3 considers Evolutionary Structural Optimisation (Xie and Steven 1997), a form of continuum-based optimisation, which has attracted interest in research and practice on account of its comprehensibility and versatility. Development in this thesis is based on more practice-orientated criteria than in previous research. The effect of variation of domain thickness is considered, both manually in a parametric study and within the context of size optimisation. Capacity for control of form is demonstrated through defining repetition patterns and symmetry lines. Finally, the importance of discrete design interpretation is discussed. Chapter 4 presents work undertaken both live and retrospectively on an Arup project, in close collaboration with the design team. The topological bracing design problem is parameterised and tackled using a discrete Pattern Search algorithm (Hooke and Jeeves 1961). Simultaneous size optimisation is introduced using the Optimality Criteria method (Rozvany 1989) at each iteration, assuming unchanged force-moment distribution in the structure. Valuable insight into complications faced in applying optimisation in practice was gained through close collaboration with the design team. Chapter 5 introduces the application of function-based Genetic Programming for the development of optimally-directed bracing designs. Branch points within the tree representation take the form of design modification operators, in a manner more closely analogous than any previous structural application to the "programme routines" on which the original Genetic Programming method (Koza 1992) was based. The potential for restricting the search space according to aesthetic style is explored. Chapter 6 concludes the thesis by summarising the results of the preceding research and discussing future work required in developing these methods. The two appendices provide notes on the integration of structural analysis into the prototype tools developed during the course of the current research, as well as discussing software development and prototyping for structural optimisation more generally.
11 2. State-of-the-art: research and practice of design optimisation in structural engineering
This chapter first presents a review of academic literature, structured to consider the areas to which this thesis contributes: size optimisation, discrete topology optimisation and continuous topology optimisation. This section also includes a discussion of research in computer-based conceptual design methods for structural design, both with and without optimisation. The subsequent section presents the current state of the building industry with regard to the use of structural optimisation. The building industry is first juxtaposed against the automotive and aeronautical industries where structural design optimisation is relatively well established. A review of commercial software is presented, followed by details of published examples of structural optimisation in building design. Potential for further use of structural optimisation is considered through a selection of case studies. Conclusions are drawn from the preceding discussion, regarding the disparity between research and practice. This leads to the establishment and justification of the design task tackled in the test-cases used in this thesis: the topological design of bracing systems for lateral stability in steel framed buildings. Finally the research contributions of the subsequent studies are stated.
2.1. STRUCTURAL DESIGN OPTIMISATION RESEARCH
Optimisation of structural shape and topology is rooted in the work of Michell (1904). His pioneering studies in the field derived conditions for limits of material economy in truss structures, developing structural concepts originally demonstrated by Maxwell (1864). In the case of a point-loaded cantilever truss with a circular support, and disregarding the weight of joints, the minimum weight is achieved by a truss-like continuum with an infinite number of joints and bars, the form of which is shown in figure 2.1.
12 F
A B
Figure 2.1: Michell truss subjected to load F at point A and fixed at a circular support at point B, after Michell (1904)
Prager (1977) refined this approach by modelling the weight of joints in his minimisation problem and employing optimality criteria and the concept of adjoint trusses to derive simple and practical cantilever structures. Nevertheless, these structures, shown in figure 2.2, are limited in their treatment of constraints and loadcases.
F F B B
A A
B B Figure 2.2: Optimal self-adjoint cantilever trusses with six and eleven joints, subjected to load F at point A and fixed at support points B (Prager 1977)
Although many popular general optimisation methods have been applied or adapted to structural design optimisation tasks, various problem specific methods also exist, in particular to tackle the unique challenges of topological design. Methods may be divided according to the representation type (discrete or continuous material), the search type (deterministic or stochastic 1) or the specific design task to which they are applied, as presented in section 1.2. Reviews of structural topology optimisation approaches are provided by Bendsøe and Sigmund (2004) and Papalambros and Shea (2002).
1 A deterministic method, under fixed parameters, will always yield the same solution from a given starting point. A stochastic search includes a probabilistic component, permitting multiple design solutions to be obtained from a fixed starting point and fixed search parameters.
13 2.1.1. Size optimisation
Vast quantities of academic literature exist testing a diverse range of optimisation methods on benchmark sizing problems. This section attempts to highlight the most pertinent of these to the issue of integrating size and topology optimisation, since this is the primary role of size optimisation in this thesis. Size optimisation problems can easily be expressed mathematically and are traditionally solved by deterministic methods. However, for statically indeterminate structures subject to multiple displacement constraints, the design space is almost invariably multi-modal.
Fully Stressed Design
Fully Stressed Design is most commonly used for structures in which strength considerations govern over stiffness, such as small and medium size frames. Maxwell (1864) recognised that in a statically indeterminate structure, in which members can be resized without influencing the load path of the structure, the minimum weight design is the one in which every member is subjected to the maximum permissible stress in at least one loadcase, i.e. fully stressed. For indeterminate structures, the number of distinct fully stressed designs can be very large. Conventional procedures increase the size of over-stressed members and reduce the size of under-stressed members, reanalysing and iterating until convergence is achieved. However, Mueller and Burns (2001) demonstrate that this approach excludes a set of repelling fully stressed designs, in which some members will respond to an increase in size by attracting greater stress. This causes an initial sizing solution in the vicinity of a repelling fully stressed design to rapidly move away from this area of the design space. Mueller and Burns (2001) employ a series of non-linear equations to define the fully stressed state and solve with a hybrid Newton-Monomial method to find sets of fully stressed designs, commencing from randomly generated initial designs. Mueller et al. (2002) consider the trade-off between material volume and maximum lateral drift in such a set and classify the load-bearing systems observed.
Mathematical Programming
Mathematical Programming methods (Borkowski and Jendo 1990) take an iterative approach in seeking an optimal solution. A search direction within the design space is
14 first calculated, then the distance to travel in this direction, often referred to as step- size, is determined. A wide range of techniques exist for determining the search direction and step-size, according to the assumed characteristics of the objective function, constraints and variables. This selection is generally, although not necessarily, based on gradient information, hence defining the set of gradient-based methods. Linear Programming may be used if functions exhibit linearity, otherwise quadratic and general non-linear methods exist. Integer Programming will account for the requirement for variables to take integer values, as required in discrete section size optimisation.
Optimality Criteria
This group of methods incorporates problem-specific knowledge, such as the principle of virtual work, into the Kuhn-Tucker conditions for optimality. A set of necessary and sufficient conditions are derived to describe optimal solutions to convex problems. Applied to structural section sizing, optimal solutions are found iteratively with reanalysis to account for changes in load distribution. A comprehensive introduction to optimality criteria methods in structural optimisation is presented by Rozvany (1989). Grierson and Chan (1993) present an approach tailored to the design of tall buildings. The efficiency of Optimality Criteria (OC) methods is strongly dependent on the number of global constraints, such as permissible displacements, and only weakly dependent on the number of design variables. OC methods hold a further advantage over mathematical programming techniques in that they are not restricted to locally optimal solutions in the vicinity of the initial design. However, in structures with a high degree of statical indeterminacy, changes in load distribution may mean the approach still fails to locate the global optimum. This led to the hybrid OC-GA method (Chan and Liu 2000), developed to combine the robustness of Genetic Algorithms (GA), discussed in section 2.1.3, with the computational efficiency of OC. OC methods sacrifice an element of generality on account of the requirement for problem-specific physical laws.
Additional considerations
The problem of handling discrete variables is recurrent in structural optimisation, since member sections must frequently be selected from catalogues, or fabricated
15 from standard gauge sheets. This issue is considered by Arora (2002) through discussion and review of recent approaches. Shea et al. (1997) consider the practical issue of dynamically assigning members to groups, based on cross-sectional area and address this in the shape annealing method. Recent trends have seen hybridisation of optimisation methods: in addition to the OC- GA method previously discussed, various approaches use one method for topology optimisation and another more appropriate technique for simultaneous size optimisation. Examples include Simulated Annealing with Fully-Stressed Design in cases with only stress constraints (Shea 1997), Genetic Algorithms with Optimality Criteria (Sakamoto and Oda 1993), (Kicinger 2004) and Genetic Programming with Optimality Criteria (Liu 2000).
2.1.2. Discrete topology optimisation methods
Ground Structure approach
The Ground Structure approach, first proposed by Dorn et al. (1964), effectively reduces the complexity of a topology optimisation problem by considering a fixed grid of nodes, initially with a high degree of connectivity (in extreme cases each node may be connected to every other node) as exemplified in figure 2.3. These links between nodes are potential structural members. They may take binary "on"/"off" states, or be iteratively assigned a section-size and removed altogether if found to be under-utilised, for example if assigned a section-size less than a prescribed minimum. Various techniques have been used in obtaining a reduced, "optimal" structural configuration from the initial system: Bendsøe and Sigmund (2003) discuss deterministic methods including Optimality Criteria and Linear Programming, Bennage and Dhingra (1995) use a meta-heuristic Tabu Search method, Xie and Steven (1997) extend the principles of Evolutionary Structural Optimisation to the design of pin- and rigid-jointed frames. Stochastic approaches include Simulated Annealing (Topping et al. 1996) and numerous examples of the application of Genetic Alogrithms, e.g. (Hajela and Lee 1995). Despite widespread interest in the ground structure approach, there exist a number of significant limitations (Ohsaki and Swan 2002): - The number of elements required for complete connectivity increases factorially with the number of nodes in the ground structure. However, performance of
16 "optimal" structures is heavily dependent on the initial design and is compromised by using simplistic ground structures. It is possible that much better solutions may exist beyond the restrictions imposed by the grid framework. Most implementations do not allow movement of nodes, or addition of nodes and elements. The method is best suited to problems of modest scale, as opposed to the design of complex structures. - Unrealistic "optimal" solutions may be generated, with the possibility of instability arising from removing too many members in a region of the structure. - Complications arise in interpreting the connectivity of overlapping members for analytical purposes. - High sensitivity to multiple loadcases can be observed, as seen in a simplified case based on the Eiffel Tower by Bendsøe and Sigmund (2004).
Figure 2.3: Fully-connected ground structure for a relatively simple (3x6) grid
In attempting to address some of the above issues, Smith (1998) has investigated methods of generation of appropriate ground structures in two and three dimensions. Bendsøe et al. (1994) construct a 3-D cantilever ground structure for which optimal topology and member sizes are found. Performing a subsequent shape optimisation, allowing nodal locations to vary, reduces grid dependence. It is worth noting that the shape-optimised structure does not necessarily constitute the globally optimal design, since it was derived indirectly and a different optimal topology may have been found if the nodal locations in the ground structure had been different. Current research by Gilbert and Tyas (2003) has used low connectivity initial designs (still based on a fixed nodal grid), introducing new members to stiffen the structure at each iteration, by considering the relative nodal displacements. This approach permits the use of
17 much larger grids and many more potential members (in excess of 100 million), but is nevertheless prone to many of the standard complications of this class of method.
Ruled-based approaches
This section discusses examples of heuristics and grammars used to govern the development of structures. Grammars are a production system capable of describing a set of designs through the transformations that map one design to another (Stiny 1980). An innovative approach to optimally directed topology design is offered by Shape Annealing (Cagan and Mitchell 1993), combining a defined shape grammar with Simulated Annealing optimisation techniques. The shape grammar is a set of permissible design transformations that may include the addition, removal or reorientation of structural members, adjustment of nodal co-ordinates and resizing of components. Shea and Cagan (1999) consider the design specifications as a syntax encoded in the structural shape grammar with objectives and constraints as semantics, to produce a language that encompasses a set of valid and meaningful structural designs. Applications include planar trusses, 3D space-trusses (domes), truss beams and practical transmission towers (Shea and Smith 2006) McKeown (1998) considers growing least-volume trusses from the simplest structure required to transfer prescribed loads to the supports. Auxiliary joints and members are sequentially added, optimising joint locations and member sizes at each stage, until the additional weight of a new joint outweighs the saving from reduced total member volume, or an alternative termination criterion is met. Rule (1994) presents a deterministic rule-based generative approach to truss design. The final structure "evolves" in a number of stages, with the level of complexity increasing at each advancement. A small base structure is generated, determined by the number and location of loads and supports and the number of design stages to be used. At each stage, the positions of free nodes (i.e. those unloaded and not acting as supports) are optimised by a Mathematical Programming method, with new members added by bisecting the longest members, thus ensuring structural members are of a similar length and that the truss remains fully triangulated. The design of a 2D representation of a high-voltage cable-support tower is detailed.
18 2.1.3. Evolutionary Algorithms in topology optimisation
Although not pure optimisation algorithms (De Jong 1993), Evolutionary Algorithms (EAs) are versatile, stochastic, problem-solving methods, alternatively classified under the name of Evolutionary Computing (EC). This class of methods (not to be confused with Evolutionary Structural Optimisation) is so-called due to its mimicry of natural biological evolution as postulated originally by Charles Darwin (1859). In general, the performance of a population of individual solutions in solving the prescribed problem is assessed according to one or more quantifiable criteria. In turn, performance, commonly referred to as fitness, influences the chances of an individual's involvement in populating the subsequent generation of solutions, by some combination of the genetic operators of reproduction, crossover (or recombination ) and mutation. Around the 1960s, three sub-classes of EA were developed independently: Genetic Algorithms (GA) (Holland 1975), Evolutionary Programming (EP) (Fogel et al. 1966), and Evolutionary Strategies (ES) (Rechenburg 1965). However, these were not brought together under the name of EAs until the 1990s. A fourth class, Genetic Programming (GP) (Koza 1992), also emerged in the 1990s. On account of operating on populations of solutions, as well as their stochastic nature, EAs tend to be computationally expensive and hence cannot compete with numerical methods in such tasks as regular continuous parametric optimisation (Eiben and Schoenauer 2002). However, there are a number of problem types for which EAs, and stochastic search methods in general, are particularly appropriate: - multi-objective design space exploration (Deb 2001) - problems with mixed (discrete and continuous) variables - problems with a discontinuous space of feasible or legal solutions - unconventional problems for which the representational flexibility of EAs can be exploited. A brief introduction to the field is presented by Eiben and Schoenauer (2002) and a more comprehensive review is given by Bäck (1996).
Genetic Algorithms
In recent years the GA has been one of the most widely used computational search techniques within the engineering research community. The genetic representation, or genotype, of a physical design, or phenotype, is central to the concept of the GA. The
19 original, simple genetic algorithm (Holland 1975) uses a binary bit-string encryption, although a wide range of alternative representations has since been used. Notable to the field of structural topology optimisation are the real-valued representation used for topology optimisation of three-dimensional trusses (Azid et al. 2002), voxel representation used for determining the optimal cross section of beams (Griffiths and Miles 2003) and graph representation (Borkowski et al. 2002) capturing structural connectivity. Kicinger (2004) uses a GA to evolve Cellular Automata for generating bracing schema in tall buildings. Rajeev and Krishnamoorthy (1997) define a variable string length Genetic Algorithm (VGA) capable of considering simultaneously a number of alternative pre-defined topologies through the use of control variables, as well as encoding nodal positioning and section sizes in a standard manner. A more conventional representation is used by Shrestha and Ghaboussi (1998), with individuals encoded as a lengthy set of sub-strings representing a node, its spatial co-ordinates and its connecting members. Significant in addressing the issue of computational intensity is the Micro-Genetic Algorithm (µGA) (Krishnakumar 1989), designed to operate on very small populations, normally of five individuals. As mentioned in section 2.1.1, GAs can and have been hybridised very effectively with other optimisation methods to combine a broad exploration of the design space in general with efficient exploitation of high performance regions. Pezeshk and Camp (2002) present a chronological survey of research work conducted in the 1990s related to the use of GAs in structural steel design.
Genetic Programming
The newest member of the class of EAs, Genetic Programming (Koza 1992) evolves "programmes" composed of a sequence of low-level functions, traditionally represented as a tree structure, to perform a prescribed task. Within the tree structure, input variables or constants appear as "leaves" at the extremities of the tree. These are operated on by functions at branch points, with the output passed to further functions closer to the root of the tree. Hence, unlike the GA, GP methods do not require an encryption scheme to convert the representation on which the genetic operators act into the true representation. Graph structures can also be used within the context of GP.
20 Genetic Programming is predominantly used in computer science for creating programmatic solutions for tasks and has also found substantial application in design of electrical circuits (Koza et al. 2003), reproducing or improving on patented solutions. A review of use of GP in civil engineering is presented by Shaw et al. (2003). Within structural engineering and topological design in particular, the work of Soh and Yang (2000) and Liu (2000) serve as the sole examples known to the author using the tree-based representation scheme. The former make appropriate use of this representation to dispense with the need for a prescribed set of variables, but retain a GA-style encryption scheme to map between genotype and phenotype. Indeed, branch points take the form of section properties rather than any form of operator, making the classification of the method as Genetic Programming debatable. Liu (2000) uses a graph structure to represent the hierarchical decomposition of structural system, again without the use of true functions at branch points. Genetic Programming will be discussed in greater detail in Chapter 5.
Application of ES and EP methods to structural topology design problems has been minimal, but these classes are mentioned below for completeness.
Evolutionary Strategies
Evolutionary Strategies (ES) (Rechenburg 1965) were developed in the 1960s as a tool for continuous parameter optimisation. With greater emphasis on mutation, offspring are generated by adding a mutation vector, with normally distributed, randomly selected components, to a current design (Beyer and Schwefel 2002), aiming to improve on previous solutions.
Evolutionary Programming
Evolutionary Programming (EP) (Fogel at al. 1966), develops populations of Finite State Automata that transform an input sequence to different output sequences. Fitness is assessed according to accuracy of response and offspring generated by mutation of parents.
2.1.4. Continuum-based optimisation methods
This class of method considers a continuous designable domain, discretised into a mesh of elements that are defined individually in a structural analysis model. The properties of the continuum elements, such as porosity or thickness, can be varied
21 individually for size optimisation, or they can be removed or considered of vanishing thickness for shape and topology optimisation. Regions of the analysis model may be designated as non-designable. Eschenauer and Olhoff (2001) and Bendsøe and Sigmund (2004) present a comprehensive review of the research advances and state-of-the-art in this field. This section presents three significant forms of continuum-based optimisation method, with particular focus on Evolutionary Structural Optimisation on account of the interest it has received in the building industry, due to its simplicity and intuitive nature, and its development in Chapter 3 of this thesis. It should be noted that other generic optimisation methods, such as GA have also been used with continuum representations.
Homogenisation
This method, pioneered by Bendsøe and Kikuchi (1988) and expounded in detail by Bendsøe and Sigmund (2004), defines individual materials for each element in the mesh, each containing infinite microscopic voids. The porosity of the medium is optimised according to some objective function. Each element-material may have its own hole-size and orientation. Commonly, intermediate densities are penalised, to encourage elements to become either fully solid or fully void. A target volume fraction is generally set and variation of such modelling parameters admits a range of types of solution from truss-like structures and plate type solutions to composites and stiffened composites. This method provides the foundations for a number of commercial packages, discussed in the next section 2.2.
Bubble method
The bubble method of Eschenauer et al. (1993) iteratively places voids or bubbles within a continuum domain through a definite function before subsequently performing a shape optimisation on each topology. Whilst this method may be suitable for small components, its applicability to structures with high topological complexity appears low, on account of the procedure required for the addition of each void.
Evolutionary Structural Optimisation
This method was originally developed by Xie and Steven (1993), (1997). In its simplest form elements that are under-utilised, as defined by some metric such as
22 strain energy density, are removed from a continuous finite element mesh, to reduce the designable domain to an efficient optimal topology. The name is misleading, since it is not evolutionary in the same sense as Evolutionary Algorithms , based on Darwinian principles, nor is it strictly optimisation , rather it is a design technique seeking uniformity of parameters within a structure, such as stress or strain energy density. However, although the term coined by Rozvany (Rozvany 2001), Sequential Element Rejection and Admission (SERA) techniques, is more accurate, it will be referred to as Evolutionary Structural Optimisation (ESO) throughout this thesis, in line with the majority of literature on the subject. ESO can be considered a hard kill method, in that a step-function is used in defining the elastic modulus of elements as opposed to the soft kill homogenisation methods where a continuous range of values is permitted. Since its conception, ESO has proven to be a versatile method, readily understood and simple to implement. There have been a number of developments on the basic ESO method: Additive Evolutionary Structural Optimisation (AESO) (Querin 2000) adds new elements adjacent to the most suitable existing perimeter elements, to solve boundary problems through shape optimisation. This method has low relevance to building design on account of its limitation to shape optimisation. Bi-directional Evolutionary Structural Optimisation (BESO) (Querin 2000a) permits removal and addition of elements, the latter either by extrapolation of "sensitivity number" (a performance parameter such as strain energy density), or by considering the sensitivity number of perimeter elements. BESO methods have the significant advantage that material that was removed early in the evolutionary process can be replaced later if found to be structurally advantageous, hence offering improved design space exploration and increasing the probability of finding globally optimal solutions. BESO also allows development of solutions from simple initial designs (Querin 2000a) that are therefore less computationally expensive. However, on account of bi-directionality, a greater number of iterations are likely to be required than in the basic algorithm. Extended Evolutionary Structural Optimisation (XESO) (Cui et al. 2003) works by constructing contour lines of stress, or some other property, within the designable domain at each iteration. Material with stress below the critical threshold is removed, with material added in areas where extrapolated contours lines predict high stress values. The finite element model is revised by remeshing for each step of the process. It is claimed that
23 XESO allows evolution of configurations that cannot be obtained by the original ESO method, such as a "suspension-arch" design for a uniformly vertically loaded beam with simple supports at each end.
2.1.5. Computer-based conceptual design methods
It is important to appreciate the iterative and multidisciplinary nature of the conceptual design of buildings. Architects, structural engineers and building services engineers all interact in developing ideas of how a structure should look and behave. The development of a design tool to support the interests of all of these groups is an ambitious goal, but one that has nevertheless been attempted in a number of research initiatives. Sisk et al. (2003) state that the focus of development for computing tools in conceptual design should be towards Decision Support Systems (DSS) rather than optimisers, suggesting the use of the GA as a search tool, but also highlighting the importance of human-computer interactivity. This point is crucial to gaining acceptance in engineering practice, since designers will inevitably be dismissive of a black-box process that produces a single "optimised" solution to a given problem, without insight into its development. Rapid interpretation and understanding of results is essential. Hence the role of computer-assisted search methods should be to enable designers to consider a wider range of design alternatives, with more indication of their projected performance. Grierson and Khajehpour (2002) present a major review of computer-based conceptual design research, both with and without optimisation, between 1989-1999. Some of the more relevant of these are mentioned below. McCarthy (2002) provides a review of the application of Knowledge-Based Expert Systems (KBESs) to structural steelwork design, used for eliciting and applying "facts" and "rules" from pre-programmed information. HI-RISE and further KBESs developed at Carnegie Mellon University are summarised by Maher (1987). The HI- RISE system (Maher 1984), amongst the most cited of early KBESs, performs the preliminary structural design of high-rise buildings, without use of optimisation. Given a 3D grid defining the space planning of the building, the system will present feasible structural systems. The aim is to support the designer by increasing the number of designs available for consideration for further development. Smith (1996) and Rafiq et al. (2003) comment on the limitations of KBESs, notably the difficulty in developing individual or novel design solutions that are unlikely to be conceived by a
24 human design team. Limitations are attributed to problems in combining computer system heuristics with human knowledge and a lack of flexibility. Park and Grierson (1999) use a Multicriteria Genetic Algorithm (MGA) to generate pareto-optimal conceptual designs of office buildings under the competing objectives of minimising building project cost and maximising flexibility of floor space. Design variables of plan dimensions and storey height were used, with the possibility of four non-rectangular floor plans, and a set of feasible floor systems. This approach was developed by Khajehpour (2001) by considering rectangular plan high-rise office buildings with multicriteria optimisation minimising capital and operating costs and maximising income revenue for a given project. Colour filtering was used to mark variable values of the pareto-optimal (Pareto 1896) design set in the three- dimensional criteria space. Substantial effort was expended in researching costs and accounting for architectural, structural, mechanical and electrical systems. Recommendations for future work include accounting for alternative floor plan shapes and changing size with height as well as development of structural design criteria. Currently material costs are based on relatively approximate member sizing techniques. A further study by Khajehpour and Grierson (2003), motivated by the progressive collapse failure of the World Trade Center Towers, extended the previous work to consider the trade-off between profitability and safety of high-rise office buildings. Load-path safety against progressive collapse is determined by the degree of force redundancy in the structural system. The SEED system ( Software Environment to support Early phases in building Design) of Rivard and Fenves (2000) is divided into three main modules, supporting the generation of an architectural program, generation of scheme layout and design of 3D building configuration. Crucial to the third of these modules, SEED-Config, is the definition of a building design representation. The following requirements are noted in such a representation: extensibility, ability to integrate multiple views and support design evolution and favouring design exploration. The authors acknowledge the tendency for designers to break down complex problems into small subproblems and this is reflected in the representation developed. Of particular note are the four structural subsystems: the foundation (transfers all loads to the ground), the vertical gravity subsystem (transfers vertical loads to foundations), vertical lateral subsystem (transfers horizontal load, e.g. wind to foundations) and horizontal subsystem (transfers live, dead and snow loads to vertical gravity subsystem). This subdivision
25 of the structural volume permits independent consideration of the subsystems. Each subsystem consists of a set of structural assemblies (e.g. floors, roofs, frames, walls, column stacks or 3D systems such as cores or tubes). These in turn are made up of basic elements.
2.2. OPTIMISATION IN BUILDING ENGINEERING DESIGN PRACTICE
2.2.1. Comparison of structural design in the automotive and aeronautical industries versus the building industry
When considering the issues associated with successful application of optimisation methods to structural design in the building industry, it is worthwhile contrasting this sector against the automotive and aeronautical industries where, in recent years, optimisation has become increasingly widespread, aided by the implementation of sophisticated optimisation methods in commercial software. Table 2.1 compares aspects of design and production of a typical automotive component and a steel building structure. Examples of topological design optimisation of automotive components are provided by Rousseau (2004), considering a steering wheel, and Wieloch and Taslim (2004), considering a pump bracket. The independent consideration of detailing and minimising weight of small components in the automotive industry stands in contrast to the standard sections used in most building projects where it is harder to define substructures and design components. Keer and Sturt (2007) discuss optimisation of the car body, or "body-in-white", as a whole. They consider gauge (panel thickness) optimisation and shape optimisation, noting that the former is better developed and more widely used. A shape optimisation example is provided by Paas and Hilman (2006). In summary, the high premium on weight in automotive and aerospace industries, coupled with the economies-of-scale offered by producing vast numbers of identical parts, mean that investment in optimisation techniques capable of saving small proportions of the total mass are generally rewarded. Additionally, aesthetics are commonly of lesser concern than in building structures and the continuous material layout task is well suited to material distribution techniques such as Homogenisation and Evolutionary Structural Optimisation.
26 An assessment and discussion of the potential for application of optimisation in the building industry will be made at the end of this chapter after considering existing and potential examples. Leubkemann and Shea (2005) have previously highlighted the benefits and potential of computational design and optimisation in building practice.
Table 2.1 Design and production of a typical automotive component versus a steel building structure Characteristic Automotive/aerospace Steel building structure component Topological Generally low, higher for High, but often ordered, complexity whole body hence parameterisation may be possible Production volume Large (up to 100,000s) Low (generally one-off) Principal cost Design, material, knock-on Design, construction, life- considerations costs of weight cycle costs Knock on effects Reduces efficiency and Extra dead-weight of excess mass vehicle speed increases loads beyond material elsewhere in structure cost Typical constraints Stiffness, strength, natural Stiffness (global), strength frequency, fatigue (local, often buckling- related) Manufacture Purpose designed process Discrete catalogued (e.g. casting) allows structural members from flexibility and irregularity of steel supplier component form Aesthetic Minimal Often critical for topology considerations design
2.2.2. Commercial optimisation software
A number of increasingly sophisticated software suites are commercially available for tackling structural optimisation problems. Material distribution approaches have attracted significant interest and are implemented in TOSCA 2 (FE-Design), GENESIS 3 (Vanderplaats Research and Development) and OptiStruct 4 (Altair), alongside other optimisation methods. A
2 http://www.fe-design.de/en/tosca/tosca.html
3 http://www.vrand.com
4 http://www.uk.altair.com/software/optistruct.htm
27 beta-capability for topology optimisation by a material distribution method is also included in Nastran 5 (MSC). Leiva (2001) presents examples of the application of GENESIS optimisation capabilities to design of automotive sub-structures. Other programs integrate with CAD/CAE systems and finite element analysis software to automate simulation tasks, with the aim of converging to optimal designs. Optimus 6 (Noesis) and LS-OPT 7 (Livermore Software Technology Corp.) offer these capabilities through Design of Experiments (DOE) and Response Surface Modelling (RSM) (Myers and Montgomery 1995) and a selection of other modules such as global optimisation through stochastic methods, discrete variable and multiobjective optimisation, robust design and parallelisation for computational efficiency. Nastran is a well-established and powerful general-purpose finite element solver, including the BIGDOT optimiser (Vanderplaats 2004), also used by GENESIS, as well as a range of gradient-based methods in the Automated Design Synthesis program. SODA 8 (Structural Optimization Design Analysis, Acronym Software Inc.) is the commercial realisation of extensive research at the University of Waterloo, e.g. (Grierson and Chan 1993), implementing the Optimality Criteria method for least- weight section-size optimisation and including code-checking capabilities. SODA has benefited from the feedback of practising engineers across North America, as well as being used in academic research, e.g. (Kicinger et al. 2007). The application of continuous design domain or distributed material methods will be discussed in more detail in Chapter 3. However, it should be noted at this point that, in general, the discrete nature and large scale of building structures means they are not as suited as an automotive component to design by distributed material methods. The use of deterministic optimisation methods and automated search for assigning structural section sizes appears to be becoming increasingly frequent in design of complex structures. In-house and small-scale commercial software, as well as
5 http://www.mscsoftware.com/support/prod_support/nastran/documentation/rg_2005.pdf
6 http://noesissolutions.com/index.php?col=/products&doc=optimus
7 http://www.lstc.com/pages/lsopt.pdf
8 http://www.acronym.ca/
28 spreadsheets are often used for this task. Section 2.2.3 discusses specific examples in more detail.
2.2.3. Published literature on industrial applications
Section-size optimisation
As discussed previously, size optimisation is the easiest structural design task to define mathematically and has been the focus of vast amounts of academic research and literature. Whilst the use of optimisation and automated search for section-sizing in industry practice is not commonplace, it is increasing, primarily using deterministic or hybrid methods. The OPTIMA system developed at HKUST (Chan 2004) has been used for the section-size optimisation of designs for a number of tall buildings in Hong Kong. The software uses techniques based on combinations of OC methods, GA and exhaustive search, interfacing with most structural analysis software. Optimal member sizes are found for tall buildings of mixed construction, subject to all design performance criteria and various codes and requirements. In the design of the Kowloon Mega Tower , due to be the world's second tallest building at 474m, optimisation was used to minimise construction material cost whilst maximising value of floor space, considering the area of vertical walls and columns. Structural layout changes were recommended for the Park Central Development by the hybrid OC-GA method, affording greater cost savings than by section-size optimisation alone. An earlier collaboration between Ove Arup and Partners Hong Kong Ltd and HKUST in the size optimisation of the North East Tower, Hong Kong Station (Chan et al. 1998) considered optimisation by OC methods for minimum overall weight, minimum material cost and minimum overall cost allowing for the benefits of increased useable floor area. An overall cost saving of 9% was reported considering the third of these objectives. Various intuitive optimisation and automated section sizing procedures have been implemented by engineers at Arup in response to the demands of highly complex design specifications, a selection of which are reported in an internal Arup seminar report (Shea and Baldock 2004) or discussed below. Each of the 25,000 steel beams in the irregular long-span cellular moment frame of the Beijing National Swimming Centre (Stansfield 2004), (Bull 2004) was assigned
29 to one of three groups and a section size chosen from the allowable set for the group, subject to strength checks and constraints on slenderness ratio. A heuristic tool was developed, combining an Excel spreadsheet with a Chinese code checker purpose- written using Visual Basic macros and the Strand7 (finite element analysis) Advanced Programming Interface (API). From an initial design, analysis results guided the increase or decrease in section sizes, with the process repeated until convergence. Since there is no explicit objective function, this is a constraint satisfaction task using a deterministic, heuristic search. Minimum tonnage was attained using the smallest available sections in the initial design and using an optimised range of cross sections for each of the three groups. An iterative graphical approach was taken in determining the most efficient material distribution for meeting stiffness requirements in the design of the 299m Commerzbank HQ, Frankfurt (Wise et al. 1996).
Evolutionary Structural Optimisation (ESO)
Of the distributed material approaches, Evolutionary Structural Optimisation has received the greatest attention in the building industry. This can be attributed to the intuitive and readily comprehensible method through which it provides the designer with an insight into load-paths within the structure. The cases below use the method for the design of free-form or "concept" structures. ESO has recently been applied to the design of the Akutagwa River Side Building at Takatsuki JR station, in Japan (Ohmori et al. 2005). This is a rectangular four- storey office building, the construction of which was completed in April 2004. The designable domain defines the free-form external concrete wall/column system. A glass façade is integrated with this structural system to form the building's enclosure. Of particular note in this project is the fact that a discrete interpretation of the ESO design was not required, although a smoothing algorithm was applied. In 2002, a Japanese collaboration between architect Arata Isozaki and engineer Mutsoro Sasaki (Sasaki 2005) resulted in the development of a competition design for the Florence New Station, Italy . The design of the supporting structure for a 150m long, 26m wide and 15m high roof was developed using the Extended ESO (XESO) method (Cui 2003). Again, the ESO-derived topology was not strictly translated into a standard section discrete design, but in this case the shape was subdivided to form a structural grid of steel hollow sections. The structural engineer (Sasaki 2005) noted
30 the importance of the "man-machine interface" in absorbing the judgements of the designers in the development of the design solution. The Convention Hall at the Qatar International Exhibition Centre was also designed using ESO, by Mutsuro Sasaki and coworkers (Cui et al. 2005). In his recent thesis, Holzer (2006) uses ESO as a focal method in a discourse on the architect-engineer interaction in the early stages of design. An architectural project brief for a 12 storey tower building is used as an exploratory study. The application of ESO to the design of discrete external bracing systems for tall buildings will be explored in greater detail in chapter 3.
Parametric optimisation
A retrospective study was carried out on the design of the pedestrian footbridge at the Selfridges store in Birmingham, UK (Maher and Burry 2003) as a collaborative effort between Arup, RMIT and Future Systems architects. This served to illustrate the potential of integrating the CATIA V5 geometric model with finite element analysis software and the in-built optimiser to generate a range of parametrically diverse solutions. Parametric variables, such as location of cable restraint, horizontal and vertical location of the main tube as well as its diameter, were used to define families of configurations of the bridge. Optimisation used a combination of a local gradient-based method and simulated annealing, provided by the CATIA module (see below). Parametric studies in three or four free parameters took around seven hours, illustrating the premium on computational efficiency in such tasks. Repeated ad hoc computer analysis was used for similar parametric analysis of the roof structure of the Wellness Center for Shaw University, North Carolina USA (Place 2001). The roof is formed from two intersecting sections cut from inclined cylinders. In over a hundred distinct analyses, issues explored included varying element spacing, section shape and size, using different enclosure materials and considering possibilities for element removal to reduce complexity and blocking of light. Whilst this is not true algorithmic optimisation, it seeks to find the best design possible within the design domain defined by the constraints.
31 Non-parametric topology optimisation
Shea and Zhao (2004) detail the design optimisation of a recently installed noon-mark cantilever, carried out in collaboration with Eric Parry architects, in London. A Structural Topology and Shape Annealing (STSA) method, using discrete representation, was used to create a selection of novel designs subject to stringent structural and architectural constraints. Solutions could then be considered according to unmodelled criteria. This project demonstrated a number of challenges faced by applying optimisation methods in practice, including changing design specifications, incorporating architectural constraints and applying design codes.
2.2.4. Facilitating structural optimisation
The purpose of this section is to briefly present examples of software suites that are likely to facilitate the application of structural optimisation in building engineering practice in the future. It also considers project cases studies to which these tools and others presented in this thesis could be applied.
Software
CATIA V5 9 (Computer Aided Three dimensional Interactive Application, Dassault Systemes) is a multi-platform Product Lifecycle Management (PLM) / Computer- Aided Design (CAD) / Computer-Aided Manufacture (CAM) / Computer-Aided Engineering (CAE) suite with 3D parametric design functionality. CATIA includes structural analysis capabilities and an optimisation module capable of gradient-based search and simulated annealing. GenerativeComponents 10 (Bentley Systems) is described as a parametric and associative design system, capable of capturing and displaying design components and their interrelationships. The intention to provide a tool for both architects and engineers is important in improving documentation exchange and common understanding, which could, in turn, aid the uptake of optimisation.
9 http://www-306.ibm.com/software/applications/plm/catiav5/
10 http://www.bentley.com/en-US/Markets/Building/GenerativeComponents.htm
32 Parametric optimisation case studies
During the design of the Swiss Re Building at 30 St Mary Axe, London , the external form was defined parametrically in terms of shape variables (Foster and Partners 2005). This approach allowed Foster and Partners to rapidly regenerate complex geometric models, which would otherwise require days to construct. Bentley Systems' GenerativeComponents software was used in this project. The method has many advantages, including aiding information exchange between architects and engineers, facilitating panel definition for ease of construction and offering the potential of integrating an iterative optimisation routine to automatically explore the defined design space. Another example of potential for parametric optimisation can be seen in the design development of the SAGE Music Centre, Gateshead (Cook et al. 2006). Parametric modelling of geometric associations between member segments facilitated exploration of primary and secondary arch profile configurations for the structure's free-flowing roof form. Although computational optimisation was not used in either of these projects, it would have been a relatively straightforward and potentially beneficial extension.
Non-parametric discrete optimisation and design generation case studies
The superstructure of the 234m tower of the CCTV Headquarters, Beijing, China is essentially a continuous braced tube, with patterned diagonal bracing on a regular grid of columns and beams (Carroll et al. 2005). The bracing was required to visually express the force distribution within the structure, whilst providing sufficient stiffness during construction and service as well as robustness and redundancy in the event of the removal of key elements. The bracing mesh was manually and iteratively modified in response to the observed force distribution in the analysis model and close consultation with the architect. Mesh density was increased in areas of high force and reduced where forces were low. The process was complicated by the highly indeterminate behaviour of the structure, with substantial changes to force distribution, stiffness and dynamic behaviour resulting from changes to the bracing patterning. With over 10,000 elements in the full structural model, analysis time was also significant. An optimisation routine or automated method of implementing the iterative, heuristic bracing pattern modifications would have been valuable on this project, although incorporating all necessary considerations would have been a major
33 challenge. A stochastic component within an automated search could have been useful in avoiding convergence to locally optimal solutions. The proposed 122 Leadenhall Street building is a 48-storey, 225m wedge-shaped tower with a braced perimeter tube stability system. Although the configuration submitted in planning applications adopted a regular pattern with a diagrid on the inclined face, in the early stages of the project a large number of bracing configurations were sketched by structural designers, as seen in figure 2.4. These were not subjected to any form of structural evaluation, but simply presented for consideration on aesthetic grounds. However, a tool for rapid generation and evaluation of diverse bracing schema could have been very informative on this project and given weight to the case for one design over another.
Figure 2.4: Concept sketches for bracing design of 122 Leadenhall St. Building (reproduced by kind permission of Chris Neighbour, Arup)
The two examples in this section were motivating factors for the research in this thesis, as they show how topology optimisation could offer significant benefit if appropriate, practical tools were available.
2.3. CONCLUSIONS
High potential cost benefits, relatively small task size and the availability of suitable commercial software have driven a significant number of cases of the application of
34 structural design optimisation in the automotive and aerospace industries. Commercial software for continuous topology optimisation is closely linked to current research and generally adopts cutting-edge techniques. However, successes in automotive and aerospace industries have not been matched in the building sector. It is apparent that assignment of member section-sizes is the most readily solvable of structural optimisation problems, due to its mathematically well-defined nature. This is reflected in the vast amount of academic research in the area and its prevalence in existing practical examples of structural optimisation. Nevertheless, this class of task is not always straightforward, with issues of local minima, discrete variables on account of section catalogues, multiple loadcases and constraints, multiple variables for a given cross-section, potentially thousands of variables and possible non-linearity (e.g. in the case of concrete structures). Despite examples in high-profile projects and increasing use in recent years, size optimisation is not common-place in structural design practice. Further, the efficient integration of size optimisation into topology optimisation routines presents a relevant challenge that is arguably under-researched. Increasing use of electronic document interchange between architects and engineers, and CAD software with appropriate capabilities, has raised the potential for parametric design investigation and ultimately computational parametric optimisation. Optimisation modules exist within tools such as CATIA, or alternatively could be purpose written. Since decisions in conceptual and topological design have a great impact on final design efficiency (Deiman et al. 1993) and the exploration of novel designs at this stage is a difficult task, research in the field of non-parameterised topological design is highly justified. With the exception of the ESO examples detailed previously, the author is unaware of cases in which this task has been successfully performed on an entire building in practice. It is likely that successful examples will fuel further interest and accelerate the uptake of design optimisation in this sector.
2.4. JUSTIFICATION OF CASE STUDY
A large proportion of the examples outlined in the above discussion have featured high profile tall buildings, generally irregular in some shape or form. Such structures are often iconic for a city and become flagship projects for the architects and engineers. The challenges presented by pushing the boundaries of structural
35 engineering demand innovative solutions. Topology optimisation in tall buildings is therefore an appropriate choice of case-study problem for the methods considered in this thesis. In very tall buildings, providing lateral stability through a central concrete core can require unacceptable loss of lettable floor space. For this reason, various forms of external stability systems are popular in this class of structure. A steel tubular system consists of columns and beams defining an orthogonal grid around the perimeter of the building. In isolation, this vierendeel framework relies primarily on its bending stiffness and is therefore susceptible to unacceptably high displacements. Diagonal bracing is generally used to triangulate the structure, so that loads are carried primarily axially, thus greatly increasing the stiffness. The bracing configuration often plays a prominent role in the aesthetic impression of the building, examples including the Swiss Re Building and the Bank of China Tower, Hong Kong (Robertson 2004). However, an efficient design can offer great savings in material and construction cost, as well as increasing the potential revenue from letting of floor space. This thesis therefore adopts as its central test problem the design of bracing for lateral stability in tubular steel building frameworks. As discussed, this choice is appropriate on account of its relevance to practical building design, but also as a common proof- of-concept problem in the academic literature. This section will highlight a selection of previous academic work tackling the bracing design problem. Continuum topology optimisation has been previously explored by Mijar et al. (1998) and Liang et al. (2000) and whilst these studies exhibit limitations, as will be discussed presently, they provide useful benchmark cases. Mention should also be made of a practice-driven study by Holzer (2006) who uses ESO for the design of a tower structure from a real architectural project brief, subject to lateral and other loads. Kicinger (2004) used the design problem to illustrate methods developed in distributed evolutionary design, considering various types of bracing and beam and support fixity. Arciszewski (1994), (Arciszewski et al. 1997) explored the learning of design rules through examples in this field, also postulating the possibility of parameterising bracing designs according to the size of a bracing unit (number of bays and storeys) and the type of bracing therein (e.g. K or X bracing) in conjunction with, for example, a Genetic Algorithm.
36 Bracing design also features as part of a number of wider-ranging tasks in optimisation research, such as those addressed by Liu (2000) and Khajehpour and Grierson (2003). Neidle-Cornejo (2004) conducted a comparative study into forms of bracing design in tall buildings.
2.5. CONTEXT OF RESEARCH CONTRIBUTIONS
The contributions of this thesis to the field of structural optimisation research are stated at this point, in advance of a full presentation of the work undertaken.
Evolutionary Structural Optimisation
The primary objective of this investigation is to demonstrate means of making ESO more useful to the building industry. The grouping of elements is a significant development on previous ESO research in topology optimisation and is done in two ways: (i) to have the same thickness. This form of group definition is required for the integration of the topological ESO process with optimisation of domain thickness. To the author's knowledge, previous integration of thickness and topology optimisation in ESO has involved only linear scaling of the thickness of a single designable domain region to meet a displacement constraint (Liang et al. 2000a). In the current research, at each iteration of the ESO process, the thickness in each group is adjusted to obtain uniform average strain energy density in each group, whilst still meeting the displacement constraint. The Bi-directional Evolutionary Structural Optimisation algorithm is modified to adapt to the constraint tracking scenario whilst moving towards efficient designs retaining a low proportion of the full set of possible elements. (ii) to be removed together. This has been done previously with pairs of elements grouped to define symmetry in a vertical centre line (Liang et al. 2000). However, defining larger groups allows designs to be developed according to preconceived aesthetic requirements such as pattern repetition or more complex symmetry. Increased practicality is also achieved by focusing on displacement constraints as opposed to overall compliance of the structure. Whilst ESO with displacement constraints is well established (Liang et al. 2000a), this was not applied to previous work in bracing design (Liang et al. 2000), where compliance was considered instead.
37 Pattern Search and Optimality Criteria
This section demonstrates how pattern search can be applied to a parameterised, grid- based topological design problem in the context of a live industrial project. To the author's knowledge the example is unique in the scale of the topology problem addressed "live". It illustrates the selection of a method appropriate to the task being considered, following the "problem-seeks-solution" model discussed by Cohn (1994) as opposed to the "solution-seeks-problem" approach more commonly seen in academic research. The documentation of topology optimisation on a "live" project is a valuable contribution in itself, especially with the associated discussion of the design issues encountered during the project development to which the optimisation model and method must adapt. The study includes using optimisation in a parametric constraint sensitivity investigation, as well as presenting a selection of high- performance, locally optimal designs at different stages of the design process. From a single starting point, the potential for generating a range of designs by randomly selecting the order of exploratory moves is seen. However, diversity of designs is shown to increase by using different starting points. An innovative method of constraint handling is developed, scheduling the convergence towards constraint boundaries through the use of penalty functions, preventing premature process termination due to acceptance of disadvantageous design modifications. Computational efficiency is highlighted as being important and the classic Hooke and Jeeves Pattern Search algorithm (1961) is modified in order to improve this. An efficient means of integrating topology and size optimisation is established, finding an approximation for the optimal set of section sizes at each topology iteration, thus minimising the amount of additional structural analysis required. This approach effectively considers strength and stiffness requirements in the member sizing operation.
Genetic Programming using design modification operators
Chapter 5 presents an innovative application of Genetic Programming to structural topology design. This approach is a unique and powerful development since the tree representation includes design modification operators as functions at internal nodes, rather than an encryption scheme for mapping genotype to phenotype (Yang and Soh 2000).
38 With appropriate optimisation parameters, consistent convergence to a global optimum is observed on a small benchmark problem, with acceptable computational efficiency. On a scale that is more realistic from an industrial perspective, novel high performance designs are obtained, including one that outperformed known designs for the defined optimisation model. Simple but powerful extensions are detailed to allow tighter control over design aesthetics, with the aim of empowering the user and improving convergence on account of the reduced design space. This form of design tool holds great potential for assisting designers in the conceptual design stage, when the ability to generate and evaluate a diverse range of design solutions is highly beneficial. The general approach shown is likely to be applicable beyond bracing design tasks.
39 3. Continuum topology optimisation of braced steel frames
3.1. INTRODUCTION
As discussed in the preceding chapter considering the state of the art in research and practice, Evolutionary Structural Optimisation (ESO), in its various forms, has gained significant interest due, in part, to its simplicity, elegance and ease of implementation. This chapter considers benchmark problems proposed by Mijar et al. (1998) and Liang et al. (2000), to address the following:
Research Question: – How can the practicality of ESO be improved to make it more useful to the building industry?
Proposals: use practical design criteria and objectives, such as displacement constraints for tall buildings, with design heuristics to meet aesthetic criteria of pattern repetition and symmetry. attempt to improve global optimality of solutions, independent of domain thickness, mesh density and initial design. introduce simultaneous topology and thickness optimisation of defined designable regions.
3.2. BACKGROUND
3.2.1. Method overview
The following description of the canonical form of ESO and the argument presented in section 3.2.3 is based on the seminal book by Xie and Steven (1997). The method offers parameter free optimisation of shape and topology by gradually removing inefficient material from a continuous representation of a structural domain.
40 Finite element analysis is performed on an initial design model, in which a material continuum is divided into a fine mesh of finite elements, generally quadrilaterals for a two dimensional problem. The model includes loads and boundary conditions. An efficiency metric, referred to as the sensitivity number, α, is defined, examples being average von Mises stress or strain energy. The sensitivity number can then be calculated for each element and a selection of the elements with the lowest sensitivity numbers, according to a defined criterion, are then removed from the structure. Examples of removal criteria are elements with sensitivity number less than a prescribed proportion of the maximum value observed within the structure, referred to by Xie and Steven (1997) as the rejection ratio, RR , or a prescribed number of elements with the lowest sensitivity number. The reduced structure is reanalysed and the process repeated. When no further elements can be removed according to the rejection ratio removal criterion, RR is increased by adding to it a defined evolutionary rate, ER , thus allowing the removal process to continue. Termination is arbitrarily defined by, for example, a set proportion of the original material having been removed or a displacement or compliance constraint being violated. Xie and Steven (1997) note that the evolutionary approach gives the designer the opportunity to select any intermediate stage in the process as a basis for further design development.
3.2.2. Addition considerations and extensions
Li et al. (1999) demonstrate equivalence in ESO between stiffness, using some form of strain energy as sensitivity number, and stress based element removal criteria. This is significant through the implication that stiffness optimisation will yield strength efficient designs. From a practical perspective, von Mises stress provides an alternative to considering strain energy, or strain energy density, for stiffness optimisation if the element stiffness matrix formulation is not known when using a commercial package. The paper further notes that some differences in evolved topology may occur, on account of numerical error, such as may arise from different numbers of Gauss points used to estimate the von Mises stress in an element. Despite the equivalence between strain energy and stress based element removal, the research
41 described in this chapter will be based on the former value, in line with previous, closely related work. Checkerboarding is a commonly observed phenomenon in continuum structural topology optimisation (Sigmund and Petersson 1998). Patterns of elements and voids may emerge in a checkerboard formation, presumed to be due to numerical errors in the finite element approximation causing the design criterion to be alternately overestimated and underestimated. Such regions are unacceptable in practice, since they inhibit interpretation of the design as a discrete or manufacturable structure. Various solutions have been proposed, notably: – use of higher order elements (Manickarajah et al. 1998), which greatly increases computational time. – cavity control (Kim et al. 2000), whereby the number of cavities within the final topology is defined. – perimeter control (Yang et al. 2003), in which, for Bi directional Evolutionary Structural Optimisation (BESO, detailed in 3.2.4), element addition and removal is restricted to prevent the total topological perimeter length from exceeding a prescribed value. Perimeter control is also capable of eliminating mesh dependency and provides influence over topological complexity, hence manufacturability and, by extension, cost of design. The current research uses a first order weighted averaging algorithm as detailed by Li et al. (2001) to avoid checkerboarding. For square elements, as used for the problem considered in this chapter, this reduces to averaging the sensitivity numbers of the element in question and its immediate neighbours. A weighting coefficient of 4 is assigned to the element itself, 2 to active elements with a common edge and 1 to active elements with a common corner, as shown in figure 3.1.
1 2 1
2 4 2
1 2 1
Figure 3.1: Weighting factors used for averaging sensitivity numbers across elements to avoid checkerboarding.
42 3.2.3. Evolutionary Structural Optimisation (ESO) for stiffness and displacement constraints
Linear static finite element analysis solves the equilibrium equation: [K]{u}= {P} Eq. 3.1 where: [K] = global stiffness matrix; {u} = nodal displacement vector; {P} = nodal load vector; The strain energy, or compliance, of the structure as a whole can be expressed as: 1 C = {}{}P T u Eq. 3.2 2 or as the sum of the strain energy of each constituent element of the finite element model.
N N N 1 i T i i C = ∑{}u []K {}u = ∑ci = ∑αi Eq. 3.3 2 i=1 i=1 i=1 where the sub or superscript i indicates that the term refers to the ith element in the structure. Thus, the sensitivity number for this problem, αi, is the element strain energy. In cases where the designable domain is divided into unequal elements, either in terms of area or thickness, strain energy density should be used as the sensitivity number on which element removals are based, obtained by dividing the element's strain energy by its volume or weight. The change in the global structural stiffness matrix by removing a single element from the structure is exactly the negative of the corresponding element stiffness matrix: [ K] = [K * ]− [K] = −[K i ] Eq. 3.4 where K* is the global stiffness matrix of the resulting structure . Making the approximation that element removal does not change the load vector { P} and by neglecting a higher order term: { u} = −[K]−1[ K]{u} Eq. 3.5 By extension, the change in compliance of the structure is thus predicted to be identical to the strain energy in the element to be removed:
43 1 1 T C = {}{}P T u = {ui } [K i ]{ui } Eq. 3.6 2 2 It is therefore proposed that by removing elements with the lowest strain energy at each iteration of the evolutionary process, optimal structures will be developed, capable of meeting a compliance constraint with minimal material volume. It is more common to consider displacement rather than compliance constraints in the design of building structures. A set of j displacement constraints applied to a structure may be expressed as:
* u j ≤ u j Eq. 3.7
* where uj is the prescribed limit for | uj|. The principle of virtual work can be used to demonstrate that the change in uj due to the removal of element i can be expressed as:
ij T i i u j = {u } [K ]{u }= α ij Eq. 3.8 where { uij } is the element displacement vector resulting from a unit load in the relevant direction, applied to the node at which displacement is constrained and { ui} is the element displacement vector corresponding to the real applied loads. It is of note that αij may be either positive or negative. Previous work (Xie and Steven 1997) suggests that the modulus of this value should be used as the sensitivity number in determining element removals, although it is actually more appropriate to select elements with the most negative value of αij for removal to effect the most negative possible change in displacement of the structure as a whole. Hereafter, this sensitivity number will be referred to as the cross strain energy with the corresponding cross strain energy density found by dividing by volume or weight. The sum of the cross strain energy in every element in the structure, including the orthogonal framework, should be equal to the displacement at the point of interest, in the direction of the virtual load. A central assumption of the method described is that the change in distribution of displacements within the structure is negligible when a small number of elements are removed, otherwise it is not possible to accurately predict which element removals are most appropriate. The number of elements to remove at each iteration therefore becomes an important process parameter, influencing the trade off between numerical accuracy and computational efficiency.
44 3.2.4. Bi directional Evolutionary Structural Optimisation (BESO)
Different algorithms have been proposed for determining element rejection and addition in BESO (Young et al. 1998), (Querin 2000a). The algorithm developed in this research adopts a fixed modification ratio, MR, (0.01), which dictates the approximate total number of removals and additions. The relative proportion of removals and additions is governed by the proximity of the maximum lateral displacement in the current design to the constraint value. The addition ratio, AR it , in iteration it is: u j AR = min ,1 Eq. 3.9 it * 2u j
The number of elements to be added, NA it , is then the product of the addition ratio, modification ratio and maximum number of elements, N max :
NA it = AR it × MR × N max Eq. 3.10 and by extension, the number of elements to be removed, NA i,, is:
NR it = (1− AR it )× MR × N max Eq. 3.11 Elements to be added are selected by working through the list of elements in decreasing order of sensitivity number, placing new elements immediately adjacent to currently active elements where possible, until the quota has been attained. According to this algorithm, if a design meets the displacement constraint exactly, an equal number of elements are added and removed.
3.3. BENCHMARK PROBLEM : STRUCTURAL MODEL SPECIFICATIONS
Figure 3.2 defines the two dimensional skeleton steel framework of the two bay, six storey structure to be considered in this benchmark problem, previously used in continuum topology optimisation research by Mijar et al. (1998) and Liang et al. (2000). Mijar et al. (1998) sized the base framework to act alone in carrying vertical live and dead loads. For the ESO process, a continuum mesh of square four noded quadrilateral elements is added to the framework, with the two components acting together to resist lateral loads. Each bay storey unit is 15 elements wide and 9 elements high. This problem is unusual in the context of ESO literature, since it includes a non designable framework. Connection to the base framework is at corner
45 nodes of the removable elements. Whilst connection of a quadrilateral to the framework is always along the side of the removable element, a string of quadrilaterals may be connected solely at a corner node, causing stress concentrations and moment free interfaces.
1 1 64.1kN 1N
7 8 7 128.2kN 2 2
8 9 8 122.7kN 3 3
10 11 10 117.3kN 4 4
11 13 11 @ 6 3.658m 103.6kN 5 5
12 13 12 90.4kN 6 6
12 14 12
6.096m 6.096m Figure 3.2: Benchmark problem specifications (Mijar et al. 1998). Real loads (left), including member groupings and geometric specifications, and virtual load (right). ASCE standard section specifications (below). 1 W8x21 2 W8x28 3 W10x26 4 W12x26 5 W14x26 6 W13x22 7 W10x17 8 W8x10 9 W12x19 10 W12x14 11 W14x22 12 W16x26 13 W16x31 14 W24x62
3.4. OPTIMISATION FOR MINIMAL MEAN COMPLIANCE
The problem addressed by Liang et al. (2000) was to maximise the performance index, PI, expressed as: C W PI = o o Eq. 3.12 Cit Wit where:
Wo, Wit = initial and current (at the it th iteration) weight of the continuum design domain;
46 Co, Cit = initial and current mean compliance of the braced framework, calculated according to equation 3.2. The continuum design domain took a thickness, t = 0.0254m, with material properties of Young's modulus E = 200GPa, Poisson's ratio = 0.3. The optimisation process is terminated when the performance index falls below unity. An attempt to reproduce the design topology published by Liang et al. (2000) is shown in figure 3.3. Although the exact shape is not identical, the topology is nearly the same and performance is close to identical. The former inconsistency may be attributed to subtle variations such as numerical inconsistencies in different analysis software. It should be noticed that in both cases the top storey has had all 2D elements removed, except for at beam column intersections, where they serve to add significant stiffness to the region and substantially reduce rotation and lateral displacement of the top corner node. This effect would not be transferred to a discrete interpretation of the design, as demonstrated in section 3.8.
Figure 3.3: Design topology of Liang et al. (2000): δ=0.024, element retention = 22% (left) Comparative result to Liang et al. (2000) topology: δ=0.024, element retention = 23% (right)
3.5. OPTIMISATION FOR DISPLACEMENT CONSTRAINT
Although the work of Liang et al. (2000) uses a performance index based on mean structural compliance, mention is made of the more practical performance metric of maximum lateral displacement, quoting a value of 0.024m for the best design. This value is substantially below any standard maximum lateral displacement limit (e.g.
47 h/500 = 0.044m), but for purposes of comparison 0.024m will be adopted in the subsequent discussion as a target or constraint value for maximum lateral displacement. Quality of solutions can then be assessed by comparing weight of bracing material required to meet the constraint. A simple optimisation model could therefore be stated as: Minimise: N Eq. 3.13 Subject to: δ ≤ δ * Eq. 3.14 where: N = total number of elements retained in the current design; δ, δ* = current and largest permissible maximum lateral displacement respectively. Careful consideration of the strain energy in pairs of elements grouped together by the horizontal symmetry condition is required. Liang et al. (2000) suggest that two load cases should be considered, identical in magnitude but acting on opposite faces. Elements with the lowest maximum strain energy in the two loadcases in either of the pair are removed. This approach may be beneficial for fully stressed design, for which every element should be at full capacity in one or more loadcases. However, as previously described, we wish to remove symmetric pairs of elements to minimise the increase in displacement (or compliance). For the current task this entails summing the strain energies of each element of the pair. It is thus only necessary to consider a single loadcase, since the strain energy in an element in loadcase 2 will be identical to that of its symmetry partner in loadcase 1. Reconsidering equation 3.8, the lowest predicted compliance increase is achieved by removing pairs of elements with the lowest combined strain energy under a single loadcase. Figure 3.4 compares the elements removed according to the requirements of Liang et al. and the requirement proposed here. Elements are removed from the same area of the structure, but with some difference in the exact elements removed. An increase in maximum lateral displacement of 6.8x10 6 m is predicted from removing elements selected on the basis of the sum of strain energies, compared to 7.4x10 6 m from removing elements selected on the basis of maximum strain energy of either of the elements in the pair. Figure 3.4: Elements removed in the top left bay unit in the first iteration, based
48 As a consequence, differences in topology occur, as observed in the designs presented in figure 3.5. Although the above argument suggests that topologies evolved by removing elements with the smallest sum of strain energies should be more efficient, this will not necessarily be true in all cases, due to differences between predicted and actual displacement and the search path that is taken.
Figure 3.5: ESO results with element removal determined by cross strain energy. 25.4mm designable domain, 8 elements removed per iteration (left: maximum of sensitivity number in pairs of elements, right: sum of sensitivity number in pairs of elements) The Bi directional Evolutionary Structural Optimisation algorithm presented previously in this chapter was used to tackle the same problem. Bi directionality has the obvious advantage that elements that were removed early in the process may be reintroduced if found to be advantageous at a later point, rather than pursuing a sub optimal path. It also allows the process to commence from virtually any design, subject to a minimal number of elements being present. Hence a more thorough design space exploration is afforded by BESO. Figure 3.6 shows the best designs evolved starting from a full initial continuum designable domain, a minimal configuration, with 2D elements only present around the 1D structure and two randomly generated initial configurations, for bracing element thicknesses of 2.5mm, 5mm, 10mm and 25.4mm (the last of these being used by Liang et al. (2000)). It is logical to expect that a change in thickness and hence the relative stiffness of frame structure and the 2D elements of the continuum domain will cause corresponding changes in optimal topology. Analysis of randomly generated initial configurations is made possible by the fact that, for the purposes of the
49 structural model, elements to be removed are actually assigned a thickness value of 10 10 m, thus avoiding floating elements causing matrix singularities, whilst not significantly affecting the results. A chevron form, or inverse 'V', in at least the lower section of the evolved bracing structure is common to all designs. The exact topology exhibits some variation with both starting point and domain thickness, but the most common and practically realisable solutions are the single chevron (25mm, Full Start; 10mm, Random Cloud 2) and double chevron designs (5mm, Full Start, Perimeter Start and Random Cloud 2; 10mm, Full Start, Perimeter Start and Random Cloud 1), the latter either being two units of equal height, or a four storey chevron above a two storey chevron. Comparing the number of elements retained in the optimal solutions derived using an element thickness of 25.4mm in figure 3.6 against the solution of Liang et al. (2000) in figure 3.3, all of the current solutions are more efficient in meeting the lateral displacement constraint on the top corner node of 0.024m. The best solution, derived from the Full Start configuration, retains 14% of original elements, compared to 22% in the benchmark solution, representing a material saving of 37%. This is primarily due to retaining bracing in the top storey. Figure 3.6 suggests that with the BESO algorithm used, although volume varies considerably according to the domain thickness, the optimal evolved topology is relatively insensitive both to change in thickness and starting point. One point of note is that although the lateral displacement of the top storey node is within the prescribed limit, it is possible that the global maximum lateral displacement may be lower in the structure. This could be addressed by placing displacement constraints at every storey level, or considering inter storey drift, in a multiple constraint formulation as presented by Xie and Steven (1997). However, this will require additional loadcase definitions, lengthening analysis times. Additionally, care must be taken in task formulation, since elements in upper storeys can have negligible sensitivity values with respect to constraints on displacement in lower storeys and be inappropriately removed.
50 Figure 3.6: Minimum volume designs satisfying the displacement constraint derived by BESO for varying domain thickness and starting configuration
3.6. INCLUDING OPTIMISATION OF DOMAIN THICKNESS
The topological diversity observed in designs resulting from variation of domain thickness indicates that solutions are not globally optimal. In this section we explore the concept of varying domain thicknesses in such a way as to maintain the maximum
51 lateral displacement of the structure at its constraint value throughout the ESO process. The continuous designable domain elements may be divided into a number of groups, with a single thickness variable used for all elements in any one group. Variation of domain thickness was previously considered by Liang et al. (2000a), but in the simple context of exploiting the linear relationship between displacement and thickness in a structure consisting purely of a single designable domain. This approach is invalid for any design which includes a non modifiable framework or multiple groups of elements which may be assigned different thicknesses. The section sizing of the orthogonal framework was previously conducted based on vertical loads. For the purposes of this study, these sections remain fixed, essentially making the conservative assumption that diagonal bracing elements do not contribute to the vertical stability of the structure. However, it would be straightforward to integrate optimisation of framework section sizes with thickness and topology optimisation of the designable domain, by minor extension to the method outlined below. Domain thickness optimisation is incorporated into the process flowchart of figure 3.7. Five iterations of thickness modification are performed to find near optimal thicknesses for the starting topology. A further single loop of thickness modifications is performed at each iteration to continuously update the domain thicknesses as elements are added and removed. This is only intended to give an approximation to the optimal thicknesses for the current topology. However, this is justified by the relatively small thickness changes occurring and the fact that this approach requires no more analysis time than for fixed thickness ESO. Performing multiple iterations with reanalysis of the thickness optimisation loop would greatly increase the run time for the ESO process. Section 4.8.2 of this thesis demonstrates the simple result that, when considering a single displacement constraint, optimal solutions require equal (cross) strain energy density in all structural members, or in the current case, equal average (cross) strain energy density in all groups of elements. This is derived by considering Optimality Criteria (Borkowski and Jendo 1990). Working on the simple approximation that the (cross) strain energy in the orthogonal framework is unaffected by modifications to thickness of designable domain groups, a discrepancy between the maximum lateral
52 displacement and the constraint value must be absorbed by changing the strain energy in the designable domain. The process adopted is outlined in the flowchart of figure 3.8 and corresponding calculations defined by equations 3.14 to 3.16. The BESO algorithm presented in section 3.2.2 determines the proportions of elements to be removed and added on the basis of the ratio of the maximum lateral displacement in the current design to the corresponding maximum permissible value. However, in the current method, including thickness optimisation, the algorithm aims to maintain displacement exactly at the constraint value, hence the previous approach is unsuitable. Since bi directionality is considered beneficial, the iterative process of element modification is split into two phases. In each of 5 iterations, a set number of elements with the lowest cross strain energy density are removed. Thereafter, for 2 iterations, the same number of elements are added. Hence a staggered bi directionality is introduced, with a general trend for reduction in the number of elements. Choice of termination criterion is arbitrary, but may be defined as the point at which a group thickness exceeds a prescribed value, a prescribed proportion of the original elements have been removed, or volume exceeds a set value or proportion of the original optimised volume.
53 START
Write structural model
Analyse structural model
Retrieve nodal displacement information
Calculate strain energy in each element, using element stiffness matrix
Calculate strain energy density in symmetric pairs and groups of elements
Adjust group thicknesses for uniform average strain energy density in all groups and meeting displacement constraint
GO TO THICKNESS OPTIMISATION LOOP
NO Initialisation complete?
YES
Topology Phase?
Removal Addition
Remove 10 element pairs with Add 10 elements adjacent to lowest strain energy density element pairs with highest (average over both elements) strain energy density
NO Termination criterion met?
YES END
Figure 3.7: Process flowchart for ESO with domain thickness optimisation
54 ENTER THICKNESS OPTIMISATION LOOP
Calculate required cross strain energy in bracing from current value and violation of constraint (equation 3.14)
Required cross strain energy < 0?
YES NO
Calculate required cross strain energy density in bracing from required cross-strain energy and total volume (equation 3.15)
Modify group thicknesses to meet target cross strain energy density requirement (equation 3.16) Double thickness of all element groups
Update total volume and hence required cross strain energy density ADJUSTMENT LOOP ADJUSTMENT ITERATIVE THICKNESS ITERATIVE NO Thickness change < 1% for all groups?
YES
EXIT THICKNESS OPTIMISATION LOOP Figure 3.8: Flowchart for domain thickness optimisation loop
* XSE bracing _ reqd = XSE bracing _ current − (u j − u j ) Eq. 3.14
XSE bracing _ reqd XSED proj = Eq. 3.15 Vtot
XSED t '= t g Eq. 3.16 g g XSED proj where:
XSE bracing_current , XSE bracing_reqd = current cross strain energy in the designable bracing domain and corresponding cross strain energy required to meet the displacement constraint, assuming the cross strain energy in the orthogonal framework to be unchanged by domain thickness modifications;
XSED proj = average cross strain energy density predicted to correspond to the required cross strain energy, calculated using the current total volume of bracing elements, Vtot ;
XSED g = current average cross strain energy density in group g; tg, tg' = current and revised thicknesses for group g, respectively.
55 The cross strain energy density corresponding to the required cross strain energy is not fixed, since it will be affected by group thickness modifications changing the domain volume. An iterative loop is therefore adopted to find the thickness required for each of the designable domain element groups, such that: – all groups have the same projected cross strain energy density – the sum of the total projected cross strain energy in the designable domain and that observed in the orthogonal framework is equal to the displacement constraint value. The progress of the process detailed using a single thickness variable for all elements is charted in figure 3.9. No elements are added for the first 5 iterations while the thickness required to meet the displacement constraint with all elements active is found. As the overall trend for element removal proceeds, the volume decreases slightly, before increasing above its original value. Until iteration 275, bracing volume remains within 10% of the initial volume. The structure adapts to removal of critical elements around iteration 280, after which, until iteration 310 the volume remains around 25% above its initial value. Topology plots beneath the graph demonstrate that it is unrealistic to use minimum volume as the sole measure of efficiency since the minimum volume designs, between iteration 50 and 100, do not offer a discrete interpretation. In practice, a chart such as this should be inspected alongside the design topologies in order to select appropriate topologies for further consideration and discrete member interpretation. In this case, the topology of iteration 300 appears a good compromise between structural efficiency and discrete interpretability.
56 10.0 Thickness (/t(0)) Displacement (/d*) 9.0 Volume (/V(0)) Elements (/E(0)) 8.0
7.0
6.0
5.0
Normalised Value 4.0
3.0
2.0
1.0
0.0 0 50 100 150Iteration 200 250 300 350
Figure 3.9: Process history for simultaneous topology and domain thickness optimisation with a single thickness group. Figure 3.10 shows the most appropriate topologies evolved by allowing domain thickness to vary in 1, 3 or 6 groups. It should be restated that these are not minimum volume topologies, but rather those that are most suitable for discrete interpretation. The designs are all similar in performance and appearance, adopting a compound 'X chevron' form, similar to that evolved from the random cloud designs with a fixed thickness of 2.5mm (figure 3.6). Material volume for the three designs in the table is similar to those evolved using a fixed domain thickness of 2.5mm, but are arguably more readily interpreted as discrete structures. It is interesting to note that there is not a continuous reduction in thickness with ascending height in the evolved topologies. For example, with 3 groups, more elements are present in the middle group than either of the others, hence thinner elements in this group achieve the same average cross strain energy density as other groups in the structure.
57 Evolved Full domain topology optimised optimised thicknesses thickness No. of groups 1 No. of iterations 310 Volume 0.196 0.00061m 0.00331m Displacement 0.0249 No. of elements 358
No. of groups 3 0.00033m 0.00320m No. of iterations 386 Volume 0.195 0.00059m 0.00289m Displacement 0.0245 No. of elements 310 0.00076m 0.00562m
0.00023m No. of groups 6 0.00311m 0.00041m No. of iterations 336 0.00415m 0.00053m Volume 0.208 0.00515m 0.00063m Displacement 0.0248 0.00427m 0.00073m No. of elements 228 0.00858m 0.00078m 0.00901m Figure 3.10: Best designs derived by simultaneous thickness and topology optimisation, with one, three and six thickness groups.
3.7. INCLUDING ARCHITECTURAL REQUIREMENTS AND PATTERN
DEFINITION
The form of solutions presented by ESO can frequently be irregular or inelegant, as seen in some of the designs presented to this point. Often an architect or client will have a preconception of the form, symmetry or degree of repetition that is sought in bracing forms. This chapter has previously considered elements in pairs defined by horizontal symmetry. It is a simple extension then to consider larger "families" of elements, defined by vertical symmetry or repetition, which will be simultaneously removed from or added to the structure. Defining groups of elements corresponding to lines of mirror or translational symmetry or patterns of repetition for simultaneous
58 optimisation of domain thickness presents the opportunity for high utilisation amongst elements retained in final solutions. The topologies evolved from defining three different symmetry systems are shown in figure 3.11: (A) reflection symmetry with mirror line at half building height, (B) translational symmetry with two bay three storey unit repeated once vertically, (C) translational symmetry with two bay two storey unit repeated twice vertically. All designs use the simultaneous topology and thickness optimisation strategy discussed in the previous section, with groups corresponding to the sections defined by the symmetry patterns. Alternative topologies are shown for case (A), occurring at different stages of the evolutionary process. When defining translational symmetry up the building, as in cases (B) and (C), the chevron patterns evolved in earlier investigations in this chapter are again observed. However, in the symmetry study with thickness optimisation, the lower chevrons are thicker with the same number of elements, as opposed to having more elements than those higher in the structure.
59 Evolved topology optimised thicknesses A No. of groups 2 Iteration 239 0.0027m Volume 0.218 Displacement 0.0246 No. of elements 352 0.0048m
No. of groups 2 Iteration 287 0.00785m Volume 0.427 Displacement 0.0242 No. of elements 204 0.01752m
B No. of groups 2 Iteration 277 0.00460m Volume 0.265 Displacement 0.0206 No. of elements 240 0.00874m
C No. of groups 3 0.00416m Iteration 302 Volume 0.204 0.00769m Displacement 0.0250 No. of elements 186 0.00936m
Figure 3.11: Evolving topologies with prescribed symmetry, using simultaneous thickness optimisation of appropriate groups
3.8. DISCRETE INTERPRETATION OF CONTINUUM TOPOLOGIES
The topological designs presented in this chapter to this point require discrete interpretation before they can be considered feasible structures. It is possible that the performance observed in the discrete interpretation may be quite different from that of the ESO design. The majority of topologies discussed, including that of Liang et al. (2000) and some notable others appear in discrete form in figure 3.12. Bracing members will all take the form of circular solid sections, with members assigned to
60 groups in the same manner as for the thickness optimisation previously presented. As is standard practice, end connections of the bracing members are modelled as having no moment capacity. Once again, uniform average strain energy density is required across the different groups and an iterative approach (similar to that shown in figure 3.8) is adopted in achieving this. Of significance is the fact that, despite achieving a maximum lateral displacement of 0.024m in the ESO process, this performance could not be repeated in the discrete interpretation of Liang et al.'s topology (M). Due to the complete absence of bracing in the top storey, rather a small strip of corner stiffening 2D elements, even infinitely large bracing members are unable to provide sufficient stiffness to meet 0.024m or 0.012m displacement constraints, without increasing the size of members in the orthogonal framework. The double and triple chevron topologies (B and C) are the most regular topologies in a high performance group that includes variations on the compound 'X chevron' form (H, I, K). The volume of steel required in bracing members to meet the δ* = 0.024m constraint for these designs varies by only around 10%. When other factors such as piece count, aesthetics, impingement on view and the effect on the rest of the structure is considered, this variation is relatively insignificant. It can also be seen that the least volume topology for one displacement constraint is not necessarily the best for another: solution C best meets the δ* = 0.044m constraint, whilst the other constraints are best met by solution I. This is indicated by large font size in figure 3.12. The material volumes required in the optimised discrete versions of solutions are generally within 25% of the corresponding design found through the thickness optimising BESO process, with the exception of the single 'X' form resulting from the definition of a horizontal mirror line.
61 δ* = 0.044 δ* = 0.024 δ* = 0.012 A Upper Diameter (m) 0.0716 0.0983 0.1402 Lower Diameter (m) 0.0828 0.1155 0.1655 Bracing Volume (m 3) 0.2142 0.4116 0.8417 Bracing XSE (m) 0.0393 0.0224 0.0113
B Upper Diameter (m) 0.0437 0.0634 0.1045 Lower Diameter (m) 0.0565 0.0831 0.1384 Bracing Volume (m 3) 0.1007 0.2155 0.5933 Bracing XSE (m) 0.0359 0.0184 0.0070
C Upper Diameter (m) 0.0345 0.0533 0.1190 Middle Diameter (m) 0.0452 0.0700 0.1571 Lower Diameter (m) 0.0493 0.0778 0.1768 Bracing Volume (m 3) 0.0847 0.2063 1.0486 Bracing XSE (m) 0.0324 0.0146 0.0030
D Upper Diameter (m) 0.0457 0.0696 0.1401 Lower Diameter (m) 0.0594 0.0918 0.1869 Bracing Volume (m 3) 0.1108 0.2618 1.0757 Bracing XSE (m) 0.0331 0.0153 0.0040
E Upper Diameter (m) 0.0537 0.0813 0.1518 Lower Diameter (m) 0.0544 0.0818 0.1524 Bracing Volume (m 3) 0.1168 0.2657 0.9211 Bracing XSE (m) 0.0342 0.0164 0.0050
F Upper Diameter (m) 0.0375 0.0631 0.2938 Lower Diameter (m) 0.0476 0.0812 0.5749 Bracing Volume (m 3) 0.0947 0.2722 10.7415 Bracing XSE (m) 0.0293 0.0112 0.0011
G Upper Diameter (m) 0.0344 0.0600 0.2526 Middle Diameter (m) 0.0447 0.0788 0.3320 Lower Diameter (m) 0.0484 0.0860 0.8468 Bracing Volume (m 3) 0.1234 0.3809 19.9 Bracing XSE (m) 0.0281 0.0101 0.0020
62 H Upper Diameter (m) 0.0354 0.0515 0.0868 Lower Diameter (m) 0.0552 0.0818 0.1390 Bracing Volume (m 3) 0.0924 0.2005 0.5752 Bracing XSE (m) 0.0353 0.0179 0.0066
I Upper Diameter (m) 0.0246 0.0357 0.0593 Middle Diameter (m) 0.0406 0.0585 0.0953 Lower Diameter (m) 0.0546 0.0804 0.1326 Bracing Volume (m 3) 0.0903 0.1928 0.5221 Bracing XSE (m) 0.0358 0.0185 0.0072
J Upper Diameter (m) 0.0408 0.0769 0.1500 Middle Diameter (m) 0.0436 0.0819 0.2924 Lower Diameter (m) 0.0599 0.0172 0.2727 Bracing Volume (m 3) 0.1241 0.4632 3.0746 Bracing XSE (m) 0.0254 0.0080 0.0033
K Upper Diameter (m) 0.0354 0.0515 0.0868 Middle Diameter (m) 0.0526 0.0788 0.1355 Lower Diameter (m) 0.0177 0.0226 0.0292 Bracing Volume (m 3) 0.0915 0.1983 0.5690 Bracing XSE (m) 0.0354 0.0179 0.0066
L Upper Diameter (m) 0.0343 0.0722 0.1740 Lower Diameter (m) 0.0448 0.0542 0.1323 Bracing Volume (m 3) 0.0954 0.2439 1.4297 Bracing XSE (m) 0.0310 0.0137 0.0027
M Diameter 5 (m) 0.1172 Diameter 4 (m) 0.1294 Diameter 3 (m) 0.0911 Diameter 2 (m) 0.1215 Diameter 1 (m) 0.1334 Bracing Volume (m 3) 0.8753 Bracing XSE (m) 0.0064 Figure 3.12: Discrete bracing topologies (with circular solid sections) optimised for minimum mass satisfaction of displacement constraint
63 3.9. CONCLUSIONS
– The bi directional form of ESO has a beneficial effect in achieving more practical solutions, that can be readily interpreted as discrete structures. However, this is accompanied by an increase in computation time on account of more analysis iterations. – Some variation occurs on account of domain thickness (if fixed) and starting point. This suggests that the result of a single process cannot be considered as globally optimal. Further variation is likely on account of numerical inconsistencies arising from process parameters such as mesh density and modification ratio. – Simultaneous optimisation of domain thickness and topology has a significant effect on the form of solutions generated. This technique works particularly well when defining symmetry patterns in the structure. – ESO has substantial limitations: the overall process is complex, requiring discrete interpretation and optimisation to assess practical performance of solutions; the structural model is expensive to analyse compared to a discrete model of the same structure using one dimensional elements, on account of the large number of two dimensional elements in each unit. – It is difficult to consider strength and buckling constraints alongside stiffness within the ESO process. Although Xie et al. (2002) consider ESO for optimisation against buckling, this is in the context of thickness optimisation only. The most practical way to consider strength and buckling is likely to be as a check in optimisation of the discrete interpretation.
3.10. GUIDELINES FOR PRACTICAL USE
Referring back to the original research question relating to improving the usefulness of ESO for the building industry and the proposals subsequently put forward, we can state the following: – Consideration of appropriate constraints is essential to successful use of any optimisation or pseudo optimisation tool, including ESO and its variants. In this case, maximum lateral displacement at the highest point of the structure is likely to govern, but the user should be aware that it is possible that displacement may
64 actually be greater elsewhere. Further, other forms of constraints, such as strength and buckling may be relevant, both in the bracing domain and the orthogonal framework. These are difficult to consider in the ESO process itself, but should be included in optimisation of the corresponding discrete structure. – BESO offers the ability to start from alternative configurations to that with all elements active. Running the process from different configurations, in the optimal thickness region (for this problem approximately between 3 and 10mm) most designs are similar (generally based on a double chevron), but consistent convergence to a single optimum is not observed. This may be beneficial in creating different design options and since performance of discrete and continuous design interpretations is often different. BESO yields higher performance and more regular designs than unidirectional ESO. – Defining all elements to be equal size and shape permits the use of a single element stiffness matrix. Different thicknesses are readily accommodated by linear factoring. – Simultaneous topology and thickness optimisation gives a reasonable indication of what material volume is likely to be required, providing there is an obvious discrete interpretation. This technique ensures appropriate thickness is used and provides a means of assigning different thicknesses to different regions of the structure, thus promoting structural efficiency. – Defining symmetry conditions with corresponding thickness grouping allows tailoring of designs to preconceived aesthetic requirements, whilst retaining high performance. Further noteworthy observations: – Using a “film” of very thin elements in place of inactive elements will stabilise the ESO process, eliminating the possibility of singularities in the global stiffness matrix causing the computational process to crash. However, this does require additional analysis time due to the extra elements. – In considering the results of an ESO process with thickness optimisation, it is valuable to inspect topologies generated throughout the history, alongside a chart of the form shown in figure 3.9. This offers the option to trade off structural
65 efficiency, as indicated by the bracing volume required, against interpretability of the design as a discrete structure. – A number of ESO solutions should be given discrete interpretation since performance of continuous and discrete solutions may vary. This also allows strength and buckling constraints to be considered.
66 4. Bracing topology and section-size optimisation by a hybrid algorithm: an industrial case-study
4.1. INTRODUCTION
This chapter presents an industrial case-study, considering issues associated with application of optimisation to the design of the lateral stability system of a specific tall building structure in the scheme design phase. The phases of research work presented include “live” project involvement, retrospective method development and convergence studies. These phases incorporate changes to the structural and optimisation models, reflecting the progression of the project. Topology optimisation is conducted by parameterising the design task and applying variants of the Pattern Search method (Hooke and Jeeves 1961). The Optimality Criteria method is used for section sizing. It is seen that minimum piece-count topologies are obtained by fixing member section sizes at their maximum value and performing topology optimisation, whilst minimum volume solutions are obtained by the simultaneous optimisation of topology and section size.
Research Questions: – How can existing optimisation techniques be adapted to practical topology problems in scheme design, accommodating considerations that are difficult to model, such as aesthetics and design intent? – How can simultaneous optimisation of size and topology be efficiently integrated? – Does simultaneous optimisation of size and topology offer improved solutions compared to performing tasks separately?
Proposals: - Use appropriate methods for the problem in question, combining these where suitable.
67 - Consider practical issues such as computation time, adaptability to changes in problem specification and model complexity. - Use stochastic methods to generate a number of designs, avoiding a single local minimum and enabling choice according to unmodelled criteria.
4.2. BACKGROUND
As discussed in chapter 2, use of optimisation techniques in building engineering practice remains low. Collaboration with Arup presented the opportunity to consider a real-world design problem, observing real-time developments and attempting to support decision-making with suitable optimised design solutions and parametric studies. It was therefore possible to assess the feasibility of applying topology optimisation in practice. A structural design team from Arup worked in conjunction with architects Kohn Pedersen Fox Associates on the design of the Pinnacle Tower, London (known previously as the DIFA Bishopsgate Tower (Baldock et al. 2005)), which is planned to stand at around 300m. As with all towers of this height, and slender structures in general, lateral stability is a key concern. This is provided by a tubular bracing system: a steel framework that wraps around the perimeter of the building and is irregular in both plan and elevation. For much of the design time, it was envisaged that individual bracing members would be grouped in spirals of fixed angle of inclination. The spirals emanate from the base of external columns, wrapping diagonally around the perimeter of the building and terminating at different, visually varied, heights up the building. Spirals rise three floors in height for every bay spanned, with individual bracing members defined by the intersections of spirals and columns. Later in the project and beyond the scope of the work presented in this chapter, the design team concluded that the requirement for continuous spirals was, in fact, an unnecessary constraint. A high premium is placed on minimising the total number of bracing members required. A high piece count will increase material cost and construction time and cost, hence losing potential letting revenue. Bracing members may also impinge on floor space and restrict view, thus reducing letting value of the corresponding floor. Adopting such a bracing system removes the need for an intrusive central core, which
68 tends to be impractical in buildings in excess of 200m. Pin-jointed internal columns serve to transfer some proportion of the vertical loads downwards. Triangulation introduced by adding diagonal bracing to the perimeter framework aims to reduce the bending moments occurring in what would otherwise be a Vierendeel framework, with a stiffening effect to reduce lateral displacement to an acceptable level. This work develops from a particular selected structural system: for a concise discussion of structural systems used in tall buildings, including tubular and horizontal load resisting systems, the reader is referred to Khajehpour (2001).
4.2.1. Overview of studies
Baldock et al. (2005) report on live work undertaken alongside the structural design team who, in collaboration with the architectural team, were seeking to develop an appropriate and efficient bracing pattern for the building. Following the “problem- seeks-design” approach (Cohn 1994), a variant on the Hooke and Jeeves “Pattern Search” method (Hooke and Jeeves 1961) was rapidly implemented to meet industrial requirements. Alongside further work to refine this method and test for convergence, studies were made to sample the landscape of the design space and the nature of the feasible region within it. The structural model used in this work was subject to minor alterations during this phase, but for the purposes of this chapter this set of models will be referred to as Model A. Almost two years later, following a submission to planning authorities and a lengthy review procedure, further optimisation studies were conducted. The structural model had been subject to a number of revisions by this point, but the model on which this second phase of optimisation research is based will be referred to as Model B for the purposes of this chapter. At this stage, section-size optimisation is considered and integrated into the pattern search algorithm.
4.3. DESIGN TASK DEFINITION
4.3.1. Structural models
Within each bracing spiral, individual pin-jointed bracing members are defined spanning adjacent columns, rising three floors in height. It is required that bracing
69 spirals should terminate at various heights up the building. Some potential spirals are omitted entirely due to peculiarities around the base of the building, with transfer areas defined at access points. However, spirals should be continuous from the base of the building to their termination point. Due to the highly iterative nature of the direct search optimisation method to be presented, a simplified finite element model is required for the evaluation of alternative bracing configurations to reduce computation time. Most internal columns are excluded from each structural model, so vertical loads applied to the model are only a proportion of the total loads. The very stiff concrete floor plates are approximated in different ways in the two structural models (see table 4.1). Constant parameters in optimisation models include: – design of orthogonal framework (topology and member sections) – definition of potential bracing spirals – member section sizes (variables in section sizing algorithm) – angle of inclination and location of potential bracing members – applied loads – nodal positions defining column locations and floor heights. The primary structural considerations in the design of the bracing system are modeled as constraints in the optimisation model. Secondary structural considerations were raised by structural designers to be checked after the optimisation process. A fully braced configuration of structural model B is shown in figure 4.1.
Figure 4.1: Fully-braced analysis model (left to right): plan view; side elevation; isometric view (shown with two spirals highlighted); isometric split sections
70 Table 4.1:Comparison of structural models
Structural model A Structural model B
“Rigid linking” constrains all nodes on a given floor of the “Spider” systems of dummy beams at each storey used to Concrete floor plate building to move together in-plane. approximate the stiffness contribution of concrete floor modelling plates
Horizontal loads applied at the centre of area of each floor Uniformly distributed loads applied to structural members Load application to a fictitious node within the relevant rigid-linking system on perimeter framework
Standard I-beam or rectangular hollow sections in all Circular hollow sections in all members, except horizontal Member section shape columns, beams and bracing members beams in the perimeter framework, which are I-sections
Six analysis cases are considered in two stages: self- Self-weight, dead load and reduceable and fixed live-load weight and superimposed dead load in a construction analysis cases combined with representative wind loads 71 stage; self-weight, live load and wind-loads in two define an ultimate limit state envelope for local strength Load cases orthogonal directions on the final structure. 23 load-case assessment. Five indicative wind loading directions for combinations defined, enveloped for "worst-case" stiffness assessment combination in each member
Strength: maximum in-plane force passing through a floor Stiffness: maximum lateral displacement of the structure at plate limited by constraint on axial force in bracing 263m AOD 1, inter-storey drift (difference in lateral Primary structural elements; moments in connections limited by constraint on displacement of central nodes at 3 storey separation) considerations bending moment in horizontal beams. Strength: buckling capacity and cross-section capacity (tension and compression) in bracing members
Out-of-plane horizontal forces on floor plates, arising from Transfer forces in floor plates, forces in pin-jointed Secondary structural sharp changes of direction of bracing elements; maximum horizontal beams in the perimeter framework and angle of considerations lateral displacements rotation of the structure as a whole under various loadcases due to asymmetry
1 AOD = Above Ordnance Datum: height relative to mean sea level at Newlyn, Cornwall 4.3.2. Topology optimisation models
Two optimisation models were developed corresponding to the distinct structural models A and B, with the same fundamental objective of minimising the total number of bracing elements, but subject to different constraints arising from different primary structural considerations.
Optimisation model A:
SP Minimise: N = ∑ nsp Eq. 4.1 sp =1
F Subject to: i ≤ 1 Eq. 4.2 Fmax M b ≤ 1 Eq. 4.3 M max where: N = number of bracing members in current design;
nsp = number of bracing members in spiral sp; SP = total number of spirals, 45;
Fi = maximum observed absolute axial force occurring in bracing member i in load case combination envelope;
Fmax = maximum permissible axial force in any bracing member;
Mb = maximum observed bending moment occurring in horizontal beam b in load case combination envelope;
Mmax = maximum permissible bending moment in any horizontal beam.
Optimisation model B:
SP Minimise: N = ∑ nsp Eq. 4.1 sp =1
Fi M i Subject to: + ≤ 1 Eq. 4.4 Ai p y M c
Fci mM i Fci + 1+ 5.0 ≤ 1 Eq. 4.5 Pc M c Pc
72 h s+1 − h s d s+1 − d s ≤ Eq. 4.6 j j 300
hmax d max ≤ Eq. 4.7 j 500 where: N = number of bracing members in current design; nsp = number of bracing members in spiral sp ; SP = total number of spirals, 48;
Ai = cross-sectional area of member i ;
Fi = maximum absolute axial force occurring in bracing member i in ultimate limit state loadcases;
Fci = maximum compressive axial force occurring in bracing member i in ultimate limit state loadcases;
Mi = maximum bending moment occurring in bracing member i in ultimate limit state loadcases; py = section design strength;
Mc = section moment capacity;
Pc = section compression resistance; m = equivalent uniform moment factor;
s+1 s dj d j = inter-storey drift between storeys s and s+1 under loadcase j; hs+1 h s = height of storey s;
max dj = maximum lateral displacement under loadcase j; hmax = total height of building. Equation 4.4 models the cross-sectional capacity requirement, equation 4.5 models the buckling resistance requirement, both of which are applicable to circular hollow sections (BSi 2000, Section 4.8.3).
4.4. PATTERN S EARCH METHOD
The Pattern Search method proposed by Hooke and Jeeves (1961) is a simple gradient-free search technique, which in its conventional form is applicable to optimisation tasks with a defined set of continuous variables. The search follows the steps outlined below.
73 1. Set base-point at initial location in the design space. Select initial distance to move in each variable direction (step-size). 2. Attempt positive and negative changes (exploratory moves) equal to the step-size to each of the variables in turn, accepting the move if the objective function is reduced. 3. After exploratory moves have been made for all variables, set a new base point at the current location in the design space. 4. Apply a pattern move, corresponding to the vector between the current and previous base-points. This move is accepted if the objective function is reduced. 5. Reduce step-size if no exploratory move was made in the last iteration. 6. Terminate if step-size has become less than a prescribed convergence value, else repeat steps 2 to 6. The use of pattern moves attempts to accelerate the search by exploiting known good directions in the search space, thus reducing computational expenditure. This basic method was adapted to the topology optimisation models defined in section 4.3, with different variants at different stages of the project. The number of bracing members in each spiral becomes the set of variables for the pattern search, with a reduction in the value of a variable corresponding to the removal of the relevant number of bracing members from the top of the spiral. Figure 4.2 shows a close-up elevation of the fully-braced upper section of structural model 2, with tip members highlighted. With step-size set to a single bracing member, the removal of these members would be attempted in the first set of exploratory moves. In all optimisation processes described in this chapter, structural analysis is performed in Oasys GSA as explained in Appendix 1 and the method implementation programmed in C++.
74 4
2 3 5
22 21 1 6 19 38 37 20 17 36 7 18 35 34 33 8 16 15 32 31 30 9 14 29 28 26 1025 13 27 24 23 11 12
Figure 4.2: Split elevation view of the upper section of structural model 1, with spiral numbering and bracing members at the tip of each element highlighted
4.5. LIVE PROJECT OPTIMISATION
4.5.1. Topology optimisation by Modified Pattern Search
The initial approach adopted for finding design solutions, subject to optimisation model A in the context of live project work, involved the piecewise removal of bracing members from a fully braced configuration. In this respect, the method bears a resemblance to the ESO methods discussed in Chapter 4. The technique can be formalised as a variant of the Pattern Search algorithm described in the previous section. In each iteration loop, the removal of a single bracing member, from the tip of a different spiral in turn, is attempted in the set of exploratory moves. The order is determined either randomly, or in order of increasing axial force in the tip elements. On completion of this phase, the pattern move attempts the removal of a further member from spirals which were reduced in length during the exploratory moves. However, an additional base point is set after successful pattern moves, so that only one bracing member is removed from any one spiral at a time. The sequence of exploratory and pattern moves is repeated until no further members can be removed without constraint violation. At this stage in the project, the step-size was fixed at one
75 bracing member and the Pattern Search was unidirectional, since members cannot be replaced. The significant challenge in this approach was in handling the constraints on axial force in bracing members and bending moment in horizontal beams, since the removal of any bracing member will improve the objective function. In the method developed, subject to the time constraints of the project, acceptance of a member removal required a change in constraint value to be less than a defined proportion of the distance to the constraint boundary. This can be stated as: (C − C ) Accept individual move if: i+1 i ≤ t Eq. 4.8 ()Clim − Ci where:
Ci = previous constraint value;
Ci+1 = new constraint value;
Clim = constraint limit; t = current tolerance (<1); The tolerance, or limit on change as a proportion of the distance from the constraint, is the same for both constraints. Initially set to a small value, this is increased when no further moves can be made at the current tolerance. When the tolerance exceeds unity, only the absolute value of the constraint function need be considered. The procedure terminates when no further moves can be made at this maximum tolerance. Moves that increase the distance to both constraint boundaries are always accepted.
4.5.2. Parametric studies
Following initial method development, the design team requested the investigation of the relative sensitivity of the number of members in minimum bracing designs to changes in parameter values governing the structural constraint limits. Such changes could be implemented in the design by reconsidering other aspects of the structure, if the new values proved to have a major positive effect on the achievable minimum number of bracing members. Figure 4.3 shows a set of designs generated using deterministic Pattern Search. The initial tolerance value, t, controlling move acceptance is set to 0.005. This is doubled
76 when no moves are possible at the current value. Fbracing and Mbeams for each final design are given in parentheses. Adjusting either parameter limit produces a significant change in the number of elements that may be removed, with the greater effect seen in adjusting the axial force limit.
7,000 kN 8,500 kN 10,000 kN Bracing force limit Bracing force limit Bracing force limit
262 216 750 kNm bracing bracing Bending limit elements elements
(8490kN (9923kN 730kNm) 725kNm)
384 296 248 500 kNm bracing bracing bracing Bending limit elements elements elements
(6976kN (8486kN (9961kN 436kNm) 480kNm) 481kNm)
303 400 kNm bracing Bending elements limit (8335kN 379kNm)
Figure 4.3: Parametric studies
4.5.3. Outline proposals
Aided by the parametric study, the design team selected limits of 750kNm for bending moment in horizontal beams and 8500kN for axial force in bracing members. Three alternative designs, shown in Figure 4.4, were generated by applying these constraints with deterministic element removal: one from a fully braced initial design and the two best feasible designs from a randomly generated set. The appearance of these designs is notably different, with lighter designs obtained from random starting points. The design team welcomed the opportunity to have alternative high performance solutions available for aesthetic consideration.
77 A feasible initial solution is a valuable asset, as it provides a starting point for design improvement, but such a design may not be known in other structural design problems.
Design A Design B Design C (from fully braced) (from design 433) (from design 1248) 271 elements 251 elements 251 elements 684kNm, 8398kN 744kNm, 8489kN 726kNm, 8475kN Figure 4.4: Designs generated for consideration for outline proposal.
4.6. CHARACTERISATION OF DESIGN SPACE
When developing an optimisation procedure, it is useful to gain an understanding of the design space as a whole, the relative size of the feasible region and the behaviour of constraint boundaries. The topology optimisation models define vast design spaces: for example in optimisation model A, with 45 potential spirals in the structural model, each with an integer number of members between 0 and at most 21, a total of
3x10 48 possible designs exist in the design space. It is immediately obvious that exhaustive search is impossible. Using structural model A, a 10,000-point domain sample was taken. For each design, a random length is assigned to each of the constituent spirals, with equal probability of all integer values between zero and the particular maximum spiral length. The performance of each design is determined by finite element analysis and a statistical overview of the designs generated is displayed in Table 4.2. Adopting the constraint values of 750kNm on maximum bending moment in horizontal beams and 8500kN on
78 maximum force in bracing members, it is observed that 0.42% of designs are feasible. The most lightly braced of this small set has 337 elements (just over half of all possible elements).
Table 4.2: Statistical analysis of 10000 randomly generated designs
Maximum bending moment (kNm) <500 <625 <750 <875 <1000 unlimited <7000 0 0 0 0 0 0
<7750 0 3 5 10 12 23
<8500 0 9 42 75 111 288
<9250 0 25 105 224 364 1087
<10000 0 36 169 398 692 2419
Maximumaxial force (kN) unlimited 0 53 274 711 1367 10000
A: Number of designs with constraint values less than the maximum
Maximum bending moment (kNm) <500 <625 <750 <875 <1000 unlimited <7000 ------
<7750 - 405 402 412 407 393
<8500 - 384 382 379 375 366
<9250 - 364 365 361 358 349
<10000 - 360 359 353 349 338
Maximumaxial force (kN) unlimited - 355 349 341 335 306
B: Mean number of elements in designs with constraint values less than the maximum
Maximum bending moment (kNm) <500 <625 <750 <875 <1000 unlimited <7000 ------
<7750 - 379 379 379 369 344
<8500 - 345 337 317 317 304
<9250 - 323 307 299 299 289
<10000 - 321 307 288 282 259
Maximumaxial force (kN ) unlimited - 305 282 256 242 174
C: Minimum number of elements in designs with constraint values less than the limit
79 An insight into constraint behaviour is gained from the simplified two-dimensional representation of the design space, shown in figure 4.5. Using a known feasible design as a starting point, two spirals were selected. The lengths of the spirals were varied such that every combination of lengths was analysed. For each combination, values of maximum axial force in bracing elements and maximum bending moment in horizontal beams were retrieved and used to put values to a set of grid points. These values were used to plot constraint boundaries in the 2D design space as shown above. The resulting space is observed to be non-convex.
Feasible Design Space
Maximum Bending Length of spiral2 Moment Constraint
Maximum Axial Force Constraint
Maximum Spiral Length Constraint
Length of spiral 1 Figure 4.5: 2D simplified representation of design domain, model A
4.7. TOPOLOGY OPTIMISATION METHOD DEVELOPMENT
Following the live project involvement detailed in section 4.5, a more rigorous investigation was undertaken, continuing to work with structural model A. This explored alternative Pattern Search strategies, assessing their effect on the number of bracing members required in the final design, computational efficiency in arriving at these designs, diversity of designs generated and reliability of finding feasible solutions. Bi-directionality was introduced to the Pattern Search, with variable step-size. In this approach, exploratory moves are made, considering each of the bracing spirals in turn, by first attempting to remove a number of elements equal to the current step-size and, if this is unsuccessful, then attempting to add the same number of elements. A move cannot take a spiral length beyond its maximum or minimum value: in this case, a move smaller than the current step-size is attempted and if successful, the attempted
80 pattern move would not include this direction. Bi-directional exploratory moves are demonstrated in figure 4.6.
43 43 43 1 40 1 40 1 40 44 44 44 6 6 6 38
42 42 42
41 41 41 10 28 10 28 10 28 38 17 17 17 27 27 27 9 9 9
22 22 22 12 14 21 12 14 21 12 14 21 32 32 38 32
2 2 2 31 23 31 23 31 23 16 39 16 39 16 39 19 19 19 37 37 37 35 35 35 33 33 33 30 30 30
7 15 7 15 7 15 20 22 324 5 26 27 829 40 34 13 11 20 22 324 5 26 27 829 40 34 13 11 20 22 324 5 26 27 829 40 34 13 11 (i) The schedule for (ii) The structure is (iii) If the element removal attempting exploratory reanalysed with 5 elements move is rejected, the moves reaches spiral 38 removed from spiral 38. structure is reanalysed with 4 (shown as double line). elements added (reaching Current step-size is 5 maximum spiral length) elements. Figure 4.6. A sample exploratory move
An alternative acceptance criterion must be formulated to include the possibility of beneficial additive moves. This is discussed in the next section. Pattern moves attempt to speed the progress of the search in a known good direction. A base point is fixed after each set of exploratory moves, then a pattern move is attempted with a vector equal to that defined between the current and previous base points, subject to constraints on spiral length. In this way, pattern moves may grow with consecutive acceptance. The current implementation modifies the classic method with regards to control of step-size. Traditionally, with continuous variables, step-size is reduced by a prescribed factor when no further moves are possible at the current size. In the current implementation, using integer variables, the step-size is adjusted through integer-valued modifications. Moreover, in seeking to minimise the total number of analyses required, step-size is reduced when the number of successful exploratory moves in an iteration falls below a critical value (nominally set to five). Similarly, step-size is increased if more than a requisite number of moves (nominally set to 10) are accepted. Figure 4.7 presents a diagrammatic representation of the evolutionary design process. In this case, structural performance criteria are
81 maximum bending moment in horizontal beams and maximum axial force in bracing members (constraints) and maximum axial force in members at the tip of each bracing spiral.
START: INITIAL DESIGN
Analyse design
Retrieve structural performance characteristics Calculate objective function
Set step-size and weighting factors Order spirals: (a) in order of increasing axial force (deterministic) or (b) in random order (stochastic)
Start of exploratory moves Select next spiral in list
NO Current Spiral Length > 0?
YES Remove relevant bracing member(s) and analyse
Retrieve structural performance characteristics Calculate objective function
YES Update objective function Move accepted? and current best design NO
NO Current spiral length < Maximum spiral length
YES Add relevant bracing member(s) and analyse
Retrieve structural performance characteristics Calculate objective function
NO YES Move accepted?
End of exploratory moves
Attempt pattern move and analyse
Retrieve structural performance characteristics Calculate objective function
YES Move accepted?
NO
Termination criteria met?
YES END: OPTIMISED DESIGN Figure 4.7: Pattern Search topology optimisation flowchart
82 4.7.1. Objective function formulation
Objective functions in optimisation are commonly formulated to include soft constraints, whereby a penalty function is included to move the search away from infeasible designs. However, in the current application, as discussed previously, due to the general correlation between member reduction and increase in constraint values, it is beneficial to move slowly towards constraint boundaries. Two formulations are proposed, both including constraint function terms in the objective function:
Formulation 1:
SP n ∑ sp F M 2 s=1 i b Eq. 4.9 X1 = + Wit + + p∑()max {},0 gi N orig Fmax M max i=1 where:
Fi − Fmax M b − M max g1 = ; g 2 = Eq. 4.10 Fmax M max
Norig = number of members in initial configuration. The first term of equation 4.9 represents the true objective of minimising number of bracing members, normalised by dividing by the original number of members. The second term represents the optimisation constraints converted to objectives in a multi- objective function with weighting, W it , scheduled to linearly reduce from 1 to 0 over 10 iterations. The final term applies a penalty to infeasible designs, with p = 10.
Formulation 2:
SP n ∑ sp 2 s=1 Eq. 4.11 X 2 = + p∑ ()max {},0 g i N orig i=1 where:
Fi − Fschedule M b − M schedule g1 = ; g2 = Eq. 4.12 Fmax M max The first term of equation 4.11 is identical to that of the objective function of formulation 1. The second term applies a penalty to designs with constraint values above prescribed targets, Fschedule and Mschedule , for the current iteration. The targets are
83 scheduled to converge linearly over 10 iterations from their values in the initial design to the values of the constraint limits. With p=10, the formulations become identical after 10 iterations. In both cases a move is accepted if the value of the relevant objective function of the resulting design is less than that of the current design. It should be noted that these formulations define search spaces which will change over the course of the optimisation process as the weighting factors are reduced. Further process parameters involved in using the above objective function formulations in a Pattern Search procedure include: – Schedule of weights in formulation 1 – Schedule of targets in formulation 2 – Value of penalty factor in both formulations – Initial step size – Conditions for decrement or increment of step size (simply decrementing step size when no moves are possible at the current size is wasteful of analysis time).
4.7.2. Comparative investigation
Table 4.3 summarises the results of five distinct algorithmic variations, with 20 optimisation runs in each case, as well as the salient details of a randomly created initial design set used as starting points in cases 3-5.
84 Table 4.3: Statistical summary of 20 runs per case
Random Case 1 Case 2 Case 3 Case 4 Case 5 design set Fully Fully Random Random Random Initial configuration(s) - braced braced set set set Optimisation formulation - 1 1 1 2 1 Exploratory move - Uni- Bi- Bi- Bi- Bi- directionality Exploratory move - Stoch. Stoch. Det. Det. Stoch. scheduling method
Optimisation parameters Optimisation Initial step size - 7 7 7 7 7
Number of Mean 314.9 260.5 251.7 244.8 253.7 245.7 elements Standard deviation 39.8 14.4 11.9 10.5 13.8 7.1 Mean maximum 1322.6 730 730.1 724.7 735 727.3 bending moment Mean maximum axial force 11471 8478 8485 8490 8479 8488
Mean number of analyses N/A 555.4 1031.5 933.6 1102.5 865.2
Failed optimisation runs N/A 0/20 0/20 1/20 2/20 2/20 Population characteristics Population Diversity metric value 5.38 4.63 4.68 5.58 5.41 5.61
Number of elements 390 243 233 213 230 234
Maximum bending moment 555.5 743.2 748.1 743.1 742.4 742.6
Maximum axial force 8246 8456 8489 8503 8447 8461 Best design
Evolving designs from fully-braced initial configuration
Two sets of twenty optimisation runs were performed using a fully braced initial design, with random ordering of exploratory moves. The first set (case 1) permitted only element removal, whilst the second set (case 2) also allowed elements to be added. Objective function formulation 1 was used, with an initial step-size of 7 elements. Unsurprisingly, the population of final designs evolved using bi-directional search exhibited better characteristics: an average of 9 fewer bracing elements; a best design with 10 fewer elements and a smaller standard deviation on element number when compared with the population evolved by uni-directional search. The only disadvantage was computational efficiency: using bi-directional search requires almost twice as many analyses on average. A gross mean number of 18 elements were added in each bi-directional optimisation run. These may be largely accounted for by multiple-element search steps in the early stages of evolution, although design domain non-linearity may also be a contributing factor.
85 Alternative objective function formulations
Two sets of twenty optimisation runs (cases 3 and 4) were performed using an identical set of randomly generated initial designs, all but one of which violate at least one of the constraints, each set adopting one of the formulations presented above. Bi- directional search is necessitated by infeasibility of initial designs. Initial step-size was set to 7 elements, with exploratory moves scheduled by increasing axial force in the tip element of the corresponding spirals. Formulation 1 performed significantly better: the corresponding population of final designs has an average of 9 fewer elements, whilst comparing best designs 17 fewer elements are observed. The standard deviation of number of bracing members is also smaller in formulation 1 and an average of 15% fewer analyses were required. It should be noted that not all optimisation runs were successful in locating a feasible design. Occasionally in intermediary infeasible designs, no exploratory move was capable of reducing the relevant objective function. One such optimisation run was encountered using formulation 1 and two using formulation 2.
Scheduling of exploratory moves
As previously discussed, a random, or stochastic, scheduling of exploratory moves is necessary to create diversity in designs evolved from a single fully-braced initial design. However, the use of randomly generated initial designs offers an alternative mode of diversification. The question therefore arises as to whether the choice of method for scheduling of exploratory moves affects the quality of final designs. The concept of attempting element removal in increasing order of axial force in tip elements is derived from the Evolutionary Structural Optimisation methods of Liang et al. (2000). However, in the current procedure, with all possible exploratory moves attempted and including constraints on bending moment in beams, there is no clear advantage in this method of scheduling, as seen by comparing the results of cases 3 and 5. One extremely light design was evolved using deterministic scheduling, but this is not statistically significant.
Performance of designs evolved from randomly generated initial configurations
Comparing the set of designs evolved through stochastic bi-directional search from a single fully-braced initial design (case 2) against the set evolved through deterministic
86 bi-directional search from randomly generated initial designs (case 3), both using formulation 1, significant advantages are observed in the latter case. On average, 7 fewer elements are present, with a best design of 213 members, compared to 233 members. Moreover, greater diversity is noted in the population evolved from different starting points. The value of this is demonstrated by the strong bias towards clockwise spirals by the best design of case 3, which may be considered unappealing. Alternative designs with slightly more members offer a more uniform distribution of elements. Diversity can be measured in a simplistic but quantitative way: two designs are compared by averaging the absolute differences in length of each spiral and the resulting sum averaged over all possible pairs of designs within the population. This diversity metric is 20% greater in the population evolved from randomly generated initial configurations (see Table 4.3).
Use of pattern moves
A pattern move attempts to move further in a direction that has been found to be profitable by previous exploratory moves. If successive pattern moves are accepted, the magnitude grows to accelerate in a direction that reduces the objective function. Pattern moves will be particularly successful in design domains with smooth objective functions and relatively small exploratory moves. The current procedure operates in a highly non-linear design domain with exploratory moves that are initially of the order of half of the permissible range of the design variables. Nevertheless, pattern moves removing up to 112 members have been accepted in case 2 (see table 4.2). On average, approximately 1 pattern move (of around 15 attempted) is accepted in each optimisation run, with a mean of around 20 elements added or removed.
4.8. TOPOLOGY OPTIMISATION : STRUCTURAL MODEL B
A later design review on the project considered a revised structural model, with different primary structural considerations, detailed in Table 4.1. The previous method development section demonstrated an efficient constraint handling method (formulation 1). The versatility of this formulation is seen through its straightforward adaptation to handle the constraints on utilisation factor, lateral displacement and inter-storey drift, defined in optimisation model 2, with no domain knowledge assumed. The revised objective function is expressed as:
87 SP ∑ nsp max s+1 s d j d j − d j X = s=1 +W U max + + + 1 it max s+1 s N orig h / 500 ()h − h / 300 max max d max d s+1 − d s max j j j pmax (),0 U −1 + max ,0 −1 + max ,0 −1 h max / 500 ()h s+1 − h s / 300 max max Eq. 4.13 where:
Norig = total number of bracing elements in the initial configuration;
Wit = weighting factor at iteration it ; Umax = maximum utilisation factor, the larger of the values of the left hand sides of equations 4.4 and 4.5, occurring in any bracing member; p = penalty factor applied to constraint violations. A range of standard circular hollow sections from STD CHS 508 15.9 (area = 0.0246m 2) to STD CHS 609 50.8 (area = 0.0891m 2) were made available for selection. A full list is presented in the following section. In the current topology optimisation problem, piece-count is of primary concern and therefore becomes the primary term in the objective function. A simple premise would be to assume that optimal solutions are obtained by assigning the largest permissible section size to each member. The STD circular hollow section 609 50.8 is therefore used for all members in the topology only optimisation process. It should be noted that if a bracing member makes a negative contribution to maximum lateral displacement or inter-storey drift, then the negative contribution is increased (and thus total displacement reduced) by reducing the section size. A negative contribution is made, as determined by the principle of virtual work, if forces in real and virtual load-cases are of opposite sign. This concept is addressed further in the next section. However, using maximum section sizes should provide near-optimal results, which can be fine-tuned if required, as well as simplifying fabrication and connection detailing. The Pattern Search topology optimisation flowchart of figure 4.7 again applies to this formulation, with structural performance characteristics of maximum lateral displacement, inter-storey drifts and forces and moments in bracing members.
88 4.8.1. Results
Two sets of optimisation runs were performed using the algorithm previously described, again comparing results obtained from a fully-braced configuration against those derived from alternative randomly generated bracing configuration. In both cases, exploratory moves were scheduling randomly. Full details are shown in tables 4.4 and 4.5, with three of the best designs displayed in figure 4.8.
From fully braced initial From randomly From randomly design: case 1 generated initial design: generated initial design: case 3 case 10 Figure 4.8: Design solutions from topology optimisation of structural model 2
89 Table 4.4: Performance of designs derived from fully braced initial configuration.
Number Inter-storey drift Successful Max lat disp (m) Max util Run of (m) Additions Removals pattern moves (limit = 0.5192) (limit = 1) members (limit = 0.038) (number (size)) Fully 648 0.0228 0.3432 1.00572 - - - braced 1 194 0.0379 0.5186 0.98131 50 500 1 (4) 2 197 0.0351 0.5182 1.00001 12 449 1 (14) 3 197 0.0374 0.5188 0.98866 33 477 1 (7) 4 207 0.0344 0.5191 0.99293 22 440 2 (6, 17) 5 205 0.0378 0.5191 0.96831 26 469 0 6 203 0.0366 0.5192 0.9829 5 450 0 7 212 0.0377 0.5182 0.98562 46 482 0 8 214 0.0353 0.5188 0.99483 15 449 0 9 206 0.0378 0.5192 0.96623 16 452 1 (6) 10 195 0.0373 0.5186 0.99035 41 477 1 (17) Mean 203 0.0367 0.5188 0.98512 26.6 464.5 0.7 (10) Standard 7.055 0.001313 0.000379 0.010934 15.35 19.28 0.675 (5.64) deviation Diversity 3.73
Table 4.5: Performance of randomly generated initial designs and solutions derived from them through bi-directional topology optimisation
Number Inter-storey drift Successful Max lat disp (m) Max util Run of (m) Additions Removals pattern moves (limit = 0.5192) (limit = 1) members (limit = 0.038) (number (size)) Start 387 0.0291 0.4271 0.99926 1 Final 213 0.0357 0.5192 0.96306 109 283 3 (62, 2, 9) Start 351 0.0575 0.4894 1.02336 2 Final 213 0.0351 0.5192 0.96854 88 226 1 (4) Start 385 0.0494 0.4917 1.12656 3 Final 189 0.0310 0.519 0.98418 125 321 0 Start 330 0.0378 0.4605 0.96264 4 Final 209 0.377 0.5190 0.98260 118 239 1(7) Start 326 0.0449 0.4871 1.00931 5 Final 196 0.0321 0.5192 0.99854 132 262 2 (16, 2) Start 326 0.146 0.8498 0.97501 6 Final 203 0.0349 0.5191 0.98034 164 287 0 Start 338 0.1165 0.6916 1.0840 7 Final 196 0.0329 0.5187 0.99889 85 227 0 Start 336 0.0531 0.5521 1.06283 8 Final 197 0.035 0.5190 0.99791 204 333 2 (83, 1) Start 306 0.1751 0.7583 0.98346 9 Final 217 0.037 0.5191 0.98729 203 292 1 (78) Start 347 0.0731 0.5969 1.18026 10 Final 193 0.0351 0.5189 0.98416 133 287 2 (3, 4) Mean 202.6 0.03465 0.51904 0.984551 136 276 1.2 (23) Standard 9.778 0.108383 0.000158 0.012144 42.16 36.94 1.033 (31.8) deviation Diversity 4.69
90 4.8.2. Observations
The pattern search algorithm using structural model 2 can be seen to perform equally well starting from a fully-braced initial configuration as from a randomly generated infeasible configuration, according to the results in table 4.5. This trend was not seen in the previous structural model (section 4.7.2). A 13% variation in number of elements in the final design is observed. Moreover, the best design (run 2 from the randomly generated initial design set), has approximately 25% fewer elements than the proposed Arup design at the corresponding stage in the design process. Each run takes around 300mins to complete on a PC Pentium 4 CPU 2.66GHz, 512MB RAM desktop computer. In general, Pattern Search moves do not make a significant contribution to the topology optimisation process, accounting for an average of just 2% of the total member removals and additions. This is likely to be due to a combination of high-dimensionality and unevenness of the search space. However, the low acceptance of pattern moves does not negate the effectiveness of the process of exploratory moves with gradual refinement.
4.8.3. Diversity
Results show that there is no discernible difference in piece-count or average structural performance between individual solutions derived from randomly generated initial starting points and those derived from a fully braced configuration. A secondary objective of the optimisation procedure is to be able to generate a diverse set of solutions. The relative diversity of the two sets of solutions can be assessed using a diversity metric, introduced in section 4.7.2. It can be seen from tables 4.4 and 4.5 that the value of the diversity metric is substantially increased, from 3.73 to 4.74, by using different randomly generated starting points, as opposed to the fully braced design. It is also of note that the average diversity value when comparing each random starting point to the corresponding final design is 4.69. This implies that optimised designs are almost as similar in appearance to each other as to their corresponding origin. The use of a set of randomly generated starting points is therefore a good way of ensuring aesthetically diverse final designs.
91 4.9. SIZE OPTIMISATION
4.9.1. Overview
This section considers the section size optimisation of bracing members for a given topological configuration through the Optimality Criteria method (Borkowski 1990). It would be possible to extend the method to include size optimisation of the remainder of the structure, but the illustrative purposes of this section are adequately served by keeping sections in the skeleton structure fixed. The subsequent argument considers pin-jointed bracing members, carrying only axial force under applied wind- loading, although they will also carry bending moment and shear force in ultimate limit state loadcases due to directly applied distributed loading. A note on application of this method to members carrying bending moment is presented in section 4.9.6.
4.9.2. Derivation of iterative approach from Optimality Criteria
For a fixed topology, the size-optimisation problem can be stated as:
N Minimise: V = ∑ vi Eq. 4.13 i=1
(vi = Ai Li )
Subject to: Ai ∈ A Eq. 4.14
Fi M i + + ≤ 1 Eq. 4.15 Ai p y M c
F mM F ci i ci + 1+ 5.0 ≤ 1 Eq. 4.16 Pc M c Pc
h s+1 − h s d s+1 − d s ≤ Eq. 4.17 j j 300 hmax d max ≤ Eq. 4.18 j 500 where: V = total volume of bracing members in the structure, equivalent to steel mass; vi = volume of individual member i; N = number of members in current design;
Ai = cross sectional area of member i ;
92 Li = length of member i; A = discrete set of cross sectional areas available from catalogue (table 4.6); S = number of bracing spirals in the structure (up to 48);
Fi = maximum absolute axial force occurring in bracing member i in ultimate limit state envelope;
Fci = maximum compressive axial force occurring in bracing member i in ultimate limit state envelope;
Mi = maximum bending moment occurring in bracing member i in ultimate limit state envelope; py = section design strength;
Mc = section moment capacity;
Pc = section compression resistance; m = equivalent uniform moment factor;
s+1 s dj d j = inter-storey drift between storeys s and s+1 under loadcase j; hs+1 h s = height of storey s;
max dj = maximum lateral displacement under loadcase j; hmax = total height of building.
Table 4.6: Catalogue of circular hollow sections and corresponding areas available for bracing members
Catalogue listing Area (m 2) Assign if continuous area (m 2): STD CHS 508 15.9 0.024581 A < 0.026 STD CHS 508 19.1 0.029336 0.026 < A < 0.031 STD CHS 508 22.2 0.033881 0.031 < A < 0.035 STD CHS 508 25.4 0.03851 0.035 < A < 0.040 STD CHS 508 31.8 0.047574 0.040 < A < 0.049 STD CHS 508 34.9 0.051871 0.049 < A < 0.053 STD CHS 508 38.1 0.056245 0.053 < A < 0.058 STD CHS 508 40.5 0.059482 0.058 < A < 0.061 STD CHS 508 44.5 0.064798 0.061 < A < 0.066 STD CHS 609 38.1 0.068334 0.066 < A < 0.070 STD CHS 609 40.5 0.072333 0.070 < A < 0.074 STD CHS 609 44.5 0.078918 0.074 < A < 0.080 STD CHS 609 47.6 0.083952 0.080 < A < 0.085 STD CHS 609 50.8 0.089085 0.085 < A
93 Equations 4.15 and 4.16 are complex constraints, since py, Mc and Fc must be calculated when considering any given prospective section size. However, considering observed peak forces and moments in member, i, these inequalities can be
min, used to set a minimum section size, Ai , above which one would expect the equations to always be satisfied. For the purposes of generating Optimality Criteria, a continuous range of cross-sectional areas is made available for each bracing
min max member, between Ai and A , the latter corresponding to the largest section size in the catalogue. Equations 4.17 and 4.18 are constraints on inter-storey drift and maximum lateral displacement respectively, both under wind loading. In practice only the three most critical drift or displacement constraints (of a total of 115) are considered at any one time. This reduces the number of virtual load cases that need to be analysed, the processing time in calculating member contributions to displacement and the complexity of the size optimisation algorithm. The critical cases are recalculated at each iteration. Inequality constraints, as required in establishing a Lagrangian function, are therefore stated as: 1 g ≡ e −1 ≤ 0 j * ∑ ij (j=1,..., L) Eq. 4.19 C j i
min Ai g L+i ≡ −1 ≤ 0 (i=1,..., E) Eq. 4.20 Ai
Ai g L+E+i ≡ −1 ≤ 0 (i=1,..., E) Eq. 4.21 Amax where:
Fij U ij li eij = Eq. 4.22 EA i (contribution to maximum lateral displacement)
Fij (U ij −U ij −1 )li or eij = Eq. 4.23 EA i (contribution to inter-storey drift)
hmax * max C j = − d j − ∑eij Eq. 4.24 500 i
94 s+1 s * h − h s+1 s or C j = − ()d j − d j − ∑eij Eq. 4.25 300 i eij = contribution of ith member (of a total of E) to inter-storey drift or displacement in case j, as calculated by the principle of virtual work . e ij will take a negative value if the forces it carries in the real load case, Fij and virtual load case, Uij or Uij U ij 1 , are of opposite sign. In this case, the negative contribution to total displacement or drift is increased by reducing the cross-sectional area of the member, which also has the benefit of reducing the volume of the structure. E = Young's modulus of steel.
* Cj = maximum permissible contribution of bracing members to inter-storey drift or displacement in the jth case (of a total of L). The Lagrangian function is then expressed as:
* V = ∑vi + ∑ j g j + ∑( L+i g L+i + L+E+i g L+E+i ) i j i min 1 Ai Ai = ∑vi + ∑ j ∑eij −1 + ∑ L+i −1 + L+E+i −1 * A Amax i jC j i i i i Eq. 4.26 where values are undetermined multipliers taking the value of zero for inactive constraints or a positive value for active constraints, where the term inside the bracket should be equal to zero. Hence for a converged, optimal design, only the first term on the right hand side of equation 4.26 contributes to the objective function. For optimality, the partial derivative of the Lagrangian function with respect to the section area of each member should be zero, i.e.: ∂V * ∂v ∂e Amin = 0 = i + j ij − L+i i + L+E+i ∑ * 2 max Eq. 4.27 ∂Ai ∂Ai j C j ∂Ai Ai Ai
Replacing vi and eij with the corresponding parameters that are independent of area: length, li, and ēij , gives: e Amin l − j ij − L+i i + L+E+i = 0 i ∑ * 2 2 max Eq. 4.28 j C j Ai Ai Ai
e e = i (v = l A ) ij ; i i i Ai
95 Hence:
e j ij + Amin ∑ * L+i i j C j Ai = Eq. 4.29 L+E+i li + Amax i It is noteworthy that in the case where there exists a single active displacement constraint and for members for which size constraints are not active, equation 4.29 can be expressed as: C * e e = i = i 2 Eq. 4.30 Ai li Aili Since the term on the right hand side of the above equation is exactly the “cross” strain energy density of the member (the displacement contribution of the member, as calculated by virtual work, per unit volume), it is apparent that cross strain energy density should be constant for all members with inactive size constraints. There is no such elegant result when multiple displacement constraints are active. The current method uses an iterative approach to solve the above problem. For each loadcase, since Ai is a function of the square root of j and Cj is a function of the reciprocal of Ai , this information can be combined in the recursive formula:
2 C v+1 = j v Eq. 4.31 j * j C j where v and v+1 denote successive iterations. Each revised value of j is used to calculate a new set of Ai values and subsequently predict Cj, on the fundamental assumption that ēij values are unchanged. Using a similar rationale, multipliers corresponding to constraints on section size are iterated according to equations 4.32 and 4.33.
2 Amin v+1 i v Eq. 4.32 L+i = L+i Ai
2 A v+1 = i v L+E+i max L+E+i Eq. 4.33 Ai
96 4.9.3. Pitfalls
Complex values of A i.
Since eij values can be negative, it is conceivable that during the course of convergence towards an optimal solution, the value to be square-rooted on the right- hand side of equation 4.29 may be negative, giving a nonsensical complex value for
Ai. Clearly this should not be the case for a converged, feasible design, since the value of the multiplier corresponding to the minimum section size constraint would become very large in order to prevent this. With reference to equation 4.29, this scenario can be satisfactorily avoided by setting the initial values of L+i and L+E+i to be substantially bigger than that for j. If the value to be square-rooted does become negative, the value of Ai is set to Amin , and the corresponding Kuhn-Tucker multiplier is doubled.
Convergence failure
For many bracing topologies, it is not possible to find a feasible set of section sizes to meet displacement constraints. In these cases, intuition suggests that for the most critical loadcase, section areas should be set to maximum for members with positive displacement contribution, or minimum for members with negative contribution. In practice, using the iterative algorithm detailed previously, section areas will increase beyond their maximum permissible value, due to continuing escalation of the j multiplier. However, after the prescribed maximum number of iterations, such members will simply be assigned the maximum section area.
* Negative values of C j and C j
According to equations 4.22 and 4.23, if the contribution of the non-designable structure to the overall displacement or lateral drift in a given loadcase is greater than
* the corresponding limiting value, Cj will be negative. It is also possible that Cj values may be negative, if there is a negative contribution to total displacement of a
* large number of bracing members. If one or both of Cj or Cj are negative, the behaviour of j may be erratic. This situation may be avoided by adding a constant
* value, k, to both Cj and Cj (calculated as the summation of eij ), to ensure both are positive at all times, without changing the fundamental problem to be solved.
97 However, selecting a value of k that is excessively large will greatly slow
* convergence of the algorithm, since the ratio of Cj to Cj in equation 4.31 will be close to unity. k is therefore assigned on a case-by-case basis according to the following equation:
* k = max [ ,0 ( 1.0 *C j _ Global )− C j ] Eq. 4.34
4.9.4. Assignment of discrete sections
On convergence of the above iterative procedure, discrete sections must be assigned to the Ai values. Extensive research has been conducted into discrete variable structural optimisation (Arora 2002) focusing on achieving strict global optima for this problem. However, this will generally involve much reanalysis, which, in practice, becomes too computationally expensive. Instead, an approximate method, which will be neither overly conservative nor cause constraint violation is required. If section size constraints are active, the choice of discrete section is obvious, otherwise the allocation method shown in table 4.6 is generally found to adequately meet the requirements.
98 INITIAL DESIGN
Analyse design
Determine critical loadcases
Reanalyse with required virtual loadcases
Retrieve: - Maximum forces in ULS envelope - Forces in real wind and virtual loadcases
Calculate for each bracing member: - Minimum permissible area - Contribution to displacement/drift in each critical case APPLICATION OF OPTIMALITY CRITERIA
Calculate for each critical displacement/drift case: Assign: - Total contribution of bracing members to - Initial multiplier values displacement/drift - Required contribution Calculate for each bracing member: - k value to ensure required contribution and projected - Revised section area contribution remain positive at all times - Revised multipliers corresponding to minimum and maximum permissible areas
Calculate for each displacement/drift case: - New projected contribution from bracing members on the basis of revised section areas - Revised multipliers
Assign: YES Convergence or NO - Catalogue sections to all bracing members maximum iterations reached?
NO Overall Convergence?
YES
END Figure 4.9: Size optimisation flowchart
4.9.5. Size optimisation of fully braced configuration
The bracing design with all members present is optimised using the above method, with various starting points: (i) all members take maximum section sizes; (ii) all members take minimum section sizes; (iii)five cases in which members take randomly assigned section sizes. The results are detailed in table 4.7.
99 Table 4.7: Size optimisation of fully-braced design from different initial distributions
CASE Initial Normalised Initial Final Normalised Final Volume Initial Max. Utilisation Volume Final Max. Utilisation (m 3) Displacement/Drift Factor (m 3) Displacement/Drift Factor (i) 880.1 0.661 1.007 298.2 0.852 0.989 (ii) 242.9 0.939 1.978 286.1 0.862 0.999 (iii).1 575.4 0.749 1.864 294.3 0.856 0.998 (iii).2 564.3 0.752 1.459 295.1 0.854 0.992 (iii).3 567.7 0.752 1.838 293.5 0.856 0.998 (iii).4 570.4 0.743 1.588 295.1 0.855 1.000 (iii).5 557.4 0.763 1.940 293.5 0.856 0.999
In all cases, despite comfortably satisfying displacement criteria, the initial design is infeasible due to the utilisation factor for buckling being greater than unity. Figures 4.10 and 4.11 show how the algorithm converges on a locally optimal solution in around 8 iterations, for cases (i) and (ii). A 4.2% variation in bracing volume in locally optimal solutions is observed, with case (i) producing the highest volume design and case (ii) producing the lowest volume design. All final designs meet the constraint on buckling capacity. This is particularly noteworthy in case (i), from which one might presume that no feasible set of section sizes exists for this topology. This is not a direct consequence of the sizing algorithm, but of unpredictable redistribution of forces in the structure resulting from section size modifications affecting local stiffnesses. There is clearly no guarantee of global optimality in sizing an indeterminate structure by this method.
100 Section Size Optimisation: Fully Braced Topology from Maximum Sections
Normalised Displacement/Drift Buckling Capacity Utilisation Normalised Volume
2
1 NormalisedValue
0 0 1 2 3 4 5 6 7 8 910 Iteration
Figure 4.10: Convergence of size optimisation algorithm from maximum section sizes in fully braced design
Section Size Optimisation: Fully Braced Topology from Minimum Sections
Normalised Displacement/Drift Buckling Capacity Utilisation Normalised Volume
2
1 Normalised Value
0 0 1 2 3 4 5 6 7 8 910 Iteration
Figure 4.11: Convergence of size optimisation algorithm from minimum section sizes in fully braced design
101 4.9.6. Size optimisation by Optimality Criteria with bending moments
Without bending moment in bracing members in critical load-cases, a major complication in the generic section-sizing task is removed, since the cross-sectional area is the only relevant section property. In a more general formulation, changes in bending stiffnesses, shear factors and torsional stiffnesses and their effect on the contribution to displacement of the member in question, must be considered, as well as the cross-sectional area. This clearly introduces a large number of semi-dependent variables that for commercial standard steel sections are linked, but not proportional to cross-sectional area. Grierson and Chan (1993) propose the use of linear regression analysis to define coefficients allowing the additional cross-sectional properties to be approximately expressed in terms of the cross-sectional area of a given member. A similar iterative method to that described above can then be adopted to perform the optimisation task.
4.10. INTEGRATION OF TOPOLOGY AND SIZE OPTIMISATION
In many structural design problems, minimising material volume or mass may be of primary concern. Although this was not the case in the original Pinnacle Tower design, minimising bracing volume is considered in this section for two reasons: – Offering designers an understanding of the trade-off between minimum mass and minimum piece count. – Research value and to provide a practical example of the potential of the proposed method. The following pertinent questions should be addressed: – Can size optimisation be incorporated into a topology optimisation algorithm without a prohibitive increase in computation time? – By optimising size and topology simultaneously, can we obtain a lower volume design than by performing size optimisation on an optimal topology? – How does simultaneous size and topology optimisation affect the resulting topology solutions? Section 4.8.5 indicates that convergence of the size optimisation algorithm requires around 8 analysis iterations using a starting point that is far from optimal. However,
102 if the starting point is closer to satisfying the optimality conditions, with very little redistribution of forces and moments in the structure required, convergence can be expected to be significantly faster and a single iteration should achieve a near optimal design. If the projected behaviour of the structure subject to revised section sizes, assuming no redistribution of forces and moments, is consistently accurate, only a single structural analysis is required per topological exploratory move. This would add very little computational expenditure to the integrated optimisation process, compared with optimisation of topology only. Unfortunately, the assumption of unchanged force and moment distribution is not always sufficiently accurate, and determining acceptance on the basis of projected behaviour can often lead to an increase in objective function value due to constraint violations. It is therefore necessary to introduce a further structural analysis with revised section sizes, to validate the projected behaviour. This also allows required changes to the set of critical loadcases to be detected. The objective function used for optimisation of topology alone is adapted for the integrated optimisation problem. However, the primary objective is now to minimise the sum of the volumes of the spirals, rather than the number of members therein. The pure topology optimisation algorithm included terms with an iteration-dependent weighting coefficient designed to keep solutions away from constraint boundaries in early iterations. In the current hybrid algorithm, we attempt to minimise the volume for each solution by meeting the most critical displacement-drift constraint exactly, hence the term favouring designs that are distant from displacement constraint boundaries is discarded. The term favouring designs that are distant from utilisation constraint boundaries is modified to consider the maximum utilisation factor value that would occur if all members took the largest section-size. This projected utilisation factor using maximum sections is calculated for each member using the current force-moment distribution in the structure. Violation of utilisation factor constraints is penalised as before, although drift and displacement constraints are considered together in a single term, since these constraints are equivalent in the section size optimisation algorithm. Hence the objective function is defined as:
103 SP ∑ vs s=1 max X 1 ()v1 ,... vSP = +Wit U cap Vorig Eq. 4.35 d max d s+1 − d s max j j j + pmax (),0 U −1 + max ,0 − ,1 −1 h max 500 ()h s+1 − h s 300 max max with the following symbols introduced: vs = volume of spiral s;
Vorig = total volume of bracing members in initial design; U max = maximum utilisation factor, as defined by the left hand sides of equations 4.15 and 4.16 . max U cap = maximum utilisation factor capacity: the largest utilisation factor that would occur in any bracing member if all took maximum section size (with current force- moment distribution). Figure 4.12 illustrates the integration of size optimisation into the topology optimisation routine.
104 INITIAL DESIGN (size optimised)
Analyse design
Retrieve structural performance characteristics: - Maximum lateral displacement in wind loadcases - Maximum interstorey drift in wind loadcases - Maximum forces and moments in ULS envelope Determine critical displacement/drift cases
Calculate objective function
Set step-size and weighting factor Generate randomly ordered list of spirals
Start of exploratory moves Select next spiral in random list
NO Current Spiral Length > 0?
YES Remove relevant bracing member(s) and analyse
Retrieve structural performance characteristics
Perform single iteration of optimality criteria SIZING ALGORITHM and reanalyse
Retrieve structural performance characteristics. Calculate objective function Update objective function, YES critical displacement/drift Move accepted? cases NO and current best design
NO Current spiral length < Maximum spiral length
Add relevant bracing member(s) and analyse
Retrieve structural performance characteristics
Perform single iteration of optimality criteria SIZING ALGORITHM and reanalyse
Retrieve structural performance characteristics Calculate objective function
NO YES Move accepted?
End of exploratory moves
Attempt pattern move and analyse
Retrieve structural performance characteristics
Perform single iteration of Optimality Criteria SIZING ALGORITHM and reanalyse
Retrieve structural performance characteristics Calculate objective function
YES Move accepted?
NO NO Termination criteria met?
YES END Figure 4.12: Flowchart for combined size and topology optimisation algorithm
105 4.10.1. Results
Two sets of optimisation runs were performed using the hybrid algorithm described. In the first set, ten runs were performed starting from an optimised fully braced configuration, each using a different random number seed to determine the order in which bracing spirals are considered for member removal or addition. In the second set, ten runs were performed each starting from a different randomly generated bracing configuration. These are the same as those considered for topology optimisation only, but in the current case full size optimisation is performed before modification of topology is introduced.
Table 4.8: Performance of optimised designs derived from fully braced initial configuration
Design Description Members Optimised Max. Normalised max. Comments volume (m 3) utilisation disp/drift
Fully Optimised 648 294.6 0.970 0.726 braced feasible 1 Final 348 194.0 0.990 0.990 2 Final 320 192.3 0.998 0.988 Fewest members 3 Final 342 193.8 0.986 0.991 4 Final 343 192.7 0.985 0.989 5 Final 330 189.2 0.989 0.990 Lowest volume 6 Final 348 196.5 0.978 0.990 7 Final 322 191.1 1.000 0.991 8 Final 377 200.6 0.968 0.996 9 Final 320 191.6 0.979 0.990 10 Final 368 197.8 0.976 0.997 Mean 341.8 194.0 0.985 0.991 Standard 11.98 2.318 0.00761 0.00107 deviation
106 Table 4.9: Performance of initial and optimised designs derived from random initial configurations.
Design Description Members Optimised Max. Normalised Max. Comments Volume (m 3) Utilisation disp/drift
1 Start: Infeasible 387 398.1 1.001 1.245 Final 294 189.8 0.989 0.989 2 Start: Infeasible 351 407.3 1.030 1.453 Final 313 198.5 0.999 0.988 3 Start: Infeasible 385 414.0 1.097 1.247 Final 357 250.5 1.055 0.989 Infeasible Solution 4 Start: Feasible 330 233.5 0.983 0.995 Final 299 191.1 0.968 0.988 5 Start: Infeasible 326 350.7 1.020 1.142 Final 303 189.9 0.993 0.989 6 Start: Infeasible 326 377.2 0.972 3.763 Final 338 372.9 1.006 1.105 Infeasible Solution 7 Start: Infeasible 338 367.4 1.101 2.997 Final 255 189.3 0.996 0.884 Fewest members 8 Start: Infeasible 336 361.2 1.003 1.350 Final 272 186.1 0.996 0.905 Lowest volume 9 Start: Infeasible 306 351.7 1.015 4.534 Final 349 227.7 0.970 0.993 10 Start: Infeasible 347 391.1 1.739 1.861 Final 329 379.9 0.997 1.076 Infeasible Solution Mean 310.9 237.6 0.997 0.991 Standard 32.75 68.45 0.0267 0.0639 deviation
4.10.2. Observations
Two solutions derived from randomly generated initial designs (7 and 8) form a Pareto-dominant set (Pareto 1896) from the solutions evolved from both fully braced and randomly generated initial designs. That is to say, compared to any other design, these two solutions have either lower volume or piece count. In fact, with the exception of solution 5 in the set starting from the fully braced solution, the two solutions discussed are superior in both respects to all other designs. Generally, starting from randomly generated infeasible designs appears to offer slightly lower volume solutions with appreciably lower piece-counts, when compared with solutions derived from the fully-braced configuration. However, in a minority of cases problems have occurred in finding feasible or high quality solutions from infeasible initial designs. Possible causes include:
107 – Inter storey drift governs at a high storey level. No additions at current step size will aid this local structural performance issue. Removal of members lower in the structure will not change the critical wind case and its value. Possible Solution: starting with a very large step-size will ensure bracing members can be added towards the top of the structure, reducing inter-storey drift at any height. – Addition of bracing members to an infeasible design may allow the new topology to meet drift constraints. In the section sizing algorithm, a major reallocation of section sizes may occur, causing a substantial redistribution of load paths within the structure. On the first iteration, this may make the new design infeasible on account of utilisation factor, despite a superior topological configuration. Possible Solutions: determine move acceptance purely on topology change or perform multiple size-optimisation iterations whenever a substantial change in constraint values occurs. – A size optimisation step gives false superiority to a design. Targets for bracing displacement contribution are intentionally conservative to avoid a design being flagged as infeasible due to load path redistribution, when a feasible solution is possible. So if bracing contribution is higher than intended, but the design is feasible (but very close to the constraint), it can be hard for further designs to improve on this. A comparison of the best designs generated by simultaneous topology and size optimisation and those generated by topology optimisation with subsequent size optimisation is presented in the following section.
4.11. SUMMARY OF RESULTS FROM OPTIMISATION MODEL B
Table 4.10 presents a summary of the characteristics of the best designs developed from structural model B and considering optimisation model B. This includes a manually evolved design proposal, shown in figure 4.13, developed by Arup designers from structural model B, but having removed the requirement for bracing members to be grouped in continuous spirals starting from the base of the vertical columns. It should be noted that 22 of the bracing members in the basement storey, that were fixed in the work previously described, have been removed. This will have
108 a detrimental effect on stiffness, hence comparison is not entirely fair. Bracing members are concentrated around the narrow edge of the building, to increase the resistance to minor-axis bending.
Figure 4.13: Arup design proposal, without requirement for bracing members to be grouped in continuous spirals.
Table 4.10: Summary of best designs from structural model 2
Optimised Normalised Solution Bracing Max. Design description bracing max disp number members utilisation volume (m 3) /drift (figure 4.14) Arup design 241 328.5 1.050 0.966 - Size-optimised fully braced 648 286.1 0.999 0.862 6 design Lowest piece-count topology-optimised solution 194 267.3 0.981 0.999 1 from fully-braced design (subsequently size- (252.5) (2) optimised) Lowest piece-count topology-optimised solution 189 261.7 0.984 1.000 4 from random initial design (subsequently size- (247.0) (5) optimised) Lowest volume topology and size optimised solution 330 189.2 0.989 0.990 7 from fully-braced design Lowest volume topology and size optimised solution 272 186.1 0.996 0.905 10 from random initial design
109 Even accounting for the removal of bracing members in the basement of the Arup design, the topology optimised designs show a significant reduction in number of bracing members, despite the design constraint requiring continuous bracing spirals. A reduction in bracing volume of over 25% is achieved in the best designs generated by simultaneous topology and size optimisation, compared to those from sequential topology and size optimisation. However, since there is no incentive towards low piece-count designs, final solutions from the hybrid method have around 50% more members. The increased reduction in bracing volume achieved by integrated size- topology optimisation over topology optimisation followed by size optimisation is illustrated in figure 4.14. Lines at 45 degrees to the axes are contours of constant volume. Movement along the y-axis (downwards), indicates reduction in volume resulting from topology reduction, whereas movement along the x-axis results from size-optimisation. Starting from an infeasible initial design, volume may be increased in an effort to locate a feasible design before the general trend of volume reduction is seen. A designer could tune trade-off between piece-count and total volume by combining these terms in the objective function of equation 4.36:
SP SP ∑ ns ∑vs s=1 s=1 max X1()v1,... vSP ;n1,... nSP = WN + WV +Wit U cap + Norig Vorig Eq. 4.36 max s+1 s max d j d j − d j pmax (),0 U −1 + max ,0 − ,1 −1 hmax 500 ()h s+1 − h s 300 max max Position on a Pareto front could be changed by adjusting the weighting of the two terms: WN and WV (=1- WN) are weighting coefficients on the total number of bracing members and the total volume of bracing members respectively (other terms in this expression have been previously used in sections 4.9 and 4.10). Alternatively a Pareto archive could be introduced, with one primary objective, such as piece-count, driving the search and a secondary objective, such as bracing volume, controlling the archive. Starling (2004) proposes the introduction of this concept to Pattern Search for grammar-based synthesis of artefacts.
110 Volume change by size-optimisation
250m 3 300m 3 350m 3 400m 3 450m 3 500m 3 550m 3 600m 3 650m 3 700m 3 750m 3 800m 3 850m 3 Datum Point Solution 0 Solution 6 Size optimisation
200m 3 850m 3
150m 3 800m 3
100m 3 750m 3
50m 3 700m 3 Simultaneous size- topology optimisation
3 3
0m optimisation Topology 650m
600m 3
550m 3 Solution 3
500m 3
Solution 9 Solution 8 optimisation topology by change Volume
Solution 7 Simultaneous 450m 3 size-topology optimisation
400m 3 Solution 10
350m 3 Topology optimisation Topology
300m 3 Solution 2 Solution 1 Solution 5 Solution 4 0m 3 50m 3 100m 3 150m 3 200m 3 250m 3
Size optimisation 0: Datum Point: Fully Braced, Maximum Sections. 648 bracing members, 880.1m 3 1: Lowest piece-count solution from fully-braced starting point (topology only, with maximum section sizes). 194 bracing members, 267.3m 3 2: Size-optimised design from solution 1. 194 bracing members 252.5m 3 3: Randomly generated infeasible starting point (maximum sections sizes) 385 bracing members, 522.8m 3 4: Lowest piece-count solution (topology only, with maximum section sizes) developed from solution 3. 189 bracing members, 261.7m 3 5: Size-optimised design from solution 4. 189 bracing members 247.0m 3 6: Size-optimised design from datum point (0). 648 bracing members, 286.1m 3 7: Lowest volume design developed from size optimised fully-braced starting point (solution 6) by simultaneous topology and size optimisation. 330 bracing elements, 189.2m 3 8: Randomly generated infeasible starting point (maximum section sizes) 336 bracing members, 456.3m 3 9: Infeasible size-optimised design from solution 8. (no feasible section-size solution exists) 336 bracing members, 361.2m 3 10: Lowest volume design developed from a size optimised random starting point (in this case solution 9) by simultaneous topology and size optimisation. 272 bracing members, 186.1m 3
Figure 4.14: Volume reduction by simultaneous versus sequential topology and size optimisation routines.
111 4.12. CONCLUSIONS
In response to the research question posed at the start of this chapter, it has been demonstrated how the established Hooke and Jeeves (1961) search method can be applied to a practical topology problem, by simplifying the task to consider a fixed set of variables. Through stochastic search and varying starting points a range of optimally-directed designs can be found, avoiding single local optima and allowing unmodelled criteria, such as aesthetics, to influence final design selection. Simultaneous optimisation of size and topology was efficiently carried out by performing a single iteration of the Optimality Criteria method at each topological step. This integrated approach offered substantial volume reductions when compared to sequential topology and size optimisation. Focusing this research on a practical problem has revealed a number of crucial considerations and obstacles relevant to the application of structural optimisation techniques in the building industry. These problems are either neglected or not apparent when considering small scale benchmarks problems and include: – Adaptability of an optimisation model to changes in geometric specifications, constraints, objectives and aesthetic requirements. This is vital since external modifications are inevitable and it would rarely be acceptable for an optimisation procedure to hinder progress of a project. During the period of involvement in and observation of the design process for the Pinnacle Tower, a number of revisions and modifications were made, influencing the structural models and constraints used in the optimisation routine. This necessitated careful reconsideration of the adaptation and subsequent behaviour of the optimisation algorithms, since approximations and simplifications must be carefully justified. – Irregularity of design spaces. The formulation of objective functions have been shown to define design spaces which are very irregular and have many local minima. Finding the global optimum, and proving it so, within a set design space is therefore virtually impossible. However, the use of stochastic methods mean that a range of locally optimal solutions, diverse in appearance but with similar high performance, can be offered for further consideration. – Constraint handling. In the early stages of topology optimisation, a means of repelling the search path from constraint boundaries was required, in order to
112 avoid acceptance of disadvantageous moves and subsequently becoming trapped in poor local optima. Addition of appropriate terms with diminishing weighting in successive iterations was successful in meeting this need. – Approximations and simplified structural models. Improving algorithm efficiency by these and other means allows more solutions to be generated in a given time. – Section sizing considerations. In considering section-size optimisation as an independent problem, best results were observed by adopting the lightest possible initial solution and allowing sizes to increase. Maximum section sizes may not always provide the best solution in terms of local strength, since larger sections attract a higher share of the load due to higher stiffness. Optimisation in industry should ideally be driven by cost models, as considered in detail by Khajehpour (2001). The holistic objective in a generic project is to minimise the total cost incurred in design, material and labour purchase, construction, provision of services and maintenance and where applicable, maximising the revenue from letting of floor space or other sources. This problem must be simplified and varying degrees of complexity may be considered in cost modelling. A crude approximation of cost being proportional to weight can be developed to consideration of component cost, construction time and cost, revenue (lettable area and corresponding value), maintenance cost and even design time costs. However, in practice, collating and establishing suitable costs for an optimisation model can prove difficult.
113 5. Structural topology optimisation of braced steel frameworks using Genetic Programming
5.1. INTRODUCTION
In the early stages of a design project it is desirable to generate and assess a number of alternative concepts. In the structural design of buildings, this might involve considering alternative structural systems, such as a concrete core or steel tubular framework for providing lateral stability. Otherwise, one might consider the design of a given structural system, for example the configuration of bracing in a steel tubular framework. Ideally assessment of solutions is from both a structural and architectural perspective. This chapter presents the application of Genetic Programming (GP) to tackle this problem, attempting to include an element of design rationale and the ability to influence design aesthetics.
Research Questions:
– How can multiple, novel, high performance designs be rapidly generated for a structural topology problem?
– How can aesthetic constraints be defined and their effect on performance be investigated?
– How can Genetic Programming be applied in a pure sense to structural topology optimisation and what benefits can this offer?
Proposals: - The tree-based representation of design development in genetic programming avoids the requirement of a fixed number of variables common to many optimisation methods. - The functional basis of Genetic Programming trees allows pattern development for large scale structures from simple representations.
114 5.2. BACKGROUND
Genetic Programming (GP) is a class of evolutionary algorithm developed in the early 1990s (Koza 1992), which in its seminal form manipulates tree structures containing instructions for solving a task, such as a design problem. Despite various attempts at using GP in civil engineering (Shaw 2003), in the field of structural topology optimisation, to the author's best knowledge, the full potential of GP has not been fully exploited. This is because the trees have taken the form of encrypted representations of a design (Yang and Soh 2002) or an assembly of lower level components without functionality defined at branch points (Liu 2000). The current research uses tree structures containing design modification operators as internal nodes, thus detailing the development of a design from fundamental components. These tree structures can be manipulated by genetic operations to evolve optimally directed solutions. Evolved programme trees can be considered as "blue-prints" for a design to play back the development of a given solution by sequential execution of the branch functions. In common with other evolutionary algorithms, GP is population-based and stochastic. This facilitates the generation of a set of optimally-directed designs for further consideration according to criteria that are difficult to model computationally, such as aesthetic value. The potential for concurrent evaluation of solutions through parallel computing can be valuable in offsetting the often prohibitive number of solution evaluations required by such methods. EAs require neither any domain knowledge, nor gradient information, but are effective at global search. Despite generally poor performance in precise local search, the ability to hybridise with domain dependent heuristics or deterministic local search methods makes EAs very versatile. Successive populations are developed through the genetic operations of reproduction, crossover and mutation .
5.3. GENETIC P ROGRAMMING METHOD
5.3.1. Introduction
Genetic Programming uses tree structures to define solutions, with fundamental components as terminals or "leaves", operated on by internal function nodes. This has
115 been applied literally in developing high performance computer programmes (Koza 1992) as well as in electronic circuit design and other fields (Koza et al. 2003). Roston (1994) presents a Genetic Design (GD) methodology applied to artifact design, adopting the tree representation and form of genetic operations associated with GP, whilst appreciating that the encoding of design information meant that this was “essentially a generalisation of GA”. In Genetic Programing, terminals take the form of constants or variables, while functions operate on given inputs from lower in the tree and pass a result up. The number of inputs on which a function operates is its arity and for a given function may be fixed or variable. Figure 5.1 demonstrates a tree representation of the mathematical equation y = 4/(X*X) + 5*(7-X) evolved to fit a set of experimental data. Terminals may be constant integer values or the variable, X. Functions are chosen from the arithmetic operators: add '+', subtract '-', multiply '*', divide '/'.
+
/ * 4 * 5 - x x 7 x Figure 5.1: Tree representation of mathematical equation: y = 4/(X*X) + 5*(7-X)
5.3.2 GP FOR BRACING DESIGN
This chapter proposes the application of Genetic Programming to the problem of bracing system design with terminals taking the form of bracing units occupying a single bay-storey cell within an orthogonal framework. A range of functions, or design modification operators, are defined to operate on terminals and outputs of functions lower in the tree. Thus the basic bracing units are developed through scaling and patterning to a state where they may represent a high-performance stability system. Each operator has an associated vector with parameters defining the direction, frequency or magnitude of the operation relative to the instance on which it acts. The set of functions is shown in figure 5.2, along with details of their associated vectors. Functions are either reversible or unidirectional, as indicated by the arrowheads. The vector(s) associated with each operator is shown above or below the arrow. As an example, the unite function combines bracing units located in a three-
116 bay, three-storey “neighbourhood”, the bottom left hand corner of which is indexed as (1,1). Every function is unary (arity = 1), with the exception of the unite function, which takes two or more inputs. The set of design modification operators can be considered as an informal form of grammar (Stiny 1980), which, along with a vocabulary of bracing types, defines a set of designs or language. Previous use of grammars within structural topology design is demonstrated by Shea (1997) through integration with simulated annealing to the optimally-directed design of planar and space trusses.
ADJUST SCALE SPLITS [1,2] [1,1] [-1,-2] [1,3] [X enlargement, [X divisions, Y enlargement] Y divisions]
ORTHOGONAL ROTATE TRANSLATE REPEAT [0,1,2] [90 o] [0,1] [0,-1] [X spacing, Y spacing, [angle of rotation] [X translation, frequency] Y translation]
10 10 10 10 IRREGULAR 9 UNITE 9 9 REFLECT 9 8 8 REPEAT 8 8 7 [1,1;3,3] 7 [1,3,2] 7 [2.5,-] 7 6 6 6 6 5 5 5 [2.5,-] 5 4 4 4 [X index, Y index; [X spacing, 4 [X centre-line, 3 X neighbourhood 3 Y spacing, 3 Y centre-line] 3 2 size, 2 frequency] 2 2 Y neighbourhood 1 1 1 1 size] 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 Figure 5.2: Function set for GP trees representing bracing designs
117 Creating initial designs
An orthogonal framework, with insufficient lateral rigidity, is taken as the starting point. An initial population of bracing system designs is created by applying the following steps to each individual: 1. Seed the framework with a number of bracing units (or instances ) each occupying a single bay-storey cell, with column (x) and row (y) indices, as seen in figure 5.3:
12 ROOT 9 X: 1,10 X: 1,2 X: 2,3 X: 3,12 4 6 10 11 3 5 7 8 2 1
1 2 3 Figure 5.3: Seeded framework
2. Sequentially apply a number of design modification operations, selected randomly, to single or united groups of instances, to assemble a tree representation of a design (figure 5.4).
118 ROOT NO CHANGE FROM ORIGINAL Unite PHENOTYPE X: 1,10 X: 3,12 (1,2; 2,2)
X: 1,2 X: 2,3 12
ROOT
Unite Orthogonal X: 1,10 (1,2; 2,2) Rep (0,-1,5)
X: 1,2 X: 2,3 X: 3,12 4 6 10 11 3 5 7 8 9 1 2 12
ROOT
Scale Unite Orthogonal (0,1) (1,2; 2,2) Rep (0,-1,5)
X: 1,10 X: 1,2 X: 2,3 X: 3,12 4 6 10 11 3 5 7 8 9 1 2 12
ROOT
Scale Irregular Orthogonal (0,1) Rep (1,-1,1) Rep (0,-1,5)
Unite X: 1,10 X: 3,12
4 6 10 11 (1,2; 2,2) 3 5 7 8 9
X: 1,2 X: 2,3 1 2
Application of design modification operators omitted
12 ROOT
Translate Irregular Rep Adjust Splits (0,-1) (0,4,1) (1,2)
Irregular Rep Orthogonal Rotate (1,-1,1) Rep (0,-1,5) 4 6 10 11
3 5 7 8 Scale Unite X: 3,12
2 9 (0,1) (1,2; 2,2) 1
X: 1,10 X: 1,2 X: 2,3 Figure 5.4: Development of an initial design by application of design modification operators
119 Analysis and fitness
In the test problems presented subsequently, all individuals in a population are analysed subject to prescribed wind-loading. Each design is then assigned a “fitness” based on bracing length and structural performance.
Generating subsequent populations
The selection process employed in the test problems in the following section is elitist, coupled with linear weighting of selection probability based on rank. Individuals are ranked relative to others in the population according to their fitness. This ranking is used to determine the individual's likelihood of selection as a parent in generating the next population. As seen from figure 5.5, the fittest individual has approximately twice the probability of selection in a given operation than the individual ranked at the fifty-percentile. Elitism ensures that the best known solution(s) is passed from one generation to the next.
2/N
1/N Probability of selection of Probability
1 2 3 ...... N-2 N-1 N Fitness-based rank (of N individuals in current generation) Figure 5.5: Linear rank-based weighting system for parent selection
Subsequent generations are populated by the application of genetic operators: – the best R% of designs from the previous generation are directly reproduced through the elitism strategy.
– Crossover is applied with probability P c to pairs of parent designs selected from the previous generation. The same design cannot be chosen as both parents. “Cuttings” from each parent are selected at random and exchanged to form new
120 trees. The cutting may be a terminal alone or a number of functions applied on a terminal(s).
– Mutation is applied with probability P m (=1-P c) to a single parent, passing one offspring design into the next generation. The mutation operator may take four forms: 1. mutation of a single terminal 2. mutation of a branch-section (terminal(s) and functions) 3. removal of a branch (subject to retaining at least one branch in the tree representation 4. addition of a branch. Mutation forms 2 and 4 involve generating a new branch. In the implementations described in this chapter, the new branch has a single terminal, with between 0 and 4 functions. A summary of the Genetic Programming search process is presented in the flowchart of figure 5.6.
START
Generate initial population
Evaluate Termination YES population and END criteria satisfied? sort by fitness
NO Reproduction YES
Population NO Select genetic complete? operation
Select parent(s)
Select crossover/ Create offspring mutation point(s) Figure 5.6: Genetic Programming evolutionary process flowchart
Handling geometrically infeasible designs
A substantial proportion of offspring from both crossover and mutation operations are found to be geometrically infeasible, either due to bracing units overlapping or extending beyond the prescribed framework. An example is shown in figure 5.7,
121 where, in attempting to accommodate a new branch into the parent tree (see figure 5.8(D)), the function “Irregular repeat (1,-1,1)” causes overlap of bracing units and the bracing system to extend outside the underlying orthogonal framework.
12 ROOT
Irregular Rep (1,-1,1)
Unite (1,2; 3,2) 4 6 10 11
3 5 7 8 Scale X: 1,2
2 9 (1,1) 1
X: 2,2 Figure 5.7: Example of geometric infeasibility, with units overlapping and extending beyond the orthogonal framework.
The Structured Genetic Algorithm (Dasgupta and McGregor 1991) includes a genetic hierarchy to account for the necessity or infeasibility of certain design combinations (Rafiq et al. 2003). Unfortunately this is not applicable in the current GP method on account of the form of the design representation. Various alternative techniques have been previously adopted for handling infeasible individuals (Michaelewicz 1996): – Rejection of infeasible solutions may help focus the search on the feasible region, but can also lead to the loss of potentially valuable “genetic” information. In the context of the current application, rejection leads to populations of sparsely braced individuals and low diversity. – Prevention of illegal solutions through careful design of evolutionary operators may be possible in some cases, though this is not applicable to the current application. – Penalisation can be applied to solutions violating constraints by reducing their fitness. This is the approach taken in the current case with designs with excessive lateral displacement under the given loads, as detailed in the following section. However, geometric infeasibility may mean that solutions cannot be assigned a fitness since they are structurally nonsensical, hence penalisation is not possible in this case. – Repair operations can be performed on infeasible solutions. Although in some applications this may be very difficult, it is readily achieved in the current case by
122 removing or making appropriate adjustments to relevant design modification operators, as illustrated in figure 5.8.
ROOT 12 Translate Irregular Rep Adjust Splits (0,-1) (0,4,1) (1,2) 9 Irregular Rep Orthogonal Rep Rotate (1,-1,1) (0,-1,5)
Unite Scale X: 3,12
4 6 10 11 (0,1) (1,2; 2,2) 3 5 7 8 2 X: 1,9 X: 1,2 X: 2,3 1 1 2 3 A: Parent Design (phenotype and genotype)
ROOT
Translate Irregular Rep Adjust Splits (0,-1) (0,4,1) (1,2)
Irregular Rep Orthogonal Rep Rotate (1,-1,1) (0,-1,5)
Scale Unite X: 3,12 (0,1) (1,2; 2,2)
X: 1,9 X: 1,2 B: A cut is made in the parent tree 12
ROOT 9 Scale (1,1)
X: 2,2 4 6 10 11 3 5 7 8 2 1
C: A new branch is created (mutation) or taken from a second parent (crossover)
(continued overleaf)
123 12 12 12 4 6 10 11 4 6 10 11 4 6 10 11 3 5 7 8 3 5 7 8 3 5 7 8 2 9 2 9 2 9 1 1 1
1 2 3 1 2 3 1 2 3
ROOT ROOT ROOT
Scale Unite Irregular Rep X: 1,2 (1,1) (1,2; 3,2) (0,4,1)
Scale Unite X: 2,2 X: 1,2 (1,1) (1,2; 3,2)
Scale X: 2,2 X: 1,2 (1,1)
X: 2,2
D: The new branch is developed by adding functions above the cut made in the original tree, where possible. N.B. “Irregular repeat (1,-1,1)” was rejected as this would have caused geometric infeasibility
ROOT 12 Irregular Rep X: 1,9 X: 3,12 (0,4,1)
Unite (1,2; 3,2)
Scale X: 1,2 4 6 10 11 (1,1) 3 5 7 8 2 9 X: 2,2 1
1 2 3 E: Other terminals appearing in the original parent tree are replaced where possible.
ROOT 12 Irregular Rep Orthogonal Rep Translate (0,-1) (0,4,1) (0,-1,4)
Unite Rotate X: 3,12 (1,2; 3,2)
Scale Scale X: 1,2
4 6 10 11 (1,1) (0,1) 3 5 7 8 2 9 X: 2,2 X: 1,9 1
1 2 3 F: These terminals are then traced back to the root of the parent tree, replacing functions where possible. N.B. In right hand branch, “Orthogonal repeat” is reduced to (0,-1,4) to avoid overlap. The subsequent “Split Adjust (1,2)” function is then infeasible and is therefore removed. Figure 5.8: Repair algorithm in the context of mutation or crossover
124 It is important to note that the mapping between physical structural representation, or phenotype, and the GP tree representation, or genotype, is not one-to-one. That is to say, a given bracing framework in the solution space can be represented by a potentially infinite number of distinct trees in the representation space. Termination of the optimisation process is determined by improvement of best-of- generation individual fitness and lowest average fitness of a generation. The method is implemented in Matlab.
5.4. BRACING DESIGN FOR A 2 X6 FRAMEWORK
Validation of the GP method will be conducted using the same two-bay, six-storey framework test problem as was used for the exploration of ESO techniques in chapter 3. By using the same loading conditions and displacement constraint, it is possible to select a fixed diameter for circular solid sections for all bracing members such that the globally optimal design solution is known with reasonable certainty. Referring to table 3.12, taking a fixed section diameter of 0.08m for bracing members of circular hollow section, it is reasonable to expect that the double echelon configuration of solution B would meet the displacement constraint of 0.024m with minimal total bracing length. It is readily demonstrated that this design is indeed acceptable, with a top corner displacement of 0.023m. As previously, bracing members are modelled with simple supports. The optimisation model for the problem established can be expressed as an unconstrained minimisation problem (using a penalty function):
n Minimise: L = ∑ Le + max (),0 p()d max − d * Eq. 5.1 e=1 where: L = design fitness (equivalent to total length of bracing members for feasible designs)
Le = length of bracing member, e n = total number of bracing members d* = limit on maximum lateral displacement dmax = maximum lateral displacement observed in structure
125 p = penalty factor imposed on designs violating constraint on top storey lateral displacement, nominally chosen to be 5000 for all studies detailed in this chapter. The total number, length and location of bracing members are variable in the evolutionary process. Fixed parameters in the structural model include framework geometry, applied loads, vocabulary of basic bracing types and section size of bracing members (A ), beams and columns. Issues of strength and buckling are recognised as e important but not included at this stage for means of comparison. A symmetry constraint is applied in this case. This simplifies the problem to one of a single bay, six storey framework, which is reflected in the horizontal centre-line and has the added advantage that only a single structural analysis loadcase need be considered for each design. An asymmetric design must be analysed in two loadcases, with identical loads applied to each face. Defining three basic bracing units: “X”, “/”, “\” will allow the majority of the designs shown in figure 3.12 to be represented (with the exception of designs F, H, J, K and M). Stochastic optimisation and search methods can often be inefficient on account of generating a large number of poor quality designs, the structural analysis of which is a waste of computational resources. A means of filtering out, or improving, such designs can greatly improve the efficiency of the search. In the case of braced steel frameworks, it is known that bracing is required in every available storey in order to maximise structural efficiency (Ji 2003). A “fill” algorithm is therefore proposed, which will add functions or fresh terminals to the tree representation until bracing is present in every storey. It is calculated that this reduces the size of the search space in this test problem by more than a third, from over 10,000 distinct designs to 3072. These values are calculated by considering the combinatorial problem of filling the six cells on one side of the structure with bracing units of different size and form. The “fill” algorithm also compensates for the fact that complexity is generally lost in implementing the “repair” algorithm. A study was made to investigate the effect of varying the optimisation parameters, using the specifications described above. Population size took values of 10, 30 and 50, combined with crossover probabilities equivalent to 0, 0.75 or 0.9 of new offspring generated by crossover operation. Twenty runs were conducted for each pairing of parameters. Since this suggested that a population size of 30 offers a good
126 compromise between accuracy and computational efficiency, further runs were then performed for a population size of 30, making up to a total of 50 runs for each crossover probability listed previously, as well as 0.5. The convergence criterion requires that no improvement is seen in either best-of-generation individual fitness or mean population fitness for 10 generations, with zero tolerance. At this point the process is terminated. The objective of this investigation is to understand sensitivity to these parameters and how maximum computational efficiency can be achieved without compromising accuracy in finding the global optimum solution. Ideally, a good level of diversity in high performance designs archived over the course of each process is still achieved. Table 5.1 shows the performance of runs with each combination of parameters. The best solution found in any of these runs was indeed the double echelon configuration of solution B in table 3.12, with a bracing length, equal to the fitness value, of 50.2m. This design is therefore referred to as the global optimum in the right hand column of table 5.1. The 119 function evaluations required, on average, to locate the optimal solution using a population size of 10 and a crossover ratio of 0.9 is less than 4% of the 3072 function evaluations required to exhaustively assess all possible design solutions defined by the current formulation with the fill algorithm.
Table 5.1: Batch characteristics in parametric study
Number of Function Final population generations to evaluations to find % runs fitness statistics (m) Popn. Crossover find local local optimum: finding optimum size probability (Population size x global Mean Mean S.D. of mean generations optimum of of S.D. means Mean S.D. to optimum) means 50 0 5.4 15.9 270 95% 29.6 114.8 3.96 (20 runs 0.75 3.6 6.2 178 100% 31.2 98.8 4.20 each) 0.9 3.8 11.3 190 100% 34.4 94.0 6.10 0 6.2 26.5 187 94% 33.0 115.6 5.64 30 0.5 7.3 45.6 220 98% 32.2 108.2 5.65 (50 runs 0.75 7.1 25.5 212 100% 35.0 99.0 8.36 each) 0.9 5.7 25.7 172 100% 32.8 93.2 6.16 10 0 11.1 81.8 110 65% 45.0 113.8 14.34 (20 runs 0.75 12.4 97.2 124 90% 33.2 98.2 7.56 each) 0.9 11.9 65.4 119 100% 32.0 88.8 8.90
127 The primary conclusion from these results is that the method exhibits relatively low sensitivity to the optimisation parameters of population size and crossover ratio. Traditionally in EAs, excessively small population size creates a danger of insufficient genetic material in the gene pool and hence convergence to a false optimum, whereas overly large population size leads to computational inefficiency. Mutation should prevent potentially important genetic information being lost, whilst crossover exploits beneficial characteristics already in the population. With reference to table 5.1, small population size or mutation-only runs may yield false optima, at least within the convergence criteria specified, as seen in 35% of runs with a population size of 10 and without use of crossover operation. It should be noted that with the use of the “fill” algorithm, new genetic information is introduced into virtually every new offspring. This may hinder convergence and cause deviation from conventional trends observed in EAs. The final three columns present details of design fitnesses in the final populations: standard deviation of fitness within a population, averaged across a batch of runs; average fitness of designs in the final populations of a batch and standard deviation of mean fitness within a run. Of these, the most significant trend is the decrease in mean fitness with increased crossover ratio, for a constant population size.
Figures 5.9 to 5.11 characterise a representative run from the set with population size of 30 individuals and crossover probability of 0.9. In the initial population a mix of the three bracing types, “X”, “/” and “\”, is observed. More than half (16 out of 30) of these randomly generated designs do not satisfy the lateral displacement constraint. Average and best individual fitness in a generation improves rapidly in the early stages, until the optimum design is found in the seventh generation, as seen in figure 5.11 (where the initial population is marked as 0). Thereafter the average fitness within a generation fluctuates, whilst the best solution is unchanged on account of the elitist strategy. The final population, seen in figure 5.10, still contains many infeasible designs (14 out of 30), and although the optimum solution is repeated several times, other good solutions are found. In this population, the '/' bracing unit dominates, although due to the mutation operator and fill algorithms, other types still appear.
128 The n:1 mapping between genotype and phenotype means that many different trees can represent the optimal design. Figure 5.12 shows that the method described finds relatively simple tree-representations, without the phenomenon of uncontrolled program growth, or “bloat” (Langdon and Poli 1997), observed in some previous applications of GP. A typical run with a population size of 30 with 18 generations, hence a total of around 500 function evaluations, took around 40 minutes to run on a PC with Pentium® 4 CPU 2.66 GHz and 512 MB RAM.
Figure 5.9: Example of initial population, penalised designs shown in grey (Population size = 30, Crossover ratio = 0.9, run number 22)
Figure 5.10: Example of final population, penalised designs shown in grey (Population size = 30, Crossover ratio = 0.9, run number 22), best-of-run design top-left
129 150 best of generation average of generation 125
100
75 Fitness value Fitness 50
25
0 0 5 10 15 20 25 30 Generation Figure 5.11: Example of evolution history (Population size = 30, Crossover ratio = 0.9, Run number 22)
ROOT ROOT
Irregular Repeat Irregular Repeat (0,-3,1) (0,-3,1)
Scale (0,2) Scale (0,1)
Adjust Splits /: 1,4 (1,1)
Irregular Repeat (0,1,1)
/: 1,4 Figure 5.12: Example of most efficient tree representation of the optimal double echelon design (left), with the actual representation found in Run number 22, Population size = 30, Crossover ratio = 0.9 (right)
5.5. BRACING DESIGN FOR A 6 X30 FRAMEWORK
The potential of the proposed method for larger structures is now demonstrated through the example of bracing design for a six-bay, thirty-storey framework. Each unit cell is 6.096m wide and 4.267m high. The orthogonal framework of beams and columns was sized from a selection of standard circular hollow sections (as per the
130 Pinnacle Tower, considered in chapter 4) by fully-stressed design. The members are grouped in three storey blocks, as shown in figure 5.13. Within each block, all horizontal beams are required to take the same section, as are the four outermost columns and the three inner columns. Table 5.2 lists the sections made available in the fully stressed design process, along with the groups assigned to each section in the converged solution. The sizing process considered the combination of two loadcases: uniform vertical loading of 40kN/m acting downwards on all horizontal beams and horizontal loading of 50kN applied as a point load at all column-beam intersections on the left hand face of the structure. Since the structure is designed to have horizontal symmetry it is not necessary to define a loadcase with horizontal loads on the opposite face. The bracing configuration is to be designed such that the maximum lateral displacement in the structure does not exceed 0.256m (1/500 of the total height of the structure) under extreme wind-loading: three times the horizontal loadcase considered in the framework sizing. Hence point loads of 150kN are applied at each storey height on the left hand face. All bracing members are to take the same circular hollow section, selected such that a known good design of repeated X-bracing, as shown in figure 5.17, satisfies the displacement constraint. This is achieved using the standard CHS 273 16.0 section, whereby maximum lateral displacement of 0.248m is observed at the top left hand corner node. Based on the assumption that larger search spaces are better explored with larger population sizes, a population size of 50 was selected for a set of three runs. The crossover ratio of 0.75 performed well in the previous study (see table 5.1) and is hence adopted for tackling the current problem. The convergence criterion was modified such that termination occurs when no improvement, again with zero tolerance, is seen in the best-of-generation fitness for 30 generations. Initial and final populations are shown for the first of these runs in figures 5.14 and 5.15 respectively. The corresponding evolution history is shown in figure 5.16. The best solution found by each run is seen in figure 5.17, alongside the datum design of repeated X-bracing, with performance characteristics included.
131 10 10 10 10 10 10
20 20 30 30 30 20 20 10 10 10 10 10 10
20 20 30 30 30 20 20 10 10 10 10 10 10
20 20 30 30 30 20 20 9 9 9 9 9 9
19 19 29 29 29 19 19 9 9 9 9 9 9
19 19 29 29 29 19 19 9 9 9 9 9 9
19 19 29 29 29 19 19 8 8 8 8 8 8
18 18 28 28 28 18 18 8 8 8 8 8 8
18 18 28 28 28 18 18 8 8 8 8 8 8
18 18 28 28 28 18 18 7 7 7 7 7 7
17 17 27 27 27 17 17 7 7 7 7 7 7
17 17 27 27 27 17 17 7 7 7 7 7 7
17 17 27 27 27 17 17 6 6 6 6 6 6
16 16 26 26 26 16 16 6 6 6 6 6 6
16 16 26 26 26 16 16 6 6 6 6 6 6
16 16 26 26 26 16 16 5 5 5 5 5 5
15 15 25 25 25 15 15
5 5 5 5 5 5 30 x4.267m
15 15 25 25 25 15 15 5 5 5 5 5 5
15 15 25 25 25 15 15 4 4 4 4 4 4
14 14 24 24 24 14 14 4 4 4 4 4 4
14 14 24 24 24 14 14 4 4 4 4 4 4
14 14 24 24 24 14 14 3 3 3 3 3 3
13 13 23 23 23 13 13 3 3 3 3 3 3
13 13 23 23 23 13 13 3 3 3 3 3 3
13 13 23 23 23 13 13 2 2 2 2 2 2
12 12 22 22 22 12 12 2 2 2 2 2 2
12 12 22 22 22 12 12 2 2 2 2 2 2
12 12 22 22 22 12 12 1 1 1 1 1 1
11 11 21 21 21 11 11 1 1 1 1 1 1
11 11 21 21 21 11 11 1 1 1 1 1 1
11 11 21 21 21 11 11
6 x 6.096m Figure 5.13: Geometry and cross-section groupings for 6x30 framework
132 Table 5.2: Cross-sections made available and selected in fully-stressed design.
Section Section catalogue Groups assigned ID listing to section 1 STD CHS 273 16.0 20, 30, bracing 2 STD CHS 273 20.0 - 3 STD CHS 273 25.0 - 4 STD CHS 323 16.0 10 5 STD CHS 323 16.0 19, 29 6 STD CHS 323 25.0 - 7 STD CHS 355 20.0 9 8 STD CHS 355 25.0 8, 28 9 STD CHS 406 20.0 - 10 STD CHS 406 25.0 6, 7 ,18, 27 11 STD CHS 406 32.0 3, 4, 5, 16, 17, 26 12 STD CHS 457 25.0 2 13 STD CHS 457 32.0 1, 15, 25 14 STD CHS 457 40.0 14, 24 15 STD CHS 508 32.0 - 16 STD CHS 508 40.0 12, 13, 22, 23 17 STD CHS 508 40.0 11, 21
133 Figure 5.14: Initial population of randomly generated designs
134 Figure 5.15: Final generation of designs (run 1), including best-of-run design (top-left).
135 1200 best of generation mean of generation 1000
800
600 Fitness Value 400
200
0 0 10 20 30 40 50 60 70 80 90 100 Generation Figure 5.16: Evolution history of run 1.
Known Projected Run 1 Run 2 Run 3 design design Total bracing length (m) 446.4 434.4 506.2 353.2 314.6 Maximum lateral 0.248 0.257* 0.252 0.258* 0.273* displacement, d max (m) Constraint violation (%) 0 0.3 0 0.8 6.8 Fitness 446.4 439.8 506.2 355.2 366.8 Generations to optimum - 68 58 113 - - Figure 5.17: Best-of-run designs and performance (* indicates displacement constraint violation)
136 From figure 5.14 it is seen that a high degree of diversity is attained in the generation of random designs. Figure 5.15 shows that diversity remains high in the final population, although occurrences of the 'X' bracing unit are greatly reduced, since intuitively this is detrimental to the objective of meeting the displacement constraint with minimal bracing material. The evolution history of figure 5.16 follows a similar trend to that of the simpler 2x6 framework example, but on a longer timescale. The mean-of-generation values reach a noisy plateau after around 15 generations, with the continuous addition of new genetic material from the “fill” algorithm preventing any further reduction. The best-of generation solution improves until generation 68. Inspection of the form and performance of best-of-run designs (figure 5.17) reveals that two of the three solutions present a reduction in steel tonnage when compared with the datum known design, despite their unconventional form. In the case of the design from run 3, a material saving of 20% is offered over a popular solution in building design practice. The penalty function has permitted these solutions, despite a violation of the displacement constraint of less than 1% in both cases. This is beneficial, since a nominal addition of material would add sufficient stiffness to the structure to meet the constraint. The best-of-run designs exhibit a high proportion of diagonal bracing units which rise two storeys for each bay spanned. Arrangement of these units into chevrons, as per the optimal 2x6 framework bracing solution, also appears highly beneficial. The right-hand design of figure 5.17 presents a design of more regular appearance than might be considered by designers as a result of the above discussion. However, despite offering a reduction in bracing length of 11% over the best design of run 3, the larger constraint violation of 6.8%, as opposed to 0.8%, means that it has a poorer fitness according to the defined optimisation model. Hence in three runs, the genetic programming method has found a design superior to the regular designs proposed, as evaluated by the fitness function used. Each run performs of the order of 5000 structural analyses in calculating the objective function value for each solution. Whilst this number of analyses would have permitted an exhaustive search of the design space defined for the 2x6 framework problem, it will only cover a minute proportion of the solutions included in the vast design space for the 6x30 framework problem. It is therefore unsurprising that a
137 stochastic method with restricted computational resources is unable to consistently locate a single optimal solution.
5.6. CONTROL OF AESTHETIC STYLE
The methodology previously described offers great versatility on account of the capacity for prescribing aesthetic style. This can be achieved through exerting control over: unit types – examples above allowed 'X', '/' and '\' units. This list could be extended or restricted. maximum and minimum bracing size – intricate designs may be generated by specifying a maximum size to which bracing units may be scaled. Alternatively, designs in the search space may be simplified by requiring bracing units to be greater than a minimum size. Search may also be simplified by, for example, considering a 6x30 framework to be reduced to a 1x10 grid, with each basic unit being 3-bay x 3- storey, reflected in the centre-line of the structure aspect ratio – scaling operations could be constrained to keep units in a certain proportion, such that, for example, units must have equal height and width. pattern definition – repetition could be constrained to follow prescribed vectors, e.g. orthogonal or rising one storey for every bay. As an example, considering the aesthetic requirements on the Pinnacle Tower on which the work in Chapter 4 is based, one might define the following constraints: – unit types: '/' or '\', must be anchored to the base of the building. – minimum bracing size: 3 storey by 1 bay units – aspect ratio: 3 storey rise for each bay spanned. This influences permissible scale and adjust splits function vectors. – Repeat vectors must be of the form (1, 3, r) or (x, 0, r) However, the pattern generating behaviour of the functions defined in the Genetic Programming methodology would not be very compatible with the desire for a randomised visual effect on this project. This form of control can also be used to restrict the design space to be searched in an attempt to improve convergence and performance of solutions. For example, figure 5.17 suggests that 'X' bracing could be removed from the list of available unit types
138 and 2-storey, 1-bay bracing used as a basic unit, due to its frequency in best-of-run solutions. This would constitute a learning strategy and could be incorporated into the algorithm as machine learning. It should also be noted that it would be very straight- forward to define further symmetry-lines and repeating units as used in Chapter 3.
5.7. FURTHER WORK
It is a notable omission that the proof-of-concept studies do not include section-size optimisation nor any form of strength consideration. Section-sizing may be overlooked in topology design if the minimisation of piece-count or bracing length rather than steel tonnage is the primary objective, as in the project-based work of Chapter 4. Alternatively, approximate section-sizing operations could be performed on every solution, or on selected individuals using a Lamarkian operator, as employed in the pseudo-Genetic Programming method of Liu (2000). The former approach would add significant computation time to each run, whilst the latter is likely to slow convergence.
5.8. CONCLUSIONS
This chapter has described a methodology for optimally directed search using genetic programming. This approach exploits the capacity of GP for evolving instructions to create a solution, as opposed to evolving solution representations. Through the use of two proof-of-concept examples, the potential of this method for generating high performance, pattern-based topological designs has been demonstrated. Further, the versatility of the method offers potential for defining subsets of solutions based on aesthetic criteria. The initial research question is answered by employing an Evolutionary Algorithm: the use of populations and a stochastic method allow multiple diverse solutions to be obtained, whilst the GP algorithm removes the need for a fixed set of variables. Section 5.6 presents a framework for defining a desired aesthetic form, allowing different styles to be explored in response to the second research question. Finally, the use of design modification operators offers a more pure interpretation of GP for structural topology design than is seen in previous studies and allows a range of complexity in the structural designs, but with simple tree-representations.
139 The form of design modification operators used offer potential well beyond bracing design, to any design problem in which pattern emergence is relevant and performance may be quantified.
140
6. Concluding Remarks
This chapter commences with a reassessment of each of the three methods presented in the preceding chapters. Recommendations are then put forward for future research. There follows a summary of issues raised relating to the application of structural optimisation in building design practice. Finally a discussion of potential future trends in structural design automation and optimisation practice is given.
6.1. REVIEW OF CONTRIBUTIONS
Table 6.1 presents a summary of the three distinct structural optimisation and search methods investigated in the preceding research chapters. The method type (e.g. stochastic or deterministic, discrete or continuous material representation) is stated, alongside the type of solution obtained (single or multiple, form of optimality). The most appropriate design stage for application of each method is also stated, along with requirements and practical implications for appropriate use. The research contribution of each was stated in Chapter 2 and is briefly summarised in the table. The wider context of these contributions, realisation of the research objectives and validity of the central hypothesis are now considered. There is no single method that is applicable to, let alone optimal for, all structural design problems, since the nature of design task specifications vary greatly from one project to another. For example, ESO is well suited to the free-form structures to which it has been applied, detailed in section 2.2.3, but when the layout of discrete structural members is more tightly controlled, as in the bracing design of the Pinnacle Tower (Chapter 4), ESO is inefficient and impractical. The Genetic Programming methodology is arguably most powerful when more concept generation and pattern emergence is desirable and in larger scale structures. Fogel (1999) states, in a discussion of the No Free Lunch theorem that: "…for an algorithm to perform better than even random search (which is simply another algorithm) it must reflect something about the structure of the problem it faces. By consequence, it mismatches the structure of some other problem…" Therefore a structural optimisation toolbox is required for structural topology, shape and size optimisation, of which the methods presented in this thesis could form a part.
141
Chapter 3 demonstrated how the practicality of Evolutionary Structural Optimisation can be improved by using appropriate design criteria, such as displacement, to govern element removal and addition. The consideration of aesthetic issues such as symmetry and repetition through element grouping also offers advances on previous work. The integration of element thickness optimisation by defined groups removes the dependence of final solutions on prescribed thickness. This method could readily be used in structural topology optimisation problems in general, especially where aesthetic considerations are concerned. Chapter 4 applied Hooke and Jeeves' Pattern Search method to tackle a parameterised topology optimisation problem, in an example of a "problem-seeks-design" scenario. It demonstrated how existing optimisation techniques may be adapted to practical topology problems in preliminary or detailed design. This study also showed how size and topology optimisation can be efficiently integrated by performing a single iteration of the optimality criteria sizing algorithm for each topology change. This only increases the number of analyses required by a factor of two, compared to the optimisation of topology alone. It was observed that simultaneously optimising size and topology offered a volume reduction of more than 20% when compared to the staged process of performing topology optimisation with maximum sections, followed by a separate size optimisation. The use of stochastic search allowed a range of distinct high-performance designs to be generated for appraisal by designers considering unmodelled criteria. The Pattern Search method, with integration of the Optimality Criteria sizing approach, has potential for any structural topology optimisation problem that can be parameterised in some way. Chapter 5 demonstrated the potential for generating multiple, novel, high performance conceptual designs for a topological design problem, through the use of a Genetic Programming method using design modification operators as internal nodes. This approach avoids the need for a fixed set of variables and allows complex bracing patterns to be developed from very simple blue-prints. Control over the form of the design modification operators allows the user to influence the form of the solutions obtained. Use of this GP method in other structural topology design problems would be eminently possible, but dependent on the definition of an appropriate set of design modification operators. The central hypothesis, as stated in chapter 1, is that optimisation can be successfully and appropriately applied in practice through consideration of industry specific issues.
142
This has been validated most directly in chapter 4, with optimised designs being used directly in outline proposals. Relevant industry specific issues included aesthetic requirements, desire for alternative proposals, adaptability to specification changes and multiple loadcases. The ESO and GP methods developed in chapters 3 and 5 respectively have shown potential for successful application in practice, through the ability to include aesthetic considerations and in the case of GP, multiple proposed solutions, with possible decision support and interactivity.
143
Table 6.1: Summary of methods used in this thesis Chapter 3: Evolutionary Chapter 4: Pattern Search Chapter 5: Grammar-based
Structural Optimisation - Optimality Criteria Genetic Programming Stochastic (or deterministic), Stochastic Evolutionary Algorithm discrete material representation, Tree-based representation Deterministic, continuous material mathematical programming (PS); without encryption. Method type representation Deterministic, continuous / Design modification operation discrete variables (OC) functions.
Appropriate design Conceptual / Preliminary design Preliminary / Detailed design Conceptual / Preliminary design stage Definition of criteria for element Ability to parameterise Grid-based framework Requirements addition and removal optimisation problem Single or multiple solutions from Local optima, improved by Multiple optimally-directed different start points and domain constraint handling method (PS). solutions. Type of solution thicknesses. Local optima on account of Global optimum for small-scale Uncertainty regarding degree of force-moment redistribution (OC). tasks. optimality Use of GP to generate Practical consideration of Efficient integration of topology "programmes" that in turn constraints, repetition and and size optimisation generate structural designs. Primary research symmetry Constraint handling contributions New tool for optimally directed Thickness optimisation of Successful industrial application and controllable bracing pattern element groups on live project generation
Modest practical applicability to General method applicability to Example of potential for optimally- bracing design due to difficulty in Practical any parameterised problem. directed, rapid design generation discrete interpretation. Large and evaluation methods implications numbers of elements required Constraint handling requires leads to large computational time. problem specific modification Currently bracing frame specific
6.2. RECOMMENDATIONS FOR FUTURE WORK
This section focuses on future work that would be appropriate in further developing the methods and themes presented in this thesis.
Evolutionary Structural Optimisation
- Increase of scale: The scale and complexity of the structures considered here is substantially less than in some studies, for example the three dimensional, multi- storey Docklands Tower model of Holzer (2006). However, the results presented in Chapter 3 are arguably more regular and more appropriate for discrete interpretation and assignment of standard sections than those found elsewhere in the literature. It would therefore be a worthwhile extension to increase the scale of the structural models considered to gain both benefits. - Frame design: The described integration of size optimisation into the ESO process considered only the thickness of the two-dimensional elements in the designable domain. A logical extension would be to also allow the section sizes of the orthogonal framework to adapt to changes in load path in meeting the displacement constraint. This would require the inclusion of the vertical load- case, originally defined by Mijar et al. (1998), to size the framework. - Inclusion of inter-storey drift constraints: Considering inter-storey drift constraints in addition to top-storey lateral displacement will complicate the ESO process, but avoid the possibility of the profile of the framework bowing out lower down the structure. - Design process integration: The integration of ESO into the overall design process also warrants further consideration. In evolving a structure to efficiently meet a stiffness requirement, one must consider how the solution is interpreted as a discrete design and at what stage to consider strength requirements, including buckling.
Pattern Search - Optimality Criteria
- Investigation of the performance compromise in hybridisation: The hybrid method described executes an Optimality Criteria sizing algorithm for each topological change, assuming the force-moment distribution within the structure to be unchanged by these modifications. Performing a full sizing operation for each
145
topological stage is likely to yield improved designs accompanied by an order of magnitude increase in the number of structural analyses required. A study into the compromise made by the approximate method would be very informative. Various possible alternative strategies exist for efficient simultaneous size and topology optimisation, such as reapplying the OC algorithm until the changes made are less than a defined threshold. - Cost modelling: The possibility of developing a cost model to quantify the trade- off between piece-count and volume minimisation was discussed with Arup designers. This would include estimates of construction time per additional member and associated costs, reduction in letting revenue on account of view impingement, material costs, etc. A cost modelling approach for conceptual design was previously adopted by Khajehpour (2001). However, it is crucial to be aware of the liability of the cost model to change throughout the project. - Wider method application: The "problem-seeks-solution" approach adopted for the design task on this industrial project means that it requires validation through use on further problems and more conventional building forms.
Genetic Programming
- More thorough testing on structures of a practical scale: The 2x6 benchmark case used in Chapter 5 is a relatively simple problem. Further validation and possible method development is required on design tasks of larger scale, such as the 6x30 framework also considered in Chapter 5, or the concept sketches of figure 2.4 that were inspirational to the development of this tool. - Strength (including buckling) considerations: In the work presented, maximum lateral displacement is the only constraint considered. This is a simplification and should be extended to include consideration of inter-storey drift and strength constraints, potentially in conjunction with the introduction of section-size optimisation. - Section-size optimisation: Although detailed consideration of section-sizing is often neglected in conceptual design, the integration of section sizing into the GP algorithm is of significant research interest. This could be achieved in a similar manner to the Pattern Search - Optimality Criteria hybrid method of chapter 4, although topology changes in the GP process are likely to be more significant, making the approximations less acceptable.
- Control over aesthetic style: Section 5.5 presents a framework for offering the user control over the aesthetic style of solutions generated, by defining available unit-types, restrictions on aspect-ratio and maximum/minimum size of bracing units, pattern repetition characteristics, symmetry and regions of structure in which bracing is prohibited. This approach offers the potential to incorporate project specific requirements and alternatives. - Graphical User Interface: For use beyond prototyping, this tool requires the development of a graphical user interface to become a "user-seductive" program (Cohn 1994) appropriate to mainstream structural designers.
6.3. APPLICATION OF STRUCTURAL OPTIMISATION IN
PRACTICE
This thesis has highlighted a number of issues relevant to the practical application of structural optimisation in industry, most significantly through the involvement in the Pinnacle Tower project of Chapter 4, as well as the discussion of drivers and barriers in section 1.4. The following points have become apparent: - Current commercially available optimisation software is more suited to design of components in automotive and aerospace engineering and is regarded as a specialist tool, rather than being accessible to the majority of structural designers. - Aesthetic issues influence a substantial proportion of structural design decisions and should generally be either incorporated into the optimisation model or used to assess a range of high-performance solutions produced by the optimisation process. These alternative solutions may be generated using a stochastic process or through parametric variation. - Decision support tools, potentially including optimisation methods such as the Genetic Programming tool presented in Chapter 5, show potential to be valuable in the early stages of structural design. Currently decisions are often based on experience and intuition, with a limited range of alternatives considered. - Substantial interest in the use of structural design optimisation exists within the building industry. However, this is tempered by the barriers discussed in section 1.4. Optimisation should be an interactive process with a high level of control offered to the user, with easy customisation and the ability to rapidly incorporate model changes.
- Design "freezes" (Eger et al. 2005), marking the end of a development stage and fixing some aspect of the design, occur in the building engineering industry in much the same way as any other design process, although the architect-engineer interaction can lead to frequent "thaws". Use of optimisation must account for these freezes, as well as offering the versatility to adapt to frequent changes in specification of geometry, constraints, objectives and variables. Questions that should be addressed when considering the use of structural design optimisation in practice, and more specifically when selecting an optimisation method, include: - Is the problem suitably well-defined to yield meaningful optimal solutions? What are the design objectives, variables, constraints and parameters in the optimisation task? - Is it possible to gauge the size and complexity of the design space? - How will constraints be incorporated? - How long will each objective function evaluation or structural analysis take? What simplifications and approximations can be made? - For a chosen method, is it possible to predict how many function evaluations are likely to be required for acceptable convergence? Hence is the total projected run time realistic? - Is the goal to find a single best solution, a range of good designs or to understand trade-offs between multiple objectives? - What starting point will be used? Is this point feasible according to the constraints? Is it appropriate to use a range of starting points?
6.4. PROJECTED TRENDS IN STRUCTURAL DESIGN
AUTOMATION AND OPTIMISATION IN PRACTICE
Over 20 years ago, Templeman (1983) stated that, "… unless researchers are prepared to step back from the research frontiers of structural optimisation and become involved in providing practical design software and unless practicing design organisations are prepared to step forward and sponsor the writing of such software, both sides of the engineering profession are likely to miss out on an exciting development within the profession…"
In the years since this observation, several commercial software packages, examples of which can be found in chapter 2, have developed directly from academic research and practicing design organisations have sponsored and collaborated in research investigations, such as that in this thesis. However the full potential of structural optimisation in design practice remains to be realised. In the context of structural design of buildings, the potential of methods using a continuum design domain is limited, on account of the difficulties discussed earlier in the thesis. However, it should continue to find use in small scale, novel structures and isolated sub-systems of larger structures as well as in providing insight into load path distribution. With the exception of certain ESO examples, significant use of optimisation in non- parameterised topological design remains a long-term goal, despite the strong incentives. However, if a small number of high-profile examples can be established, this could serve as proof or potential and increase interest and further use. It is likely that the use of section-size automation and optimisation will continue to increase, in response to the demands of complex design tasks, aided by dissemination of rigorous methods, e.g. Optimality Criteria and software, and inspired by existing examples. Naturally, this advancement could be hastened by the inclusion of relevant material in undergraduate structural engineering teaching to spread awareness of methods. Increasing use of appropriate supporting software, such as parametric CAD modelling, and digitalisation of document interchange between architects and engineers mean that the path has been laid for parametric shape and topology optimisation. This could become widely practised in the near future on account of the modest risk and high potential return.
6.5. CLOSING NOTES
In summary, it has been shown that this research has satisfied the objective of contributing towards reducing the gap between research and industry, through the steps made towards establishing a structural optimisation toolbox for the building industry, as discussed in section 6.1. The thesis has demonstrated, primarily in Chapter 4, the potential benefits of applying structural topology optimisation to "live", full-scale projects, hence validating the central hypothesis. These include rapid
generation and optimisation of a range of solutions, customisation of methods such that results are directly applicable and gaining understanding of feasible design spaces. As demonstrated in chapters 3 to 5, using appropriate methods it is possible to successfully accommodate aesthetic design considerations, amongst other practical issues, either explicitly in an optimisation model, or by generating multiple optimally directed designs.
Appendix 1: Structural analysis
Common to each of the methods presented in this thesis is the requirement for structural analysis to evaluate the performance of a proposed design. In all cases, finite element analysis is carried out using Oasys 1 GSA (General Structural Analysis) (Oasys 2003). In the prototype implementation of each optimisation algorithm, structural models are automatically written as GSA text input files, to include requests for the required results files to be written when analysis is conducted. From the prototype programme, GSA is called, the relevant file is opened and analysed and results files are written. These can then be read by the programme and processed as a required. This process is illustrated in figure A1.1.
Write structural model as GSA text input file,
Call batch file
Read and process results files
Figure A1.1: Structural analysis flowchart
1 Oasys Limited is the software house of Arup.
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Appendix 2: Software development and prototyping
Each of the three search methods investigated in this thesis was implemented in prototype form in either C++ (Pattern Search with Optimality Criteria) or MATLAB (Evolutionary Structural Optimisation and Genetic Programming). As previously discussed, minimising the time spent on software development is crucial to the successful application of optimisation methods in industry. Although the precise nature of the design problem may require tailoring of algorithms and corresponding code, it is clearly desirable to recycle and adapt existing code where possible in an effort to drive down development time. This applies to adapting to changes in specification within a project, as seen in Chapter 4, as well as transfer from one project to another. In the author's experience, MATLAB is an excellent numerical computing environment and programming language for development of prototype tools. It is also optimised for matrix manipulation and is easy to learn and intuitive to use. Plotting of results and development of Graphical User Interfaces (GUIs) is very straightforward. MATLAB is widely used within the research community. Licensing for industry is an additional overhead, although free open-source alternatives such as SciLab 2 and Octave 3 are available. C++, Java etc. are free, although licensing may be required for Microsoft development environments. They are potentially considerably more powerful than MATLAB for general programming purposes, but harder to learn and more time consuming for tool development. These programming languages are arguably better suited to development of full-scale systems.
2 http://www.scilab.org/
3 http://www.gnu.org/software/octave/
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