Form-finding and patterning of fabric structures using shape optimization techniques

Thomas Linthout

Supervisors: Prof. dr. ir. Wim Van Paepegem, Dr. Ali Rezaei Counsellor: Tien Dung Dinh

Master's dissertation submitted in order to obtain the academic degree of Master of Science in Civil Engineering

Department of Materials Science and Engineering Chair: Prof. dr. ir. Joris Degrieck Faculty of Engineering and Architecture Academic year 2015-2016

Form-finding and patterning of fabric structures using shape optimization techniques

Thomas Linthout

Supervisors: Prof. dr. ir. Wim Van Paepegem, Dr. Ali Rezaei Counsellor: Tien Dung Dinh

Master's dissertation submitted in order to obtain the academic degree of Master of Science in Civil Engineering

Department of Materials Science and Engineering Chair: Prof. dr. ir. Joris Degrieck Faculty of Engineering and Architecture Academic year 2015-2016 Acknowledgements

“Rome wasn’t built in a day”, and similarly, this dissertation is the result of a year of hard work and research. However, the completion of this dissertation would not have been possible without the advice and support of some people who I would like to thank.

I would like to thank professor Van Paepegem for his efforts as supervisor, and for intro- ducing me to the wonderful world of optimization and tensile architecture.

I am grateful for the advice of my counselor Tien Dung Dinh. Whenever I had a question about membrane materials, about optimization, or when a software bug slipped past my eye, he was always ready to assist.

Special thanks to Ali Rezaei for helping me out numerous times.

I am thankful for my friends, who provided distraction whenever I couldn’t see the forest for the trees.

And finally, I would like to thank my parents, for supporting me throughout my studies.

Thomas Linthout, June 2016 Permission

“The author gives permission to make this master dissertation available for consultation and to copy parts of this master dissertation for personal use. In the case of any other use, the limitations of the copyright law have to be respected, in particular with regard to the obligation to state expressly the source when quoting results from this master dissertation”

Thomas Linthout, June 2016 Form-finding and patterning of fabric structures using shape optimization techniques

by Thomas Linthout Master’s dissertation submitted in order to obtain the academic degree of Master of Science in Civil Engineering Academic year 2015–2016 Supervisors: Prof. Dr. Ir. W. Van Paepegem, Dr. A. Rezaei Counsellor: M. Sc. T.D. Dinh Faculty of Engineering and Architecture Ghent University Department of Materials Science and Engineering

Abstract Fabric materials used for tent structures are form-active, behave non-linearly and anisotrop- ically, and can only transfer loads through tension. Due to this, designing and modelling such structures is quite complicated and different from traditional structures. The so- called conventional method is the most frequently used design method for designing tensile structures. In this method, a form-finding is carried out, followed by the development of a cutting pattern on the surface. Another design method is the integrated method, in which flat panels are iteratively generated, assembled and subjected to tension, until a structure with a permissible stress distribution is reached. This thesis is a continuation on the method proposed by Dinh et al., based on the integrated approach. In this method, the shape of the assembled panels is iteratively adjusted until the final structure has a permissible stress distribution. It is assumed that the shape of the assembled panels, also called the intermediate configuration, is stress-free. Expressing the shape of the assembled panels by parameters, a SolidWorks model is created, which is then imported in Abaqus. After calculation, the stresses in the membrane, along with the forces in the boundary cables is compared to a specified stress and cable force, and the parameters are adjusted. This process is repeated until the stress and force deviations are minimal. Keywords Tensile architecture, shape optimization, fabric materials, Bayesian optimization, optimiza- tion Form-finding and patterning of fabric structures using shape optimization techniques

Thomas Linthout

Supervisors: Prof. Dr. Ir. Wim Van Paepegem, Dr. Ali Rezaei, M. Sc. Tien Dung Dinh

Abstract Multiple methods exist to design fabric does not correctly represent the material. A material structures. The most commonly used method is the so- model was proposed by Dinh et al.[1], taking into called conventional method, which has some deficiencies. account these characteristics. However, time- However, an accurate method, taking into accounts the dependent behavior is not included in this model. This material characteristics of the membrane is non-existent. model is further referred to as the fabric plasticity In this thesis, a new method is proposed, taking into account the correct loading path the material follows. model. The proposed method assumes that the assembled panels form a similar shape as the final structure, albeit with a III. DESIGN METHODS smaller curvature. Using shape optimization, the shape of this intermediate shape is determined in such a way A. Conventional method that the stress distribution in the final structure is permissible. The most frequently used method in the industry is the so-called conventional method. The method Keywords Tensile architecture, shape optimization, consists of a form-finding step, in which a fictitious fabric materials, Bayesian optimization, optimization membrane material is subject to a prestress, resulting in an equilibrium shape. This shape is then subjected I. INTRODUCTION to a structural analysis, using the actual material For over 50 years, coated woven fabrics have been characteristics, in order to assess the impact of used in modern lightweight structures. However, clear environmental loads such as wind and snow loads. guidelines for such architecture do not exist, and their Finally, a cutting pattern is generated on the design is mainly reliant on experience. Fabric prestressed shape. This cutting pattern is then materials used for tent structures are form-active, flattened, resulting in flat panels. In this step, the behave non-linearly and anisotropically, and can only deformations caused by this flattening operation are transfer loads through tension. neglected. The shape of the stress-free panels is then The aim of this thesis is to propose a new method to determined by unloading these panels. Due to the non- design tensile structures, taking into account the linear behavior of the membrane and the difference characteristics of the material and the loading path it between the stress-strain curves for loading and follows. To this extent, shape optimization methods unloading, this is not straightforward. As such, a may provide an answer. linear orthotropic material model is often used in this step, and the shape of the panels is adjusted to II. MEMBRANE MATERIALS compensate for the inaccuracy of the model. Textile fabrics used for tensile structures are a Due to this compensation, and because deformations combination of a system of orthogonally woven yarns occurring during flattening are ignored, this method is and a coating applied to withstand weathering and less than optimal. ageing. The coating is applied continuously, by applying the liquid coating while the fabric is moving B. Integrated approach under a knife. During this process, the fabric moved While the conventional method starts with a through drums by pulling the warp fibres. Due to this, prestressed shape, the integrated approach includes the the warp fibres are more stretched than the fill fibres, influence of the cutting pattern in the form-finding and which will have an impact on the behavior of the analysis. Cutting patterns are iteratively adjusted such material. that, after applying the deformations resulting in the As a result of the production process and the final shape, the structure has a permissible stress- composite action, the fabric has a quite complex distribution, minimizing the difference between the material behaviour. The material behaves design prestress and the actual stress in the membrane. anisotropically, inelastically and nonlinearly. In a study conducted by Dinh et al.[2], a new design Additionally, the material behavior is dependent on method was proposed. This method is a variation of the load ratio between warp and fill threads. As a the integrated approach. In the method, the shape of result, a simple orthotropic elasticity material model the assembled panels is assumed to be flat, and three parameters describing the intermediate shape are The result of this optimization is a certain adjusted until the stress distribution in the final intermediate shape. As this shape consists of fabric construction is permissible. The method proposed in panels assembled together, this shape should be stress- this study is a continuation on this method. free. This assumption can be validated by cutting the three-dimensional intermediate shape into panels, IV. PROPOSED METHOD computationally flatten these panels using Rhino3D’s ‘Squish’ algorithm, and check the deformations. If A. Proposed design method these deformations are small enough to neglect, the In this case study, the proposed method is tested on method is deemed valid, and the panels can correctly a 4 m x 4 m hypar, with a height of 80 cm once assemble the hyperstructure. assembled and tensioned. First, the shape of the adjoined panels is considered. It is assumed that this B. Objective function intermediate stress-free shape will have a similar The objective function is used to convert the results shape of that of the final hypar, albeit with a smaller from the FEM analysis to one value, which is to be height. To describe this intermediate shape, a set of minimized. In order to do this, the average squared five state variables P is used, as seen in Figure 1. deviation from a predefined membrane stress and cable force is calculated. This results in two values, of which a weighted average is taken resulting in one value, which is the value of the objective function, as seen in the formulae below.

� � = �! ⋅ �! + �! ⋅ �!

1 !"!#$%&" !"!#$%&" �! = � − �!"# �!"!#$%&" !!"!#$%&"

Figure 1 Five state variables describing the intermediate 1 !"#$% !"#$% shape of the membrane �! = � − �!"# �!"#$% In Abaqus, the membrane material characteristics !!"#$% are assigned to the shape, and a cable element is modeled in sleeves at the edges of the membrane. The cable has an effective stiffness of 2000 kN. In a first C. Optimization algorithm step, an orthotropic material model is used for the Many capable algorithms exist to perform the membrane, and next the fabric plasticity model will be optimization of the objective value, however it is not used. The characteristics of the orthotropic material clear which algorithm is the most suited for this model can be found in table 1. Next, the purpose. To know the most appropriate algorithm, displacements are described, resulting in a stress three algorithms are compared to each other: a genetic distribution in the membrane and a force in the cable algorithm, a Bayesian optimization algorithm adopted element. from a study conducted by Snoek et al.[3] and a Pattern Search algorithm. All the algorithms are Table 1: Parameters of the orthotropic elasticity model applied to the objective function described above, and Ew Ef �!" �!" G the algorithms are compared based on their number of 975.6 MPa 716.1 MPa 0.23 0.17 87.7 MPa function evaluations and the resulting value of the objective function. This comparison is done for five Using an Abaqus Python script, the average values of the weight factors. absolute deviation from a predefined membrane stress In this comparison, the pattern search algorithm and cable force is calculated. This results in two returns the fastest results. However, the Bayesian values, of which a weighted average is taken resulting optimization always returns a better value, albeit with in one value. The goal of this design method is to more function evaluations. The genetic algorithm uses minimize this value, and as such bringing the stress in the most function evaluations, and never returns a the membrane as close to the design stress as possible. better value than the Bayesian optimization. As a The target stress in the membrane is 5 MPa, while the result, the Bayesian optimization is deemed the most target force in the cable is 20 kN. suitable for this project. To minimize this value, an optimization algorithm is used. The goal of optimization is to find the variables D. Verification of stress-free hypothesis at which the objective function reaches a minimum. In Finally, it has to be verified that the resulting this case, the objective function is the result from the intermediate shape can be assembled from flat panels post processing, which is a function of the state without inducing stresses that are too high. In variables. Rhino3D, the intermediate shape is divided into six panels, along geodesic lines. Using the ‘Squish’ feature, these panels are then flattened, while returning maximum and average deformations caused by the flattening. A clear overview of the process can be seen in figure 2.

Figure 3 Warp stresses in the optimal configuration, using the orthotropic elasticity model (w1=0.7, w2=0.3)

Figure 2 Flowchart of the proposed design method

E. Benchmark As the method proposed in this thesis is a continuation of the method proposed by Dinh et al.[2], results from this study can be compared to results Figure 4 Fill stresses in the optimal configuration, using the from the earlier work. As the intermediate shape is orthotropic elasticity model (w1=0.7, w2=0.3) controlled by five parameters instead of three, better Table 2: Parameters of the optimal configuration, using the stress distributions and cable forces are expected. orthotropic elasticity model (w1=0.7, w2=0.3) p (m) p (m) p (m) p (m) p (m) F (kN) V. RESULTS AND CONCLUSION 1 2 3 4 5 cable -0.011 0.367 0.310 0.000 0.242 19.998 Varying the weight factors, intermediate shapes causing a lower stress and force deviation were indeed found, both using the orthotropic elasticity model and the fabric plasticity model. After dividing the found intermediate configurations into six panels of equal width and flattening these panels, deformations up to 0.15% occur. These deformations correspond with negligible stresses, so the intermediate configuration can indeed be assembled without inducing stresses. Results of the shape optimization can be seen in figures 3-4 and table 2 for the orthotropic elasticity material model, and in figures 5-6 and table 3 for the fabric plasticity material model.

Figure 5 Warp stresses in the optimal configuration, using the fabric plasticity model (w1=0.7, w2=0.3) ACKNOWLEDGEMENTS The author would like to thank the suggestions of his supervisors and counsellors.

REFERENCES [1] T.D. Dinh, A. Rezaei, L. De Laet, M. Mollaert, D. Van Hemelrijck, and W. Van Paepegem. A new elasto-plastic material model for coated fabric. Engineering Structures, 71:222–233, 2014. [2] T.D. Dinh, A. Rezaei, L. De Laet, M. Mollaert, D. Van Helmelrijck, and W. Van Paepegem. A shape optimization approach to integrated design and nonlinear analysis of cable reinforced fabric membrane structures. [3] J. Snoek, H. Larochelle, and R.P. Adams. Practical Bayesian optimization of machine learning algorithms. Figure 6 Fill stresses in the optimal configuration, using the fabric plasticity model (w1=0.7, w2=0.3) Table 3: Parameters of the optimal configuration, using the fabric plasticity model (w1=0.7, w2=0.3)

p1 (m) p2 (m) p3 (m) p4 (m) p5 (m) Fcable (kN) -0.033 0.314 0.350 0.000 0.349 19.993

It can be concluded that the method can indeed provide form-finding and patterning of fabric structures, taking into account their material characteristics. The difference between the used material model manifests itself in the resulting shape of the panels. As seen in figure 7, the resulting panels when using the orthotropic elasticity model (blue) are bigger than those resulting when using the fabric plasticity model (red).

Figure 7 Panels from the orthotropic elasticity result (blue) compared to panels from the fabric plasticity result (red) However, some further research could be done on this method. First, only a handful of optimization algorithms were tested, so it is possible that a better algorithm exists for this purpose. Only parametric optimization was considered, and non-parametric optimization might provide even better results. Also, due to SolidWorks API restrictions, parallelization was not possible, which could have provided faster solutions. Finally, the method was only tested on a single type of structure, a hyperstructure. It remains to be confirmed whether the method can also be applied to other shapes, such as conical membranes or more complex structures. CONTENTS xi

Contents

Acknowledgements iv

Permission v

Abstract vi

Extended abstract vii

Contents xi

List of Figures xiv

List of Tables xvii

Used abbreviations xviii

Used symbols xix

1 Introduction 1

2 Fabric architecture 3 2.1 History of fabric architecture ...... 3 2.2 Analysis of fabric tension structures ...... 6 2.2.1 Conventional method ...... 7 2.2.2 Integrated design method ...... 12

3 Membrane materials 14 3.1 Textile fabrics ...... 14 3.2 Coating ...... 16 3.3 Mechanical Properties ...... 16 3.3.1 Tensile strength ...... 17 3.3.2 Anisotropy ...... 17 3.3.3 Nonlinear behaviour ...... 18 CONTENTS xii

3.3.4 Inelastic behaviour ...... 18 3.3.5 Shear stiffness ...... 18 3.3.6 Kink resistance ...... 19 3.3.7 Material model ...... 19 3.4 Structural behaviour ...... 21

4 Optimization 23 4.1 Optimization problems ...... 24 4.1.1 Linear vs. Nonlinear ...... 24 4.1.2 Constrained vs. Unconstrained ...... 24 4.1.3 Discrete vs. Continuous ...... 25 4.1.4 Single vs. Multiobjective ...... 25 4.1.5 Single vs. Multiple Minima ...... 26 4.1.6 Deterministic vs. Nondeterministic ...... 27 4.1.7 Simple vs. Complex Problem ...... 27 4.2 Solution approaches ...... 27 4.2.1 Experimental Optimization ...... 27 4.2.2 Analytical Optimization ...... 27 4.2.3 Numerical Optimization ...... 28 4.3 Summary of optimization algorithms ...... 28 4.3.1 Gradient-based methods ...... 28 4.3.2 Genetic algorithms ...... 29 4.3.3 Pattern Search ...... 30 4.3.4 Simulated annealing ...... 31 4.3.5 Bayesian optimization with Gaussian process priors ...... 31 4.4 Comparison of optimization algorithms ...... 32

5 Shape optimization 34 5.1 Parametric shape optimization ...... 34 5.1.1 Design model ...... 34 5.1.2 Analysis model ...... 35 5.1.3 Optimization algorithm ...... 36 5.2 Non-parametric shape optimization ...... 36

6 Shape optimization project 39 6.1 Proposed design method ...... 39 6.2 Design model ...... 41 6.3 Analysis model ...... 42 6.3.1 Creating the model ...... 42 6.3.2 Computation and post-processing ...... 43 CONTENTS xiii

6.4 Optimization algorithm ...... 44 6.5 Patterning and verifying the stress-free hypothesis ...... 45 6.6 Overview ...... 45 6.7 Benchmark ...... 46 6.7.1 Orthotropic elasticity material model ...... 47 6.7.2 Fabric plasticity material model ...... 47

7 Results 49 7.1 Comparing optimization algorithms using the orthotropic elasticity model . 49 7.2 Shape optimization using the orthotropic elasticity material model . . . . . 52

7.2.1 w1 = 0.7 and w2 = 0.3 ...... 52 7.2.2 w1 = 0.9 and w2 = 0.1 ...... 54 7.2.3 w1 = 0.95 and w2 = 0.05 ...... 55 7.2.4 Conclusion ...... 58 7.3 Shape optimization using the fabric plasticity material model ...... 58

7.3.1 w1 = 0.7 and w2 = 0.3 ...... 58 7.3.2 w1 = 0.9 and w2 = 0.1 ...... 60 7.3.3 w1 = 0.95 and w2 = 0.05 ...... 62 7.3.4 Conclusion ...... 63

8 Conclusion and recommendations for future work 65 8.1 Conclusion ...... 65 8.2 Recommendations for future work ...... 65

Bibliography 67 LIST OF FIGURES xiv

List of Figures

2.1 A reconstruction of a 10,000-year-old tent from remains found at Pincevent, Northern France, adapted from [2]...... 4 2.2 Artist’s render of the Colosseum’s velarium, while deploying, adapted from [3].5 2.3 The Olympia Stadium in Munich, Germany, designed by Frei Otto, adapted from [4]...... 5 2.4 A membrane structure from the ’10 Expo in Shanghai, China, adapted from [5].6 2.5 The air-supported roof of the Tokyo Dome, adapted from [6]...... 7 2.6 The different steps in creating a membrane structure, adapted from [7]. .8 2.7 Tangential surface stress field, adapted from [9]...... 9 2.8 Example of cloth unfolding, adapted from [10]...... 11 2.9 An illustration of the patterning in the conventional method, adapted from [7]...... 12

3.1 Diagram of different weaving methods, adapted from [16]...... 15 3.2 Typical stress-strain curves from a uni-axial test in different orientations, adapted from [17]...... 17 3.3 Stress strain curves for uniaxial test in warp and fill directions in case of cyclic load, adapted from [15]...... 19 3.4 A proposed elasto-plastic material model for coated fabrics, adapted from [19]. 20

4.1 Traditional vs. Optimal Design Process, adapted from [20]...... 23 4.2 Multiobjective optimization, adapted from [20]...... 25 4.3 Pareto frontier, adapted from [20]...... 26 4.4 Procedure for a Genetic Algorithm, adapted from [20]...... 30 4.5 An example of a Bayesian optimization on a 1D optimization problem, adapter from [25]. The figure also contains the acquisition function, which is highest in the points where the Gaussian process predicts a high objective along with a high uncertainty...... 32 LIST OF FIGURES xv

5.1 General shape optimization process, adapted from [28]...... 35 5.2 Example of cloth unfolding, adapted from [10]...... 37

6.1 The different steps in creating a membrane structure, adapted from [7]. . 40 6.2 Final configuration of the hypar, adapted from [19]...... 40 6.3 Intermediate configuration of the hypar, including its state variables . . . . 41 6.4 Flowchart depicting the proposed design method ...... 46 6.5 Membrane stresses in the optimized shape found by Dinh. et al. [15] using the orthotropic elasticity model ...... 47 6.6 Membrane stresses in the optimized shape found by Dinh. et al. [15] using the fabric plasticity model ...... 48

7.1 Membrane stresses in the optimized shape ...... 51 7.2 Membrane stresses in the optimized shape using the orthotropic elasticity

model (w1 = 0.7 and w2 = 0.3)...... 52 7.3 Optimization process ...... 53 7.4 Optimized shape divided in panels ...... 53 7.5 Membrane stresses in the optimized shape using the orthotropic elasticity

model (w1 = 0.9 and w2 = 0.1)...... 54 7.6 Optimization process ...... 55 7.7 Optimized shape divided in panels ...... 55 7.8 Membrane stresses in the optimized shape using the orthotropic elasticity

model (w1 = 0.95 and w2 = 0.05) ...... 56 7.9 Optimization process ...... 56 7.10 Optimized shape divided in panels ...... 57 7.11 Membrane stresses in the optimized shape using the fabric plasticity model

(w1 = 0.7 and w2 = 0.3) ...... 58 7.12 Optimization process ...... 59 7.13 Optimized shape divided in panels ...... 59 7.14 Membrane stresses in the optimized shape using the fabric plasticity model

(w1 = 0.9 and w2 = 0.1) ...... 60 7.15 Optimization process ...... 60 7.16 Optimized shape divided in panels ...... 61 7.17 Membrane stresses in the optimized shape using the fabric plasticity model

(w1 = 0.95 and w2 = 0.05) ...... 62 7.18 Optimization process ...... 62 LIST OF FIGURES xvi

7.19 Optimized shape divided in panels ...... 63 7.20 Difference between the panels when using the orthotropic elasticity model

(blue) or the fabric plasticity model (red), when using weight factors w1 =

0.7 and w2 = 0.3 ...... 64 LIST OF TABLES xvii

List of Tables

6.1 Boundaries of the state variables ...... 42 6.2 Parameters of the orthotropic elasticity material model ...... 43 6.3 Genetic algorithm parameters ...... 44

6.4 Parameters and results (w1 = 0.7 and w2 = 0.3) ...... 47

6.5 Parameters and results (w1 = 0.7 and w2 = 0.3) ...... 48

7.1 Different weight factor combinations ...... 49

7.2 Algorithm comparison for the case w1 = 0.1 and w2 = 0.9...... 50

7.3 Algorithm comparison for the case w1 = 0.3 and w2 = 0.7...... 50

7.4 Algorithm comparison for the case w1 = 0.5 and w2 = 0.5...... 50

7.5 Algorithm comparison for the case w1 = 0.7 and w2 = 0.3...... 50

7.6 Algorithm comparison for the case w1 = 0.9 and w2 = 0.1...... 50 7.7 Results for each weight factor combination ...... 51

7.8 Parameters and results (w1 = 0.7 and w2 = 0.3) ...... 52 7.9 Equivalent strains caused by assembly, panels numbered from left to right . 53

7.10 Parameters and results (w1 = 0.9 and w2 = 0.1) ...... 54 7.11 Equivalent strains caused by assembly, panels numbered from left to right . 54

7.12 Parameters and results (w1 = 0.95 and w2 = 0.05) ...... 55 7.13 Equivalent strains caused by assembly, panels numbered from left to right . 56

7.14 Parameters and results (w1 = 0.7 and w2 = 0.3) ...... 59 7.15 Equivalent strains caused by assembly, panels numbered from left to right . 59

7.16 Parameters and results (w1 = 0.9 and w2 = 0.1) ...... 60 7.17 Equivalent strains caused by assembly, panels numbered from left to right . 61

7.18 Parameters and results (w1 = 0.95 and w2 = 0.05) ...... 62 7.19 Equivalent strains caused by assembly, panels numbered from left to right . 63 xviii

Used abbreviations

2D Two-dimensional 3D Three-dimensional ALLIE Total strain energy ALLSD Viscous damping energy CAD Computer-aided design FEM Finite element method GA Genetic algorithm MSAJ Membrane Structure Association of Japan PTFE Polytetrafluoroethylene PVC Polyvinyl chloride SLS Serviceability limit state UV Ultraviolet VBA Visual Basic for Applications xix

Used symbols

 Strain ν Poisson’s ratio F Force σ Stress

σ11 or S11 Stress in the warp direction

σ22 or S22 Stress in the fill direction c Damping factor

nmembrane Number of finite element nodes in the membrane

ncable Number of finite element nodes in the cable

D1 Averaged absolute stress deviation in the membrane

D2 Averaged absolute force deviation in the cable

w1 Weight factor regarding to D1

w2 Weight factor regarding to D2 1

Chapter 1

Introduction

For over 50 years, coated woven fabrics have been used in modern lightweight structures. The membrane materials have a strong architectural value, can be installed quickly and are environmentally friendly. In comparison with traditional construction materials such as concrete and steel, architectural fabrics are lighter and cheaper, making them an ideal choice when a large area needs to be covered.

Fabric materials used for tent structures are form-active, behave non-linearly and anisotrop- ically, and lack bending or compressive stiffness. Because of this, they can only transfer loads through tension, while compression stresses will lead to wrinkling, which can eventu- ally cause the failure of the structure. These material characteristics make the design and modelling of tensile structure difficult.

A fabric tensile structure is constructed in three steps: first, panels are cut out from the rolls which the fabric is supplied from. These panels are then welded together, and the assembled membrane is tensioned. Due to the complex nature of such structures, along with the material characteristics of the membrane, designing tensile structures is more difficult than designing traditional structures.

Usually, fabric tensile structures are designed using the conventional method. This method consists of finding a shape of the tensioned structure, followed by developing a pattern on the surface. Next, the flat shapes of the 2D panels are to be determined. Due to the non-linear and inelastic behaviour of the membrane material, this is hard to do, and in practice, an orthotropic material law is used during the flattening, and the resulting panels are ‘compensated’ to account for this simplification. The flaws of this method are apparent, as the correct material characteristics are not used throughout the process, and because 2 the compensation is poorly documented.

A second method is the integrated method. In this method, flat panels are iteratively generated, assembled, and subjected to tensioning. This is repeated until a permissible stress distribution in the membrane is obtained. Throughout the design, the correct material characteristics are used.

The aim of this master’s dissertation is to propose a new design method for tensile architec- ture, continuing on the research by Dinh et al. and on the integrated approach in general. The purpose of this method is to find a cutting pattern of a fabric in such a way that the difference between the final stress distribution in the membrane is negligible compared to the design. A combination of finite element analysis and shape optimization is used to iteratively find the optimal shape of the structure, performing a nonlinear analysis in each iteration. 3

Chapter 2

Fabric architecture

When thinking about fabric structures, tents directly come to mind, being a small and portable solution. However, due to modern construction technology and materials, con- temporary tents can be enormous and can provide shelter for huge crowds. Two types of tensile architecture are commonly used: air-supported fabric structures, and fabric tension structures. These structures are incredibly light when compared to structures made of traditional construction materials, such as steel and concrete. Fabric tension structures have a low maintenance cost, no need for air-tightness, and a large variety of double-curved surfaces [1]. In this master’s dissertation, the focus lies on fabric tension structures.

2.1 History of fabric architecture

The earliest structures made by mankind were, like tents, both functional and transient. These humans were hunters, and had the ability to use animal by-products to make shelters, as well as weapons, tools and clothing. In Northern France, archaeologists have discovered 4.5 m wide tents, with wooden poles covered by animal skin, as seen in figure 2.1. Some of these prehistoric structures are still recognizable today, such as the Lapp Keti, a conical structure clad in hides, used by Siberian nomads and hunters. The skins used have a low tensile strength, and are relatively heavy, so when available, they were replaced by fabrics [2].

Small tents were also used by more developed nations in commercial, civic and military settings. The Roman military used ’papilio’, or butterfly tents that were constructed from a calf leather membrane. These tents were 2 m high at their central ridge, with 0.3 m walls that could be lifted for ventilation. Sometimes, tents had greater importance. 2.1. HISTORY OF FABRIC ARCHITECTURE 4

Figure 2.1: A reconstruction of a 10,000-year-old tent from remains found at Pincevent, North- ern France, adapted from [2].

For example, Ottoman rulers considered their tents as main residences. Their tents were arranged in complex structures including many different buildings and were larger than Roman tents [2].

To increase in size, membrane materials had to be tested in another area of design, such as ships. From 2000 BC, animal skins were used as sails by the Minoans to propel their ships using wind force. The curved sail can be compared by a pneumatic structure, which got optimized over the ages. This technology reached a peak in the 1869 tea clipper Cutty Sark, which could sail at 17 knots with its 3000 m2 of sail [2].

This technology was transferred back to buildings throughout history. Roman architec- ture used featured temporary and retractable shading called vela (Latin for sail). The Colosseum in Rome featured a tensile canvas roof of 189 m by 156 m, which was probably suspended from 7.5 m high masts around the building, as seen in figure 2.1. At a political meeting in 1520 called ‘Camp du Drap d’Or’ in France, a circular, 16-sided structure with a 40 m high central mast was used as main building [2].

At the beginning of the twentieth century, temporary fabric structures were used for other purposes, such as performance venues. This was pioneered by the company Stromeyer, which worked in close collaboration with the great innovator in fabric architecture design, 2.1. HISTORY OF FABRIC ARCHITECTURE 5

Figure 2.2: Artist’s render of the Colosseum’s velarium, while deploying, adapted from [3].

Frei Otto. From the 1950’s, Otto designed membrane structures with the combination of engineering, research, and sensitivity to the beauty and sustainability attributes of lightweight structures. Frei Otto invented the weighted cable and soap bubble method to produce natural physical models resulting in an optimal form [2].

Fabric architecture at this time was suited to big events and was therefore temporary. A precedent for permanent fabric structures was set by Otto’s German Pavilion on the Expo ’67 in Montreal, Canada. This however was a building with a steel cable-net mesh, from which polyester fabric was hung. Nowadays, vinyl-coated polyesters (PVCs) are the main material for fabric architecture, due to their low cost and 15-20-year lifespan [2].

Figure 2.3: The Olympia Stadium in Munich, Germany, designed by Frei Otto, adapted from [4]. 2.2. ANALYSIS OF FABRIC TENSION STRUCTURES 6

Figure 2.4: A membrane structure from the ’10 Expo in Shanghai, China, adapted from [5].

While the majority of fabric structures are fabric tension structures, air-supported struc- tures are another trend in fabric architecture. These fabrics achieve their tensioning by a difference in air pressure at both sides of the membrane [2]. The average air pressure at kN sea level is 1.013 25 bar, or about 100 m2 . As a comparison, this is fifty times the usual live load on a floor. Because of this value, small variations in air pressure can easily lift and tension a lightweight membrane [6].

Air-supported structures are commonly shaped as a dome, or as a half cylinder. By realizing an air pressure difference of about 0.001 MPa, or 10 mbar compared to the outside air pressure, the roof of the structure is held in place. As this is only a 1 % difference compared to the average air pressure, it is hardly noticeable by people inside of the building. However, such a structure is never completely air-tight, due to entrances, exits, and gaps in the membrane. Because of this, fans are used to guarantee a constant airflow. An example of this type of structure is the Tokyo Dome, seen in figure 2.1.

2.2 Analysis of fabric tension structures

Modern fabric tension structures are constructed from numerous flat panels that are welded together and tensioned. The panels are cut from rolls of fabric, and the shape of the assembled stress-free panels is called the intermediate configuration, as seen in figure 2.6. By tensioning this intermediate configuration, the final configuration is reached. Due to the complex assembly of membrane structures, along with the complex behaviour of membrane materials, designing these structures is not straightforward. 2.2. ANALYSIS OF FABRIC TENSION STRUCTURES 7

Figure 2.5: The air-supported roof of the Tokyo Dome, adapted from [6].

The most frequently used method to design fabric tension structures is the so-called con- ventional method. An important property of the conventional method is that the form- finding and the patterning are decoupled. By contrast, the patterning is incorporated in the form-finding in the so-called integrated approach. Both methods will be discussed in what follows.

2.2.1 Conventional method

The conventional method is generally used when designing and patterning fabric struc- tures. This method consists of four steps: the form-finding analysis, the load analysis, the patterning analysis and the fabrication analysis [1]. A commonly used software pro- gram following the conventional method is EASY, which incorporates every step from the form-finding up to the patterning analysis.

Form-finding analysis

Traditional structures, often made of steel and concrete, are designed to guarantee ser- viceability. In limit state design, a structure that satisfies the Serviceability Limit State (SLS) does not exceed a defined limit for deflections or local deformations [8]. As such, the shape of the structure under loads, assuming it has not yet reached failure, is not that different from the initial shape. By contrast, membranes tend to deform more than traditional structures, as they lack bending stiffness. As such, the final shape is different from the initial configuration, and should be determined beforehand. 2.2. ANALYSIS OF FABRIC TENSION STRUCTURES 8

Figure 2.6: The different steps in creating a membrane structure, adapted from [7].

When no loads are acting on the structure, except for the prestress, a force equilibrium

can be expressed. As seen in figure 2.7, stresses σx and σy act on the membrane. The force equilibrium in the z-direction is given by:

σ σ x + y = 0 (2.1) Rx Ry

As the stresses σ and σ are both positive, the curvatures 1 and 1 should have an x y Rx Ry opposite sign, resulting in a negative Gaussian curvature K = 1 1 , which is a typical Rx Ry property of a so-called anticlastic surface. If the curvatures 1 and 1 are equal but Rx Ry opposite in every point of the surface, the membrane is subjected to a uniform stress in both directions. Additionally, the membrane describes a so-called minimum surface.

Minimum surfaces are frequently used in the design of tension structures. The advantages of using minimum surfaces are their aesthetically pleasing shapes, and the uniform stress in the membrane. However, minimum surfaces tend to be quite flat due to the mean curvature being zero. Because the load-bearing capacities of the membrane are dependent on its curvature, this is not desirable [10].

To create such a surface, a constant prestress is maintained, independent of changes in strain, as seen in figure 2.7. This corresponds in a Young’s modulus of zero. However, in calculations, a very small Young’s modulus is used, to avoid numerical singularities. This process is an iterative process, and mostly Newton-Raphson iterations are used to 2.2. ANALYSIS OF FABRIC TENSION STRUCTURES 9

Figure 2.7: Tangential surface stress field, adapted from [9]. determine an equilibrium shape in which the specified prestresses are achieved [10]. Ex- perimentally, minimal surfaces can be found using soap bubbles (as discussed in 2.1).

One of the most frequently used shapes in fabric tension architecture is the hyperbolic paraboloid, also known as a hypar or a saddle shape. This shape can be constructed by lifting two opposite corners of a square membrane, while lowering the other two corners. Another standard shape is a conical shape, which can be realized by lifting an internal point of a membrane while keeping the edges in place.

However, form-finding ignores any external effects, such as the dead load of the membrane, wind loads, crimp, creep and temperature effects, and material anisotropy. To counteract deformation resulting from external loads, the prestress in the membrane has to be high enough to guarantee a sufficient out-of-plane stiffness. This is where the load analysis comes into play.

Load analysis

After the equilibrium state of the membrane is found, a structural analysis can be con- ducted to study the behaviour under environmental loads. In this analysis, the prestress is taken into account, along with the external loads such as loads from wind and snow. 2.2. ANALYSIS OF FABRIC TENSION STRUCTURES 10

Due to the form-active behaviour of fabric tension structures, changes in the geometry can occur that cannot be neglected. Therefore, a nonlinear analysis should be conducted. Additionally, the membrane will buckle or wrinkle under compressive stresses, so the correct material characteristics should be included. Again, this step can be performed in EASY, but also in general FEM software such as Abaqus or ADINA [1].

Cutting pattern analysis

Finally, a cutting pattern has to be generated on the prestressed membrane. Because archi- tectural fabrics are made in rolls of 2 m-3 m wide, the 3D structure has to be constructed out of 2D panels of this width or smaller. Cutting patterns are usually created along geodesic lines, both for aesthetic reasons and to minimize fabric usage [11]. Tension fabric structures are highly varied in size, material characteristics and curvature, and the cutting pattern depends on all these factors. In plane cloth geometry determination, there is al- ways some approximation, as double-curved surfaces are by definition undevelopable [7]. However, in order to produce reliable data, this approximation should be minimized. In other words, the flat panels resulting from the flattening should resemble the curved strips as good as possible [10].

The patterning process consists of two steps. First, a global layout of individual cloths has to be developed. The mesh used for the shape-finding can be used for this purpose. Afterwards, 3D-data for each cloth has to be converted in usable plane form. If a fabric cloth consists of only one layout of elements, the cloth may be developed to a distortion- free plane form. This method is called cloth unfolding and is the simplest one. It is a geometric technique and is therefore reliable. An example is shown in figure 2.8 [10].

However cloth unfolding is a major approximation, as every cloth is only one element wide. In order to accurately simulate double-curved membranes, the cloths have to be flattened. To minimize stresses caused by the flattening, the change in link length is minimized, using a least-squares approach. In this approach, the sum of squared differences between the link lengths in the flat situation and the link lengths in the curved situation is minimized, as seen in equation 2.2-2.4 [10].

m X 2 S(x) = φi (x) (2.2) i=1 q 2 2 φi = (xi,1 − xi,2) + (yi,1 − yi,2) − di (2.3) 2.2. ANALYSIS OF FABRIC TENSION STRUCTURES 11

(a) Double curved membrane structure (b) Cloth unfolding of (a)

Figure 2.8: Example of cloth unfolding, adapted from [10].

q 2 2 2 di = (Xi,1 − Xi,2) + (Yi,1 − Yi,2) + (Zi,1 − Zi,2) (2.4)

To ensure that the cloth strips remain compatible after flattening, the seam boundaries

have to remain unchanged. To achieve this, weights ωi are are assigned to links lying on the edge of a cloth strip. Instead of minimizing equation 2.2, equation 2.5 is minimized. [10]

m X 2 S(x) = ωiφi (x) (2.5) i=1

The choice in location of the panel’s edges, or seams, requires artistic judgement, so that the seaming does not visually corrupt the structure. It should also be economic, to minimize waste of material, and it should guarantee safe transfer of stresses between panels. For these reasons, it is optimal that the seams follow geodesic lines on the surface. [7]

The traditional patterning method described above is a geometrical method, and does not take into account the prestress of the membrane, nor its material characteristics. To take this into account, the panels are ‘compensated’ afterwards. This is an ad hoc method, so there is ample room for improvement. Figure 2.9 describes the traditional patterning process as a whole [7].

After the creation of the pattern, the shape of the stress-free panels are determined by unloading the prestress. However, these materials have a strongly non-linear behaviour, and the stress-strain curves for loading and unloading are different, resulting in multiple strain levels for a certain stress level. As a result, a correct unloading is impossible. 2.2. ANALYSIS OF FABRIC TENSION STRUCTURES 12

Figure 2.9: An illustration of the patterning in the conventional method, adapted from [7].

In practice, a linear orthotropic elasticity model is used, and the cutting patterns are compensated afterwards.

The conventional method obviously has its disadvantages. The compensation of the panels is badly documented and the stresses caused by deforming the flat panels are not taken into account, although they may be significant. As a consequence, wrinkles can occur in the structure, and the final shape of the membrane may differ from the shape found by the form-finding [7].

2.2.2 Integrated design method

While the conventional method can be considered a shape optimization problem, in which the patterning is done independently of the form-finding, the integrated design method, proposed by Abel et al. [12], tries to optimize the shape of the flat panels, in such a way that the shape of the structure and the stresses in the membrane are permissible.

The approach begins with the shape found by the conventional method, followed by a initial cutting pattern generation using the technique used in the conventional method. 2.2. ANALYSIS OF FABRIC TENSION STRUCTURES 13

Next, the flat panels are deformed to form the 3D shape from the beginning, and the stresses in the membrane are compared to the desired prestress, and a new cutting pattern is generated [13]. Iteratively, cutting patterns are adjusted such that the final structure has a stress distribution that has a minimal difference from the designed prestress [14].

In the integrated design method, the influence of patterning is incorporated in the form- finding, and the correct non-linear mechanics of the membrane are used. Naturally, this method is more accurate than the conventional method, as there are no approximations, and the correct material model is used throughout the process. However, as it is an iterative process, it takes longer to reach a solution.

Recently, a new design method based on the integrated approach was proposed by Dinh et al. [15], based on shape optimization techniques. The method assumes that the intermedi- ate shape of the assembled panels is flat, and uses shape optimization to find the optimal intermediate shape, such that the final structure has a permissible stress distribution. Due to this simplification, a solution could be found faster.

The method proposed in this thesis will continue on the work of Dinh et al. [15]. However, the intermediate shape is not assumed to be flat anymore, but it is assumed that the inter- mediate shape has a similar shape of the final structure, albeit with a smaller curvature. The method will be discussed in detail in Chapter 6. 14

Chapter 3

Membrane materials

Membrane materials are very thin and lightweight, and therefore suited to applications where large areas are to be covered. They are mostly classified in technical textiles and technical plastics. Technical textiles consist out of large membrane parts, joined together at the seam, and can only transfer loads by tension. Architectural fabrics are composites, consisting of woven yarns coated to withstand weathering, loads and temperature effects. Non-coated fabrics also exist for other purposes. Another option for tensile architecture is to use technical plastics. These films are made of fluorothermoplastics and are transparent. Because of this, the material is highly popular among architects. However, the strength is far lower than that of coated fabrics, and frequently multiple layers are used. This has the added benefit of providing isolation. Still, the lower strength of the material inhibits its use in larger spans [16]. In what follows, coated textile fabrics will be discussed.

Coated textile fabrics are a combination of multiple materials, and are therefore composite materials. They form a system of woven yarns, arranged orthogonally when unstressed, which consist of single threads, parallel or twisted together. Two types of materials are mostly used as architectural membranes: PVC-coated polyester, and PTFE-coated high- strength glass fiber. Less frequently used are silicone-coated glass fiber fabrics [16].

3.1 Textile fabrics

A fabric is made by interlacing warp and fill threads. Multiple methods exist to weave these fibers, namely the plain weave, basket weave and modified plain weave. Usually in polymers, warp and fill threads are equally curved, as is the case in figure 3.1. However, if large rolls of the fabric have to be manufactured, the coating process requires that the 3.1. TEXTILE FABRICS 15

fabric is pulled through drums along its warp direction. Because of this, the warp threads are more stretched than the fill threads. This waviness is called crimp [16].

Figure 3.1: Diagram of different weaving methods, adapted from [16].

Polyester fabrics are cheap, and offer great resistance during production and erection. Before yielding, the fabric can deform significantly, enabling timely corrections. However, the material properties decrease under UV light, and it is subject to ageing. Polyester fabrics are often combined with a PVC coating, and is mostly used for temporary structures. The PVC coating also gives it fire-resistant properties.

Fibreglass fabrics have a higher tensile strength, but fail in a brittle way, with low elastic straining. Because of this, the manufacturing process needs to be very accurate, and the material should be handled with proper care. On the other hand, the material is almost insusceptible to the effects of ageing, making it a preferred choice for long-time solutions. In combination with a PTFE coating, the fabric is non-combustible, and relatively translucent [17].

Finally, aramid fibers, most notably Kevlar, are used in tensile structures. Kevlar is used for its fire-resistive properties, however it is far more expensive than its competitors. Again, UV light can diminish its properties, requiring a coating [6]. 3.2. COATING 16

3.2 Coating

Textile fabrics used in architecture have to be resistant against alternating loads, creep, ageing, climatic and atmospheric effects, and should be non-flammable. In order to guar- antee these characteristics, a suitable coating has to be applied to the fabric. The applied coating has no load bearing function, only a protective function. However, applying the coating gives the fabric a certain shear stiffness [16]. Additionally, the coating influences the flexibility, the weldability, and the price of the fabric [6].

A commonly used coating material is polyvinyl chloride, or PVC. Usually, a PVC layer alone is not sufficient, and an extra protection layer has to be applied to prevent ageing under UV light. Mostly, a polyvinyl fluoride film is used for this purpose. A second coating material is polytetrafluorethylene or PTFE, commercially available under the name Teflon. PTFE is resistant against most chemicals and solvents, humidity, and UV light. It is dirt repellent, and can face high temperature fluctuations. However, it has a lower flexibility than PVC [6].

Recently, synthetic rubber coatings have been developed, such as Hypalon. These materials have better sustainability and strength properties, along with a lower manufacturing price. Finally, also polyurethane is used as a coating material [6].

Different methods exist to apply a coating. A pre-heated film can be pressed on the fabric, or the coating can be applied in liquid form, after which the fabric has to be heated in an oven [6]. However, in order to produce large quantities, a continuous method is required. In the industry process, the coating material is heated so it starts flowing. Using a knife, the coating is spread on the fabric while it is rolling under the knife. Afterwards, the coating is treated, to give additional protection and prevent moisture penetration. To guarantee that the coating provides sufficient protection, it should be thick enough. Generally, the thicker the coating is, the better the protection of the thread. Usually, the thickness is between 0.08 mm and 0.25 mm [16].

3.3 Mechanical Properties

The weaving process described above results in a specific material behaviour. Traditional materials used in construction show linear elastic and isotropic behaviour, while membrane materials exhibit non-linear, anisotropic and non-elastic characteristics [17]. 3.3. MECHANICAL PROPERTIES 17

3.3.1 Tensile strength

The tensile strength of a fabric material is dependent on the tensile strength of its fibres. If a single thread in the fabric fails, the load it initially carried is transferred to its neighbours. If this happens to multiple threads, threads in the neighbourhood will also fail due to the force distribution, and a rupture will occur [6].

3.3.2 Anisotropy

When a uniaxial tension test is conducted for strips that differ from each other in yarn orientation, results such as in figure 3.2 are obtained. It is apparent that yarn orientation has a noticeable influence on the material characteristics. When the yarns are oriented perpendicular to eachother, the material behaviour is described as orthogonally anisotropic or orthotropic.

Figure 3.2: Typical stress-strain curves from a uni-axial test in different orientations, adapted from [17].

This anisotropy can be explained by the woven base of the fabric, and the way it is 3.3. MECHANICAL PROPERTIES 18

produced. As mentioned above, the warp threads are tensioned during production, while the fill threads are woven in between. As a result, when the membrane is subjected to tension in the fill direction, the fill fibres first straighten before they elongate. In contrast, when the warp fibres are tensioned, they will deform less as they are already straight [17].

Additionally, the warp and fill yarns interact with eachother. When the membrane is tensioned in its fill direction, the curvature of the warp threads changes at the intersection points, causing a shortening in the warp direction. Similarly, when both directions are tensioned, the tensioned warp threads exert a resistance on the fill threads. In other words, the crimp in the fill direction is complemented by the crimp in the warp direction. This behaviour is called crimp interchange. As a result, fabrics behave differently under varying

load ratios. The ratio of the strain 1 in the direction of the applied load to the resulting crimp in the transverse direction 2 is Poisson’s ratio, or µ. In traditional materials, Poisson’s ratio is constant and related to the modulus of elasticity of the material. In contrast, the Poisson’s ratio of a fabric is dependant on the force applied [16].

3.3.3 Nonlinear behaviour

Unlike traditional materials used in construction, no linear relation can be found between the stresses and strains in the material, as seen in figure 3.2, and as a result Hooke’s law of elasticity cannot be used to describe material behaviour.

3.3.4 Inelastic behaviour

When fabric materials are subjected to repetitive, cyclic loads, some hysteresis is shown, as seen in figure 3.3. It is clear that all loading and unloading paths are different, indicating that the textile undergoes permanent strains, and that its material characteristics are dependent on its loading history. Additionally, the extent, speed, duration and number of load cycles have an influence on the material behaviour.

3.3.5 Shear stiffness

The shear stiffness of architectural fabrics is typically quite low. As a rule of thumb, 1/20th of the tensile elastic modulus is commonly used as an estimation [18]. Due to this, the detailed shear behaviour is often neglected in calculations. The shear stiffness is low at low shear angles, since the yarns can rotate freely and only the coating provides some shear resistance. However, starting from a certain angle, called the limiting angle, warp and fill 3.3. MECHANICAL PROPERTIES 19

Figure 3.3: Stress strain curves for uniaxial test in warp and fill directions in case of cyclic load, adapted from [15]. yarns lock eachother, increasing the resistance against further shear displacement. [16].

3.3.6 Kink resistance

During erection, fabric materials are often subjected to significant mechanical effects. Local damage to the coating of the fabric caused by folding or lifting should be prevented. Dam- age to the coating can locally expose the fabric threads, which are subsequently subjected to environmental influences. This can lead to premature failure of the construction [16].

3.3.7 Material model

Because of the previous material characteristics, it is a difficult task to model the material behaviour. Despite the fact that the yarns in the fabric are oriented perpendicularly, the material doesn’t follow an orthotropic elasticity material law. A full description of the load response of these fabrics for biaxial and shear loading has not been carried out yet, and even if it were, few techniques exist to interpret this data [18].

In the Commentary to the Membrane Structures Association of Japan (MSAJ) Testing Method for Elastic Constants of Membrane Materials, a standardized method is described for testing fabrics. For plain weave fabrics, a method consisting of biaxial and uniaxial tests is proposed to define mechanical constants. An orthotropic elasticity model is often assumed, since calculation software is often based on elastic parameters.

An elasto-plastic material model for coated fabrics was proposed by Dinh et al. in [19]. In this paper, uniaxial tension, biaxial tension and uniaxial bias extension tests were carried 3.3. MECHANICAL PROPERTIES 20

out. The model handles interaction between warp and fill yarns, as well as the influence of the coating. Different load ratios were considered, in order to capture the material characteristics as correctly as possible. A stress-strain diagram of this model is shown in figure 3.4.

Figure 3.4: A proposed elasto-plastic material model for coated fabrics, adapted from [19].

It is apparent that this material model is more complex than a simple orthotropic elasticity law. The material follows a different stress-strain behaviour in the warp and fill direction, as is the case in real life. In the warp direction, a linear correlation exists between the

stress and the strain, with a modulus Ew1 as long as the stress is lower than the yield stress y σw. After this, linear hardening will occur, modeled with the plastic modulus Hw. When unloading the textile, the relationship between stress and strain is again linear, albeit with

a different modulus Ew3. Reloading the textile again follows a different material law, with

a modulus Ew4.

The fabric acts similar in the fill direction, although the relationship between stress and strain is nonlinear in the elastic region. Additionally, linear hardening happens in two stages, with two different plastic moduli Hf1 and Hf2. For the membrane used in this thesis, the following moduli are used:

In the warp direction: 3.4. STRUCTURAL BEHAVIOUR 21

Ew1 = 1364 MPa Hw = 448 MPa

Ew3Hw y Ew2 = σ = 15 MPa Hw+Ew3 w

Ew3 = 1130 MPa

In the fill direction:

2 Ef1 = 10 000 MPa(14.1309(f ) + 0.04f + 0.0115) Hf1 = 593 MPa

Hf1Ef4 Ef2 = Hf2 = 236.82 MPa Hf1+Ef4

Hf2Ef4 y Ef3 = σ = 5 MPa Hf2+Ef4 f1

y Ef4 = 825 MPa σf2 = 15 MPa

The warp and fill threads interact with each other according to two Poisson’s ratios ν12 =

0.09076 and ν21 = 0.46923. To take into account the load ratio dependency, two moduli

Ew4 = 852.241 MPa and Ef5 = 618.402 MPa are used. Using a user material subroutine (UMAT), this material model can be implemented in Abaqus.

This material model accurately represents the nonlinear behaviour in both warp and fill directions, along with the permanent strains and dependency on load ratio. However, the model is only verified for load ratios 1:1, 2:1, 1:2, 1:0 and 0:1. Also, time dependent behaviour and resistance to shear are not considered.

3.4 Structural behaviour

As membrane materials are thin and flexible, they have low, negligible shear, flexural and compressive stiffnesses. Due to prestress and geometry, some out-of-plane stiffness can be achieved. However, if this prestress is too low, this out-of-plane stiffness is too low, and wrinkling can occur [16].

Clearly, the prestress of the membrane is of the utmost importance for the structural integrity of the structure. However, fabric materials exhibit creep behaviour, of which very 3.4. STRUCTURAL BEHAVIOUR 22 little is documented. As a result, prestress may vanish over time in some locations, along with the out-of-plane stiffness. One can imagine that if rainfall occurs, out-of-plane stiffness guarantees that water simply flows of the structure. However, over time, it can happen that a certain portion of the structure forms a pond, in which water can accumulate. This ponding phenomenon is to be avoided, as it can cause ruptures in the fabric. As the creep behaviour is tough to predict, creep is usually compensated by re-tensioning the boundary elements [6] [18].

To conceive the prestress in the membrane, boundary elements such as cables are used. These boundary elements are commonly installed in sleeves at the edges of the membrane, and by tensioning the elements, a prestress is introduced in the membrane. These flexible boundary elements usually are made out of spiral steel wire ropes, or band-shaped textile belts. Additionally, rigid boundary elements exist, which can be implemented to reinforce the edge of the surface.

The deformation capability of flexible membranes is an important property of tensile structure. While traditional structures mostly act by bending, requiring a large cross- sectional area, membrane materials only transfer loads by tensile forces, resulting in a lightweight design. 23

Chapter 4

Optimization

In [20], optimization is defined as “the art of making things the best”, or more correctly “the process of maximizing and/or minimizing one or more objectives without violating specified design constraints, by regulating a set of variable parameters that influence both the objectives and the design constraints”. Without optimization, ideal solutions are searched for using experience, intuition and plain luck. It goes without saying that this method is suboptimal, ineffective, and dated. Using optimization, these solutions are found in a systematic way, using the power and speed of a computer. Additionally, optimization algorithms make sure this search is done in an efficient fashion.

Figure 4.1: Traditional vs. Optimal Design Process, adapted from [20].

Figure 4.1 illustrates the difference between traditional design approaches and optimal design approaches. Box A represents the input, consisting of an initial design, a desired performance, for example a minimal cost or material usage, and any constraints that may 4.1. OPTIMIZATION PROBLEMS 24 apply. Box B includes the analysis, or post-processing, which returns the output results for any given set of variables. The optimization cycle begins at box C, which evaluates the analysis results, and assesses whether the performance is acceptable. If this is not the case, a design change will occur using optimization (Box D), or via intuition, experience, ... (Box E).

4.1 Optimization problems

Optimization problems can be classified by seven major characteristics. These character- istics are important, as they can indicate which algorithms are applicable to the problem at hand, and its complexity. These characteristics are as follows [20]:

• Linear vs. Nonlinear

• Constrained vs. Unconstrained

• Discrete vs. Continuous

• Single vs. Multiobjective

• Single vs. Multiple Minima

• Deterministic vs. Nondeterministic

• Simple vs. Complex

4.1.1 Linear vs. Nonlinear

A linear optimization or linear programming problem is a problem in which both the objective function and the constraints are linear. When either the objective function or any of the constraints is nonlinear, the problem is called a nonlinear problem. In general, linear problems are more easy to solve. Solving a large nonlinear problem is also less reliable than solving a large linear problem, and this difference is more pronounced when the number of variables increases.

4.1.2 Constrained vs. Unconstrained

A problem that has constraints, such as most practical problems, is called a constrained problem. In linear programming, constraints are required, as the solution would otherwise not be finite. 4.1. OPTIMIZATION PROBLEMS 25

4.1.3 Discrete vs. Continuous

When any of the design variables is discrete, the optimization problem is no longer contin- uous. Several cases exist. If the design variables can only take on the values 0 or 1, this is called 0-1 programming or binary programming. If the design variables can only take on integer values, this is called integer programming. When the design variables can only take on a given set of values, this is called discrete optimization. Often, a problem consists of a mixture of these cases. Usually, continuous optimization problems are easier to deal with.

4.1.4 Single vs. Multiobjective

In some cases, more than one objective is to be optimized. However, improving one objective often involves compromising one or more of the other objectives. In figure 4.2, an optimization problem with 2 objectives is shown.

Figure 4.2: Multiobjective optimization, adapted from [20].

If only Objective 1 is to be optimized, a minimum is found in point M1, and similar with Objective 2. However, if a better value for Objective 2 is to be found than the one in M1, we have to move to the right, and compromise the value of Objective 1. This is true for every point between M1 and M2, which are called the Pareto optimal solutions, or non- dominated solutions. Points outside this region are called dominated solutions, as there 4.1. OPTIMIZATION PROBLEMS 26 are always points in which both objectives perform better; for example, point A can be optimized by moving to the right. If the Pareto optimal solutions are plotted with a design objective on each axis, a line known as the Pareto frontier appears.

Figure 4.3: Pareto frontier, adapted from [20].

It is clear that to solve a multiobjective problem, a Pareto optimal point is needed. In two-objective problems, this is done with the weighted sum method, as seen in equation 4.1. With this method, a weighted average is taken from the objectives, which is the new value to be minimized. Minimizing this value yields a Pareto point, and by changing the weights, a series of Pareto points can be found.

J(x) = w1µ1(x) + w2µ2(x) (4.1)

4.1.5 Single vs. Multiple Minima

Some optimization problems have only one minimum or maximum (optimum), while others have have to find the best solution among multiple local optimum values. These problems are called unimodal and multimodal, respectively. Solving multimodal problems is referred to as global optimization, which is a lot more difficult than single optimum optimization. 4.2. SOLUTION APPROACHES 27

4.1.6 Deterministic vs. Nondeterministic

In practical problems, for example when manufacturing a part, tolerances are very impor- tant. Low tolerances often go hand in hand with a high cost. Because of this, information is almost never exact. Similarly, some variables are simply unknown, and can only be estimated. These variables are called nondeterministic.

4.1.7 Simple vs. Complex Problem

Assessing whether a problem is simple or complex involves all of the above characteristics. A problem is simple if it involves only continuous variables, if it is linear, when local optimization will suffice, ... In practice, each of above items only gives an indication of the complexity of the problem.

4.2 Solution approaches

Three main solution approaches exist: experimental, analytical, and numerical. These are briefly discussed in the following section.

4.2.1 Experimental Optimization

As mentioned before, without optimization, ideal solutions are searched for using experi- ence, intuition and plain luck. Experimental optimization involves building a version of the system, evaluating its performance, and determining whether the performance is accept- able. If this is not the case, changes are made, and the process is repeated (see Figure 4.1). It is very clear that this approach is obsolete, as it can be very costly and time-consuming, with no guarantee to convergence at all.

4.2.2 Analytical Optimization

If a mathematical function exists that represents the performance of the design, an ana- lytical approach is the most commonly used. From calculus, if a point is found where the derivative is zero, this point is a local optimum. The second derivative of the function can be used to find out whether the optimum is a minimum, a maximum or a saddle point. If the problem contains constraints, Lagrange multipliers are used, which will not be explained in further detail. 4.3. SUMMARY OF OPTIMIZATION ALGORITHMS 28

Before the use of computers, an analytical approach was the most viable. However, for most practical issues, this approach is way too complicated, especially when compared with a numerical approach. Furthermore, the majority of problems cannot be described by a mathematical function.

4.2.3 Numerical Optimization

Indeed, a numerical optimization approach is the most modern way to optimize a design. Using a computer, it can explore far more possibilities than any manual way. A numerical approach starts with one (or more) design(s), evaluates the objective function, and esti- mates a next design, based on an algorithm. This process is repeated until the stopping criteria are met, hopefully reaching an optimal design.

This type of optimization uses iterations to reach an optimal solution. Compared to the experimental approach, the next iteration is based on an algorithm and not on sheer luck, and the use of a computer can result in a fast result.

A large assembly of algorithms and software exists to this purpose. In this thesis, the MATLAB Toolbox is used, along with third-party algorithms implemented in MATLAB.

4.3 Summary of optimization algorithms

In this section, some frequently used global optimization algorithms implementable in MATLAB are reviewed.

4.3.1 Gradient-based methods

Gradient-based methods find local extrema using first and second order derivatives, compa- rable to analytical optimization as described above. In MATLAB, two gradient-based solvers are incorporated: MultiStart and GlobalSearch. Both generate a series of starting points, from which a MATLAB function called fmincon is used to find the local minima. The global minimum is simply the local minimum with the lowest objective value.

At default, MultiStart uses uniformly distributed starting points, within the specified bounds. User-supplied starting points are also supported. In contrast, GlobalSearch generates the starting points by a scatter-search mechanism. GlobalSearch then analyzes 4.3. SUMMARY OF OPTIMIZATION ALGORITHMS 29 these points, and dismisses of those that are unlikely to generate a better minimum than the best already found [21].

The fmincon algorithm evaluates the first- and second-order derivatives of the objective function at a certain point, and uses this information to obtain the direction in which to evaluate a new point. However, once a local optimum is found, the algorithm stops, regardless of the existence of superior optima.

4.3.2 Genetic algorithms

Genetic algorithms, or GA, are inspired by Darwin’s principle of survival of the fittest, using the objective function as a measure of fitness, and the so-called genes of an individual to represent the solutions. Usually, the parameters of each solution are encoded, resulting in the genes. For example, the number 21 can be represented in binary as 10101. The five digits of this binary number can be treated as genes [20].

During each iteration, a new generation is created, consisting of individuals created from the genes of so-called ‘parents’, which are randomly selected from the previous iteration. Children are created using crossover, mutation or elitism. Elite children are unchanged across generations, mutation involves a random change in the genes of a parent, and crossover randomly combines the genes of two parents. Using the objective function, the function values of the individuals are calculated. After each iteration, the individ- uals of that iteration are assigned a fitness value, which is based on the objective value. Afterwards, a new generation is created, and the process is repeated.

In GA, it is important that the population has a large genetic diversity. For example, if only crossover is considered, new children can only exist out of genes from the previous iteration. The only way to introduces new genes is through mutation.

A genetic algorithm does not require information about the objective function, in contrast to gradient-based methods, so the objective function is treated as a black-box function. As all individuals of a generations are known when the generation is created, multiple objective values can be calculated at once, increasing the performance of the algorithm. This technique is called parallelization. Genetic algorithms are powerful and computationally simple, making it a useful algorithm for a myriad of optimization problems [21] [22]. 4.3. SUMMARY OF OPTIMIZATION ALGORITHMS 30

Figure 4.4: Procedure for a Genetic Algorithm, adapted from [20].

4.3.3 Pattern Search

The term pattern search refers to a family of direct search algorithms. These types of algorithms do not require information about the gradient or objective function, and are therefore more suited to non-smooth or discontinuous objective functions. The pattern search algorithm constructs a pattern of multiple points around a starting point, using a specified poll method. Next, the function values at each of these points are calculated.. A new pattern is then constructed around the point with the best function value, and the process is repeated.

However, as the searching direction is unknown, these methods can be computationally expensive [21]. The global convergence of pattern search algorithms was proven by V. Torczon [23]. However, if the objective function to be minimized has many local minima, the algorithms might get stuck at one of the local minima, instead of continuing to find the global minimum. 4.3. SUMMARY OF OPTIMIZATION ALGORITHMS 31

4.3.4 Simulated annealing

Simulated annealing (SA) is an optimization process inspired by the physical process of annealing, or reducing defects in a material by heating and cooling it. It is a direct search function, meaning it does not require information about the gradient or the objective function. As a result, this process is more suited to non-smooth or discontinuous objective functions. However, problesimms occur when applying this function to a problem with constraints. Combining SA with another solver can alleviate these problems. This is called hybrid simulated annealing (HSA) [21].

4.3.5 Bayesian optimization with Gaussian process priors

In computer science, machine learning algorithms are often dependent on parameters that significantly impact the performance of the algorithm. To enhance the quality of the algorithm, these parameters have to be optimized. In this, the performance of the algorithm is a so-called black-box function, meaning only the input and output of the function are known, without any knowledge of its internal workings. In this optimization, the performance evaluation is usually computationally expensive, so the number of function evaluations should be minimal [24].

When function evaluations are expensive, more computational time should be spent on determining the next point where the function should be evaluated. For this kind of prob- lem, Bayesian optimization provides an elegant approach, and outperforms some other state of the art global optimization algorithms. For continuous functions, Bayesian opti- mization assumes the unknown black-box function was sampled from a Gaussian process, and maintains a posterior distribution for this function after observations [24].

Bayesian optimization constructs a probabilistic model for f(x), and uses this model to choose the location of the next function evaluation, while integrating out uncertainty. In contrast to traditional optimization algorithms, Bayesian optimization uses information from previous evaluations of f(x), and it does not rely solely on a local gradient or Hessian. In this procedure, a minimum of difficult functions can be found using a relatively small amount of function evaluations, which is especially interesting when a function evaluation is expensive to perform [24]. An illustration of the optimization algorithm can be found in figure 4.5.

In Bayesian optimization a prior has to be chosen, which has to give assumptions about the function that is to be optimized. In this case, a Gaussian process prior is chosen. This 4.4. COMPARISON OF OPTIMIZATION ALGORITHMS 32

Figure 4.5: An example of a Bayesian optimization on a 1D optimization problem, adapter from [25]. The figure also contains the acquisition function, which is highest in the points where the Gaussian process predicts a high objective along with a high uncertainty. prior is responsible for the blue zone in figure 4.5. Additionally, an acquisition function has to be chosen, which has to allows us to determining the ideal location for the next function evaluation. This function determines the green graph in figure 4.5.

4.4 Comparison of optimization algorithms

The genetic algorithm, pattern search and simulated annealing algorithms were compared in [21], optimizing a hot rolling process. The study found that Pattern Search and Multi- Start provide on average the fastest result, but that MultiStart performs more consistently. It is also noted that the GA is too slow and not accurate.

In MATLAB documentation, several solvers are used to optimize Rastrigin’s function, a 4.4. COMPARISON OF OPTIMIZATION ALGORITHMS 33 function with a lot of local minima [26]. Among the solvers used are the GA, GlobalSearch and Pattern Search algorithms. The Pattern Search algorithm returned a value within 174 function evaluations, compared to 5400 evaluations for the GA and 2178 evaluations for the GlobalSearch. However, only GlobalSearch and GA reached the global minimum of the function, while the Pattern Search algorithm got stuck in one of the local minima.

Clearly, no optimization algorithm stands out as ‘the best’. Instead, some algorithms are more suited to certain problems. For the project discussed in this thesis, an optimization algorithm is required. As it is not known beforehand which optimization technique is the most suited for this issue, the different algorithms have to be compared against eachother. In this comparison, the Pattern Search algorithm will be considered along with the GA and the Bayesian algorithm. 34

Chapter 5

Shape optimization

Shape optimization is a mathematical approach that optimizes a material layout within a given design space, for a given set of loads and boundary conditions, such that the resulting layout meets a prescribed set of performance targets. Using shape optimization, engineers can find the best concept design that meets the design requirements. The method is of great importance in aircraft design, bridge construction, and many other industries. Shape optimization problems can also be described as the minimization of a certain objective function within certain boundaries, where the objective function depicts the quality of the design [27]. Two main types of shape optimization are parametric and nonparametric shape optimization.

5.1 Parametric shape optimization

In general, a parametric shape optimization process consists of three steps. A parametrized geometry, also known as the design model, the analysis of the structural response, also known as the analysis model, and finally the optimization algorithm. Some drawbacks herein are the exchange between the CAD model and the FEM solver, and limited vari- ability of the possible shapes, as they are described by a finite number of parameters [28].

5.1.1 Design model

In the design model, the potential variations of the considered shape are governed by certain parameters, or design variables. This design model is often described by a CAD model. Several CAD applications exist to create a design model, such as SolidWorks or AutoCAD. 5.1. PARAMETRIC SHAPE OPTIMIZATION 35

Figure 5.1: General shape optimization process, adapted from [28].

Many methods exist to represent a shape by parameters. For example, Bezier curves or B-splines can be used to define the edges of a certain shape. These curves are variation diminishing, meaning that they lie within the convex hull of the control points that define it. They are independent of the coordinate system used to measure the location of the control points, and they can be used to create lots of shapes. If the edges of a shape that is to be optimized are represented by these curves, smoothness is guaranteed, resolving some problems of the methods above [29]. When considering 3D shells, or membrane structures, splines are often used [19] [30].

5.1.2 Analysis model

In the analysis model, the considered shape is subjected to its boundary conditions and external loads. Afterwards, the structural response of said shape is computed using nu- merical methods such as FEM. If the analysis model is not linked with the design model, as is the case when using Bezier curves or B-splines, the approach requires an automated meshing procedure [31].

Numerous FEM applications exist for this purpose, such as Ansys and Abaqus. However, not all FEM software is compatible with all CAD software, which is one of the negative aspects of this method. 5.2. NON-PARAMETRIC SHAPE OPTIMIZATION 36

5.1.3 Optimization algorithm

Lastly, the optimization algorithms handles the iterative process of reaching an optimum. As discussed before, a large variety of optimization algorithms exists. One can make the difference between gradient algorithms, in which the gradient information can be accessed by the optimization software, or response-surface-based algorithms, in which this is not the case [31]. When using a commercial CAD system, it is often not straightforward to calculate the gradient.

In practice, genetic algorithms are frequently used in shape optimization problems in which the gradient cannot be calculated. For example, in the study conducted by Woon et al., a genetic algorithm was used to optimize the shape of a simple spanner head. The shape of the spanner head was described by 20 parameters, and a GA with a population of 18 was used for 807 generations [32]. Similarly, in the study conducted by Gilbert et al., a shape optimization for thin-walled steel cold-formed steel profiles was conducted using a genetic algorithm. The algorithm reached an optimal solution after 50 generations with a population size of 50 [33]. Finally, in a study conducted by Annicchiarico et al, the shape of a perforated cantilever is optimized using GA. The shape is described by 5 parameters, and 50 generations were used with a population of 100 [34].

However, this does not take away that other algorithms are not suited for shape optimiza- tion. In contrary, similar to any optimization problem, some algorithms perform better than others, depending on the nature of the problem.

5.2 Non-parametric shape optimization

The main disadvantage of the parametric approach are that only shapes within the so- lution space are considered, meaning that limiting the number of parameters will lead to suboptimal results. Additionally, a CAD geometry is required, along with automatic mesh generation, limiting the approach to simple triangle and tetrahedron meshes.

If these limitations are considered a problem, a non-parametric shape optimization can be carried out instead. In this method, the design space is built up by implicit parameters from a chosen set of surface nodes. These implicit parameters are the displacements of the design nodes, along a local optimization displacement vector. As each design node can be modified, all possible solutions lie in the design space. In non-parametric shape optimization, there is no real difference between the design model and the analysis model. 5.2. NON-PARAMETRIC SHAPE OPTIMIZATION 37

If the edges of the considered shape are to be optimized, one could use the finite element method to mesh the geometry, and vary the coordinates of nodes on the design boundary. This method is called independent node movement. As proven by Neuber [35], the stress in an edge is only influenced by the geometry in the vicinity of the stress peak. Non- parametric shape optimization can change the displacements of nodes in this vicinity in order to eliminate these stress peaks.

However, because the stresses are sensitive to geometric distortion of the elements, nu- merical instabilities can occur when using this method. For example, the boundary could become jagged after optimization, which is of course undesirable. Additionally, this method uses a large number of design variables, which inhibits fast results [36].

To overcome these numerical instabilities, design elements could be used. These design elements consist out of multiple finite elements.. The shape of these design elements is also controlled by its nodes, however the number of nodes is less than the sum of the number of nodes of its components. As shown in figure 5.2, these design elements exist in both two-dimensional [37] and three-dimensional cases [38].

(a) Some two-dimensional design elements (b) 20-noded three- [37]. dimensional design element, adapted from [38].

Figure 5.2: Example of cloth unfolding, adapted from [10].

To further improve non-parametric shape optimization, optimality criteria methods should be implemented. This means that an optimization displacement for each node is calculated 5.2. NON-PARAMETRIC SHAPE OPTIMIZATION 38 as a result of the stress in the FEM analysis. This method can solve problems with a large number of design variables in a timely fashion, as the convergence speed is independent on the number of variables [31].

One of the advantages of this method is that a bigger number of solutions can be evaluated when compared to parametric shape optimization. 39

Chapter 6

Shape optimization project

As discussed earlier, the conventional method of designing and patterning has its limita- tions. The method neglects certain aspects of the used fabric material, such as its non-linear and non-elastic behaviour. An alternative is to use the so-called integrated approach. In this thesis, a variation on the integrated method for designing and patterning hyperbolic paraboloid structures, or hypars, is proposed. The method, which is based on shape opti- mization, is a continuation of the work done by Dinh et al. [15], and takes into account the material characteristics of the membrane material by using the material model proposed by Dinh et al. [19].

6.1 Proposed design method

As described earlier, the integrated method consists of iteratively generating cutting pat- terns, which are then assembled to achieve a so-called intermediate configuration, as seen in figure 6.1. This intermediate configuration is then subjected to tensioning using FEM, and after computation the stresses are studied. However, this method can be quite com- putationally intensive, as the whole assembly process is simulated at every iteration.

In the method proposed in this thesis, the integrated method will be simplified by postpon- ing the first step in this process. Instead, an intermediate configuration will be iteratively generated and subjected to tensioning. It is assumed that the intermediate configuration of the structure has a similar shape, albeit with a smaller curvature.

In this case, the method will be tested using a hyperstructure or hypar, a commonly used shape in tensile architecture. The hypar considered in this thesis has the same dimensions as the hypar used in the study conducted by Dinh et al. [19], and is a square hypar with a 6.1. PROPOSED DESIGN METHOD 40

Figure 6.1: The different steps in creating a membrane structure, adapted from [7]. side of 4 m and a height of 0.8 m. Figure 6.2 is a depiction of the final shape of the hypar. The hypar will be reinforced in its edges with cable elements, installed through sleeves in the membrane. The target stress in the membrane is 5 MPa, while the target force in the boundary elements is 20 kN.

Figure 6.2: Final configuration of the hypar, adapted from [19].

First, an intermediate configuration is described by a set of parameters, which is then subjected to deformations leading to the final shape. Next, analysis is done using FEM, leading to stresses in the membrane and forces in the edge cables. After this, these stresses and forces are compared to a certain reference stress and force. This process is repeated until the difference between the reference stress/force and the actual stress/force 6.2. DESIGN MODEL 41

is minimal. In this way, an optimal intermediate shape is found using an optimization algorithm. As the shape of the intermediate configuration is prescribed by parameters, this is a case of parametric optimization. It is assumed that this intermediate shape is stress-free. After the optimal shape is found, the patterning can be performed. Finally, the stress-free assumption is controlled by flattening the patterned shape, and looking at the deformations. If these are sufficiently low, the stress-free assumption is verified. The influence of the welds is neglected, and environmental loads are not taken into account.

6.2 Design model

To build the model used in this study, an intermediate shape has to be prescribed by parameters. In figure 6.3, five state variables are shown which determine the initial shape. p2 and p5 indicate the vertical coordinates of the corners relative to a flat plane, while p3 indicates the curvature of the edge of the membrane. The middle point of the edge vertically lies in the middle the two corners. p1 and p4 indicate the extra lengthening or shortening of the membrane; this might be necessary to prevent negative stresses in the membrane.

Figure 6.3: Intermediate configuration of the hypar, including its state variables

These parameters determine the location of eight points: four points at the corners of the membrane, and four points at the middle of the edges. Using SolidWorks, the edges are drawn as splines through these points. With the boundary surface feature, a minimum 6.3. ANALYSIS MODEL 42 surface is created using these splines as edges. The shape is then exported as a STEP-file. This step of the process is done using a SolidWorks VBA script. To prevent issues in creating the CAD model, or singularities in the FEM model, the parameters are limited between certain boundaries. These boundaries can be found in table 6.1.

Table 6.1: Boundaries of the state variables p1 p2 p3 p4 p5 Lower boundary (m) -0.05 0 0 -0.05 0 Upper boundary (m) 0.05 0.40 0.75 0.05 0.40

6.3 Analysis model

6.3.1 Creating the model

In order to calculate the stresses in the membrane after erection, a FEM model has to be created. This can be done by importing the previously created STEP-file as a part in Abaqus. In order to make computation possible, boundary conditions have to be created, along with material descriptions.

Cable elements are installed in a sleeve along the edge of the membrane. To control the displacements, a tie constraint is put in place, linking the displacements of the cable to those of the membrane. To represent the cable-sleeve interaction, SLIPRING connectors are created along the cable. These connectors represent a frictionless pulley system, allowing the cable to slip through the sleeve without resistance. The cable elements have an effective stiffness of 2000 kN. In order to prevent any singularities during computation, the corners of the membrane are chamfered using a sphere with a 5 cm radius.

Next, displacements are applied at the corners of the membrane, in order to transform the intermediate configuration to the final configuration. The deformations are vertical, to reach the final height of the structure, and horizontal, to compensate for the length difference of the diagonals (p1 and p4 in figure 6.3). Consecutively, the job is submitted. This step of the process is done using an Abaqus Python script.

Finally, the material characteristics of the membrane are assigned. In a first step, an orthotropic elasticity material model is used for computations. The material characteristics 6.3. ANALYSIS MODEL 43

of this model can be found in table 6.2. Later, the model proposed by Dinh et al. [19] will be used. The characteristics of this model are described in 3.3.7.

Table 6.2: Parameters of the orthotropic elasticity material model

Ew Ef ν12 ν21 G 975.6 MPa 716.1 MPa 0.23 0.17 87.7 MPa

6.3.2 Computation and post-processing

As the problem studied here is a nonlinear one, Abaqus uses a volume-proportional damp-

ing to prevent instabilities. This implies adding viscous forces Fv to the global equilibrium equations.

P − I − Fv = 0 (6.1)

∗ Fv = c · M · v (6.2)

∗ ∆u In these formulas, M is an artificial mass matrix, c is a certain damping factor, v = ∆t is the nodal velocity vector and ∆t is the time increment. As a rule of thumb, the viscous damping energy (ALLSD) should be smaller than about 5% or 10% of the total strain energy (ALLIE), in order to guarantee a correct solution. After experimenting with some values of the damping factor, a value of c = 10−8 returned good results for the orthotropic elasticity material model, while a value of c = 2.5 · 10−7 is used when using the fabric plasticity model.

After completion of the job, the results are available for post-processing. In this case, the stress distribution in warp and fill direction in the membrane is used, together with the force in the cable. In order to optimize the shape of the membrane, these values have to be converted into a single value, also known as the objective. This step of the process is done using an Abaqus Python script.

To do this, an average of absolute deviations over all the nodes of the model is calculated, compared to a certain reference stress/force. For the membrane, this reference stress is 5 MPa, while for the cable, the reference force is 20 kN. The formula used is formula 6.3. As shown, the formula consists out of two parts: a segment for the membrane, and a segment for the cable. Using weight factors these two segments are combined into a single value. 6.4. OPTIMIZATION ALGORITHM 44

W (P ) = w1 · D1 + w2 · D2 (6.3) where

1 X membrane membrane D1 = |σ − σref | (6.4) nmembrane nmembrane and

1 X cable cable D2 = |F − Fref | (6.5) ncable ncable

Of course, these weight factors are not known beforehand, and can be arbitrarily chosen. However, different choices for the weight factors will, after optimization, result in different solutions. To resolve this, different combinations of weight factors will be used.

Additionally, to enhance the quality of the final solution, some penalty factors are intro- duced. If the viscous damping energy is more than 10% of the total strain energy, the result is not accurate as describe above. Therefore, if this occurs, or when the calculations do not converge, a high value is returned. This way, the optimization algorithm will steer away from points in which this is the case.

6.4 Optimization algorithm

Finally, an optimization algorithms is necessary in order to find the configuration in which the objective function has the lowest value. As mentioned before, some good candidates for this purpose are the genetic algorithm, and the pattern search algorithm, both available using MATLAB. Additionally, a Bayesian optimization algorithm with 1000 iterations based on the study conducted by Kawaguchi et al. [39] was used. For the genetic algorithm, the same parameters as in [34] are used, as that study also considers a 5-parameter problem. These parameters can be found in table 6.3.

Table 6.3: Genetic algorithm parameters Population size 100 Number of generations 50 Selection scheme RSSWR (Remainder stochastic sampling without replacement) Crossover probability 0.8 6.5. PATTERNING AND VERIFYING THE STRESS-FREE HYPOTHESIS 45

6.5 Patterning and verifying the stress-free hypothe- sis

As discussed, the intermediate shape of the membrane is assumed to be stress free. In reality, this shape consists out of several flat panels welded together. However, the inter- mediate shape of the membrane is a double-curved surface, and is therefore undevelopable to a flat surface. Because of this, some deformation will always occur when assembling the panels. The goal is to minimize this deformation, and as a result to minimize the stress in the intermediate shape. In order to validate this stress-free hypothesis, the intermediate configuration found in the shape optimization will be divided into six pieces of equal width, with the edges of the panels laying on geodesic lines, as discussed earlier. These panels should be able to be flattened without significant strains.

To computationally flatten these panels, Rhino3D’s Squish algorithm is used. This algo- rithm first meshes the surface, and then flattens the mesh by minimizing the changes in facet area and changes in facet edge lengths. The panels are flattened in such a way that they only stretch during assembly, to prevent wrinkling. The algorithm returns the average and maximal equivalent strain caused by the flattening. If this maximal equivalent strain is small enough, it can be assumed that the intermediate shape can be assembled without introducing significant stresses.

6.6 Overview

In a first step, a shape optimization will be carried out, using the orthotropic elasticity material model. The GA, Pattern Search and Bayesian algorithms will be used for this. From these optimization processes, it will be clear which algorithm is the most suited to this type of problem. Subsequently, a shape optimization will be done using the algorithm determined in the first step, using both the orthotropic elasticity and the fabric plasticity model. In these shape optimizations, the stress-free hypothesis has to be confirmed on the found shapes. A clear overview of the method is given in figure 6.4. 6.7. BENCHMARK 46

Figure 6.4: Flowchart depicting the proposed design method

6.7 Benchmark

As this study is a continuation on the study conducted by Dinh et al. [15], the results obtained in that study can be used as a benchmark for the results in this study. As the intermediate shape used here has more parameters than the shape used by Dinh et al., lower stress and force deviations are expected. Therefore, obtained results will be compared

with the results from said research. A weight factor combination of w1 = 0.7 and w2 = 0.3 was used to obtain the benchmark results. 6.7. BENCHMARK 47

6.7.1 Orthotropic elasticity material model

As shown in figure 6.5 and table 6.4, a reasonable stress distribution was found in the membrane along with a permissible cable force. However, there are some zones where the stress in one of the directions is lower than 3 MPa, indicated in blue, which might inhibit the out-of-plane stiffness of the membrane.

(a) S11 (b) S22

Figure 6.5: Membrane stresses in the optimized shape found by Dinh. et al. [15] using the orthotropic elasticity model

Table 6.4: Parameters and results (w1 = 0.7 and w2 = 0.3)

p1(m) p2(m) p3(m) p4(m) p5(m) Fcable(kN) D1 D2 W 0.020 0.000 0.350 0.021 0.000 20.247 1.13079 0.12361 0.82863

6.7.2 Fabric plasticity material model

As shown in figure 6.6 and table 6.5, a reasonable stress distribution was found in the membrane along with a permissible cable force. However, the stresses are more varied in this case, and the force in the cable is also further apart from the target value. Again, zones are visible where the stress in one of the directions is lower than 3 MPa. These zones are larger than in the previous case. 6.7. BENCHMARK 48

(a) S11 (b) S22

Figure 6.6: Membrane stresses in the optimized shape found by Dinh. et al. [15] using the fabric plasticity model

Table 6.5: Parameters and results (w1 = 0.7 and w2 = 0.3)

p1(m) p2(m) p3(m) p4(m) p5(m) Fcable(kN) D1 D2 W -0.011 0.000 0.482 0.032 0.000 19.575 1.81705 0.21243 1.33566 49

Chapter 7

Results

7.1 Comparing optimization algorithms using the or- thotropic elasticity model

First, the algorithm used for the shape optimization has to be determined. To do this, the shape optimization is carried out using three different algorithms. The results obtained from these optimizations are then compared based on their speed and quality. Simulations are carried out for the five combinations of weight factors listed in table 7.1.

Table 7.1: Different weight factor combinations

Combination w1 w2 1 0.1 0.9 2 0.3 0.7 3 0.5 0.5 4 0.7 0.3 5 0.9 0.1

From the results it is visible that Pattern Search is by far the fastest algorithm. However, its results are inferior to those of the GA and the Bayesian algorithm. From those two algorithms, the Bayesian algorithm is both faster and more reliable than the genetic algorithm. While the Pattern Search algorithm is faster than the Bayesian algorithms, the Bayesian algorithm is chosen for further optimization procedures as it returned the best result in a reasonable number of function evaluations. In some cases the genetic 7.1. COMPARING OPTIMIZATION ALGORITHMS USING THE ORTHOTROPIC ELASTICITY MODEL 50 algorithm did not provide results at all due to a software error preventing the optimization from completing.

Table 7.2: Algorithm comparison for the case w1 = 0.1 and w2 = 0.9 Function evaluations D1 D2 W Bayesian 1000 5.4230 0.0126 0.5597 GA NA NA NA NA Pattern Search 378 12.5622 0.0004 1.2565

Table 7.3: Algorithm comparison for the case w1 = 0.3 and w2 = 0.7 Function evaluations D1 D2 W Bayesian 1000 5.6420 0.0180 1.7527 GA NA NA NA NA Pattern Search 289 9.3203 0.0033 2.7771

Table 7.4: Algorithm comparison for the case w1 = 0.5 and w2 = 0.5 Function evaluations D1 D2 W Bayesian 1000 5.6568 0.6396 2.7611 GA 4300 6.2179 0.0005 3.0000 Pattern Search 272 6.8295 0.2210 3.5293

Table 7.5: Algorithm comparison for the case w1 = 0.7 and w2 = 0.3 Function evaluations D1 D2 W Bayesian 1000 5.4137 0.0038 3.8372 GA 4600 5.7680 0.0678 3.8995 Pattern Search 280 5.6954 0.0479 3.9725

Table 7.6: Algorithm comparison for the case w1 = 0.9 and w2 = 0.1 Function evaluations D1 D2 W Bayesian 1000 5.6711 0.0796 4.8988 GA NA NA NA NA Pattern Search 291 4.1361 11.1634 5.1495 7.1. COMPARING OPTIMIZATION ALGORITHMS USING THE ORTHOTROPIC ELASTICITY MODEL 51 The five obtained results should lie on Pareto points or non-dominated points. However, this is not the case here. In this case, as seen in table 7.7, one result clearly stands out from the rest in both objectives - result 4. This might be explained by the difference in

magnitude of the objectives. In the solutions, D1 is two orders of magnitude bigger than

D2. As a result, the optimizations converge to similar results, regardless of the weight factors.

Table 7.7: Results for each weight factor combination

Result w1 w2 p1 p2 p3 p4 p5 D1 D2 1 0.1 0.9 0.0067 0.3778 0.4154 0.0000 0.3333 5.4230 0.0126 2 0.3 0.7 0.0082 0.3333 0.3648 0.0000 0.3778 5.6420 0.0180 3 0.5 0.5 -0.0030 0.3778 0.4500 0.0000 0.3778 5.6568 0.6396 4 0.7 0.3 0.0073 0.3752 0.4130 0.0000 0.3333 5.4137 0.0038 5 0.9 0.1 -0.0030 0.3926 0.4167 0.0067 0.3333 5.6711 0.0796

(a) S11 (b) S22

Figure 7.1: Membrane stresses in the optimized shape

Fcable=20.008 kN

However, after determining the ideal algorithm, a bug was discovered in the code creating the model. The result obtained above will therefore be disregarded. Still, the algorithm comparison is useful, and it is assumed that the Bayesian algorithm still provides the best solution out of the tested algorithms. In what follows, the Bayesian algorithm is used for shape optimization. 7.2. SHAPE OPTIMIZATION USING THE ORTHOTROPIC ELASTICITY MATERIAL MODEL 52 7.2 Shape optimization using the orthotropic elastic- ity material model

7.2.1w 1 = 0.7 and w2 = 0.3 First, a shape optimization is carried out using the orthotropic elasticity material model.

Initially, the weight factors will be w1 = 0.7 and w2 = 0.3, so a good comparison can be made with the benchmark result. Clearly, this result is better than the benchmark result, as both D1 and D2 are lower. A better stress distribution is found, and there are no large zones with small stresses. The stress-free hypothesis has to be confirmed, so the membrane is split symmetrically into six panels along geodesic lines, which are then flattened. The deformations resulting from this flattening can be seen in table 7.9. These deformations are low enough to assume a stress-free state after assembly.

(a) S11 (b) S22

Figure 7.2: Membrane stresses in the optimized shape using the orthotropic elasticity model

(w1 = 0.7 and w2 = 0.3)

Table 7.8: Parameters and results (w1 = 0.7 and w2 = 0.3)

p1(m) p2(m) p3(m) p4(m) p5(m) Fcable(kN) D1 D2 W -0.011 0.367 0.310 0.000 0.242 19.998 0.77947 0.00100 0.54953

While D2 is already very close to zero, and with it the cable force close to 20 kN, the stress deviation in the membrane D1 can still be improved. Better results for D1 can be found 7.2. SHAPE OPTIMIZATION USING THE ORTHOTROPIC ELASTICITY MATERIAL MODEL 53

10 2 Function evaluations 9 Best found value 1.8

8 1.6

7 1.4

6 1.2

5 1

4 0.8 Function value Function value 3 0.6

2 0.4

1 0.2

0 0 0 100 200 300 400 500 600 700 800 900 1000 0 100 200 300 400 500 600 700 800 900 1000 Function evaluation Function evaluation

Figure 7.3: Optimization process

Figure 7.4: Optimized shape divided in panels

Table 7.9: Equivalent strains caused by assembly, panels numbered from left to right Panel 1 Panel 2 Panel 3 Maximum 0.27% 0.11% 0.02% Average 0.06% 0.03% 0.01% by shifting the weight towards it. Therefore, the optimization will be repeated with weight factor combinations w1 = 0.9 and w2 = 0.1, and w1 = 0.95 and w2 = 0.05. 7.2. SHAPE OPTIMIZATION USING THE ORTHOTROPIC ELASTICITY MATERIAL MODEL 54

7.2.2w 1 = 0.9 and w2 = 0.1

Indeed, by moving the weight more towards D1, a lower value for D1 is reached, while compromising on D2. The force in the cable is still very close to the target value, and the stress distribution in the membrane is allowable. Once more, no large zones with low stresses are observed. To verify the stress-free hypothesis, the membrane is again split into panels and flattened. The maximum deformation is low enough to neglect and thus to verify the stress-free state of the intermediate configuration.

(a) S11 (b) S22

Figure 7.5: Membrane stresses in the optimized shape using the orthotropic elasticity model

(w1 = 0.9 and w2 = 0.1)

Table 7.10: Parameters and results (w1 = 0.9 and w2 = 0.1)

p1(m) p2(m) p3(m) p4(m) p5(m) Fcable(kN) D1 D2 W -0.012 0.379 0.312 -0.004 0.251 19.983 0.76071 0.00841 0.68548

Table 7.11: Equivalent strains caused by assembly, panels numbered from left to right Panel 1 Panel 2 Panel 3 Maximum 0.19% 0.12% 0.01% Average 0.05% 0.03% 0.01% 7.2. SHAPE OPTIMIZATION USING THE ORTHOTROPIC ELASTICITY MATERIAL MODEL 55

10 2 Function evaluations 9 Best found value 1.8

8 1.6

7 1.4

6 1.2

5 1

4 0.8 Function value Function value 3 0.6

2 0.4

1 0.2

0 0 0 100 200 300 400 500 600 700 800 900 1000 0 100 200 300 400 500 600 700 800 900 1000 Function evaluation Function evaluation

Figure 7.6: Optimization process

Figure 7.7: Optimized shape divided in panels

7.2.3w 1 = 0.95 and w2 = 0.05

While the obtained cable force deviation is equal to the previous result, the membrane stress deviation is actually higher. As such, this result is dominated by the previous result and is therefore not useful.

Table 7.12: Parameters and results (w1 = 0.95 and w2 = 0.05)

p1(m) p2(m) p3(m) p4(m) p5(m) Fcable(kN) D1 D2 W -0.011 0.249 0.315 -0.004 0.378 20.017 0.77789 0.00841 0.73657 7.2. SHAPE OPTIMIZATION USING THE ORTHOTROPIC ELASTICITY MATERIAL MODEL 56

(a) S11 (b) S22

Figure 7.8: Membrane stresses in the optimized shape using the orthotropic elasticity model

(w1 = 0.95 and w2 = 0.05)

10 2 Function evaluations 9 Best found value 1.8

8 1.6

7 1.4

6 1.2

5 1

4 0.8 Function value Function value 3 0.6

2 0.4

1 0.2

0 0 0 100 200 300 400 500 600 700 800 900 1000 0 100 200 300 400 500 600 700 800 900 1000 Function evaluation Function evaluation

Figure 7.9: Optimization process

Table 7.13: Equivalent strains caused by assembly, panels numbered from left to right Panel 1 Panel 2 Panel 3 Maximum 0.16% 0.11% 0.02% Average 0.04% 0.03% 0.01% 7.2. SHAPE OPTIMIZATION USING THE ORTHOTROPIC ELASTICITY MATERIAL MODEL 57

Figure 7.10: Optimized shape divided in panels 7.3. SHAPE OPTIMIZATION USING THE FABRIC PLASTICITY MATERIAL MODEL 58

7.2.4 Conclusion

Using the orthotropic elasticity model, the method provides good results, not only for the stress distribution in the membrane, but also for the force in the cable and the deformation due to flattening. As the first two obtained configurations are Pareto solutions, they are equal solutions to the design problem. However, the second result showed the smallest deformation during flattening, and is therefore deemed the optimal solution.

7.3 Shape optimization using the fabric plasticity ma- terial model

7.3.1w 1 = 0.7 and w2 = 0.3 The same optimizations are repeated using the fabric plasticity material model. Again, the obtained result is better than the benchmark result, as both the stress deviation and the force deviation are smaller. Compared to the orthotropic elasticity solutions, a larger zone with low fill stresses is apparent. Similar to the orthotropic elasticity optimization, there is room to improve D1, while compromising on D2. When flattening the panels, deformation up to 0.57% occurs. The panels resulting from this shape optimization are smaller than those resulting from the orthotropic elasticity optimization.

(a) S11 (b) S22

Figure 7.11: Membrane stresses in the optimized shape using the fabric plasticity model (w1 =

0.7 and w2 = 0.3) 7.3. SHAPE OPTIMIZATION USING THE FABRIC PLASTICITY MATERIAL MODEL 59

Table 7.14: Parameters and results (w1 = 0.7 and w2 = 0.3)

p1(m) p2(m) p3(m) p4(m) p5(m) Fcable(kN) D1 D2 W -0.033 0.314 0.35 0.000 0.349 19.993 1.26782 0.00355 0.88854

10 2 Function evaluations 9 Best found value 1.8

8 1.6

7 1.4

6 1.2

5 1

4 0.8 Function value Function value 3 0.6

2 0.4

1 0.2

0 0 0 100 200 300 400 500 600 700 800 900 1000 0 100 200 300 400 500 600 700 800 900 1000 Function evaluation Function evaluation

Figure 7.12: Optimization process

Figure 7.13: Optimized shape divided in panels

Table 7.15: Equivalent strains caused by assembly, panels numbered from left to right Panel 1 Panel 2 Panel 3 Maximum 0.57% 0.08% 0.02% Average 0.10% 0.02% 0.01% 7.3. SHAPE OPTIMIZATION USING THE FABRIC PLASTICITY MATERIAL MODEL 60

7.3.2w 1 = 0.9 and w2 = 0.1

(a) S11 (b) S22

Figure 7.14: Membrane stresses in the optimized shape using the fabric plasticity model (w1 =

0.9 and w2 = 0.1)

Table 7.16: Parameters and results (w1 = 0.9 and w2 = 0.1)

p1(m) p2(m) p3(m) p4(m) p5(m) Fcable(kN) D1 D2 W -0.0163 0.253 0.338 0.005 0.289 20.127 1.34759 0.06371 1.21920

10 2 Function evaluations 9 Best found value 1.8

8 1.6

7 1.4

6 1.2

5 1

4 0.8 Function value Function value 3 0.6

2 0.4

1 0.2

0 0 0 100 200 300 400 500 600 700 800 900 1000 0 100 200 300 400 500 600 700 800 900 1000 Function evaluation Function evaluation

Figure 7.15: Optimization process

Using weight factors w1 = 0.9 and w2 = 0.1, a solution is obtained that is dominated by the previous solution. While this configuration is inferior to the previous solution regarding 7.3. SHAPE OPTIMIZATION USING THE FABRIC PLASTICITY MATERIAL MODEL 61

Figure 7.16: Optimized shape divided in panels

Table 7.17: Equivalent strains caused by assembly, panels numbered from left to right Panel 1 Panel 2 Panel 3 Maximum 0.19% 0.15% 0.03% Average 0.04% 0.03% 0.02% stress and force deviations, deformations caused by assembling the panels are lower, causing the result to be more accurate. 7.3. SHAPE OPTIMIZATION USING THE FABRIC PLASTICITY MATERIAL MODEL 62

7.3.3w 1 = 0.95 and w2 = 0.05

The solution obtained by using weight factors w1 = 0.95 and w2 = 0.05 is a Pareto solution, as D1 is lower compared to the first solution, while D2 is higher. Again, the deformations caused by assembling the panels are negligible.

(a) S11 (b) S22

Figure 7.17: Membrane stresses in the optimized shape using the fabric plasticity model (w1 =

0.95 and w2 = 0.05)

Table 7.18: Parameters and results (w1 = 0.95 and w2 = 0.05)

p1(m) p2(m) p3(m) p4(m) p5(m) Fcable(kN) D1 D2 W -0.035 0.353 0.324 0.006 0.264 21.949 1.22046 0.97395 1.20813

10 2 Function evaluations 9 Best found value 1.8

8 1.6

7 1.4

6 1.2

5 1

4 0.8 Function value Function value 3 0.6

2 0.4

1 0.2

0 0 0 100 200 300 400 500 600 700 800 900 1000 0 100 200 300 400 500 600 700 800 900 1000 Function evaluation Function evaluation

Figure 7.18: Optimization process 7.3. SHAPE OPTIMIZATION USING THE FABRIC PLASTICITY MATERIAL MODEL 63

Figure 7.19: Optimized shape divided in panels

Table 7.19: Equivalent strains caused by assembly, panels numbered from left to right Panel 1 Panel 2 Panel 3 Maximum 0.29% 0.18% 0.02% Average 0.06% 0.04% 0.01%

7.3.4 Conclusion

Using the fabric plasticity material model, the method again provides good results, not only for the stress distribution in the membrane, but also for the force in the cable and the deformation due to flattening. While both the first and the third solution are Pareto solution, the latter is more accurate as the panels deform the least during assembly. The difference in material model manifests itself in the shape of the patterns, as seen in figure 7.20.

However, the results are inferior to those of the orthotropic elasticity model, as the so- lutions contain large zones where S22 is quite low. With the chosen design model, all possible configurations are relatively rotationally symmetric: when turning the shape 90 degrees about its Z-axis, a similar shape appears. As the fabric plasticity model is highly anisotropic, this explains the regions where S22 is low. Further improvements can be made by preventing this rotational symmetry, by introducing more shape parameters. 7.3. SHAPE OPTIMIZATION USING THE FABRIC PLASTICITY MATERIAL MODEL 64

Figure 7.20: Difference between the panels when using the orthotropic elasticity model (blue) or

the fabric plasticity model (red), when using weight factors w1 = 0.7 and w2 = 0.3 65

Chapter 8

Conclusion and recommendations for future work

8.1 Conclusion

Compared to previous work, some improvements are made regarding the design method. As more parameters were used to describe the intermediate configuration, better stress distributions can be found. The Bayesian algorithm also provides faster and better results than the genetic algorithm used in previous work. The obtained results are able to be split into panels and flattened without substantial deformations, and are therefore considered reliable.

8.2 Recommendations for future work

Some recommendations can be made for future work regarding this design method. For starters, only parametric optimization was considered, and non-parametric optimization might also provide a solution. Regarding parametric optimization, one could use more than the current number (5) of parameters to describe the intermediate shape of the membrane. Due to restrictions regarding the SolidWorks API, parallellization was not possible. If this can be resolved, or if other CAD software is used to create the design model, a faster conclusion can be obtained.

A myriad of optimization algorithms exists, and only a few of those have been considered for this specific shape optimization. Therefore it is reasonable to believe that a better optimization algorithm exists for this purpose. While a new algorithm might provide a 8.2. RECOMMENDATIONS FOR FUTURE WORK 66 faster solution, it might also return solutions that perform better than the results from this study.

A next step in developing the design method can be to integrate the structural analysis. Similarly, an intermediate shape can be prescribed by parameters, followed by its erec- tion. If then the environmental forces such as wind and snow are considered, the shape optimization could prevent rupturing of the fabric due to these loads. BIBLIOGRAPHY 67

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