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Approximation of Topology Optimization Problems Using Sizing Optimization Problems THESIS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY Approximation of topology optimization problems using sizing optimization problems Anton Evgrafov Department of Mathematics Chalmers University of Technology and Göteborg University Göteborg, Sweden 2004 Approximation of topology optimization problems using sizing optimization problems Anton Evgrafov ISBN 91–7291–466–1 © Anton Evgrafov, 2004 Doktorsavhandlingar vid Chalmers tekniska högskola Ny serie nr 2148 ISSN 0346–718X Department of Mathematics Chalmers University of Technology and Göteborg University SE-412 96 Göteborg Sweden Telephone +46 (0)31–772 1000 Printed in Göteborg, Sweden 2004 Approximation of topology optimization problems using sizing optimization problems Anton Evgrafov Department of Mathematics Chalmers University of Technology and Göteborg University ABSTRACT The present work is devoted to approximation techniques for singular extremal problems arising from optimal design problems in structural and fluid mechanics. The thesis con- sists of an introductory part and four independent papers, which however are united by the common idea of approximation and the related application areas. In the first half of the thesis we are concerned with finding the optimal topology of truss- like structures. This class of optimal design problems arises when in order to find the op- timal truss not only are we allowed to redistribute the material among the structural mem- bers (bars), but also to completely remove some parts altering the connectivity (topology) of the structure. The other half of the thesis addresses the question of the optimal topolog- ical design of flow domains for Stokes and Navier–Stokes fluids. For flows, optimizing topology means finding the optimal partition of the given design domain into disjoint parts occupied by the fluid and the impenetrable walls, given the in-flow and the out-flow boundaries. In particular, impenetrable walls change the shape and the connectivity of the flow domain. In the first paper we construct an example demonstrating the singular behaviour of truss topology optimization problems including a linearized global buckling (linear elastic sta- bility) constraint. This singularity phenomenon has not been known before and affects the choice of numerical methods that can be applied to the optimization problem. We propose a simple approximation strategy and establish the convergence of globally op- timal solutions to perturbed problems towards globally optimal solutions to the original singular problem. In the second paper we are concerned with the construction of finer approximating prob- lems that allow us to reconstruct the local behaviour of a general class of singular truss topology optimization problems, namely to approximate stationary points to the limiting problem with sequences of stationary points to the regular approximating problems. We do so on the classic problem of weight minimization under stress constraints for trusses in unilateral contact with rigid obstacles. In the third paper we extend a design parametrization previously proposed for the topo- logical design of flow domains for Stokes flows to also include the limiting case of porous materials—completely impenetrable walls. We demonstrate that, in general, the resulting design-to-flow mapping is not closed, yet under mild assumptions it is possible to approx- imate globally optimal minimal-power-dissipation domains using porous materials with diminishing permeability. In the fourth and last paper we consider the optimal design of flow domains for Navier– Stokes flows. We illustrate the discontinuous behaviour of the design-to-flow mapping caused by the topological changes in the design, and propose “minor” changes to the design parametrization and the equations that allow us to rigorously establish the closed- ness of the design-to-flow mapping. The existence of optimal solutions as well as the convergence of approximation schemes then easily follows from the closedness result. ii LIST OF PUBLICATIONS This thesis consists of an introductory part and the following papers: Paper 1 A. Evgrafov, On globally stable singular topologies, to appear in Structural and Multidisciplinary Optimization, 2004. Paper 2 A. Evgrafov and M. Patriksson, On the convergence of station- ary sequences in topology optimization, submitted to International Journal for Numerical Methods in Engineering, 2004. Paper 3 A. Evgrafov, On the limits of porous materials in the topology optimization of Stokes flows, submitted to Applied Mathematics and Optimization, 2003. Paper 4 A. Evgrafov, Topology optimization of slightly compressible flu- ids, submitted to Zeitschrift für Angewandte Mathematik und Mechanik, 2004. iv CONTENTS Abstract ................................... i List of publications ............................. iii Acknowledgement ............................. vii Introduction and overview ......................... ix 1. On globally stable singular topologies ................. 1 2. On the convergence of stationary sequences in topology optimiza- tion ...................................... 15 3. On the limits of porous materials in the topology optimization of Stokes flows ................................. 39 4. Topology optimization of slightly compressible fluids ........ 55 vi ACKNOWLEDGEMENT URING the time spent on writing this dissertation, as well as on many other occasions, DI have always been feeling the unflagging support of many people, including my supervisor, Prof. Michael Patriksson; ◦ my wife Elena; ◦ my friends; ◦ my parents. ◦ I will ever be indebted to all of you. Many thanks to the School of Mathematical Sciences for being such a welcoming and friendly place to work, and to the Swedish Research Council for financially supporting me under the grant VR-621-2002-5780. I dedicate my dissertation to the memory of Joakim Petersson, who untimely passed away on September 30, 2002, at the mere age of 33. This work is to a great extent inspired by his research. Anton Evgrafov Göteborg, May 2004 viii INTRODUCTION AND OVERVIEW N the present thesis we study approximation techniques for singular extremal problems Iarising from optimal design problems in structural and fluid mechanics. The purpose of this chapter is to provide an introduction to some of the models and methods coming from bi-level programming, topology optimization, and sensitivity analysis for non-linear programs, as well as to give a summary of the results established in the four appended papers, placing them in a proper perspective. Mathematical programming with equilibrium constraints Hierarchical decision-making problems are encountered in a wide variety of domains in the engineering and experimental natural sciences, and in regional planning, management, and economics. These problems are all defined by the presence of two or more objectives with a prescribed order of priority or information. In many applications it is sufficient to consider a sub-class of these problems having two levels, or objectives. We refer to the upper-level as the objective having the highest priority and/or information level; it is defined in terms of an optimization with respect to one set of variables. The lower-level problem, which in the most general case is described by a variational inequality, is then a supplementary problem parameterized by the upper-level variables. These models are known as generalized bi-level programming problems, or mathematical programs with equilibrium constraints (MPEC); see, for example, Luo et al. [LPR96]. Many extremal problems arising from applications in mechanics of solids, structures, and fluids have an inherent bi-level form. The upper-level objective function measures some performance of the system, such as its weight, stiffness, maximal contact force, pressure drop, drag, or power dissipation. This objective function is optimized by select- ing design parameters, which in our case will be related to the geometry of the system and the amount of material being used. Further, the upper-level optimization is subject to design constraints, such as limits on the amount of available material, and to behavioural constraints, such as bounds on the displacements and stresses, or buckling/stability con- straints. The lower-level problem describes the behaviour of the system given the choices of the design variables, the external forces acting on it, and the boundary conditions. For linear elastic structures the behaviour is governed by the equilibrium law of min- imal potential energy, which determines the values of the state variables (displacements) at the lower level. Equivalently, the equilibrium law can be expressed as a (dual) principle of the minimum of complementary energy, determining the stresses and contact forces. Similarly, for slow (creep, Stokes) flows the flow velocity is determined by the princi- ple of minimal potential power. In the case of faster flow, the non-linear convection effects x must be taken into account in the lower-level problem, leading to the Navier–Stokes equa- tions that (in a weak form) can be formulated as a variational inequality. Mathematical programs with equilibrium constraints are known to be an especially difficult subclass of non-smooth non-convex NP-hard mathematical problems, which in addition violates standard non-linear programming constraint qualifications [LPR96, Chapter 1]. Furthermore, MPEC problems coming from the topology optimization of structures, solids and fluids typically violate even the novel qualifications, such as the strict complementarity conditions, or strong regularity
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