Variational Methods for Structural Optimization

In recent decades, it has become possible to turn the design process into computer algorithms. By applying different computer oriented methods the topology and shape of structures can be optimized and thus designs systematically improved. These possibilities have stimulated an interest in the mathematical foundations of structural optimization. The challenge of this book is to bridge a gap between a rigorous mathematical approach to variational problems and the practical use of algorithms of structural optimization in engineering applications. The foundations of structural optimization are presented in a sufficiently simple form to make them available for practical use and to allow their critical appraisal for improving and adapting these results to specific models. Special attention is to pay to the description of optimal structures of composites; to deal with this problem, novel mathematical methods of nonconvex calculus of variation are developed. The exposition is accompanied by examples.

I Preliminaries.- 1 Relaxation of One-Dimensional Variational Problems.- 1.1 An Optimal Design by Means of Composites.- 1.2 Stability of Minimizers and the Weierstrass Test.- 1.2.1 Necessary and Sufficient Conditions.- 1.2.2 Variational Methods: Weierstrass Test.- 1.3 Relaxation.- 1.3.1 Nonconvex Variational Problems.- 1.3.2 Convex Envelope.- 1.3.3 Minimal Extension and Minimizing Sequences.- 1.3.4 Examples: Solutions to Nonconvex Problems.- 1.3.5 Null-Lagrangians and Convexity.- 1.3.6 Duality.- 1.4 Conclusion and Problems.- 2 Conducting Composites.- 2.1 Conductivity of Inhomogeneous Media.- 2.1.1 Equations for Conductivity.- 2.1.2 Continuity Conditions in Inhomogeneous Materials.- 2.1.3 Energy, Variational Principles.- 2.2 Composites.- 2.2.1 Homogenization and Effective Tensor.- 2.2.2 Effective Properties of Laminates.- 2.2.3 Effective Medium Theory: Coated Circles.- 2.3 Conclusion and Problems.- 3 Bounds and G-Closures.- 3.1 Effective Tensors: Variational Approach.- 3.1.1 Calculation of Effective Tensors.- 3.1.2 Wiener Bounds.- 3.2 G-Closure Problem.- 3.2.1 G-convergence.- 3.2.2 G-Closure: Definition and Properties.- 3.2.3 Example: The G-Closure of Isotropic Materials.- 3.2.4 Weak G-Closure (Range of Attainability).- 3.3 Conclusion and Problems.- II Optimization of Conducting Composites.- 4 Domains of Extremal Conductivity.- 4.1 Statement of the Problem.- 4.2 Relaxation Based on the G-Closure.- 4.2.1 Relaxation.- 4.2.2 Sufficient Conditions.- 4.2.3 A Dual Problem.- 4.2.4 Convex Envelope and Compatibility Conditions..- 4.3 Weierstrass Test.- 4.3.1 Variation in a Strip.- 4.3.2 The Minimal Extension.- 4.3.3 Summary.- 4.4 Dual Problem with Nonsmooth Lagrangian.- 4.5 Example: The Annulus of Extremal Conductivity.- 4.6 Optimal Multiphase Composites.- 4.6.1 An Elastic Bar of Extremal Torsion Stiffness.- 4.6.2 Multimaterial Design.- 4.7 Problems.- 5 Optimal Conducting Structures.- 5.1 Relaxation and G-Convergence.- 5.1.1 Weak Continuity and Weak Lower Semicontinuity.- 5.1.2 Relaxation of Constrained Problems by G-Closure..- 5.2 Solution to an Optimal Design Problem.- 5.2.1 Augmented Functional.- 5.2.2 The Local Problem.- 5.2.3 Solution in the Large Scale.- 5.3 Reducing to a Minimum Variational Problem.- 5.4 Examples.- 5.5 Conclusion and Problems.- III Quasiconvexity and Relaxation.- 6 Quasiconvexity.- 6.1 Structural Optimization Problems.- 6.1.1 Statements of Problems of Optimal Design.- 6.1.2 Fields and Differential Constraints.- 6.2 Convexity of Lagrangians and Stability of Solutions.- 6.2.1 Necessary Conditions: Weierstrass Test.- 6.2.2 Attainability of the Convex Envelope.- 6.3 Quasiconvexity.- 6.3.1 Definition of Quasiconvexity.- 6.3.2 Quasiconvex Envelope.- 6.3.3 Bounds.- 6.4 Piecewise Quadratic Lagrangians.- 6.5 Problems.- 7 Optimal Structures and Laminates.- 7.1 Laminate Bounds.- 7.1.1 The Laminate Bound.- 7.1.2 Bounds of High Rank.- 7.2 Effective Properties of Simple Laminates.- 7.2.1 Laminates from Two Materials.- 7.2.2 Laminate from a Family of Materials.- 7.3 Laminates of Higher Rank.- 7.3.1 Differential Scheme.- 7.3.2 Matrix Laminates.- 7.3.3 Y-Transform.- 7.3.4 Calculation of the Fields Inside the Laminates.- 7.4 Properties of Complicated Structures.- 7.4.1 Multicoated and Self-Repeating Structures.- 7.4.2 Structures of Contrast Properties.- 7.5 Optimization in the Class of Matrix Composites.- 7.6 Discussion and Problems.- 8 Lower Bound: Translation Method.- 8.1 Translation Bound.- 8.2 Quadratic Translators.- 8.2.1 Compensated Compactness.- 8.2.2 Determination of Quadratic Translators.- 8.3 Translation Bounds for Two-Well Lagrangians.- 8.3.1 Basic Formulas.- 8.3.2 Extremal Translations.- 8.3.3 Example: Lower Bound for the Sum of Energies.- 8.3.4 Translation Bounds and Laminate Structures..- 8.4 Problems.- 9 Necessary Conditions and Minimal Extensions.- 9.1 Variational Methods for Nonquasiconvex Lagrangians.- 9.2 Variations.- 9.2.1 Variation of Properties.- 9.2.2 Increment.- 9.2.3 Minimal Extension.- 9.3 Necessary Conditions for Two-Phase Composites.- 9.3.1 Regions of Stable Solutions.- 9.3.2 Minimal Extension.- 9.3.3 Necessary Conditions and Compatibility.- 9.3.4 Necessary Conditions and Optimal Structures.- 9.4 Discussion and Problems.- IV G-Closures.- 10 Obtaining G-Closures.- 10.1 Variational Formulation.- 10.1.1 Variational Problem for Gm-Closure.- 10.1.2 G-Closures.- 10.2 The Bounds from Inside by Laminations.- 10.2.1 The L-Closure in Two Dimensions.- 11 Examples of G-Closures.- 11.1 The Gm-Closure of Two Conducting Materials.- 11.1.1 The Variational Problem.- 11.1.2 The Gm-Closure in Two Dimensions.- 11.1.3 Three-Dimensional Problem.- 11.2 G-Closures.- 11.2.1 Two Isotropic Materials.- 11.2.2 Polycrystals.- 11.2.3 Two-Dimensional Polycrystal.- 11.2.4 Three-Dimensional Isotropic Polycrystal.- 11.3 Coupled Bounds.- 11.3.1 Statement of the Problem.- 11.3.2 Translation Bounds of Gm-Closure.- 11.3.3 The Use of Coupled Bounds.- 11.4 Problems.- 12 Multimaterial Composites.- 12.1 Special Features of Multicomponent Composites.- 12.1.1 Attainability of the Wiener Bound.- 12.1.2 Attainability of the Translation Bounds.- 12.1.3 The Compatibility of Incompatible Phases.- 12.2 Necessary Conditions.- 12.2.1 Single Variations.- 12.2.2 Composite Variations.- 12.3 Optimal Structures for Three-Component Composites.- 12.3.1 Range of Values of the Lagrange Multiplier.- 12.3.2 Examples of Optimal Microstructures.- 12.4 Discussion.- 13 Supplement: Variational Principles for Dissipative Media.- 13.1 Equations of Complex Conductivity.- 13.1.1 The Constitutive Relations.- 13.1.2 Real Second-Order Equations.- 13.2 Variational Principles.- 13.2.1 Minimax Variational Principles.- 13.2.2 Minimal Variational Principles.- 13.3 Legendre Transform.- 13.4 Application to G-Closure.- V Optimization of Elastic Structures.- 14 Elasticity of Inhomogeneous Media.- 14.1 The Plane Problem.- 14.1.1 Basic Equations.- 14.1.2 Rotation of Fourth-Rank Tensors.- 14.1.3 Classes of Equivalency of Elasticity Tensors.- 14.2 Three-Dimensional Elasticity.- 14.2.1 Equations.- 14.2.2 Inhomogeneous Medium. Continuity Conditions.- 14.2.3 Energy, Variational Principles.- 14.3 Elastic Structures.- 14.3.1 Elastic Composites.- 14.3.2 Effective Properties of Elastic Laminates.- 14.3.3 Matrix Laminates, Plane Problem.- 14.3.4 Three-Dimensional Matrix Laminates.- 14.3.5 Ideal Rigid-Soft Structures.- 14.4 Problems.- 15 Elastic Composites of Extremal Energy.- 15.1 Composites of Minimal Compliance.- 15.1.1 The Problem.- 15.1.2 Translation Bounds.- 15.1.3 Structures.- 15.1.4 The Quasiconvex Envelope.- 15.1.5 Three-Dimensional Problem.- 15.2 Composites of Minimal Stiffness.- 15.2.1 Translation Bounds.- 15.2.2 The Attainability of the Convex Envelope.- 15.3 Optimal Structures Different from Laminates.- 15.3.1 Optimal Structures by Vigdergauz.- 15.3.2 Optimal Shapes under Shear Loading.- 15.4 Problems.- 16 Bounds on Effective Properties.- 16.1 Gm-Closures of Special Sets of Materials.- 16.2 Coupled Bounds for Isotropic Moduli.- 16.2.1 The Hashin―Shtrikman Bounds.- 16.2.2 The Translation Bounds.- 16.2.3 Functionals.- 16.2.4 Translators.- 16.2.5 Modification of the Translation Method.- 16.2.6 Appendix: Calculation of the Bounds.- 16.3 Isotropic Planar Polycrystals.- 16.3.1 Bounds.- 16.3.2 Extremal Structures: Differential Scheme.- 16.3.3 Extremal Structures: Fixed-Point Scheme.- 17 Some Problems of Structural Optimization.- 17.1 Properties of Optimal Layouts.- 17.1.1 Necessary Conditions.- 17.1.2 Remarks on Instabilities.- 17.2 Optimization of the Sum of Elastic Energies.- 17.2.1 Minimization of the Sum of Elastic Energies.- 17.2.2 Optimal Design of Periodic Structures.- 17.3 Arbitrary Goal Functionals.- 17.3.1 Statement.- 17.3.2 Local Problem.- 17.3.3 Asymptotics.- 17.4 Optimization under Uncertain Loading.- 17.4.1 The Formulation.- 17.4.2 Eigenvalue Problem.- 17.4.3 Multiple Eigenvalues.- 17.5 Conclusion.- References.- Author/Editor Index.