Variational Methods for Structural Optimization

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.