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 C