Andrej Cherkaev

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.

Topics in the mathematical modelling of composite materials

On the control of partial differential equations estimation of homogenized coefficients H-convergence a strange term coming from nowhere design of composite plates of extremal rigidity calculus of variations and homogenization.

Which Elasticity Tensors are Realizable

It is shown that any given positive definite fourth order tensor satisfying the usual symmetries of elasticity tensors can be realized as the effective elasticity tensor of a two-phase composite comprised of a sufficiently compliant isotropic phase and a sufficiently rigid isotropic phase configured in an suitable microstructure. The building blocks for constructing this composite are what we call extremal materials. These are composites of the two phases which are extremely stiff to a set of arbitrary given stresses and, at the same time, are extremely compliant to any orthogonal stress. An appropriately chosen subset of the extremal materials are layered together to form the composite with elasticity tensor matching the given tensor.

On the effective conductivity of polycrystals and a three‐dimensional phase‐interchange inequality

We derive optimal bounds on the effective conductivity tensor of polycrystalline aggregates by introducing an appropriate null‐Lagrangian that is rotationally invariant. For isotropic aggregates of uniaxial crystals an outstanding conjecture of Schulgasser is proven, namely that the lowest possible effective conductivity of isotropic aggregates of uniaxial crystals is attained by a composite sphere assemblage, in which the crystal axis is directed radially outwards in each sphere. By laminating this sphere assemblage with the original crystal we obtain anisotropic composites that are extremal, i.e., attaining our bounds. These, together with other results established here, give a partial characterization of the set of all possible effective tensors of polycrystalline aggregates. The same general method is used to prove a conjectured phase interchange inequality for isotropic composites of two isotropic phases. This inequality correlates the effective conductivity of the composite with the effective tensor...

Regularization of optimal design problems for bars and plates, part 1

In this paper, we consider a number of optimal design problems for elastic bars and plates. The material characteristics of rigidity of an elastic nonhomogeneous medium are taken as the control variables. A linear functional of the solutions to the equilibrium boundary-value problem is minimized under additional restrictions upon the control variables.Special variations of the control within a narrow strip provide a necessary condition for a strong local minimum (Weierstrass test). This necessary condition can be used for the detailed analysis of the following problems: bar of extremal torsional rigidity; optimal distribution of isotropic material with variable shear modulus within a plate; and optimal orientation of principal axes of elasticity in an orthotropic plate. For all of these cases, the stationary solutions violate the Weierstrass test and therefore are not optimal. This is because, in the course of the approximation of the optimal solution, the curve dividing zones occupied by materials with different rigidities displays rapid oscillations sweeping over a two-dimensional region. Within this region, the material behaves as a composite medium assembled of materials of the initial class.

Variational principles for complex conductivity, viscoelasticity, and similar problems in media with complex moduli

Linear processes in media with dissipation arising in conductivity, optics, viscoelasticity, etc. are considered. Time‐periodic fields in such media are described by linear differential equations for complex‐valued potentials. The properties of the media are characterized by complex valued tensors, for example, by complex conductivity or complex elasticity tensors. Variational formulations are suggested for such problems: The functionals whose Euler equations coincide with the original ones are constructed. Four equivalent variational principles are obtained: two minimax and two minimal ones. The functionals of the obtained minimal variational principles are proportional to the energy dissipation averaged over the period of oscillation. The last principles can be used in the homogenization theory to obtain the bounds on the effective properties of composite materials with complex valued properties tensors.

Dynamics of chains with non-monotone stress–strain relations. I. Model and numerical experiments

Abstract We discuss dynamic processes in materials with non-monotonic constitutive relations. We introduce a model of a chain of masses joined by springs with a non-monotone strain–stress relation. Numerical experiments are conducted to find the dynamics of that chain under slow external excitation. We find that the dynamics leads either to a vibrating steady state (twinkling phase) with radiation of energy, or (if dissipation is introduced) to a hysteresis, rather than to an unique stress–strain dependence that would correspond to the energy minimization.

Sliding regimes and anisotropy in optimal design of vibrating axisymmetric plates

Abstract This paper deals with optimal design of solid, elastic, axisymmetric plates performing free, transverse vibrations. It is the objective to determine the plate thickness distribution from the condition that the plate volume is minimized for a given value of the fundamental natural frequency, or for a given higher order natural frequency that corresponds to a vibration mode with a prescribed number of nodal diameters. It is found that the Weierstrass necessary condition for optimality is generally not satisfied for a traditional formulation of this problem, and that the optimal design is characterized by a sliding regime of control where the plate thickness exhibits an infinite number of discontinuities, as a system of infinitely thin, circumferential stiffeners are formed on the optimal axisymmetric plate. This inherent anisotropy of the optimal design is taken into account in a regularization of the initial optimization problem by establishing the tensorial character of the plate bending rigidity and using the concentration of thin, circumferential stiffeners as a new design variable (control). It is shown that the new formulation of the problem can be solved numerically, and examples of optimal designs are presented in the paper.

The cavity of the optimal shape under the shear stresses

The problem of optimal shape of a single cavity in an infinite 2-D elastic domain is analyzed. An elastic plane is subjected to a uniform load at infinity. The cavity of the fixed area is said to be optimal if it provides the minimal energy change between the homogeneous plane and the plane with the cavity. We show that for the case of shear loading the contour of the optimal cavity is not smooth but is shaped as a curved quadrilateral. The shape is specified in terms of conformal mapping coefficients, and explicit analytical representations for components of the dipole tensor associated with the cavity are employed. We also find the exact values of angles at the corners of the optimal contour. The applications include the problems of optimal design for dilute composites.

Design of Composite Plates of Extremal Rigidity

We consider design problems for plates possessing an extremal rigidity. The plate is assumed to be assembled from two isotropic materials characterized by different values of their elastic moduli; the amount of each material is given. We look for a distribution of the materials which renders the plate’s rigidity for either its maximal or minimal value. The rigidity is defined here as work produced by an extremal load on deflection of the points of the plate. The optimal distribution of the materials is characterized by some infinitely often alternating sequences of domains occupied by each of the materials (see [1], [2]). This leads to the appearance of anisotropic composites; their structures are to be determined at each point of the plate.