Nonquasiconvex Variational Problems Analysis of Problems that do

Nonquasiconvex Variational Problems: Analysis of Problems that do not have Solutions Andrej Cherkaev Department of Mathematics University of Utah cherk@math. utah. edu The work supported by NSF and ARO UConn, April 2004

Plan • Non-quasiconvex Lagrangian – Motivations and applications – Specifics of multivariable problems • Developments: – Bounds (Variational formulation of several design problems) – Minimizing sequences – Detection of instabilities (Variational conditions) and Detection of zones of instability and sorting of structures – Suboptimal projects Dynamics UConn, April 2004

Why do structures appear in Nature and Engineering? UConn, April 2004

Energy of equilibrium and constitutive relations Equilibrium in an elastic body corresponds to solution of a variational problem corresponding constitutive relations (Euler-Lagrange eqns) are Here: W is the energy density, w is displacement vector, q is an external load If the BVP is elliptic, the Lagrangian W is (quasi)convex. UConn, April 2004

Convexity of the Lagrangian • In “classical” (unstructured) materials, Lagrangian W(A) is quasiconvex – The constitutive relations are elliptic. – The solution w(x) is regular with respect to a variation of the domain O and load q. • However, problems of optimal design, composites, natural polymorphic materials (martensites), polycrystals, “smart materials, ” biomaterials, etc. yield to non(quasi)convex variational problems. In the region of nonconvexity, – The Euler equation loses elliptiticity, – The minimizing sequence tends to an infinitely-fast-oscillating limit. UConn, April 2004

Optimal design and multiwell Lagrangians Problem: Find a layout c(x) that minimizes the total energy of an elastic body with the constraint on the used amount of materials. An optimal layout adapts itself on the applied stress. Energy cost UConn, April 2004

Examples of Optimal Design: Optimal layout is a fine-scale structure Thermal lens: Optimal wheel: Structure maximizes the stiffness against a pair of forces, applied in the hub and the felly. Optimal geometry: radial spokes and/or two twin systems of spirals. A structure that optimally concentrates the current. Optimal structure is an inhomogeneous laminate that directs the current. Concentration of the good conductor is variable to attract the current or to repulse it. A. Ch, Elena Cherkaev, 1998 UConn, April 2004 A. Ch, L. Gibiansky, K. Lurie, 1986

Structures perfected by Evolution A leaf A Dynosaur bone Dragonfly’s wing Dűrer’s rhino The structures are known, the goal functional is unknown! UConn, April 2004

Polymorphic materials • Smart materials, martensite alloys, polycrystals and similar materials can exist in several forms ( phases). The Gibbs principle states that the phase with minimal energy is realized. Optimality + nonconvexity =structured materials UConn, April 2004

Alloys and Minerals A martensite alloy with “twin” monocrystals Polycrystals of granulate Coal Steel UConn, April 2004

All good things are structured! Mozzarella cheese Chocolate UConn, April 2004

Nonmonotone constitutive relations: Instabilities • Nonconvex energy leads to nonmonotone constitutive relations and to nonuniqueness of constitutive relations. • Variational principle selects the solution with the least energy. UConn, April 2004

Oscillatory solutions and relaxation (1 D) (from optimal control theory) Young, Gamkrelidge, Warga, …. from 1960 s F(w’) Convex envelope: Definition CF(w’) Relaxation of the variational problem – replacement the Lagrangian with its convex envelope: UConn, April 2004

Example Relaxation f w(x) w Euler equations for an extremal w’ 1 x -1 w’(x) UConn, April 2004

Optimal oscillatory solutions in 1 D problems 1. When the solutions are smooth/oscillatory? (The Lagrangian is convex/nonconvex function of w’) 2. What are minimizing sequences? (Trivial in 1 D -- alternation) 3. What are the pointwise values of optimal solution? (They belong to the common boundary of the Lagrangian and its convex envelope) 4. How to compute or bound the Lagrangian on the oscillating solutions? Replace the Lagrangian with its convex envelope w. r. t. w’ 5. How to obtain or evaluate suboptimal solutions? By forcing a finite scale of oscillations or requiring an additional smoothness of minimizers. UConn, April 2004

What is special in the multivariable case: Integrability conditions Magnitudes of jumps depend on the normal n; Therefore the properties depend on the structure. UConn, April 2004 + - n

Consequences of integrability conditions w’(x) x • In a one-dimensional problem, w’ -the strain in a stretched composed bar -- is discontinuous • In a multidimensional problem, the tangential components of the strain must be continuous. • If the only mode of deformation is the uniform contraction (Material made from Hoberman spheres), then – No discontinuities of the strain field are possible UConn, April 2004

Quasiconvex envelope • • Minimum over all periodic trial minimizers with allowed discontinuities is called the quasiconvex envelope. Quasiconvex envelope equals to the minimal energy of periodic oscillating sequences, it is a pointwise transform of the Lagrangian Without this constraint, the definition coincides With the definition of the convex envelope Here O is a cube in UConn, April 2004 Murray (1956), Ball, Lurie, Kohn, Strang, Ch, Milton, Gibiansky, Murat, Francfort, Tartar, Dacorogna, Miller, Kinderlehrer, Pedregal.

Approaches to calculation of the Quasiconvex envelope • Sufficient conditions (Translation bounds) replace the variational problem with a minorant finite-dimensional optimization problem (analog of Lyapunov function). Generally, they are better that the lower bound by the convex envelope. • Minimizing sequences correspond to special multiscale fractal-type partitions. Generally, the optimal nesting partitions (microstructures) are not unique and based on a priori conjectures. • Structural variations is a variational method that analyses pointwise values of the minimizer or the fields in optimal structures: It provides an upper bound of the quasiconvex envelope stable to a class of variations UConn, April 2004

Sufficient conditions: Translation method Sufficient conditions use: Constancy of the potentials Periodicity of the fields in the definition of the quasiconvex envelope UConn, April 2004

Duality bound of a constrained problem The differential constraint is replaced with a weaker (integral) constraint . UConn, April 2004 Lurie, Cherkaev, Kohn, Strang, Tartar, Murat, Milton, Francfort, Gibiansky, Torquato, . .

Calculation of PF(v) for a piece-wise quadratic Lagrangian: Observe that the first term in PF(v) is a homogeneous second order function of v but not a quadratic form of v. UConn, April 2004

Translation bound for effective properties The energy of a heterogeneous mixture equals to the energy W* of the effective medium; quasiconvex envelope QF is the lower bound of all effective energies Comparing, we obtain Inequalities are observed UConn, April 2004

Development of translation-type bounds • The bounds for effective properties are applied to various problems: – Optimal conducting structures. (Lurie, Cherkaev, 1982, 84, Murat, Tartar, 1985) – Optimal elastic structures. (Cherkaev, Gibiansky, 1985, 87, Arraire, Kohn 1987) – Complex conductivity and viscous-elasticity. (Cherkaev, Gibiansky, 1996, ). (Milton, Gibiansky, Berryman ) – Minimization of the sum of energies in all directions. – (Avellaneda, Milton, 1993 (2 d), Francfort, Murat, Tartar, 1998 3 d) – Minimization of the sum of the stiffness and compliance. (Ch. , 1999) – Expansion tensor (eigenstrain). (Ch. , Sigmund, Vinogradov, 2004) – Multiphase mixtures (Nesi bounds, 1997), UConn, April 2004

Application: Bounds for effective conductivity tensor The bounds of the set of all effective tensors with prescribed volume fractions is found from a variational problem of the layout that minimizes the energy. The result: UConn, April 2004 Lurie, Cherkaev, 1982, 84, Tartar, Murat, 1985

Improvement of classical Hashin-Shtrikman bounds for elasticity (2 d) Gibiansky &Ch, 1994 • • Hashin and Shtikman in 1963 suggested a bond for isotropic elastic moduli of a composite Set of possible pairs of the moduli Translation method allowed to establish the coupled bounds between these quantities and the bounds for the moduli of anisotropic composites. UConn, April 2004

Bounds for effective expansion tensor (eigenstrain) • Energy (Lagrangian) for each phase • Effective energy for a mixture • Problem: Bounds for – Using the Legende transform and the technique of polyconvex envelope, we obtain the bounds for where • Cherkaev, Sigmund, Vinogradov 2004 (in progress) UConn, April 2004

Bruggeman, Hashin, Shtrickman, Milton, Lurie, Cherkaev, Gibiansky, Noris, Avellaneda, Murat, Tartar, Francfort, Bendsoe, Kikuchi, Sigmund. Minimizing sequences Algebra of laminates: Lego of laminate structures UConn, April 2004

Minimizers: w is a scalar: Two phases -“wells”: • As in one-dimensional case, the fields are constant at each phase; • Every two fields can be neighbors, if layouts are properly oriented laminates. • Quasiconvex envelope coincides with the convex envelope. Continuity constraint [v] t=0 serves to define tangent t to layers UConn, April 2004

Minimizers: more than two wells (phases) Again, the quasiconvex envelope coincides with the convex envelope. • Fields are constant within each phase. • Minimizing sequences: Laminates of (N-1)-th rank. Optimal structure is not unique: For instance, a permutation of materials is possible. UConn, April 2004

Properties: the minimizing field is constant in each phase, the structure is a laminate. the field is not constant within each phase because of too many continuity conditions. The quasiconvex envelope is not smaller than the convex envelope and not larger than the function itself: UConn, April 2004

Laminates of a rank Properties of laminates are explicit functions of mixed materials S, their fractions c, and the normal to layers n. Si are the properties tensors, ci are the volume fractions, and p(n) is the projection on the subspace of discontinuous fields components Next step – laminates of a rank (repeated laminates) They are optimal if the fields inside the structure are either constant within each material, or a projection of the fields is constant. UConn, April 2004

Differential scheme and control of fractal layouts • Consider the process of adding a new material in infinitesimal quantity and obtain the differential equation of evolution of the effective properties: A robust scheme, applicable to large class of problem. Constraints on geometry. The control problem: Choose the order, orientation and structure of added materials, in order to maximize the objective at the end of the process. The energy of laminate (or other specialized) structures is an upper bound of the quasiconvex envelope. UConn, April 2004

Optimal fractal geometry: Elastic polycrystal with extreme properties Sometimes, the nesting sequence is complicated, it can be found as a stable point of a set of transforms. The minimizing layouts are generally not unique. Avellaneda, Cherkaev, Gibiansky, Milton, Rudelson, 1997 UConn, April 2004

Example: Optimal structures of conducting composites: Minimize a functional of a conductivity potential Using the conjugate variables and the Legendre transform, the functional can be transformed to the form j Conductivity: Laminates are always optimal v UConn, April 2004 Elasticity: Laminates of a rank are optimal in an asymptotic case

Structural variations – Analysis of the fields in optimal structures • • • Based on classical Weierstrass test and Eshelby approach First version was suggested by K. Lurie, 1972; Sokolovsky, Telega, Fedorov, Ch, Presented version- Cherkaev 2000. UConn, April 2004

Structural variations Aim: Description of the discontinuous minimizer in the multiwell nonconvex problems, or Fields in an optimal structure. Consider an infinitesimal variation of the layout: • Place an infinitesimal elliptical inclusion of one of the admissible phases Sg into the tested phase Sh : • • d S= (Sg – Sh )cincl (x) Compute the perturbation of minimizer caused by this variation and the increment of the functional. UConn, April 2004

Weierstrass-type test • • Increment of the field depends on the shape of the trial ellipse. The increment is maximized by choosing – Shape – Orientation – Composition (for multi-material mixtures) of the trial ellipse, finding the “most dangereos variation” Solving we obtain the region of optimality of the tested material UConn, April 2004

Example: Minimization of the energy of a layout of linear elastic materials. • The perturbed fields can be explicitly calculated if the energy is piece-wise quadratic: materials are linear. • To compute the increment we may either use modified Eshelby formulas or simply compute effective properties of a matrix-laminate. structure, when send the volume fraction of the inclusions to zero. UConn, April 2004

Cherkaev, Kucuk, 2004 Forbidden Two-well problem: Permitted regions of the fields A norm of the stress in the weak and cheap material in an optimal structure is bounded from above Weak are the eigenvalues of the stress tensor Strong • There is a region where the NONE of materials is optimal. If the applied filed belongs to this region, the structure appears and the point-wise fields in the materials are sent away from the forbidden region. This phenomenon explains the appearance of composites. A norm of the stress in the strong and expensive material in an optimal structure is bounded from below: UConn, April 2004

Cherkaev, Kucuk, 2004 Optimal fields and optimal structures • The jump over “forbidden region” is only possible if the composite has a special microstructure. • The necessary conditions are examined together with the conditions on the boundary between materials – The field in the nuclei is hydrostatic and constant. – The field in the inner layer of the envelope are on the boundary of the regimes – The fields in the external layer lies on the straight component of the boundary • The optimal structures are not unique. UConn, April 2004

Interpretation of the optimality conditions In order to keep the fields on the boundary of the permitted regions, the design forms a microstructure that adjusts itself to the loading conditions In the zone of nonquasiconvexity, a norm of the field is each phase is constant everywhere no matter what are the external conditions. This feature extends the known engineering principle of “equally stressed” designs to the tensor of stresses. UConn, April 2004

Cherkaev, Kucuk, 2004 Suboptimal projects • Checking the fields in a design, we can find out how close these fields are to the boundaries of the permitted domains of optimal field Color shows the distance from the boundary of optimality. Remark Similar coloring is used in the ANSYS to warn about closeness to limits of carrying capacity UConn, April 2004

Cherkaev, Kucuk, 2004 3 D problem: Permitted regions UConn, April 2004

Permitted range of fields in a three-material mixture • The field in the intermediate material is constrained from zero and infinity. This implies that – the three materials in an optimal structure cannot meet in a singular point. Intermediate and the weakest material do not have a common boundary UConn, April 2004

The use of necessary. conditions: Optimal mixtures of three materials Large fraction of the best material Small fraction of the best material No contacts points between the three phases The set of contact points is dense UConn, April 2004

Atomistic models and Dynamics In collaboration with Leonid Slepyan, Elena Cherkaev, Alexander Balk, 2001 -2004 UConn, April 2004

Dynamic problems for multiwell energies • Formulation: Lagrangian for a continuous medium If W is (quasi)convex • • If W is not quasiconvex Questions: Radiation and other losses Dynamic homogenization – There are infinitely many local minima; each corresponds to an equilibrium. How to choose “the right one” ? – The realization of a particular local minimum depends on the existence of a path to it. What are initial conditions that lead to a particular local minimum? – How to account for dissipation and radiation? UConn, April 2004

Set of unstrained configurations • The geometrical problem of description of all possible unstrained configuration is still unsolved. • Some sophisticated configurations can be found. • Because of nonuniqueness, the expansion problem requires dynamic consideration. Random lattices: Nothing known UConn, April 2004

Waves in active materials • • • Links store additional energy remaining stable. Particles are inertial. When an instability develops, the excessive energy is transmitted to the next particle, originating the wave. Kinetic energy of excited waves takes away the energy, the transition looks like a domino or an explosion. Active materials: Kinetic energy is bounded from below Homogenization: Accounting for radiation and the energy of high-frequency modes is needed. Extra energy UConn, April 2004

Unstable reversible links Force • • • Each link consists of two parallel elastic rods, one of which is longer. Initially, only the longer road resists the load. If the load is larger than a critical (buckling)value: Elongation ü The longer bar looses stability (buckling), and ü the shorter bar assumes the load. The process is reversible. H is the Heaviside function No parameters! UConn, April 2004

Chain dynamics. Generation of a spontaneous transition wave x 0 Initial position: (linear regime, close to the critical point) UConn, April 2004

Observed spontaneous waves in a chain “Twinkling” phase “Chaotic” phase Under a smooth excitation, the chain develops intensive oscillations and waves. Sonic wave Wave of phase transition UConn, April 2004

Application: Structures that can withstand an impact Tao of Damage o Damage happens! o Uncontrolled, damage concentrates and destroys o Dispersed damage absorbs energy o Design is the art of scattering the damage UConn, April 2004

Conclusion Every variational problem has a solution provided that the word “solution” is properly understood. David Hilbert UConn, April 2004