some remarks on quasiconvexity and rank one convexity we explore some necessary conditions for quasiconvexity in an attempt to show that rank one convexity does not imply quasiconvexity when the target space for deformations is two dimensional an interesting construction is presented showing how rank one directions may fit with each other making the task harder than in higher dimensions

multiscale resolution in the computation of crystalline microstructure this paper addresses the numerical approximation of microstructures in crystalline phase transitions without surface energy it is shown that branching of different variants near interfaces of twinned martensite and austenite phases leads to reduced energies in finite element approximations such behavior of minimizing deformations is understood for an extended model that involves surface energies moreover the closely related question of the role of different growth conditions of the employed bulk energy is discussed by explicit construction of discrete deformations in lowest order finite element spaces we prove upper bounds for the energy and thereby clarify the question of the dependence of the convergence rate upon growth conditions and lamination orders for first order laminates the estimates are optimal

approximation of a laminated microstructure for a rotationally invariant double well energy density we give error estimates for the approximation of a laminated microstructure which minimizes the energy int omega phi nabla v x dx for a rotationally invariant double well energy density phi a we present error estimates for the convergence of the deformation in l 2 the convergence of directional derivatives of the deformation in the twin planes the weak convergence of the deformation gradient the convergence of the microstructure or young measure of the deformation gradients and the convergence of nonlinear integrals of the deformation gradient

a higher order beam finite element for bending and vibration problems abstract the finite element equations for a variationally consistent higher order beam theory are presented for the static and dynamic behavior of rectangular beams the higher order theory correctly accounts for the stress free conditions on the upper and lower surfaces of the beam while retaining the parabolic shear strain distribution the need for a shear correction coefficient is therefore eliminated full integration of the shear stiffness terms is shown to result in the recovery of the kirchhoff constraint for thin beams without introducing spurious locking constraints the accuracy of this formulation is demonstrated by using several numerical examples for the cases of small and large displacements for a hinged hinged beam the linear thickness shear mode frequency can be matched with the timoshenko frequency to yield a shear coefficient of 0 824 matching the bending frequencies between the two theories indicates a shear coefficient for the timoshenko theory that changes with mode number and slenderness ratio the influence of in plane inertia and slenderness ratio on the non linear frequency is examined for beams with a number of different support conditions

numerical approximation of young measures in non convex variational problems summary in non convex optimisation problems in particular in non convex variational problems there usually does not exist any classical solution but only generalised solutions which involve young measures in this paper first a suitable relaxation and approximation theory is developed together with optimality conditions and then an adaptive scheme is proposed for the efficient numerical treatment the young measures solving the approximate problems are usually composed only from a few atoms this is the main argument our effective active set type algorithm is based on the support of those atoms is estimated from the weierstrass maximum principle which involves a hamiltonian whose good guess is obtained by a multilevel technique numerical experiments are performed in a one dimensional variational problem and support efficiency of the algorithm

numerical approximation of the solution of variational problem with a double well potential variational problems with a double well potential are not lower semicontinuous and can fail to attain a minimum value rather the gradients of minimizing sequences do not converge pointwise and can have oscillations however the gradients do converge in the weak topology i e their local spatial averages converge such functionals arise in the description of equilibria of crystals or other ordered materials stable configurations for solid crystals which have symmetry related martensitic energy wells have a fine scale microstructure which can be related to the oscillations that energy minimizing sequences for the bulk energy exhibit an analysis is given of approximation methods for variational problems with a double well potential to give a rigorous justification for the use of such numerical methods to model the behavior of this class of solid crystals

optimal order error estimates for the finite element approximation of the solution of a nonconvex variational problem nonconvex variational problems arise in models for the equilibria of crystals and other ordered materials the solution of these variational prob lems must be described in terms of a microstructure rather than in terms of a deformation moreover the numerical approximation of the deformation gra dient often does not converge strongly as the mesh is refined nevertheless the probability distribution of the deformation gradients near each material point does converge recently we introduced a metric to analyze this convergence in this paper we give an optimal order error estimate for the convergence of the deformation gradient in a norm which is stronger than the metric used earlier

the simply laminated microstructure in martensitic crystals that undergo a cubic to orthorhombic phase transformation we study simply laminated microstructures of a martensitic crystal capable of undergoing a cubic to orthorhombic transformation of type mathcal p 432 to mathcal p 222 the free energy density modeling such a crystal is minimized on six energy wells that are pairwise rank one connected we consider the energy minimization problem with dirichlet boundary data compatible with an arbitrary but fixed simple laminate we first show that for all but a few isolated values of transformation strains this problem has a unique young measure solution solely characterized by the boundary data that represents the simply laminated microstructure we then present a theory of stability for such a microstructure and apply it to the conforming finite element approximation to obtain the corresponding error estimates for the finite element energy minimizers

a condition for the holder regularity of local minimizers of a nonlinear elastic energy in two dimensions we prove the local holder continuity of strong local minimizers of the stored energy functional e u int omega lambda nabla u 2 h rm det nabla u rm d x subject to a condition of positive twist the latter turns out to be equivalent to requiring that u maps circles to suitably star shaped sets the convex function h s grows logarithmically as s to 0 linearly as s to infty and satisfies h s infty if s leqq 0 these properties encode a constitutive condition which ensures that material does not interpenetrate during a deformation and is one of the principal obstacles to proving the regularity of local or global minimizers the main innovation is to prove that if a local minimizer has positive twist a e on a ball then an euler lagrange type inequality holds and a caccioppoli inequality can be derived from it the claimed holder continuity then follows by adapting some well known elliptic regularity theory we also demonstrate the regularizing effect that the term int omega h rm det nabla u rm d x can have by analysing the regularity of local minimizers in the class of shear maps in this setting a more easily verifiable condition than that of positive twist is imposed with the result that local minimizers are holder continuous

quasiconvex elastodynamics weak strong uniqueness for measure valued solutions a weak strong uniqueness result is proved for measure valued solutions to the system of conservation laws arising in elastodynamics the main novelty brought forward by the present work is that the underlying stored energy function of the material is assumed strongly quasiconvex the proof employs tools from the calculus of variations to establish general convexity type bounds on quasiconvex functions and recasts them in order to adapt the relative entropy method to quasiconvex elastodynamics

global c regularity for second order quasilinear elliptic euqations in divergence form on considere des equations elliptiques quasilineaires d ordre 2 sous forme divergence sous des conditions aux limites de dirihlet

invertibility and weak continuity of the determinant for the modelling of cavitation and fracture in nonlinear elasticity in this paper we present and analyze a variational model in nonlinear elasticity that allows for cavitation and fracture the main idea in unifying the theories of cavitation and fracture is to regard both cavities and cracks as phenomena of the creation of a new surface accordingly we define a functional that measures the area of the created surface this functional has relationships with the theory of cartesian currents we show that the boundedness of that functional implies sequential weak continuity of the determinant of the deformation gradient and that the weak limit of one to one almost everywhere deformations is also one to one almost everywhere we then use these results to obtain the existence of minimizers of variational models that incorporate elastic energy and this created surface energy taking into account orientation preserving and non interpenetration conditions

on the theory of relaxation in nonlinear elasticity with constraints on the determinant we consider vectorial variational problems in nonlinear elasticity of the form i u int w du rm d x where w is continuous on matrices with a positive determinant and diverges to infinity along sequences of matrices whose determinant is positive and tends to zero we show that under suitable growth assumptions the functional int w rm qc du rm d x is an upper bound on the relaxation of i and coincides with the relaxation if the quasiconvex envelope wqc of w is polyconvex and has p growth from below with p geqq n this includes several physically relevant examples we also show how a constraint of incompressibility can be incorporated in our results

on the computation of the rank one convex hull of a function we report on several numerical experiments where the rank one convexification of an energy density is computed the explicit examples cover a whole spectrum of typical situations one may encounter one of those is especially relevant for the computation of microstructures in crystalline solids

classical refined and component wise analysis of reinforced shell structures this paper compares early and very recent approaches to the static analysis of reinforced shell wing structures early approaches were those based on the pure semimonocoque theory along with the beam assumptions of the euler bernoulli and timoshenko type the recent approaches are based on a hierarchical one dimensional formulation these are obtained by adopting various polynomial expansions of the displacement field above the cross section of the structure according to the unified formulation which was recently proposed by the first author two classes were developed in the unified formulation framework in the first class taylor expansion models were developed by exploiting n order taylor like polynomials classical beam theories euler bernoulli and timoshenko were obtained as special cases of taylor expansion in the second class lagrange expansion models were built by means of four and nine point lagrange type polynomials over the cross section of the wing the component wise approach was obtai

local and global norm comparison theorems for solutions to the nonhomogeneous a harmonic equation abstract we establish local and global norm inequalities for solutions of the nonhomogeneous a harmonic equation a x g d u h d v for differential forms as applications of these inequalities we prove the sobolev poincare type imbedding theorems and obtain l p estimates for the gradient operator and the homotopy operator t from the banach space l s d l to the sobolev space w 1 s d l 1 l 1 2 n these results can be used to study both qualitative and quantitative properties of solutions of the a harmonic equations and the related differential systems

integral estimates for null lagrangians sobolev spaces of differential forms are studied lp projections onto exact forms are introduced as a tool to obtain integral estimates for null lagrangians new results on compensated compactness are given and mean coercive variational integrals are found existence of minima of certain mean coercive functionals is established

semicontinuity problems in the calculus of variations soit f r 3 r une fonction deux fois continument differentiable et pour u w 1 1 a b f u a b a b f x u x u x dx si f r n r m r nm r satisfait 1 f est mesurable en x continue en cs et 2 0 f x s a x c s p p ou p 1 a est une fonction localement sommable non negative et c une constante non negative alors la fonctionnelle f u a b est sequentiellement faiblement semi continue inferieurement sur w 1 p r m si et seulement si f x s est quasi convexe pour chaque s et presque chaque x

gradient polyconvex material models and their numerical treatment abstract gradient polyconvex materials are nonsimple materials where we do not assume smoothness of the elastic strain but instead regularity of minors of the strain is required this allows for a larger class of admissible deformations than in the case of second grade materials we describe a possible implementation of gradient polyconvex elastic energies in nonlinear finite strain elastostatics besides a new geometric interpretation of gradient polyconvexity is given and it is compared with standard second grade materials finally we demonstrate application of the proposed approach using two different models namely a saint venant kirchhoff material and a double well stored energy density

quasiconvexity and relaxation of nonconvex problems in the calculus of variations keywords quasiconvexity relaxation reference caa article 1982 002doi 10 1016 0022 1236 82 90046 5view record in web of science record created on 2008 11 25 modified on 2017 05 12

w1 p quasiconvexity and variational problems for multiple integrals abstract variational problems for the multiple integral i u g u x dx where r m and u r n are studied a new condition on g called w1 p quasiconvexity is introduced which generalizes in a natural way the quasiconvexity condition of c b morrey it being shown in particular to be necessary for sequential weak lower semicontinuity of i in w 1 p r n and for the existence of minimizers for certain related integrals counterexamples are given concerning the weak continuity properties of jacobians in w 1 p r n p n m an existence theorem for nonlinear elastostatics is proved under optimal growth hypotheses

optimal control of a class of variational inequalities of the second kind optimal control problems governed by a class of elliptic variational inequalities of the second kind are investigated applications include the optimal control of viscoplastic fluid flow and of simplified friction based on a tikhonov regularization of the dual problem a family of primal dual regularized control problems is introduced and convergence of the regularized solutions towards a solution of the original control problem is verified for each regularized problem an optimality condition is derived and an optimality system for the original control problem is obtained as a limit of the regularized ones thanks to the structure of the proposed regularization complementarity relations between the variables involved are derived since the regularized optimality systems involve newton differentiable functions a semismooth newton algorithm is proposed and its numerical performance investigated

numerical analysis of oscillations in multiple well problems variational problems which fail to be convex occur often in the study of ordered materials such as crystals in these problems the energy density for the material has multiple potential wells in this paper we study multiple well problems by first determining the analytic properties of energy minimizing sequences and then by estimating the continuous problem by an approximation using piecewise linear finite elements we show that even when there is no minimizer of the energy the approximations still take on a predictable structure

accurate response of wing structures to free vibration load factors and nonstructural masses based on the carrera unified formulation this work extends variable kinematic finite beam elements to include load factors and nonstructural masses for the static and vibration analyses of complex metallic wing structures according to the carrera unified formulation variable kinematic beam theories are formulated in an automatic and hierarchical manner by expressing the displacement field as an arbitrary expansion through generic cross sectional functions both taylor like and lagrange polynomials are used in this paper to develop refined beam kinematics and the related theories are referred to as taylor expansion and lagrange expansion respectively the generalized unknowns of taylor expansion models are the beam axis displacements and the n order displacement derivatives with n being a free parameter of the analysis classical beam theories are clearly particular cases of the linear n 1 taylor expansion model on the other hand lagrange expansion models have only pure translational displacemen

on algorithms for nonconvex optimization in the calculus of variations a simple algorithm for the computation of local minima of nonconvex problems in one dimension is proposed this algorithm avoids certain well known local minima that are distant from the global minimum and can be used to improve the minima obtained using classical descent methods the calculated minima have mesh scale oscillations typical of weakly converging sequences it is shown that these local minima converge weakly to the correct limit for certain simple problems however numerical evidence suggests that the gradients may not converge to the correct weak limits and this is verified using asymptotic expansions extensions to multiple dimensions and some numerical examples in two dimensions are considered

numerical approximations in variational problems with potential wells in this paper some numerical aspects of variational problems which fail to be convex are studied it is well known that for such a problem in general the infimum of the energy the functional that has to be minimized fails to be attained instead minimizing sequences develop oscillations which allow them to decrease the energy it is shown that there exists a minimizes for an approximation of the problem and the oscillations in the minimizing sequence are analyzed it is also shown that these minimizing sequences choose their gradients in the vicinity of the wells with a probability which tends to be constant an estimate of the approximate deformation as it approximates a measure and some numerical results are also given

quasiconvexity at the boundary positivity of the second variation and elastic stability for one dimensional problems of the calculus of variations there is a well developed theory relating the positivity of the second variation at a critical point i e linearized stability to strong relative minima these results largely due to weierstrass are usually proved by the method of fields of extremals as is described in bolza 1904 and morrey 1966 or by an argument involving a careful use of taylor series as in hestenes 1966 chapter 3 14

approximated convex envelope of a function the goal of this paper is to introduce the approximated convex envelope of a function and to estimate how it differs from its convex envelope such a problem arises in various physical situations where the function considered is some energy that has to be minimized this study is a first step toward understanding how to approximate the quasi convex envelope of a function the importance of this issue is due to the various applications that are encountered in particular in the field of material science 1994 copyright society for industrial and applied mathematics

computation of the lower semicontinuous envelope of integral functionals and non homogeneous microstructures abstract a numerical method is established for computing the weakly lower semicontinuous envelope of integral functionals with non quasiconvex integrands the convergence of the method is proved and it is shown that the method is capable of capturing curved and non homogeneous microstructures numerical examples are given to show the effectiveness of the method for capturing curved and non homogeneous laminated microstructures

on rank one convex and polyconvex conformal energy functions with slow growth we make some remarks about rank one convex and polyconvex functions on the set of all real n n matrices that vanish on the subset k n consisting of all conformal matrices and grow like a power function at infinity we prove that every non negative rank one convex function that vanishes on k n and grows below a power of degree n 2 must vanish identically in odd dimensions n 3 we prove that every non negative polyconvex function that vanishes on k n must vanish identically if it grows below a power of degree n while in even dimensions such polyconvex functions can exist that also grow like a power of half dimension degree

numerical approach to double well problems this paper deals with two sided numerical energy estimates of a solution to two dimensional double well problems these estimates are obtained by computing the energy appropriate for the polyconvex envelope and an upper estimate of the energy appropriate for the rank one convex envelope of the energy density examples of computed laminated microstructures are shown

nonconforming finite element approximation of crystalline microstructure we consider a class of nonconforming nite element approxima tions of a simply laminated microstructure which minimizes the nonconvex variational problem for the deformation of martensitic crystals which can un dergo either an orthorhombic to monoclinic double well or a cubic to tetrag onal triple well transformation we rst establish a series of error bounds in terms of elastic energies for the l 2 approximation of derivatives of the de formation in the direction tangential to parallel layers of the laminate for the l 2 approximation of the deformation for the weak approximation of the de formation gradient for the approximation of volume fractions of deformation gradients and for the approximation of nonlinear integrals of the deformation gradient we then use these bounds to give corresponding convergence rates for quasi optimal nite element approximations

numerical analysis of a nonconvex variational problem related to solid solid phase transitions the description of equilibria of shape memory alloys or other ordered materials gives rise to nonconvex variational problems in this paper a two dimensional model of such materials is studied due to the fact that the corresponding functional has two symmetry related martensitic energy wells the numerical approximation of the deformation gradient does not converge but tends to oscillate between the two wells as the size of the mesh is refined these oscillations may be interpreted in terms of microstructures using a nonconforming mathcal p 1 finite element an estimate is given for the rate of convergence of the probability for the approximated deformation to have its gradient near one of the two martensitic wells

finite element error estimates for non linear elliptic equations of monotone type in this paper we shall consider the application of the finite element method to a class of second order elliptic boundary value problems of divergence form and with gradient nonlinearity in the principal coefficient and the derivation of error estimates for the finite element approximations such problems arise in many practical situations for example in shock free airfoil design seepage through coarse grained porous media and in some glaciological problems by making use of certain properties of the nonlinear coefficients we shall demonstrate that the variational formulations associated with these boundary value problems are well posed we shall also prove that the abstract operators accompanying such problems satisfy certain continuity and monotonicity inequalities with the aid of these inequalities and some standard results from approximation theory we show how one may derive error estimates for the finite element approximations in the energy norm

error estimates for optimal control problems of a class of quasilinear equations arising in variable viscosity fluid flow we consider optimal control problems of quasilinear elliptic equations with gradient coefficients arising in variable viscosity fluid flow the state equation is monotone and the controls are of distributed type we prove that the control to state operator is twice frechet differentiable for this class of equations a finite element approximation is studied and an estimate of optimal order h is obtained for the control the result makes use of the distributed structure of the controls together with a regularity estimate for elliptic equations with holder coefficients and a second order sufficient optimality condition the paper ends with a numerical experiment where the approximation order is computationally tested

null lagrangians weak continuity and variational problems of arbitrary order abstract we consider the problem of minimizing integral functionals of the form i u f x k u x dx where r p u r and k u denotes the set of all partial derivatives of u with orders k the method is based on a characterization of null lagrangians l ku depending only on derivatives of order k applications to elasticity and other theories of mechanics are given

a new computational framework for electro activation in cardiac mechanics abstract this paper presents a novel computational framework for the numerical simulation of the electromechanical response of the myocardium during the cardiac cycle the paper presents the following main novelties 1 two new mixed formulations tailor made for active stress and active strain coupling approaches have been developed and used in conjunction with two different ionic models namely bueno orovio et al 2008 and ten tusscher et al 2004 taking as a reference the mixed formulations introduced by bonet et al 2015 in the context of nonlinear elasticity the proposed formulations include as unknown fields the geometry and the transmembrane potential and possibly a lagrange multiplier enforcing weakly the incompressibility constraint as well as the deformation gradient tensor its cofactor its determinant the gradient of the transmembrane potential and their respective work conjugates the finite element implementation of these formulations is shown in this paper where a static condensation procedure is presented in order to yield an extremely competitive computational approach 2 a comprehensive and rigorous study of different ionic models i e bueno orovio and ten tusscher and electromechanical activation couplings i e active strain and active stress has been carried out 3 an analytical and numerical analysis of the possible loss of ellipticity and polyconvexity of one of the most widely used constitutive models in the context of cardiac mechanics is carried out in this paper putting forward possible polyconvexifications of the existing model 4 in addition an invariant representation of guccione s constitutive model is proposed finally a series of numerical examples are included in order to demonstrate the applicability and robustness of the proposed formulations

linear programming approach to nonconvex variational problems in nonconvex optimization problems in particular in nonconvex variational problems there usually does not exist any classical solution but only generalized solutions which involve young measures in this paper after reviewing briefly the relaxation theory for such problems an iterative scheme leading to a sequential linear programming slp scheme is introduced and its convergence is proved by a banach fixed point technique then an approximation scheme is proposed and analyzed and calculations of an illustrative 2d broken extremal example are presented

aw1 p estimate for solutions to mixed boundary value problems for second order elliptic differential equations on demontre que sous des hypotheses plutot faibles une solution d un probleme aux valeurs limites mixte pour une equation elliptique d ordre 2 est dans l espace de sobolev w 1 p pour p 2

analysis of a class of nonconforming finite elements for crystalline microstructures an analysis is given for a class of nonconforming lagrange type finite elements which have been successfully utilized to approximate the solution of a variational problem modeling the deformation of martensitic crystals with microstructure these elements were first proposed and analyzed in 1992 by rannacher and turek for the stokes equation our analysis highlights the features of these elements which make them effective for the computation of microstructure new results for superconvergence and numerical quadrature are also given

error estimates for the numerical approximation of a distributed control problem for the steady state navier stokes equations we obtain error estimates for the numerical approximation of a distributed control problem governed by the stationary navier stokes equations with pointwise control constraints we show that the l 2 norm of the error for the control is of order h 2 if the control set is not discretized while it is of order h if it is discretized by piecewise constant functions these error estimates are obtained for local solutions of the control problem which are nonsingular in the sense that the linearized navier stokes equations around these solutions define some isomorphisms and which satisfy a second order sufficient optimality condition we establish a second order necessary optimality condition the gap between the necessary and sufficient second order optimality conditions is the usual gap known for finite dimensional optimization problems

some properties of the degree for a class of sobolev maps we prove that a map u w loc1 1 m n whose differential and adjoint are integrable with exponents n 1 p n and n n 1 q respectively possesses an integer valued degree with the usual properties here m c 0 1 and n c 1 1 are n dimensional oriented riemannian manifolds with m possibly non empty

one dimensional variational problems whose minimizers do not satisfy the euler lagrange equation in this paper we consider the problem of minimizing i u text int a b text f x u x u x dx 1 1 in the set a of absolutely continuous function u a b satisfying the end conditions u left a right text alpha text quad u left b right text beta text 1 2 where and are given constant in 1 1 a b is a finite interval denotes frac d dx and the integrand f f x u p is assumed to be smooth nonnegative and to satisfy the regularity condition f pp text 0 text 1 3

finite element analysis of microstructure for the cubic to tetragonal transformation martensitic crystals which can undergo a cubic to tetragonal phase transformation have a nonconvex energy density with three symmetry related rotationally invariant energy wells we give estimates for the numerical approximation of a first order laminate for such martensitic crystals we give bounds for the l2 convergence of directional derivatives in the twin plane for the l2 convergence of the deformation for the weak convergence of the deformation gradient for the convergence of the microstructure and for the convergence of nonlinear integrals of the deformation gradient

convergence of a reduced integration method for computing microstructures recent interest in materials which form microstructures has led to efforts to determine the material configurations computationally an approximation method using bilinear finite elements with a reduced integration scheme and a local postprocessing projection has been found in numerical experiments to have the flexibility to represent microstructures on various scales and for various orientations with respect to the grid and has produced results which agree with the theory in this paper we give a rigorous proof of the convergence of the microstructure for this method

distributed control of systems governed by a general class of quasilinear elliptic equations abstract in this paper we are concerned with optimal control problems governed by a general class of quasilinear elliptic equations our aim is to prove existence of solution and derive optimality conditions in order to deal with this last question we begin investigating differentiability of state solution of equation with respect to the control second term of equation obtaining it only in certain cases a perturbation method is used to treat other cases as a consequence of the optimality system regularity of optimal control and state is establised

numerical analysis of oscillations in nonconvex problems we study numerically the pattern of the minimizing sequences of nonconvex problems which do not admit a minimizer

approximation of infima in the calculus of variations abstract the goal of this paper is to give numerical estimates for some problems of the calculus of variations in the nonhomogeneous scalar case the stored energy function considered is then a function r n r we try to compare the infimum of the energy defined by on a sobolev space with the infimum of the same energy on a finite element space in terms of the mesh size