arXiv subjects

Konstantinos Koumatos

Publications and source records attributed to Konstantinos Koumatos.

At least 19 recordsLinked to original sources

arXiv2025

A class of Discontinuous Galerkin methods for nonlinear variational problems

In the context of Discontinuous Galerkin methods, we study approximations of nonlinear variational problems associated with convex energies. We propose element-wise nonconforming finite element methods to discretize the continuous minimisation problem. Using $Γ$-convergence arguments we show that the discrete minimisers converge to the unique minimiser of the continuous problem as the mesh parameter tends to zero, under the additional contribution of appropriately defined penalty terms at the level of the discrete energies. We finally substantiate the feasibility of our methods by numerical examples.

math.NA↗

arXiv2025

Convergence of Discontinuous Galerkin Methods for Quasiconvex and Relaxed Variational Problems

In this work, we establish that discontinuous Galerkin methods are capable of producing reliable approximations for a broad class of nonlinear variational problems. In particular, we demonstrate that these schemes provide essential flexibility by removing inter-element continuity while also guaranteeing convergent approximations in the quasiconvex case. Notably, quasiconvexity is the weakest form of convexity pertinent to elasticity. Furthermore, we show that in the non-convex case discrete minimisers converge to minimisers of the relaxed problem. In this case, the minimisation problem corresponds to the energy defined by the quasiconvex envelope of the original energy. Our approach covers all discontinuous Galerkin formulations known to converge for convex energies. This work addresses an open challenge in the vectorial calculus of variations: developing and rigorously justifying numerical schemes capable of reliably approximating nonlinear energy minimization problems with potentially singular solutions, which are frequently encountered in materials science.

math.NA↗

arXiv2021

Approximations of energy minimization in cell-induced phase transitions of fibrous biomaterials: $Γ$-convergence analysis

We consider a model of energy minimization arising in the study of the mechanical behavior caused by cell contraction within a fibrous biological medium. The macroscopic model is based on the theory of non rank-one convex nonlinear elasticity for phase transitions. We study appropriate numerical approximations based on the discontinuous Galerkin treatment of higher gradients and used succesfully in numerical simulations of experiments. We show that the discrete minimizers converge in the limit to minimizers of the continuous problem. This is achieved by employing the theory of $Γ$-convergence of the approximate energy functionals to the continuous model when the discretization parameter tends to zero. The analysis is involved due to the structure of numerical approximations which are defined in spaces with lower regularity than the space where the minimizers of the continuous variational problem are sought. This fact leads to the development of a new approach to $Γ$-convergence, appropriate for discontinuous finite element discretizations, which can be applied to quite general energy minimization problems. Furthermore, the adoption of exponential terms penalising the interpenetration of matter requires a new framework based on Orlicz spaces for discontinuous Galerkin methods which is developed in this paper as well.

math.NA↗

arXiv2021

Weak-strong uniqueness for measure-valued solutions to the equations of quasiconvex adiabatic thermoelasticity

This article studies the equations of adiabatic thermoelasticity endowed with an internal energy satisfying an appropriate quasiconvexity assumption which is associated to the symmetrisability condition for the system. A Gårding-type inequality for these quasiconvex functions is proved and used to establish a weak-strong uniqueness result for a class of dissipative measure-valued solutions.

math.AP↗

arXiv2020

Existence and uniqueness for a viscoelastic Kelvin-Voigt model with nonconvex stored energy

We consider nonlinear viscoelastic materials of Kelvin-Voigt type with stored energies satisfying an Andrews-Ball condition, allowing for non convexity in a compact set. Existence of weak solutions with deformation gradients in $H^1$ is established for energies of any superquadratic growth. In two space dimensions, weak solutions notably turn out to be unique in this class. Conservation of energy for weak solutions in two and three dimensions, as well as global regularity for smooth initial data in two dimensions are established under additional mild restrictions on the growth of the stored energy.

math.AP↗

arXiv2020

On the uniqueness of solutions to hyperbolic systems of conservation laws

For general hyperbolic systems of conservation laws we show that dissipative weak solutions belonging to an appropriate Besov space $B^{α,\infty}_q$ and satisfying a one-sided bound condition are unique within the class of dissipative solutions. The exponent $α>1/2$ is universal independently of the nature of the nonlinearity and the Besov regularity need only be imposed in space when the system is expressed in appropriate variables. The proof utilises a commutator estimate which allows for an extension of the relative entropy method to the required regularity setting. The systems of elasticity, shallow water magnetohydrodynamics, and isentropic Euler are investigated, recovering recent results for the latter. Moreover, the article explores a triangular system motivated by studies in chromatography and constructs an explicit solution which fails to be Lipschitz, yet satisfies the conditions of the presented uniqueness result.

math.AP↗

arXiv2020

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.

math.AP↗

arXiv2020

$\mathcal{A}$-quasiconvexity, Gårding inequalities and applications in PDE constrained problems in dynamics and statics

A Gårding-type inequality is proved for a quadratic form associated to $\mathcal{A}$-quasiconvex functions. This quadratic form appears as the relative entropy in the theory of conservation laws and it is related to the Weierstrass excess function in the calculus of variations. The former provides weak-strong uniqueness results, whereas the latter has been used to provide sufficiency theorems for local minimisers. Using this new Gårding inequality we provide an extension of these results to PDE constrained problems in dynamics and statics under $\mathcal{A}$-quasiconvexity assumptions. The application in statics improves existing results by proving uniqueness of $L^p$ local minimisers in the classical $\mathcal{A}= {\rm curl}$ case.

math.AP↗

arXiv2019

Necessary and sufficient conditions for the strong local minimality of $C^1$ extremals on a class of non-smooth domains

Motivated by applications in materials science, a set of quasiconvexity at the boundary conditions is introduced for domains that are locally diffeomorphic to cones. These conditions are shown to be necessary for strong local minimisers in the vectorial Calculus of Variations and a quasiconvexity-based sufficiency theorem is established for ${\rm C}^1$ extremals defined on this class of non-smooth domains. The sufficiency result presented here thus extends the seminal theorem by Grabovsky \& Mengesha (2009), where smoothness assumptions are made on the boundary.

math.AP↗

arXiv2016

Optimality of General Lattice Transformations with Applications to the Bain Strain in Steel

This article provides a rigorous proof of a conjecture by E.C. Bain in 1924 on the optimality of the so-called "Bain strain" based on a criterion of least atomic movement. A general framework that explores several such optimality criteria is introduced and employed to show the existence of optimal transformations between any two Bravais lattices. A precise algorithm and a GUI to determine this optimal transformation is provided. Apart from the Bain conjecture concerning the transformation from face-centred cubic to body-centred cubic, applications include the face-centred cubic to body-centred tetragonal transition as well as the transformation between two triclinic phases of Terephthalic Acid. The GUI can be accessed under http://uk.mathworks.com/matlabcentral/fileexchange/55554-optlat

math-ph↗

arXiv2016

A Theoretical Investigation of Orientation Relationships and Transformation Strains in Steels

The identification of orientation relationships (ORs) plays a crucial rôle in the understanding of solid phase transformations. In steels, the most common models of ORs are the ones by Nishiyama-Wassermann (NW) and Kurdjumov-Sachs (KS). The defining feature of these and other OR models is the matching of directions and planes in the parent face-centred cubic $γ$-phase to ones in the product body-centred cubic/tetragonal $α/α'$-phase. In this paper a novel method that identifies transformation strains with ORs is introduced and used to develop a new strain-based approach to phase transformation models in steels. Using this approach, it is shown that the transformation strains that leave a close packed plane in the $γ$-phase and a close packed direction within that plane unrotated are precisely those giving rise to the NW and KS ORs when a cubic product phase is considered. Further, it is outlined how, by choosing different pairs of unrotated planes and directions, other common ORs such as the ones by Pitsch (PT) and Greninger-Troiano (GT) can be derived. One of the advantages of our approach is that it leads to a natural generalisation of the NW, KS and other ORs for different ratios of tetragonality $r$ of the product bct $α'$-phase. These generalised ORs predict a sharpening of the transformation textures with increasing tetragonality and are thus in qualitative agreement with experiments on steels with varying alloy concentration.

cond-mat.mtrl-sci↗

arXiv2016

Shape programming for narrow ribbons of nematic elastomers

Using the theory of $Γ$-convergence, we derive from three-dimensional elasticity new one-dimensional models for non-Euclidean elastic ribbons, i.e. ribbons exhibiting spontaneous curvature and twist. We apply the models to shape-selection problems for thin films of nematic elastomers with twist and splay-bend texture of the nematic director. For the former, we discuss the possibility of helicoid-like shapes as an alternative to spiral ribbons.

math.AP↗

arXiv2015

The morphology of lath martensite: a new perspective

A mathematical framework is proposed to predict the features of the (5 5 7) lath transformation in low-carbon steels based on energy minimisation. This theory generates a one-parameter family of possible habit planes and a selection mechanism then identifies the (5 5 7) normals as those arising from a deformation with small atomic movement and maximal compatibility. While the calculations bear some resemblance to those of double shear theories, the assumptions and conclusions are different. Interestingly, the predicted microstructure morphology resembles that of plate martensite, in the sense that a type of twinning mechanism is involved.

cond-mat.mtrl-sci↗

arXiv2015

Quasiconvexity at the boundary and the nucleation of austenite

Motivated by experimental observations of H. Seiner et al., we study the nucleation of austenite in a single crystal of a CuAlNi shape-memory alloy stabilized as a single variant of martensite. In the experiments the nucleation process was induced by localized heating and it was observed that, regardless of where the localized heating was applied, the nucleation points were always located at one of the corners of the sample - a rectangular parallelepiped in the austenite. Using a simplified nonlinear elasticity model, we propose an explanation for the location of the nucleation points by showing that the martensite is a local minimizer of the energy with respect to localized variations in the interior, on faces and edges of the sample, but not at some corners, where a localized microstructure, involving austenite and a simple laminate of martensite, can lower the energy. The result for the interior, faces and edges is established by showing that the free-energy function satisfies a set of quasiconvexity conditions at the stabilized variant in the interior, faces and edges, respectively, provided the specimen is suitably cut.

math.AP↗

arXiv2014

Differential inclusions and Young measures involving prescribed Jacobian

This work presents a general principle, in the spirit of convex integration, leading to a method for the characterization of Young measures generated by gradients of maps in $W^{1,p}$ with $p$ less than the space dimension, whose Jacobian determinant is subjected to a range of constraints. Two special cases are particularly important in the theories of elasticity and fluid dynamics: (a) the generating gradients have positive Jacobians that are uniformly bounded away from zero and (b) the underlying deformations are incompressible, corresponding to their Jacobian determinants being constantly one. This characterization result, along with its various corollaries, underlines the flexibility of the Jacobian determinant in subcritical Sobolev spaces and gives a more systematic and general perspective on previously known pathologies of the pointwise Jacobian. Finally, we show that, for $p$ less than the dimension, $W^{1,p}$-quasiconvexity and $W^{1,p}$-orientation-preserving quasiconvexity are both unsuitable convexity conditions for nonlinear elasticity where the energy is assumed to blow up as the Jacobian approaches zero.

math.AP↗

arXiv2014

Orientation-preserving Young measures

We prove a characterization result in the spirit of the Kinderlehrer-Pedregal Theorem for Young measures generated by gradients of Sobolev maps satisfying the orientation-preserving constraint, that is the pointwise Jacobian is positive almost everywhere. The argument to construct the appropriate generating sequences from such Young measures is based on a variant of convex integration in conjunction with an explicit lamination construction in matrix space. Our generating sequence is bounded in $L^p$ for $p$ less than the space dimension, a regime in which the pointwise Jacobian loses some of its important properties. On the other hand, for $p$ larger than, or equal to, the space dimension the situation necessarily becomes rigid and a construction as presented here cannot succeed. Applications to relaxation of integral functionals, the theory of semiconvex hulls, and approximation of weakly orientation-preserving maps by strictly orientation-preserving ones in Sobolev spaces are given.

math.AP↗

arXiv2014

From nonlinear to linearized elasticity via Gamma-convergence: the case of multiwell energies satisfying weak coercivity conditions

Linearized elasticity models are derived, via Gamma-convergence, from suitably rescaled nonlinear energies when the corresponding energy densities have a multiwell structure and satisfy a weak coercivity condition, in the sense that the typical quadratic bound from below is replaced by a weaker p bound, 1<p <2, away from the wells. This study is motivated by, and our results are applied to, energies arising in the modeling of nematic elastomers.

math.AP↗

arXiv2013

An investigation of non-planar austenite-martensite interfaces

Motivated by experimental observations on CuAlNi single crystals, we present a theoretical investigation of non-planar austenite-martensite interfaces. Our analysis is based on the nonlinear elasticity model for martensitic transformations and we show that, under suitable assumptions on the lattice parameters, non-planar interfaces are possible, in particular for transitions with cubic austenite.

math.AP↗