Minimising movements for the motion of discrete screw dislocations along glide directions

In Alicandro et al. (J Mech Phys Solids 92:87–104, 2016) a simple discrete scheme for the motion of screw dislocations toward low energy configurations has been proposed. There, a formal limit of such a scheme, as the lattice spacing and the time step tend to zero, has been described. The limiting dynamics agrees with the maximal dissipation criterion introduced in Cermelli and Gurtin (Arch Ration Mech Anal 148, 1999) and predicts motion along the glide directions of the crystal. In this paper, we provide rigorous proofs of the results in [3], and in particular of the passage from the discrete to the continuous dynamics. The proofs are based on \(\Gamma \)-convergence techniques.     

A Variational Model for Dislocations at Semi-coherent Interfaces

We propose and analyze a simple variational model for dislocations at semi-coherent interfaces. The energy functional describes the competition between two terms: a surface energy induced by dislocations and a bulk elastic energy, spent to decrease the amount of dislocations needed to compensate the lattice misfit. We prove that, for minimizers, the former scales like the surface area of the interface, the latter like its diameter. The proposed continuum model is built on some explicit computations done in the framework of the semi-discrete theory of dislocations. Even if we deal with finite elasticity, linearized elasticity naturally emerges in our analysis since the far-field strain vanishes as the interface size increases.     

Dislocations in nanowire heterostructures: from discrete to continuum

We discuss an atomistic model for heterogeneous nanowires, allowing for dislocations at the interface. We study the limit as the atomic distance converges to zero, considering simultaneously a dimension reduction and the passage from discrete to continuum. Employing the notion of Gamma-convergence, we establish the minimal energies associated to defect-free configurations and configurations with dislocations at the interface, respectively. It turns out that dislocations are favoured if the thickness of the wire is sufficiently large. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Twinned martensite configurations arising as ground states of a two-well discrete Hamiltonian

In this paper we construct and analyze a two-well Hamiltonian on 2D atomic lattice considered with nonconvex interactions. Two wells of the Hamiltonian are given by two rank-one connected martensitic twins, respectively. Our combined analytical and numerical results show that the structure of ground states under appropriate boundary conditions is close to the macroscopically expected twinned configuration plus additional exponential boundary layers localized near the twinning interface. Besides, we proceed to continuum limit, show asymptotic piece-wise rigidity of minimizing sequences and derive the limiting form of their surface energy. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the Spectrum of 1D Quantum Ising Quasicrystal

We consider one-dimensional quantum Ising spin-1/2 chains with two-valued nearest neighbor couplings arranged in a quasi-periodic sequence, with uniform, transverse magnetic field. By employing the Jordan–Wigner transformation of the spin operators to spinless fermions, the energy spectrum can be computed exactly on a finite lattice. By employing the transfer matrix technique and investigating the dynamics of the corresponding trace map, we show that in the thermodynamic limit the energy spectrum is a Cantor set of zero Lebesgue measure. Moreover, we show that local Hausdorff dimension is continuous and non-constant over the spectrum. This forms a rigorous counterpart of numerous numerical studies.     

A variational arbitrary Lagrangian-Eulerian finite element formulation for standard dissipative solids at finite strains

A novel Arbitrary Lagrangian-Eulerian (ALE) finite element formulation for standard dissipative media at finite strains is presented. In contrast to previously published ALE approaches accounting for dissipative phenomena, the proposed scheme is fully variational. Consequently, no error estimates are necessary and thus, linearity of the problem and the corresponding Hilbert-space are not required. Hence, the resulting Variational Arbitrary Lagrangian-Eulerian (VALE) finite element method can be applied to highly nonlinear phenomena as well. In case of standard dissipative solids, so-called variational constitutive updates provide a variational principle. Based on these updates, the deformation mapping follows from minimizing an incrementally defined (pseudo) potential, i.e., energy minimization is the overriding criterion that governs every aspect of the system. Therefore, it is natural to allow the variational principle to drive mesh adaption as well. Thus, in the present paper, the discretizations of the deformed as well as the undeformed configuration are optimized jointly by minimizing the respective incremental energy of the considered mechanical system. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Mesh optimization for singular axisymmetric harmonic maps from the disc into the sphere

We describe in a mathematical setting the singular energy minimizing axisymmetric harmonic maps from the unit disc into the unit sphere; then, we use this as a test case to compute optimal meshes in presence of sharp boundary layers. For the well-posedness of the continuous minimizing problem, we introduce a lower semicontinuous extension of the energy with respect to weak convergence in BV, and we prove that the extended minimization problem has a unique singular solution. We then show how a moving finite element method, in which the mesh is an unknown of the discrete minimization problem obtained by finite element discretization, mimics this geometric point of view. Finally, we present numerical computations with boundary layers of zero thickness, and we give numerical evidence of the convergence of the method. This last aspect is proved in another paper. This work was supported by the Centre de Mathématiques et de Leurs Applications, Ecole Normale Supérieure de Cachan, 61 av. du Président Wilson, 94235 Cachan Cedex, France     

Gamma-convergence for Ginzburg-Landau functional with degenerate 3-well potential in one dimension

We consider the Ginzburg-Landau functional in one dimension, endowed with epsilon-dependent 3-well potential which degenerates as small parameter epsilon tends to zero. By using the approach in G. Alberti, S. Muller: A new approach to variational problems with multiple scales, Comm. Pure Appl. Math. 54 , 761-825 (2001), we obtain Gamma-convergence as small parameter epsilon tends to zero. We also recover the underlying geometric properties shared by all minimizing sequences. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A novel approach to the solution of boundary-layer problems

By replacing a differential equation boundary-layer problem by its discrete lattice equivalent we are able to treat the resulting equation as a regular perturbation problem. We obtain the solution on the lattice as a regular perturbation series in inverse powers of the lattice spacing. To obtain the answer to the continuum problem we extrapolate the solution to the lattice problem to zero lattice spacing. This extrapolation, which is a Padé-like procedure, yields good numerical results for a wide range of problems.     

A fully variational algorithmic formulation for wrinkling at finite strains

This contribution is concerned with an efficient novel algorithmic formulation for wrinkling at finite strains. In contrast to previously published numerical implementations, the advocated method is fully variational. More precisely, the parameters describing wrinkles or slacks, together with the unknown deformation mapping, are computed jointly by minimizing the potential energy of the considered mechanical system. Furthermore, the wrinkling criteria are naturally included within the presented variational framework. The presented approach allows to employ three–dimensional constitutive models directly, i.e., plane stress conditions characterizing membranes are variationally enforced by minimizing the potential energy with respect to the transversal strains. Since the proposed formulation for wrinkling in membranes is fully variational, it can be conveniently combined with other variational methods (based on energy minimization). As an example, a variationally consistent framework for finite strain plasticity theory is considered. More precisely, the minimization principle characterizing wrinkling in elastic membranes and that describing plasticity in inelastic solids are coupled leading to a novel variational approach for inelastic membranes. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Existence and regularity of minimizers of a functional for unsupervised multiphase segmentation

We consider a variational model for image segmentation proposed in Sandberg et al. (2010) [12]. In such a model the image domain is partitioned into a finite collection of subsets denoted as phases. The segmentation is unsupervised, i.e., the model finds automatically an optimal number of phases, which are not required to be connected subsets. Unsupervised segmentation is obtained by minimizing a functional of the Mumford–Shah type (Mumford and Shah, 1989 [1]), but modifying the geometric part of the Mumford–Shah energy with the introduction of a suitable scale term. The results of computer experiments discussed in [12] show that the resulting variational model has several properties which are relevant for applications. In this paper we investigate the theoretical properties of the model. We study the existence of minimizers of the corresponding functional, first looking for a weak solution in a class of phases constituted by sets of finite perimeter. Then we find various regularity properties of such minimizers, particularly we study the structure of triple junctions by determining their optimal angles.     

Solving discrete systems of nonlinear equations

We study the existence problem of a zero point of a function defined on a finite set of elements of the integer lattice Zⁿ of the n-dimensional Euclidean space Rⁿ. It is assumed that the set is integrally convex, which implies that the convex hull of the set can be subdivided in simplices such that every vertex is an element of Zⁿ and each simplex of the triangulation lies in an n-dimensional cube of size one. With respect to this triangulation we assume that the function satisfies some property that replaces continuity. Under this property and some boundary condition the function has a zero point. To prove this we use a simplicial algorithm that terminates with a zero point within a finite number of iterations. The standard technique of applying a fixed point theorem to a piecewise linear approximation cannot be applied, because the ‘continuity property’ is too weak to assure that a zero point of the piecewise linear approximation induces a zero point of the function itself. We apply the main existence result to prove the existence of a pure Cournot-Nash equilibrium in a Cournot oligopoly model. We further obtain a discrete analogue of the well-known Borsuk-Ulam theorem and a theorem for the existence of a solution for the discrete nonlinear complementarity problem.     

Finiteness of the discrete spectrum of the Schrödinger operator of three particles on a lattice

We consider a system of three quantum particles interacting by pairwise short-range attraction potentials on a three-dimensional lattice (one of the particles has an infinite mass). We prove that the number of bound states of the corresponding Schrödinger operator is finite in the case where the potentials satisfy certain conditions, the two two-particle sub-Hamiltonians with infinite mass have a resonance at zero, and zero is a regular point for the two-particle sub-Hamiltonian with finite mass.     

Dipole Interactions in Doubly Periodic Domains

We consider the interactions of finite dipoles in a doubly periodic domain. A finite dipole is a pair of equal and opposite strength point vortices separated by a finite distance. The dynamics of multiple finite dipoles in an unbounded inviscid fluid was first proposed by Tchieu, Kanso, and Newton in Tchieu et al. (Proc. R. Soc. Lond. Ser. A, Math. Phys. Eng. Sci. 468(2146):3006–3026, 2012) as a model that captures the “far-field” hydrodynamic interactions in fish schools. In this paper, we formulate the equations of motion governing the dynamics of finite dipoles in a doubly periodic domain. We show that a single dipole in a doubly periodic domain exhibits periodic and aperiodic behavior, in contrast to a single dipole in an unbounded domain. In the case of two dipoles in a doubly periodic domain, we identify a number of interesting trajectories including collision, collision avoidance, and passive synchronization of the dipoles. We then examine two types of dipole lattices: rectangular and diamond. We verify that these lattices are in a state of relative equilibrium and show that the rectangular lattice is unstable, while the diamond lattice is linearly stable for a range of perturbations. We conclude by commenting on the insight these models provide in the context of fish schooling.     

Variational methods for a singular SPDE yielding the universality of the magnetization ripple

The magnetization ripple is a microstructure formed in thin ferromagnetic films. It can be described by minimizers of a nonconvex energy functional leading to a nonlocal and nonlinear elliptic SPDE in two dimensions driven by white noise, which is singular. We address the universal character of the magnetization ripple using variational methods based on Γ-convergence. Due to the infinite energy of the system, the (random) energy functional has to be renormalized. Using the topology of Γ-convergence, we give a sense to the law of the renormalized functional that is independent of the way white noise is approximated. More precisely, this universality holds in the class of (not necessarily Gaussian) approximations to white noise satisfying the spectral gap inequality, which allows us to obtain sharp stochastic estimates. As a corollary, we obtain the existence of minimizers with optimal regularity.     

An evolution equation based approach to topology optimization

The objective of topology optimization is to find a mechanical structure with maximum stiffness and minimal amount of used material for given boundary conditions [2]. There are different approaches. Either the structure mass is held constant and the structure stiffness is increased or the amount of used material is constantly reduced while specific conditions are fulfilled. In contrast, we focus on the growth of a optimal structure from a void model space and solve this problem by introducing a variational problem considering the spatial distribution of structure mass (or density field) as variable [3]. By minimizing the Gibbs free energy according to Hamilton's principle in dynamics for dissipative processes, we are able to find an evolution equation for the internal variable describing the density field. Hence, our approach belongs to the growth strategies used for topology optimization. We introduce a Lagrange multiplier to control the total mass within the model space [1]. Thus, the numerical solution can be provided in a single finite element environment as known from material modeling. A regularization with a discontinuous Galerkin approach for the density field enables us to suppress the well-known checkerboarding phenomena while evaluating the evolution equation within each finite element separately [4]. Therefore, the density field is no additional field unknown but a Gauß-point quantity and the calculation effort is strongly reduced. Finally, we present solutions of optimized structures for different boundary problems. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Characterization of energy minimizers in micromagnetics

We present a new characterization of minimizing sequences and possible minimizers (all called the minimizing magnetizations) for a nonlocal micromagnetic-like energy (without the exchange energy). Our method is to replace the nonlocal energy functional and its relaxation with certain local integral functionals on divergence-free fields obtained by a two-step minimization of some auxiliary augmented functionals. Through this procedure, the minimization problem becomes equivalent to the minimization of a new local variational functional, called the dual variational functional, which has a unique minimizer. We then precisely characterize the minimizing magnetizations of original nonlocal functionals in terms of the unique minimizer of the dual variational functional. Finally, we give some remarks and ideas on solving the dual minimization problem.     

Regular congruence-preserving extensions of lattices

In this paper, we prove that every lattice L has a congruence-preserving extension into a regular lattice , moreover, every compact congruence of is principal. We construct by iterating a construction of the first author and F. Wehrung and taking direct limits.? We also discuss the case of a finite lattice L, in which case can be chosen to be finite, and of a lattice L with zero, in which case can be chosen to have zero and the extension can be chosen to preserve zero. Received September 10, 1999; accepted in final form October 16, 2000.     

Modeling of coupled deformation-induced twinning and dislocation slip using incremental energy minimization

The complex interplay between dislocations and deformation-induced twinning leads to a relatively poor formability of magnesium at room temperature. For understanding the complicated behavior of this metal, a novel model is presented. It is based on a variational principle. Within this principle based on energy minimization, dislocation slip is modeled by crystal plasticity theory, while the phase decomposition associated with twinning is considered by sequential laminates. The proposed model captures the transformation of the crystal lattice due to twinning in a continuous fashion by simultaneously taking dislocation slip within both, possibly co-existent, phases into account. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Solutions to five problems on tensor products of lattices and related matters

The notion of a capped tensor product, introduced by G. Grätzer and the author, provides a convenient framework for the study of tensor products of lattices that makes it possible to extend many results from the finite case to the infinite case. In this paper, we answer several open questions about tensor products of lattices. Among the results that we obtain are the following:¶¶Theorem 2. Let A be a lattice with zero. If A ?B A \oplus B is a lattice for every lattice L with zero, then A is locally finite and A ?B A \oplus B is a capped tensor product for every lattice L with zero.¶¶Theorem 5. There exists an infinite, three-generated, 2-modular lattice K with zero such that K ?K K \oplus K is a capped tensor product.¶¶Here, 2-modularity is a weaker identity than modularity, introduced earlier by G. Grätzer and the author.