A Geometrically Exact Incremental Variational Formulation for Phase Field Models in Micromagnetics

Magnetic materials have been finding increasingly wide areas of application. We focus here on the continuum modeling of such materials and present an incremental variational principle for a dissipative micro-magneto-elastic model. It describes the quasi-static evolution of both magnetically as well as mechanically driven magnetic domains, which also incorporates the surrounding free space. Furthermore, the algorithmic preservation of the geometrical nature of the variables is an important challenge from the numerical perspective and to this end we present a novel FE discretization whereby the geometric property of the magnetization director is pointwise exactly preserved by nonlinear rotational updates at the nodes. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Variational Homogenization of Micro-Electro-Elasticity

Understanding of micromechanical mechanisms in functional materials with electro-mechanical coupling is a highly demanding area of simulation technology and increasing interest has been shown in the last decades. Smart materials are characterized by microstructural properties, which can be changed by external stress and electric field stimuli, and hence find use as the active components in sensors and actuators. In this context, a key challenge is to combine models for microscopic electric domain evolution with variational principles of homogenization. We outline a variational-based micro-electro-elastic model for the micro-structural evolution of electric domains in ferroelectric ceramics. The micro-to-macro transition is performed on the basis of variational principles, extending purely mechanical formulations to coupled electro-mechanics. We focus on an electro-mechanical Boltzmann continuum on the macro-scale with mechanical displacement and electric potential as primary variables. The material model on the micro-scale is described by a gradient-extended continuum formulation taking into account the polarization vector field and its gradient, see Landis [1] and Schrade et al. [2] for conceptually similar approaches. A crucial aspect of the proposed homogenization analysis is the derivation of appropriate boundary conditions on the surface of the representative volume element. In this work we derive stiff Dirichlet-type, soft Neumann-type, and periodic boundary constraints starting from a generalized Hill-Mandel macrohomogeneity condition. Furthermore, we propose techniques to incorporate these boundary conditions in the variational principles of homogenization by means of Lagrange multiplier methods. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A variational theory for monotone vector fields

Monotone vector fields were introduced almost 40 years ago as nonlinear extensions of positive definite linear operators, but also as natural extensions of gradients of convex potentials. These vector fields are not always derived from potentials in the classical sense, and as such they are not always amenable to the standard methods of the calculus of variations. We describe here how the selfdual variational calculus, developed recently by the author, provides a variational approach to PDEs and evolution equations driven by maximal monotone operators. To any such vector field T on a reflexive Banach space X, one can associate a convex selfdual Lagrangian L _T on the phase space X × X ^* that can be seen as a “potential” for T, in the sense that the problem of inverting T reduces to minimizing a convex energy functional derived from L _T . This variational approach to maximal monotone operators allows their theory to be analyzed with the full range of methods—computational or not—that are available for variational settings. Standard convex analysis (on phase space) can then be used to establish many old and new results concerned with the identification, superposition, and resolution of such vector fields. Dedicated to Felix Browder on his 80th birthday     

A Variational Homogenization Approach to Electromechanical Hystereses Caused by Domain Wall Movement

In recent years, increasing interest in so-called smart materials such as ferroelectric polymers and ceramics has been shown. Those materials are used in various actuators, sensors, and also in medical devices. In this paper, we outline a micro-macro approach to the modeling of macroscopic hystereses which directly takes into account the microstructural evolution of electrically poled domains. To this end, an incremental variational formulation for a gradient-type phase field model is developed and exploited for the simulation of electromechanically coupled problems. The formulation determines the hysteretic response of the material in terms of an energy-enthalpy and a dissipation function which both depend on the microscopic remanent polarization treated as an order parameter. The gradient-type balance law for the phase field can be considered as a generalization of Biot's equation for standard dissipative materials and may be related to the classical Ginzburg-Landau equation. Furthermore, the variational formulation serves as natural starting point for a compact and symmetric finite element implementation of the coupled micromechanical problem covering the displacement, the electric potential, and the microscopic polarization vector. For this three-field scenario we develop a variational-based homogenization method which determines the overall macroscopic hysteretic properties of a polycrystalline aggregate. The proposed computational method can be used as a numerical laboratory for the improvement of microstructural properties. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Estimates of the deviations from the exact solutions for variational inequalities describing the stationary flow of certain viscous incompressible fluids

This paper is concerned with computable and guaranteed upper bounds of the difference between exact solutions of variational inequalities arising in the theory of viscous fluids and arbitrary approximations in the corresponding energy space. Such estimates (also called error majorants of functional type) have been derived for the considered class of nonlinear boundary‐value problems in (Math. Meth. Appl. Sci. 2006; 29:2225–2244) with the help of variational methods based on duality theory from convex analysis. In the present paper, it is shown that error majorants can be derived in a different way by certain transformations of the variational inequalities that define generalized solutions. The error bounds derived by this techniques for the velocity function differ from those obtained by the variational method. These estimates involve only global constants coming from Korn‐ and Friedrichs‐type inequalities, which are not difficult to evaluate in case of Dirichlet boundary conditions. For the case of mixed boundary conditions, we also derive another form of the estimate that contains only one constant coming from the following assertion: the L² norm of a vector‐valued function from H¹(Ω) in the factor space generated by the equivalence with respect to rigid motions is bounded by the L² norm of the symmetric part of the gradient tensor. As for some ‘simple’ domains such as squares or cubes, the constants in this inequality can be found analytically (or numerically), we obtain a unified form of an error majorant for any domain that admits a decomposition into such subdomains. Copyright © 2009 John Wiley & Sons, Ltd.     

A Dynamical System Associated with Newton's Method for Parametric Approximations of Convex Minimization Problems

We study the existence and asymptotic convergence when t→+∞ for the trajectories generated by where is a parametric family of convex functions which approximates a given convex function f we want to minimize, and ε(t) is a parametrization such that ε(t)→ 0 when t→+∞ . This method is obtained from the following variational characterization of Newton's method: where H is a real Hilbert space. We find conditions on the approximating family and the parametrization to ensure the norm convergence of the solution trajectories u(t) toward a particular minimizer of f . The asymptotic estimates obtained allow us to study the rate of convergence as well. The results are illustrated through some applications to barrier and penalty methods for linear programming, and to viscosity methods for an abstract noncoercive variational problem. Comparisons with the steepest descent method are also provided. Accepted 5 December 1996     

A Galerkin Approach for Solving Matrix Equations with Hierarchical Matrices

We consider a new approach for computing solutions of certain matrix equations, for example AXA = C, AX + XB = C or AX = I. This approach is based on a variational formulation in the matrix space, employing the Frobenius inner product. Using the space of ℋ²-matrices as trial space leads to a sparse linear system that can be solved by iterative methods to compute an approximate solution of the underlying matrix equation. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On Configurational-Force-Driven Crack Propagation and Adaptive Mesh Refinement in Finite Inelasticity

We introduce a consistent variational framework for inelasticity at finite strains, yielding dual balances in physical and material space as the Euler equations. The formulation is employed for the simultaneous usage of configurational forces as both driving forces for crack propagation as well as h-adaptive mesh refinement. The theoretical basis builds upon a global balance of internal and external power, where the mechanical response is exclusively governed by two scalar functions, the free energy function and a dissipation potential. The resulting variational structure is exploited in the context of fracture mechanics and yields evolution equations for internal variables. In the discrete setting, we present a geometry model fully separated from the finite element mesh structure that represents structural changes of the material configuration due to crack propagation. Advanced meshing algorithms provide an optimal discretization at the crack tip. Local and global criteria are obtained via error estimators based on configurational forces being interpreted as indicators of an energetic misfit due to an insufficient discretization. The numerical handling is decomposed into a staggered algorithm scheme for the dual set of equilibrium equations in material and physical space and efficient mesh generation tools. Exemplary numerical examples are considered to illustrate the method and to underline the effects of inelastic material behaviour in the presented context. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On Fictitious Domain Formulations for Maxwell''s Equations

We consider fictitious domain-Lagrange multiplier formulations for variational problems in the space H(curl: Ω{\bf)} derived from Maxwell's equations. Boundary conditions and the divergence constraint are imposed weakly by using Lagrange multipliers. Both the time dependent and time harmonic formulations of the Maxwell's equations are considered, and we derive well-posed formulations for both cases. The variational problem that arises can be discretized by functions that do not satisfy an a-priori divergence constraint.     

A Large-Deformation Phase-Field Approach for the Modeling of Magneto-Sensitive Elastomers

Magneto-sensitive materials show magneto-mechanical coupled response and are thus of increasing interest in the recent age of smart functional materials. Ferromagnetic particles suspended in an elastomeric matrix show realignment under the influence of an external applied field, in turn causing large deformations of the substrate material. The magneto-mechanical coupling in this case is governed by the magnetic properties of the inclusion and the mechancial properties of the matrix. The magnetic phenomenon in ferromagnetic materials is governed by the formation and evolution of domains on the micro scale. A better understanding of the behavior of these particles under the influence of an external applied field is required to accurately predict the behavior of such materials. In this context it is of particular importance to model the macro scopic magneto-mechanically coupled behavior based on the micro-magnetic domain evolution. The key aspect of this work is to develop a large-deformation micro-magnetic model that can accurately capture the microscopic response of such materials. Rigorous exploitation of appropriate rate-type variational principles and consequent incremental variational principles directly give us field equations including the time evolution equation of the magnetization, which acts as the order parameter in our formulation. The theory presented here is the continuation of the work presented in [1, 7] for small deformations. A summary of magneto-mechanical theories spanning over multiple scales has been presented in [4]. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Asymptotics of negative eigenvalues of the Dirichlet problem with the density changing sign

In a three-dimensional anisotropic elastic space with either a bounded foreign inclusion or a void, we derive asymptotic formulas for the increment of the polarization tensor of a defect caused by a smooth variation of the defect boundary. The formulas involve weighted integrals of jumps of the surface enthalpy evaluated for solutions to the problem about deformation of an unperturbed composite space by constant stress at infinity. The study of the positiveness/negativeness of the polarization matrix increment leads to inferences with a clear physical interpretation, in particular, for elastic solids admitting phase transitions. For homogeneous ellipsoid shaped inclusions we derive a relation between the polarization tensor and the Eshelby tensor and obtain miscellaneous consequences of this relation as well. In particular, we introduce the notion of the link tensor which is symmetric and positive definite for any elastic properties of homogeneous materials of the composite space. Bibliography: 60 titles. Illustrations: 5 figures. Dedicated to Nina Nikolaevna Uraltseva Translated from Problemy Matematicheskogo Analiza, 41, May 2009, pp. 3–36.     

A Route to Routh—The Parametric Problem

There is a well known principle in classical mechanic stating that a variational problem independent of a space variable w (so called cyclic variable), but dependent on the velocity w′ can be expressed without both w and w′. This is the Routh reduction principle. We develop a geometrical approach to the problem and deal with general first order variational integrals admitting a Lie symmetry group of point transformations. In the classical setting the Routh reduction is applied only to the symmetries that preserve the independent variable. In this article we remove such restriction.     

A 1D constitutive law for hysteresis effects in piezoceramics

This paper is concerned with a macroscopic constitutive law for domain switching effects, which occur in piezoelectric ceramics. The thermodynamical framework of the law is based on two scalar valued functions: the Helmholtz free energy and a switching surface. In common usage, the remanent polarization and the remanent strain are employed as internal variables. The novel aspect of the present work is to introduce an irreversible electric field, which serves besides the irreversible strain as internal variable. The irreversible electric field has only theoretical meaning, but leads to advantages within the finite element implementation, where displacement and the electric potential are the nodal degrees of freedoms. A common assumption is a one-to-one relation between the irreversible strain and the polarization. This simplification is not employed in the present paper. To accomplish enough space for the polarization, resulting from an applied electric field, the irreversible strains are additively split and a special hardening function is introduced. This balances the amount of space and the domain switching due to polarization. The constitutive model reproduces the ferroelastic and the ferroelectric hysteresis as well as the butterfly hysteresis for piezoelectric ceramics and it accounts for the mechanical depolarization effect. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Commutative free loop space models at large primes

Let p be a prime number and a natural number. If E is a r-connected finite CW-complex of dimension at most pr, then E is an example of a p -Anick space. For p > 2 we construct a commutative cochain algebra over that is an -model of the free loop space on a p-Anick space, i.e., its cohomology algebra is isomorphic to the mod p cohomology of the free loop space. For p-Anick spaces that are p-formal, such as spheres and projective spaces, we define an even simpler commutative free loop space model that applies for all primes p. We then use the simplified model to compute the cohomology algebras of a number of free loop spaces explicitly. Received: 23 June 1999; in final form: 8 September 2000 // Published online: 7 April 2003     

Computational Homogenization in Micro-Magneto-Elasticity

The overall macroscopic response of magneto-mechanically coupled materials stems from complex magnetization evolution and corresponding domain wall motion occurring on a lower length scale. In order to account for such effects we propose a computational homogenization approach that incorporates a ferromagnetic phase-field formulation into a macroscopic Boltzmann continuum. This scale-bridging is obtained by rigorous definition of rate-type and incremental variational principles. An extended version of the classical Hill-Mandel macro-homogeneity condition is obtained as a consequence. In order to satisfy the unity constraint of the magnetization on the micro-scale, an efficient operator-split method is proposed. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Algorithm for solving a new class of general mixed variational inequalities in Banach spaces

In this paper, a new concept of η-proximal mapping for a proper subdifferentiable functional (which may not be convex) on a Banach space is introduced. An existence and Lipschitz continuity of the η-proximal mapping are proved. By using properties of the η-proximal mapping, a new class of general mixed variational inequalities is introduced and studied in Banach spaces. An existence theorem of solutions is established and a new iterative algorithm for solving the general mixed variational inequality is suggested. A convergence criteria of the iterative sequence generated by the new algorithm is also given.     

Phase coexistence in Ising, Potts and percolation models

We study phase coexistence (separation) phenomena in Ising, Potts and random cluster models in dimensions d3 below the critical temperature. The simultaneous occurrence of several phases is typical for systems with appropriately arranged (mixed) boundary conditions or for systems satisfying certain physically natural constraints (canonical ensembles). The various phases emerging in these models define a partition, called the empirical phase partition, of the space. Our main results are large deviations principles for (the shape of) the empirical phase partition. More specifically, we establish a general large deviation principle for the partition induced by large (macroscopic) clusters in the Fortuin–Kasteleyn model and transfer it to the Ising–Potts model where we obtain a large deviation principle for the empirical phase partition induced by the various phases. The rate function turns out to be the total surface free energy (associated with the surface tension of the model and with boundary conditions) which can be naturally assigned to each reasonable partition. These LDP-s imply a weak law of large numbers: asymptotically, the law of the phase partition is determined by an appropriate variational problem. More precisely, the empirical phase partition will be close to some partition which is compatible with the constraints imposed on the system and which minimizes the total surface free energy. A general compactness argument guarantees the existence of at least one such minimizing partition. Our results are valid for temperatures T below a limit of slab-thresholds conjectured to agree with the critical point T_c. Moreover, T should be such that there exists only one translation invariant infinite volume state in the corresponding Fortuin–Kasteleyn model; a property which can fail for at most countably many values and which is conjectured to be true for every T≠T_c.     

A mixed finite element formulation for piezoelectric shells

A finite element formulation to analyze piezoelectric shell problems is presented. A reference surface of the shell is modelled with a four node element. Each node possesses six mechanical degrees of freedom, three displacements and three rotations, and one electric degree of freedom, which is the difference of the electric potential in thickness direction. The formulation is based on the mixed field variational principle of Hu-Washizu. The independent fields are displacements u , electric potential φ, strains E , electric field E , stresses S and dielectric displacements D . The mixed formulation allows an interpolation of the strains and the electric field in thickness direction. Accordingly a three-dimensional material law is incorporated in the variational formulation. It is remarked that no simplification regarding the constitutive law is assumed. The formulation allows the consideration of arbitrary constitutive relations. The normal zero stress condition and the normal zero dielectric displacement condition are enforced by the independent stress and dielectric displacement fields. They are defined as zero in thickness direction. The present shell element fulfills the important patch tests: the in-plane, bending and shear test. Some numerical examples demonstrate the applicability of the present piezoelectric shell element. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Mixed projection methods for systems of variational inequalities

Let H be a real Hilbert space. Let be bounded and continuous mappings where D(F) and D(K) are closed convex subsets of H. We introduce and consider the following system of variational inequalities: find [u ^*,v ^*]∈D(F) × D(K) such that This system of variational inequalities is closely related to a pseudomonotone variational inequality. The well-known projection method is extended to develop a mixed projection method for solving this system of variational inequalities. No invertibility assumption is imposed on F and K. The operators K and F also need not be defined on compact subsets of H.