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)     

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)     

Deformation Driven Homogenization of Fracturing Solids

The paper discusses numerical formulations of the homogenization for solids with discrete crack development. We focus on multi–phase microstructures of heterogeneous materials, where fracture occurs in the form of debonding mechanisms as well as matrix cracking. The definition of overall properties critically depends on the developing discontinuities. To this end, we extend continuous formulations [1] to microstructures with discontinuities [2]. The basic underlying structure is a canonical variational formulation in the fully nonlinear range based on incremental energy minimization. We develop algorithms for numerical homogenization of fracturing solids in a deformation–driven context with non–trivial formulations of boundary conditions for (i) linear deformation and (ii) uniform tractions. The overall response of composite materials with fracturing microstructures are investigated. As a key result, we show the significance of the proposed non–trivial formulation of a traction–type boundary condition in the deformation–driven context. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Lax Pairs for the Deformed Kowalevski and Goryachev–Chaplygin Tops

We consider a quadratic deformation of the Kowalevski top. This deformation includes a new integrable case for the Kirchhoff equations recently found by one of the authors as a degeneration. A 5×5 matrix Lax pair for the deformed Kowalevski top is proposed. We also find similar deformations of the two-field Kowalevski gyrostat and the so(p,q) Kowalevski top. All our Lax pairs are deformations of the corresponding Lax representations found by Reyman and Semenov-Tian-Shansky. A similar deformation of the Goryachev–Chaplygin top and its 3×3 matrix Lax representation is also constructed.     

The product-product singular value decomposition of matrix triplets

A new decomposition of a matrix triplet (A, B, C) corresponding to the singular value decomposition of the matrix productABC is developed in this paper, which will be termed theProduct-Product Singular Value Decomposition (PPSVD). An orthogonal variant of the decomposition which is more suitable for the purpose of numerical computation is also proposed. Some geometric and algebraic issues of the PPSVD, such as the variational and geometric interpretations, and uniqueness properties are discussed. A numerical algorithm for stably computing the PPSVD is given based on the implicit Kogbetliantz technique. A numerical example is outlined to demonstrate the accuracy of the proposed algorithm.The work was partially supported by NSF grant DCR-8412314.     

A reduced theory for thin‐film micromagnetics

Micromagnetics is a nonlocal, nonconvex variational problem. Its minimizer represents the ground‐state magnetization pattern of a ferromagnetic body under a specified external field. This paper identifies a physically relevant thin‐film limit and shows that the limiting behavior is described by a certain “reduced” variational problem. Our main result is the Γ‐convergence of suitably scaled three‐dimensional micromagnetic problems to a two‐dimensional reduced problem; this implies, in particular, convergence of minimizers for any value of the external field. The reduced problem is degenerate but convex; as a result, it determines some (but not all) features of the ground‐state magnetization pattern in the associated thin‐film limit. © 2002 Wiley Periodicals, Inc.     

A globally and superlinearly convergent quasi-Newton method for general box constrained variational inequalities without smoothing approximation

A new quasi-Newton algorithm for the solution of general box constrained variational inequality problem (GVI(l, u, F, f)) is proposed in this paper. It is based on a reformulation of the variational inequality problem as a nonsmooth system of equations by using the median operator. Without smoothing approximation, the proposed quasi-Newton algorithm is directly applied to solve this class of nonsmooth equations. Under appropriate assumptions, it is proved that the algorithmic sequence globally and superlinearly converges to a solution of the equation reformulation and also of GVI(l, u, F, f). Numerical results show that our new algorithm works quite well.     

Multiscale Modeling of Nanocrystalline Materials: A Variational Approach

This paper presents a variational multi-scale constitutive model in the finite deformation regime capable of capturing the mechanical behavior of nanocrystalline (nc) fcc metals. The nc-material is modeled as a two-phase material consisting of a grain interior (GI) phase and a grain boundary (GB) phase. A rate-independent isotropic porous plasticity model is employed to describe the GB phase, whereas a crystal-plasticity model which accounts for the transition from partial dislocation to full dislocation mediated plasticity is employed for the GI phase. Assuming the rule of mixtures, the overall behavior of a given grain is obtained via volume averaging. The scale transition from a single grain to a polycrystal is achieved by Taylor-type homogenization. It is shown that the proposed model is able to capture the inverse Hall-Petch effect. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Canonical Moments and Random Spectral Measures

We study some connections between the random moment problem and random matrix theory. A uniform draw in a space of moments can be lifted into the spectral probability measure of the pair (A,e), where A is a random matrix from a classical ensemble, and e is a fixed unit vector. This random measure is a weighted sampling among the eigenvalues of A. We also study the large deviations properties of this random measure when the dimension of the matrix increases. The rate function for these large deviations involves the reversed Kullback information.     

Computational characterization of ferroelectric solids with a 3D Preisach model based on an orientation distribution function

Ferroelectric or ferromagnetic materials show an interaction between mechanical deformations and polarization or magnetization. A few multiferroic materials possess both ferroic properties and exhibit a magneto-electric (ME) coupling. These ME properties can be achieved in two-phase composites, which combine ferroelectric and ferromagnetic characteristics. To predict a realistic material behavior and a more precise ME coefficient, the application of suitable material models which describe the nonlinear hysteretic behavior is of particular importance. In the present contribution we focus on the characterization of a nonlinear ferroelectric material behavior, in terms of a 3D Preisach model based on an orientation distribution function. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Computing Eigenelements of Real Symmetric Matrices via Optimization

In certain circumstances, it is advantageous to use an optimization approach in order to solve the generalized eigenproblem, Ax = Bx, where A and B are real symmetric matrices and B is positive definite. In particular, this is the case when the matrices A and B are very large the computational cost, prohibitive, of solving, with high accuracy, systems of equations involving these matrices. Usually, the optimization approach involves optimizing the Rayleigh quotient.We first propose alternative objective functions to solve the (generalized) eigenproblem via (unconstrained) optimization, and we describe the variational properties of these functions.We then introduce some optimization algorithms (based on one of these formulations) designed to compute the largest eigenpair. According to preliminary numerical experiments, this work could lead the way to practical methods for computing the largest eigenpair of a (very) large symmetric matrix (pair).     

Lie symmetry of the Landau-Lifshitz-Gilbert equation and exact linearization in the Minkowski space

In this paper we present a linear representation of the Landau-Lifshitz-Gilbert equation for describing the magnetization of ferromagnetic materials. According to Lie P \Bbb M³⁺¹,P {\Bbb M}^{3+1}, of which the projective proper orthochronous Lorentz group PSO ^o(3,1) left acts. By the Lie symmetry a group preserving scheme is developed, which improves the computational accuracy and efficiency.     

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)     

Optimisation of Structures with Inelastic Deformations

In context of design optimisation, the treatment of inelastic, path-dependent materials is a topic of interest. As opposed to purely elastic materials, it is necessary to store and analyse the deformation history in order to appropriately describe inelastic material behaviour. For design optimisation of structures sensitivities of all quantities of influence have to be computed so as to use gradient based optimisation algorithms. Considering path-dependent materials the sensitivities of internal variables that represent the deformation history have to be additionally calculated. A numerical effective way of determining sensitivities is the variational sensitivity analysis. This approach is applied to develop a powerful and effective algorithm to compute sensitivities of the structural response with respect to their geometric design. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the generalized Euler-Frobenius polynomial

In this paper, the properties of the generalized Euler-Frobenius polynomial Π_n(·, q) are studied. It is proved that its zeroes are separated by the factor q and their asymptotic behavior, as q → ∞, is given. As a consequence, it is shown that least squares spline approximation on a biinfinite geometric mesh can be bounded independently of the (local) mesh ratio q and that the norm of the inverse of the corresponding order kB-spline Gram matrix decreases monotonically to 2k − 1 for large q, as q → ∞.     

Existence of travelling wave solutions for the heat equation in infinite cylinders with a nonlinear boundary condition

The existence of travelling wave solutions for the heat equation ∂_t u –Δu = 0 in an infinite cylinder subject to the nonlinear Neumann boundary condition (∂u /∂n) = f (u) is investigated. We show existence of nontrivial solutions for a large class of nonlinearities f. Additionally, the asymptotic behavior at ∞ is studied and regularity properties are established. We use a variational approach in exponentially weighted Sobolev spaces. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Structure of principal eigenvectors and genetic diversity

The main concern of this paper is long-term genotypic diversity. Genotypes are represented as finite sequences (s₁,s₂,…,s_n), where the entries {s_i} are drawn from a finite alphabet. The mutation matrix is given in terms of Hamming distances. It is proved that the long time behavior of solutions for a class of genotype evolution models is governed by the principal eigenvectors of the sum of the mutation and fitness matrices. It is proved that the components of principal eigenvectors are symmetric and monotonely decreasing in terms of Hamming distances whenever the fitness matrix has those properties.     

Preserving geometric properties of the exponential matrix by block Krylov subspace methods

Given a large square real matrix A and a rectangular tall matrix Q, many application problems require the approximation of the operation . Under certain hypotheses on A, the matrix preserves the orthogonality characteristics of Q; this property is particularly attractive when the associated application problem requires some geometric constraints to be satisfied. For small size problems numerical methods have been devised to approximate while maintaining the structure properties. On the other hand, no algorithm for large A has been derived with similar preservation properties. In this paper we show that an appropriate use of the block Lanczos method allows one to obtain a structure preserving approximation to when A is skew-symmetric or skew-symmetric and Hamiltonian. Moreover, for A Hamiltonian we derive a new variant of the block Lanczos method that again preserves the geometric properties of the exact scheme. Numerical results are reported to support our theoretical findings, with particular attention to the numerical solution of linear dynamical systems by means of structure preserving integrators. AMS subject classification (2000) 65F10, 65F30, 65D30     

The existence and uniqueness of solution and the convergence of a multi-step iterative algorithm for a system of variational inclusions with (<Emphasis Type="Italic">A</Emphasis>, <Emphasis Type="Italic">η</Emphasis>, <Emphasis Type="Italic">m</Emphasis>)-accretive operators

In this paper, we introduce and study a new system of variational inclusions with (A, η, m)-accretive operators which contains variational inequalities, variational inclusions, systems of variational inequalities and systems of variational inclusions in the literature as special cases. By using the resolvent technique for the (A, η, m)-accretive operators, we prove the existence and uniqueness of solution and the convergence of a new multi-step iterative algorithm for this system of variational inclusions in real q-uniformly smooth Banach spaces. The results in this paper unifies, extends and improves some known results in the literature.      

The equations of the thermodynamic model of magnetoelasticity of dielectric ferromagnetic bodies

We develop a thermodynamic approach to the mathematical modeling of magnetoelastic processes in dielectric ferromagnetic bodies that are subject to the action of force loading, heating, and an external electromagnetic field. The basis for the construction of the physical relations is the principle of local thermodynamic state. In the computations of the momentum balance equation of the magnetization process account is taken of its tensor character.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 36, 1992, pp. 30–34.