Aspects of a direct homogenization procedure for electro-mechanically coupled problems

The aim of this work is to discuss a micro–macro homogenization procedure for electro–mechanically coupled problems. In this context a two–scale homogenization ansatz for ferroelectric ceramics based on an FE²-approach is presented. The microscopic discretization of the heterogeneous structure of the polycrystalline material allows for the incorporation of microscopic effects, which are necessary to determine the corresponding overall macroscopic material response. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A two-scale model for the periodic homogenization of the wave equation

We present a new mathematical object designed to analyze the oscillations occurring on both microscopic and macroscopic scales in a wave equation with oscillating coefficients and data. Through a Bloch wave homogenization method, our study addresses typical problems of two-scale convergence in the interior of the domain, and sheds some light on the behavior near the boundary. A decoupled system of (systems of) transport equations is derived in each energy band, and the total energy field is approximated. We also recover previously known results in homogenization as a restricted part of our model.     

A coupled two-scale shell model for comb-like sandwich structures

In this work a coupled two-scale sandwich shell model is proposed, where 4-node quadrilaterals are employed both on the global and the local scale. The coupled global-local boundary value problem is derived by means of a variational formulation and ensuing linearization. A numerical simulation is carried out for linear elastic and elasto-plastic material behavior with small strains. The resulting coupled nonlinear boundary value problem is solved simultaneously in a Newton iteration with incremental load steps. Various types of sandwich models are investigated in the form of uni- and bidirectionally stiffened structures. For the unidirectionally stiffened beam, an analytical reference solution is present by means of classical beam theory. In addition, the numerical results of all coupled calculations are compared to full scale shell models, showing very good agreement while significantly reducing the size of occurring system matrices. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Weak turbulence plasma induced by two-scale homogenization

This paper is devoted to the homogenization of the quasilinear theory of the plasma turbulence described by the Vlasov–Poisson system. It is shown that the homogenization limit, in the sense of two-scale limit, of the distribution function satisfies the linear Vlasov–Poisson equations. Moreover, the limit distribution function can be decomposed into the mean and the fluctuation parts and the mean part (the equilibrium distribution function) is shown to be the solution of the nonlocal quasilinear velocity-space diffusion equation. We also investigate the Landau damping from the point of view of homogenization through the two-scale limit.     

A note on concentrations for integral two-scale problems

This work is devoted to the characterization of the asymptotic behavior, as \({\{\varepsilon_n\}}\) goes to zero, of a family of integral functionals of the form \({\int_{\Omega}f(x,\langle x/\varepsilon_n \rangle, \nabla u_{n}(x))\, dx}\) in terms of measures of oscillation and concentration associated to the sequence \({\{(\langle x/\varepsilon_n\rangle, \nabla u_{n}(x))\}}\).     

A posteriori error analysis of the heterogeneous multiscale method for homogenization problems

In this Note we derive a posteriori error estimates for a multiscale method, the so-called heterogeneous multiscale method, applied to elliptic homogenization problems. The multiscale method is based on a macro-to-micro formulation. The macroscopic method discretizes the physical problem in a macroscopic finite element space, while the microscopic method recovers the unknown macroscopic data on the fly during the macroscopic stiffness matrix assembly process. We propose a framework for the analysis allowing to take advantage of standard techniques for a posteriori error estimates at the macroscopic level and to derive residual-based indicators in the macroscopic domain for adaptive mesh refinement. To cite this article: A. Abdulle, A. Nonnenmacher, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Multiscale methods for elliptic homogenization problems

In this article we study two families of multiscale methods for numerically solving elliptic homogenization problems. The recently developed multiscale finite element method [Hou and Wu, J Comp Phys 134 (1997), 169–189] captures the effect of microscales on macroscales through modification of finite element basis functions. Here we reformulate this method that captures the same effect through modification of bilinear forms in the finite element formulation. This new formulation is a general approach that can handle a large variety of differential problems and numerical methods. It can be easily extended to nonlinear problems and mixed finite element methods, for example. The latter extension is carried out in this article. The recently introduced heterogeneous multiscale method [Engquist and Engquist, Comm Math Sci 1 (2003), 87–132] is designed for efficient numerical solution of problems with multiscales and multiphysics. In the second part of this article, we study this method in mixed form (we call it the mixed heterogeneous multiscale method). We present a detailed analysis for stability and convergence of this new method. Estimates are obtained for the error between the homogenized and numerical multiscale solutions. Strategies for retrieving the microstructural information from the numerical solution are provided and analyzed. Relationship between the multiscale finite element and heterogeneous multiscale methods is discussed. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

A posteriori error majorants of the modeling errors for elliptic homogenization problems

In this paper, we derive new two-sided a posteriori estimates of the modeling errors for linear elliptic boundary value problems with periodic coefficients solved by homogenization. Our approach is based on the concept of functional a posteriori error estimation. The estimates are obtained for the energy norm and use solely the global flux of the non-oscillatory solution of the homogenized model and solution of a boundary value problem on the cell of periodicity.     

A class of homogenization problems in the calculus of variations

We study a class of integral functionals for which the integrand f^e(x, u, ?u) is an oscillatory function of both x and u. Our method is based on the concept of Γ-convergenee. Technical difficulties arise because f^e(x, u, ?u) is not convex or equi-continuous in u with respect to e. Two somewhat different approaches, based respectively on abstract convergence theorems and the study of affine functions, are exploited together to overcome these technical difficulties. As an application, we give another proof of a homogenization result of P. L. Lions, G. Papanicolaou, and S. R. S. Varadhan for Hamilton-Jacobi equations.     

Substantiation of two-scale homogenization of the equations governing the longitudinal vibrations of a viscoelastoplastic Ishlinskii material

Initial-boundary value problems for the system of quasilinear operator-differential equations governing the longitudinal vibrations of a viscoelastoplastic Ishlinskii material with nonsmooth rapidly oscillating coefficients and initial data are investigated. The system involves the hysteresis Prandtl-Ishlinskii operator. Passage to the limit to initial-boundary value problems for the corresponding system of two-scale homogenized operator integro-differential equations is strictly substantiated globally in time without assuming that the data are small.     

On the homogenization of noncoercive variational problems

The paper deals with variational problems of the form $$\mathop {\inf }\limits_{u \in W^{1,p} (\Omega )} \int\limits_\Omega {a(\varepsilon ^{ - 1} x)(\left| {\nabla u} \right|^p + \left| {u - g} \right|^p )} dx,$$ where Ω is a bounded Lipschitzian domain in ? ^N , g∈L^p(Ω). The function a(x) is assumed to satisfy the following conditions:

1. a(x) is periodic and lower semicontinuous;
2. 0≤a(x)≤1 and the set {∈? ^N , a(x)>0} is connected in ? ^N Under these conditions, basic properties of homogenization (convergence of energies and generalized solutions) and properties of Г-convergence type are proved. Bibliography: 3 titles.

     

Analysis of the heterogeneous multiscale method for parabolic homogenization problems

The heterogeneous multiscale method (HMM) is applied to various parabolic problems with multiscale coefficients. These problems can be either linear or nonlinear. Optimal estimates are proved for the error between the HMM solution and the homogenized solution.

     

Analysis of the heterogeneous multiscale method for elliptic homogenization problems

A comprehensive analysis is presented for the heterogeneous multiscale method (HMM for short) applied to various elliptic homogenization problems. These problems can be either linear or nonlinear, with deterministic or random coefficients. In most cases considered, optimal estimates are proved for the error between the HMM solutions and the homogenized solutions. Strategies for retrieving the microstructural information from the HMM solutions are discussed and analyzed.

     

A new dominance procedure for combinatorial optimization problems

It is known that the effectiveness of the branch and bound algorithms for combinatorial optimization problems can be improved through dominance criteria which allow fathomings of large solution subsets. We describe a new dominance procedure which overcomes some of the drawbacks of the commonly used dominance criteria. An application to the Multiple Knapsack Problem and some computational results are also reported.     

A coupled NMM-SPH method for fluid-structure interaction problems

The main challenges in the numerical simulation of fluid–structure interaction (FSI) problems include the solid fracture, the free surface fluid flow, and the interactions between the solid and the fluid. Aiming to improve the treatment of these issues, a new coupled scheme is developed in this paper. For the solid structure, the Numerical Manifold Method (NMM) is adopted, in which the solid is allowed to change from continuum to discontinuum. The Smoothed Particle Hydrodynamics (SPH) method, which is suitable for free interface flow problem, is used to model the motion of fluids. A contact algorithm is then developed to handle the interaction between NMM elements and SPH particles. Three numerical examples are tested to validate the coupled NMM-SPH method, including the hydrostatic pressure test, dam-break simulation and crack propagation of a gravity dam under hydraulic pressure. Numerical modeling results indicate that the coupled NMM-SPH method can not only simulate the interaction of the solid structure and the fluid as in conventional methods, but also can predict the failure of the solid structure.     

Reiterated homogenization of monotone parabolic problems

Reiterated homogenization is studied for divergence structure parabolic problems of the form . It is shown that under standard assumptions on the function a(y ₁,y ₂,t,ξ) the sequence of solutions converges weakly in to the solution u of the homogenized problem .