Multiscale analysis and numerical simulation for stability of the incompressible flow of a Maxwell fluid

How to predict the stability of a small-scale flow subject to perturbations is a significant multiscale problem. It is difficult to directly study the stability by the theoretical analysis for the incompressible flow of a Maxwell fluid because of its analytical complexity. Here, we develop the multiscale analysis method based on the mathematical homogenization theory in the stress–stream function formulation. This method is used to derive the homogenized equation which governs the transport of the large-scale perturbations. The linear stabilities of the large-scale perturbations are analyzed theoretically based on the linearized homogenized equation, while the effect of the nonlinear terms on the linear stability results is discussed numerically based on the nonlinear homogenized equation. The agreements between the multiscale predictions and the direct numerical simulations demonstrate the multiscale analysis method is effective and credible to predict stabilities of flows.     

On a multiscale analysis of an inverse problem of nonlinear transfer law identification in periodic microstructure

This paper is devoted to the multiscale analysis of a homogenization inverse problem of the heat exchange law identification, which is governed by parabolic equations with nonlinear transmission conditions in a periodic heterogeneous medium. The aim of this work is to transform this inverse problem with nonlinear transmission conditions into a new one governed by a less complex nonlinear parabolic equation, while preserving the same form and physical properties of the heat exchange law that it will be identified, based on periodic homogenization theory. For this, we reformulate first the encountered homogenization inverse problem to an optimal control one. Then, we study the well-posedness of the state problem using the Leray–Schauder topological degrees and we also check the existence of the solution for the obtained optimal control problem. Finally, using the periodic homogenization theory and priori estimates, with justified choise of test functions, we reduce our inverse problem to a less complex one in a homogeneous medium.     

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     

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.

     

Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients

We propose a multiscale finite element method for solving second order elliptic equations with rapidly oscillating coefficients. The main purpose is to design a numerical method which is capable of correctly capturing the large scale components of the solution on a coarse grid without accurately resolving all the small scale features in the solution. This is accomplished by incorporating the local microstructures of the differential operator into the finite element base functions. As a consequence, the base functions are adapted to the local properties of the differential operator. In this paper, we provide a detailed convergence analysis of our method under the assumption that the oscillating coefficient is of two scales and is periodic in the fast scale. While such a simplifying assumption is not required by our method, it allows us to use homogenization theory to obtain a useful asymptotic solution structure. The issue of boundary conditions for the base functions will be discussed. Our numerical experiments demonstrate convincingly that our multiscale method indeed converges to the correct solution, independently of the small scale in the homogenization limit. Application of our method to problems with continuous scales is also considered.

     

Periodic homogenization of nonlinear non-monotone parabolic operators with three time scales

Multiscale homogenization of nonlinear non-monotone degenerated parabolic operators is investigated. Under a periodicity assumption on the coefficients of the operators under consideration, we obtain by means of multiscale convergence method, an accurate homogenization result. It is also shown that in spite of the presence of several time scales the global homogenized problem is not a reiterated one.     

A mixed multiscale finite element method for elliptic problems with oscillating coefficients

The recently introduced multiscale finite element method for solving elliptic equations with oscillating coefficients is designed to capture the large-scale structure of the solutions without resolving all the fine-scale structures. Motivated by the numerical simulation of flow transport in highly heterogeneous porous media, we propose a mixed multiscale finite element method with an over-sampling technique for solving second order elliptic equations with rapidly oscillating coefficients. The multiscale finite element bases are constructed by locally solving Neumann boundary value problems. We provide a detailed convergence analysis of the method under the assumption that the oscillating coefficients are locally periodic. While such a simplifying assumption is not required by our method, it allows us to use homogenization theory to obtain the asymptotic structure of the solutions. Numerical experiments are carried out for flow transport in a porous medium with a random log-normal relative permeability to demonstrate the efficiency and accuracy of the proposed method.     

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).     

Generalized Besicovitch spaces and applications to deterministic homogenization

The purpose of the present work is to introduce a framework which enables us to study nonlinear homogenization problems. The starting point is the theory of algebras with mean value. Very often in physics, from very simple experimental data, one gets complicated structure phenomena. These phenomena are represented by functions which are permanent in mean, but complicated in detail. In addition the functions are subject to the verification of a functional equation which in general is nonlinear. The problem is therefore to give an interpretation of these phenomena using functions having the following qualitative properties: they are functions that represent a phenomenon on a large scale, and which vary irregularly, undergoing nonperiodic oscillations on a fine scale. In this work we study the qualitative properties of spaces of such functions, which we call generalized Besicovitch spaces, and we prove general compactness results related to these spaces. We then apply these results in order to study some new homogenization problems. One important achievement of this work is the resolution of the generalized weakly almost periodic homogenization problem for a nonlinear pseudo-monotone parabolic-type operator. We also give the answer to the question raised by Frid and Silva in their paper [35] [H. Frid, J. Silva, Homogenization of nonlinear pde’s in the Fourier-Stieltjes algebras, SIAM J. Math. Anal, 41 (4) (2009) 1589-1620] as regards whether there exist or do not exist ergodic algebras that are not subalgebras of the Fourier-Stieltjes algebra.     

Deterministic homogenization of nonlinear degenerate elliptic operators with nonstandard growth

We study the deterministic homogenization of nonlinear degenerate elliptic equations with nonstandard growth.One fundamental in this topic is to extend the classical compactness results of theΣ-convergence method to the Orlicz spaces.We also show that one can homogenize nonlinear Dirichlet problems in a general way by leaning on a simple abstract hypothesis contrary to what has been done in the determinstic homogenization theory.     

The heterogeneous multiscale finite element method for elliptic homogenization problems in perforated domains

In this contribution we analyze a generalization of the heterogeneous multiscale finite element method for elliptic homogenization problems in perforated domains. The method was originally introduced by E and Engquist (Commun Math Sci 1(1):87–132, 2003) for homogenization problems in fixed domains. It is based on a standard finite element approach on the macroscale, where the stiffness matrix is computed by solving local cell problems on the microscale. A-posteriori error estimates are derived in L ²(Ω) by reformulating the problem into a discrete two-scale formulation (see also, Ohlberger in Multiscale Model Simul 4(1):88–114, 2005) and using duality methods afterwards. Numerical experiments are given in order to numerically evaluate the efficiency of the error estimate.     

A posteriori error estimates in quantities of interest for the finite element heterogeneous multiscale method

We present an “a posteriori” error analysis in quantities of interest for elliptic homogenization problems discretized by the finite element heterogeneous multiscale method. The multiscale method is based on a macro‐to‐micro formulation, where the macroscopic physical problem is discretized in a macroscopic finite element space, and the missing macroscopic data are recovered on‐the‐fly using the solutions of corresponding microscopic problems. We propose a new framework that allows to follow the concept of the (single‐scale) dual‐weighted residual method at the macroscopic level in order to derive a posteriori error estimates in quantities of interests for multiscale problems. Local error indicators, derived in the macroscopic domain, can be used for adaptive goal‐oriented mesh refinement. These error indicators rely only on available macroscopic and microscopic solutions. We further provide a detailed analysis of the data approximation error, including the quadrature errors. Numerical experiments confirm the efficiency of the adaptive method and the effectivity of our error estimates in the quantities of interest. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Semilinear parabolic problems in thin domains with a highly oscillatory boundary

In this paper, we study the behavior of the solutions of nonlinear parabolic problems posed in a domain that degenerates into a line segment (thin domain) which has an oscillating boundary. We combine methods from linear homogenization theory for reticulated structures and from the theory on nonlinear dynamics of dissipative systems to obtain the limit problem for the elliptic and parabolic problems and analyze the convergence properties of the solutions and attractors of the evolutionary equations.     

矩阵形式二次修正Maxwell-Dirac系统的多尺度算法

带二次修正项的Dirac方程在拓扑绝缘体、石墨烯、超导等新材料电磁光特性分析中有着十分广泛的应用.本文工作的创新点有：一是首次提出了矩阵形式带有二次修正项的Dirac方程,它是比较一般的数学框架,涵盖了上述材料体系很多重要的物理模型,具体见附录A;二是针对上述材料体系的电磁响应问题,提出了有界区域Weyl规范下具有周期间断系数矩阵形式带二次修正项Maxwell-Dirac系统的多尺度渐近方法,结合Crank-Nicolson有限差分方法和自适应棱单元方法,发展了一类多尺度算法.数值试验结果验证了多尺度渐近方法的正确性和算法的有效性.     

MULTISCALE HOMOGENIZATION OF NONLINEAR HYPERBOLIC EQUATIONS WITH SEVERAL TIME SCALES

We study the multiscale homogenization of a nonlinear hyperbolic equation in a periodic setting.We obtain an accurate homogenization result.We also show that as the nonlinear term depends on the microscopic time variable,the global homogenized problem thus obtained is a system consisting of two hyperbolic equations.It is also shown that in spite of the presence of several time scales,the global homogenized problem is not a reiterated one.     

Fast homogenization algorithm based on asymptotic theory and multiscale schemes

A time-frequency interpretation of the classical asymptotic theory of homogenization for elliptic PDE with periodic coefficients is presented and the relations with known multilevel/multiscale numerical schemes are investigated. We formulate a new fast iterative algorithm for the approximation of homogenized solutions based on the combination of these two apparently different approaches. The asymptotic homogenization process is interpreted as a migration to infinity of the frequencies related to microscale contributions and the discovering of those related to the homogenized solution. At different scale/frequency of the periodic coefficients of the operator, band-pass filters select only the contributions of the homogenized solution which is then composed as the limit of an iterative procedure. This novel method can be interpreted in case of finite difference discretizations as a generalized nonstationary subdivision scheme and its convergence and stability are discussed. In particular, stable compositions of the homogenized solution are investigated in relation with the contracting behavior of specific operators generated by reduction processes and Schur's complements of suitable matrices produced by discretizations via wavelets and multiscale bases. AMS subject classification 35B27, 35J25, 65N55, 65M99, 65T60, 78M25, 78M30 Maria Morandi Cecchi: The support of the italian MIUR under project “Numerical Modeling for Scientific Computing and Advanced Applications” (COFIN 2003) is gratefully acknowledged. Massimo Fornasier: The author acknowledges the financial support provided through the Intra-European Individual Marie Curie Fellowship, project FTFDORF-FP6-501018, and the hospitality of NuHAG (Numerical Harmonic Analysis Group), Facutly of Mathematics, University of Vienna, Austria.     

Matrix completion for cost reduction in finite element simulations under hybrid uncertainties

In recent years, uncertainty appears in different aspects of physical simulations including probabilistic boundary, stochastic loading, and multiscale modeling. Stretching across engineering domains and applied mathematics, uncertainty quantification is a multi-disciplinary field which is an inseparable part of risk analysis. However, many real-world problems deal with large number of simulations or experiments. Considering the limited budget and time to perform all these efforts (specially for practitioners), an essential task is to reduce the computational cost in an uncertain environment.This paper proposes to use a matrix completion technique for reducing the overall computational cost of engineering systems when they are subjected to the simultaneous effects of aleatory and epistemic uncertainties with high dimensions. The proposed method is further improved using hidden information in the uncertain variables based on clustering techniques. Several parametric and Monte Carlo simulations were performed to demonstrate the accuracy of our method with different compression ratios. Experimental results show a decent overall performance of our technique for high-dimensional hybrid uncertain systems.     

On the Convergence of a Non-incremental Homogenization Method for Nonlinear Elastic Composite Materials

In order to simulate the nonlinear behaviour of elastomer composite materials, we use a homogenization technique which induces nonlinear problems. This paper presents a non-incremental solving method which allows the reduction of computational costs. In this paper the equivalence between the proposed solving method and a Newton-type method is proved, which allows us to prove the convergence under realistic assumptions. Numerical results on a composite illustrate the performances of this method.     

发展型H—半变分不等式解的存在性

本文研究非线性发展型H－半变分不等式,即具有非凸泛函的抛物型变分不等式,这类问题的研究起源于力学。利用Clarke广义梯度和（S＋）型多值映象的不动点理论,我们证明了这类问题解的存在性。并利用这一理论,研究了具间断项的非线性抛物型方程解的存在性。     

Multiscale modeling and analysis for some special additive noises driven stochastic partial differential equations

This paper is concerned with some special additive noises driven stochastic partial differential equations with multiscale parameters. Then, the constraint energy minimizing generalized multiscale finite element method with a novel multiscale spectral representation of the noise is constructed to solve the multiscale models. The corresponding convergence analysis and error estimates are derived, and the effects of noises on the accuracy of the multiscale computation are demonstrated. Some numerical examples are provided to validate our theoretic analysis, and numerical results show the highly efficient computational performance of our method, which is a beneficial attempt to deal with the noises in the complex multiscale stochastic physical system.