Heterogeneous Multiscale Finite Element Research Articles

Finite element modeling of a multi-physics poro-elastic problem in multiscale media

We present an iterative algorithm for mathematical modeling of an elastic deformation process in a fluid-saturated fractured-porous medium. A three-dimensional multi-physics problem describes the coupled isothermal processes of the solid elastic deformation and slightly compressible fluid flow under external pressure. Mathematical models of these processes are connected via interface conditions for the pressure and density fields on the surface of a fractured-porous medium. For solving the multi-physics problem, a special multiscale procedure was developed. We use a heterogeneous multiscale finite element discretization on coarse polyhedral grids for the solid elastic deformation problem. Multiscale shape functions are constructed using special interface conditions for a hydrodynamic pressure on the surface of pores. We apply a discontinuous Galerkin method and a stabilized finite element discretization on fine tetrahedral grids for solving the hydrodynamics problem in fluid-saturated pores. In this case, we can realize an effective parallel procedure for solving the multi-physics problem. In each pore, hydrodynamics problems can be solved in parallel and independently. Verifications of the computational schemes are presented. We consider three-dimensional media with a different volume concentration of cracks and pores. Computational modeling results are presented. A time of solving the multi-physics problem using fine and coarse grids is shown.

The heterogeneous multiscale finite element method for the homogenization of linear elastic solids and a comparison with the FE[formula omitted] method

The Heterogeneous Multiscale Finite Element Method (FE-HMM) is a two-scale FEM based on asymptotic homogenization for solving multiscale partial differential equations. It was introduced in [W. E, B. Engquist, Commun. Math. Sci. 1 (2003) 87–132]. The objective of the present work is an FE-HMM formulation for the homogenization of linear elastic solids in a geometrical linear frame, and doing so, of a vector-valued field problem. A key ingredient of FE-HMM is that macrostiffness is estimated by stiffness sampling on heterogeneous microdomains in terms of a modified quadrature formula, which implies an equivalence of energy densities of the microscale with the macroscale. Beyond this coincidence with the Hill–Mandel condition, which is the cornerstone of the FE2 method, we elaborate a conceptual comparison with the latter method. After developing an algorithmic framework we (i) assess the existing a priori convergence estimates for the micro- and macro-errors in various norms, (ii) verify optimal strategies in uniform micro–macromesh refinements based on the estimates, (iii) analyze superconvergence properties of FE-HMM, and (iv) compare FE-HMM with FE2 by numerical results.

Computation of effective resistivity in materials with microinclusions by a heterogeneous multiscale finite element method

In this paper, we propose a numerical method to obtain an effective electrical resistivity of heterogeneous media under the influence of a direct current. The heterogeneous multiscale finite element method is used to solve the direct problem of simulation of an electrostatic field. The computational experiments using the developed software complex showed that even the small inclusion concentrations define the effective resistivity of the media. In addition, the change in the localization, orientation, and geometrical shape of inclusions also leads to a significant change of the effective properties of the media.

The heterogeneous multiscale finite element method FE‐HMM for the homogenization of linear elastic solids

AbstractThe objective of the present paper is a formulation of the Heterogeneous Multiscale Finite Element Method (FE‐HMM) for the homogenization of linear elastic solids in a geometrical linear frame, and doing so, for the first time, of a vector‐valued field problem. The macro stiffness is estimated by stiffness sampling on heterogeneous microdomains in terms of a modified quadrature formula, which implies an equivalence of energy densities of the microscale with the macroscale. Existing a‐priori estimates are assessed and used for optimal micro‐macro refinement strategies in the H1‐norm and the L2‐norm. (© 2016 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A-posteriori error estimate for a heterogeneous multiscale approximation of advection-diffusion problems with large expected drift

In this contribution we address a-posteriori error estimation in $L^\infty(L^2)$ for a heterogeneous multiscale finite element approximation of time-dependent advection-diffusion problems with rapidly oscillating coefficient functions and with a large expected drift. Based on the error estimate, we derive an algorithm for an adaptive mesh refinement. The estimate and the algorithm are validated in numerical experiments, showing applicability and good results even for heterogeneous microstructures.

Duality-based adaptivity in finite element discretization of heterogeneous multiscale problems

AbstractThis paper introduces an framework for adaptivity for a class of heterogeneous multiscale finite element methods for elliptic problems, which is suitable for

Corrector Analysis of a Heterogeneous Multi-scale Scheme for Elliptic Equations with Random Potential

This paper analyzes the random fluctuations obtained by a heterogeneous multi-scale first-order finite element method applied to solve elliptic equations with a random potential. We show that the random fluctuations of such solutions are correctly estimated by the heterogeneous multi-scale algorithm when appropriate fine-scale problems are solved on subsets that cover the whole computational domain. However, when the fine-scale problems are solved over patches that do not cover the entire domain, the random fluctuations may or may not be estimated accurately. In the case of random potentials with short-range interactions, the variance of the random fluctuations is amplified as the inverse of the fraction of the medium covered by the patches. In the case of random potentials with long-range interactions, however, such an amplification does not occur and random fluctuations are correctly captured independent of the (macroscopic) size of the patches. These results are consistent with those obtained by the authors for more general equations in the one-dimensional setting and provide indications on the loss in accuracy that results from using coarser, and hence less computationally intensive, algorithms.

An Efficient High Order Heterogeneous Multiscale Method for Elliptic Problems

We propose an efficient heterogeneous multiscale finite element method based on a local least-squares reconstruction of the effective matrix using the data retrieved from the solution of cell problems posed on the vertices of the triangulation. The method achieves high order accuracy for high order macroscopic solver with essentially the same cost as the linear macroscopic solver. Optimal error bounds are proved for the elliptic problem. Numerical results demonstrate that the new method significantly reduces the cost without loss of accuracy.

Convergence of the Heterogeneous Multiscale Finite Element Method for Elliptic Problems with Nonsmooth Microstructures

We propose a condition under which the heterogeneous multiscale finite element method converges for elliptic problems with nonsmooth coefficients, and we obtain the optimal convergence rate for elliptic problems with nonsymmetric periodic coefficients that allow for nonsmooth microstructures.

Heterogeneous multiscale finite element method with novel numerical integration schemes

In this paper we introduce two novel numerical integration schemes, within the framework of the heterogeneous multiscale method (HMM) when finite element method is used as the macroscopic solver, to resolve the elliptic problem with multiscale coefficient. For non-self-adjoint elliptic problems, optimal convergence rate is proved for the proposed methods, which naturally yields a new strategy for refining the macro-micro meshes and a criterion for determining the size of the microcell. Numerical results following this strategy show that the new methods significantly reduce the computational cost without loss of accuracy.

The heterogeneous multiscale finite element method for advection-diffusion problems with rapidly oscillating coefficients and large expected drift

This contribution is concerned with the formulation of a heterogeneous multiscale finite elements method (HMM) for solving linear advection-diffusion problems with rapidly oscillating coefficient functions and a large expected drift. We show that, in the case of periodic coefficient functions, this approach is equivalent to a discretization of the two-scale homogenized equation by means of a Discontinuous Galerkin Time Stepping Method with quadrature. We then derive an optimal order a-priori error estimate for this version of the HMM and finally provide numerical experiments to validate the method.

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.

Multiscale finite element analysis of bulk nano-particle fabrication by a mechanical method

A heterogeneous multiscale finite element approach is used to study the bulk mechanical alloying (BMA) process for nano-particles. Mechanical characteristics such as the load versus displacement relations and the heterogeneous distribution of maximum principle stress are calculated from the global analysis to support the subsequent local, small-scale, analysis. A simple local quasi-microscopic analysis of a six-ball contacting problem is conducted to determine the stress distribution in a particular particle. The Griffith fracture criterion is adopted to determine whether a particle will be forced to break into smaller fragments. To simulate the mixing process in the extruding stroke, particles in each element are randomly redistributed according to the uniform random distribution rule. The volume shrinkage effect is also considered. The relationships between the particle size, the fracture stress, and the applied stress in BMA are thus established. The goal of this paper is to explore fundamental mechanism for the nano-particle formation in BMA of brittle materials such as NaCl and Li 2NH.

On A Priori Error Analysis of Fully Discrete Heterogeneous Multiscale FEM

Heterogeneous multiscale methods have been introduced by E and Engquist [Commun. Math. Sci., 1 (2003), pp. 87--132] as a methodology for the numerical computation of problems with multiple scales. Analyses of the methods for various homogenization problems have been done by several authors. These results were obtained under the assumption that the microscopic models (the cell problems in the homogenization context) are analytically given. For numerical computations, these microscopic models have to be solved numerically. Therefore, it is important to analyze the error transmitted on the macroscale by discretizing the fine scale. We give in this paper H1 and L2 a priori estimates of the fully discrete heterogeneous multiscale finite element method. Numerical experiments confirm that the obtained a priori estimates are sharp.

A Posteriori Error Estimates for the Heterogeneous Multiscale Finite Element Method for Elliptic Homogenization Problems

In this paper, we derive a new approach for the numerical analysis of the heterogeneous multiscale finite element method (HM-FEM) for elliptic homogenization problems. The HM-FEM was introduced in [W. E and B. Engquist, Commun. Math. Sci., 1 (2003), pp. 87--132], and a priori error estimates were derived in [W. E, P. Ming, and P. Zhang, J. Amer. Math. Soc., 18 (2005), pp. 121--156]. The main idea in our approach is to reformulate the heterogeneous multiscale method as a direct finite element discretization of the two scale homogenized equation. This approach puts the problem into a variational form and allows us in a straightforward way to derive a priori and a posteriori error estimates between the HM-FEM solution and the solution of the two scale homogenized equation. The derived a posteriori error estimate is a new result which cannot be obtained with the previous approaches. The theoretical results are verified in numerical experiments, and the usability of the a posteriori error estimate for local adaptive mesh refinement of the HM-FEM is shown.