Finite Element Heterogeneous Multiscale Method Research Articles

P1$$ {P}_1 $$‐nonconforming quadrilateral finite element space with periodic boundary conditions: Part II. Application to the nonconforming heterogeneous multiscale method

AbstractA homogenization approach is one of effective strategies to solve multiscale elliptic problems approximately. The finite element heterogeneous multiscale method (FEHMM) which is based on the finite element makes possible to simulate such process numerically. In this paper we introduce a FEHMM scheme for multiscale elliptic problems based on nonconforming elements. In particular we use the noconforming element with the periodic boundary condition introduced in the companion paper. Theoretical analysis derives a priori error estimates in the standard Sobolev norms. Several numerical results which confirm our analysis are provided.

The Heterogeneous Multiscale Method for Dispersive Maxwell Systems

In this work, we apply the finite element heterogeneous multiscale method to a class of dispersive first-order time-dependent Maxwell systems. For this purpose, we use an analytic homogenization result, which shows that the effective system contains additional dispersive effects. We provide a careful study of the (time-dependent) micro problems, including $\mathrm{H}^2$ and micro errors estimates. Eventually, we prove a semi-discrete error estimate for the method.

Analysis of the effect of surface topography on lubrication using heterogeneous multiscale method

In this paper, flow factors considering the roughness of the different orientation angles and different wavelengths are obtained from the derived form of the homogenization coefficient, which is calculated by a heterogeneous multiscale inverse approach. The influence of anisotropic roughness on the flow factors is investigated. When the angle is equal to 45°, the span flow is maximum. The flow factor curves happen to overlap when the wavelength ratio is less than 4 with double cosine roughness. To verify the flow factors, film pressures are calculated by a finite element heterogeneous multiscale method (FE-HMM) without considering the flow factors. Then, by comparing the results with solutions of a finite difference method, the FE-HMM method gets the results that are needed, and with a shorter time of calculation. The pressure curve is greatest when the orientation angle is 45° with a double cosine roughness, which is caused by the asperity number and the span flow. The pressure distributions of the cosine roughness and double cosine roughness are different as the orientation angle changes.

Convergence and error analysis of FE-HMM/FE[formula omitted] for energetically consistent micro-coupling conditions in linear elastic solids

A cornerstone of numerical homogenization is the equivalence of the microscopic and the macroscopic energy densities, which is referred to as Hill–Mandel condition. Among these coupling conditions, the cases of periodic, linear displacement and constant traction conditions are most prominent in engineering applications. While the stiffness hierarchy of these coupling conditions is a theoretically established and numerically verified result, very little is known about the numerical errors and convergence properties for each of them in various norms. The present work addresses these aspects both on the macroscale and the microscale for linear as well as quadratic finite element shape functions. The analysis addresses aspects of (i) regularity and how its loss affects the convergence behavior on both scales compared with the a priori estimates, of (ii) error propagation from micro to macro and of (iii) optimal micro–macro mesh refinement strategy. For constant traction conditions two different approaches are compared. The performance of a recovery-type error estimation based on superconvergence is assessed. All results of the present work are valid for both the Finite Element Heterogeneous Multiscale Method FE-HMM and for FE2.

Heterogeneous Multiscale Method for Maxwell's Equations

We present a finite element heterogeneous multiscale method for time-dependent Maxwell's equations in first-order formulation in highly oscillatory materials using Nédélec's edge elements. Based on a uniform approach for the error analysis of nonconforming space discretizations [D. Hipp, M. Hochbruck, and C. Stohrer, IMA J. Numer. Anal., 39 (2019), pp. 1206--1245], we prove an error bound for the semidiscrete scheme. We further present error bounds for the fully discrete scheme, where we consider time discretization using algebraically stable Runge--Kutta methods, the Crank--Nicolson method, and the leapfrog method. These error bounds are confirmed by numerical experiments.

A nonlinear FE‐HMM formulation along with a novel algorithmic structure for finite deformation elasticity

AbstractThis contribution presents a nonlinear formulation of the Finite Element Heterogeneous Multiscale Method (FE‐HMM) for the homogenization of microheterogeneous solids. Moreover, a novel algorithmic structure is proposed for FE‐HMM, FE2 and alike, which enables for hyperelastic constitutive laws a considerable speed‐up compared to the existing standard.

Finite element heterogeneous multiscale method for elastic waves in heterogeneous media

A finite element heterogeneous multiscale method (FE-HMM) is proposed for the simulation of time-dependent elastic waves in a rapidly varying heterogeneous elastic medium. It is based on a standard finite element discretization of an effective wave equation at the macro scale, whose a priori unknown effective material coefficients are computed on sampling domains at the micro scale within each macro finite element. Hence the computational effort becomes independent of the highly heterogeneous elastic medium at the smallest scale. Optimal error estimates and convergence rates in the energy and the L2 norm are derived, which are explicit in the macro and micro discretization errors. Numerical experiments verify the sharpness of the error bounds and illustrate the versatility of the method for non-periodic, layered or stochastic media.

Application of finite elements heterogeneous multi-scale method to eddy currents non destructive testing of carbon composites material

In this present paper, a simulation of eddy current non-destructive testing (EC NDT) on unidirectional carbon fiber reinforced polymer is performed; for this magneto-dynamic formulation in term of magnetic vector potential is solved using finite element heterogeneous multi-scale method (FE HMM). FE HMM has as goal to compute the homogenized solution without calculating the homogenized tensor explicitly, the solution is based only on the physical characteristic known in micro domain. This feature is well adapted to EC NDT to evaluate defect in carbon composite material in microscopic scale, where the defect detection is performed by coil impedance measurement; the measurement value is intimately linked to material characteristic in microscopic level. Based on this, our model can handle different defects such as: cracks, inclusion, internal electrical conductivity changes, heterogeneities, etc. The simulation results were compared with the solution obtained with homogenized material using mixture law, a good agreement was found.

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.

Numerical homogenization method for parabolic advection–diffusion multiscale problems with large compressible flows

We introduce a numerical homogenization method based on a discontinuous Galerkin finite element heterogeneous multiscale method to efficiently approximate the effective solution of parabolic advection---diffusion problems with rapidly varying coefficients, large Peclet number and compressible flows. To estimate the missing data of an effective model, numerical upscaling is performed which accurately captures the effects of microscopic solenoidal or gradient flow at a macroscopic scale such as enhancement or depletion of the effective diffusion. For compressible flow with periodic data, we derive sharp a priori error estimates for the macro and micro discretization errors which are robust in the advection dominated regime. Numerical tests confirm the error estimates for problems with periodic data and illustrate the applicability of our method for problems with non-periodic data.

Finite element heterogeneous multiscale method for nonlinear monotone parabolic homogenization problems

We propose a multiscale method based on a finite element heterogeneous multiscale method (in space) and the implicit Euler integrator (in time) to solve nonlinear monotone parabolic problems with multiple scales due to spatial heterogeneities varying rapidly at a microscopic scale. The multiscale method approximates the homogenized solution at computational cost independent of the small scale by performing numerical upscaling (coupling of macro and micro finite element methods). Taking into account the error due to time discretization as well as macro and micro spatial discretizations, the convergence of the method is proved in the general Lp(W1,p) setting. For p = 2, optimal convergence rates in the L2(H1) and C0(L2) norm are derived. Numerical experiments illustrate the theoretical error estimates and the applicability of the multiscale method to practical problems.

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 reduced basis finite element heterogeneous multiscale method for Stokes flow in porous media

A reduced basis Darcy–Stokes finite element heterogeneous multiscale method (RB-DS-FE-HMM) is proposed for the Stokes problem in porous media. The multiscale method is based on the Darcy–Stokes finite element heterogeneous multiscale method (DS-FE-HMM) introduced in Abdulle and Budáč (2015) that couples a Darcy equation solved on a macroscopic mesh, with missing permeability data extracted from the solutions of Stokes micro problems at each macroscopic quadrature point. To overcome the increasingly growing cost of repeatedly solving the Stokes micro problems as the macroscopic mesh is refined, we parametrize the microscopic solid geometry and approximate the infinite-dimensional manifold of parameter dependent solutions of Stokes problems by a low-dimensional space. This low-dimensional (reduced basis) space is obtained in an offline stage by a greedy algorithm and used in an online stage to compute the effective Darcy permeability at a cost independent of the microscopic mesh. The discretization of the parametrized Stokes problems relies on a Petrov–Galerkin formulation that allows for a stable and fast online evaluation of the required permeabilities. A priori and a posteriori estimates of the RB-DS-FE-HMM are derived and a residual-based adaptive algorithm is proposed. Two- and three-dimensional numerical experiments confirm the accuracy of the RB-DS-FE-HMM and illustrate the speedup compared to the DS-FE-HMM.

An Optimization Based Coupling Method for Multiscale Problems

A new multiscale coupling method is proposed for elliptic problems with highly oscillatory coefficients with a continuum of scales in a subset of the computational domain and scale separation in complementary regions of the computational domain. A discontinuous Galerkin (DG) finite element heterogeneous multiscale method (FE-HMM) is used in the region with scale separation, while a continuous standard finite element method is used in the region without scale separation. The use of a DG-FE-HMM method allows for a flexible meshing of the different models in the overlapping region. The unknown boundary conditions at the interfaces are obtained by minimizing the error of the two models in the overlapping region. We prove the well-posedness of both the continuous and discrete coupling problems and establish convergence of the multiscale method towards the fine scale solution. Since in the region with scale separation we obtain an approximation at a cost independent of the smallest scale in the problem, the computational cost of the multiscale method is significantly smaller than a fine scale solver over the whole computational domain, while the algorithm allows us to treat situations for which standard numerical homogenization methods do not apply.

A Priori Error Analysis of the Finite Element Heterogeneous Multiscale Method for the Wave Equation over Long Time

A fully discrete a priori analysis of the finite element heterogeneous multiscale method introduced in [A. Abdulle, M. Grote, and C. Stohrer, Multiscale Model. Simul., 12, 2014, pp. 1135--1166] for the wave equation with highly oscillatory coefficients over long time is presented. A sharp a priori convergence rate for the numerical method is derived for long time intervals. The effective model over long time is a Boussinesq-type equation that has been shown to approximate the one-dimensional multiscale wave equation with $\epsilon$-periodic coefficients up to time ${\cal O}(\epsilon^{-2})$ in [A. Lamacz, Math. Models Methods Appl. Sci., 21, 2011, pp. 1871--1899]. In this paper we also revisit this result by deriving and analyzing a family of effective Boussinesq-type equations for the approximation of the multiscale wave equation that depends on the normalization chosen for certain micro functions used to define the macroscopic models.

A finite element heterogeneous multiscale method with improved control over the modeling error

Multiscale partial differential equations (PDEs) are difficult to solve by traditional numerical methods due to the need to resolve the small wavelengths in the media over the entire computational ...

An Adaptive Finite Element Heterogeneous Multiscale Method for Stokes Flow in Porous Media

A finite element heterogeneous multiscale method is proposed for solving the Stokes problem in porous media. The method is based on the coupling of an effective Darcy equation on a macroscopic mesh with unknown permeabilities recovered from micro finite element calculations for Stokes problems on sampling domains centered at quadrature points in each macro element. The numerical method accounts for nonperiodic microscopic geometry that can be obtained from a smooth deformation of a reference pore sampling domain. The computational work is nevertheless independent of the small size of the pore structure. A priori error estimates reveal that the overall accuracy of the numerical scheme is limited by the regularity of the solutions of the Stokes microproblems. This regularity is low for a typical situation of nonconvex microscopic pore geometries. We therefore propose an adaptive scheme with micro-macro mesh refinement driven by residual-based indicators that quantify both the macro- and microerrors. A posteriori error analysis is derived for the new method. Two- and three-dimensional numerical experiments confirm the robustness and the accuracy of the adaptive method.

Finite-element heterogeneous multiscale method for the Helmholtz equation

We show that the standard Finite Element Heterogeneous Multiscale Method (FE-HMM) can be used to approximate the effective behavior of solutions to the classical Helmholtz equation in highly oscillatory media. Using a novel combination of well-known results about FE-HMM and the notion of T-coercivity, we derive an a priori error bound. Numerical experiments corroborate the analytical findings.

Error control and adaptivity for heterogeneous multiscale approximations of nonlinear monotone problems

In this work we introduce and analyse a new adaptive Petrov-Galerkin heterogeneous multiscale finiteelement method (HMM) for monotone elliptic operators with rapid oscillations.In a general heterogeneous setting we prove convergence of theHMM approximations to the solution of a macroscopic limit equation.The major new contribution of this work is an a-posteriori error estimatefor the $L^2$-error between the HMM approximation and the solution of themacroscopic limit equation.The a posteriori error estimate is obtained in a general heterogeneous settingwith scale separation without assuming periodicity or stochastic ergodicity.The applicability of the method and the usage of the a posteriori error estimatefor adaptive local mesh refinement is demonstrated in numerical experiments.The experimental results underline the applicability of the a posteriori errorestimate in non-periodic homogenization settings.