A Generalized Multiscale Finite Element Method (GMsFEM) for perforated domain flows with Robin boundary conditions

Spiridonov, Denis; Vasilyeva, Maria; Leung, Wing Tat

doi:10.1016/j.cam.2019.03.007

Cited by 27 publications

(9 citation statements)

References 19 publications

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“…Additional multiscale basis for Γ C . Inspired by the technique used in [27] when it deals with the boundary condition, we also need additional multiscale basis to tackle the unilateral boundary condition. In order to ensure the multiscale solution is positive on Γ C we introduce a class of additional basis functions, namely v (i) ex1 , which are the solutions of the following Dirichlet problems (3.2).…”

Section: Multiscale Spacementioning

confidence: 99%

A Multiscale Method for the Heterogeneous Signorini Problem

Su¹,

Pun²

2021

Preprint

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In this paper, we develop a multiscale method for solving the Signorini problem with a heterogeneous field. The Signorini problem is encountered in many applications, such as hydrostatics, thermics, and solid mechanics. It is well-known that numerically solving this problem requires a fine computational mesh, which can lead to a large number of degrees of freedom. The aim of this work is to develop a new hybrid multiscale method based on the framework of the generalized multiscale finite element method (GMsFEM). The construction of multiscale basis functions requires local spectral decomposition. Additional multiscale basis functions related to the contact boundary are required so that our method can handle the unilateral condition of Signorini type naturally. A complete analysis of the proposed method is provided and a result of the spectral convergence is shown. Numerical results are provided to validate our theoretical findings.

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Section: Multiscale Spacementioning

confidence: 99%

A Multiscale Method for the Heterogeneous Signorini Problem

Su¹,

Pun²

2021

Preprint

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“…In this work, we continue developing the multiscale model reduction techniques for problems with multiscale features and developing the generalization of the techniques for DG-GMsFEM. In our previous work [35,36], we considered elliptic problems in perforated domains and constructed additional basis functions to capture non-homogeneous boundary conditions on perforations. In [35], we proposed a non-local multi-continua (NLMC) method for Laplace, elasticity, and parabolic equations with non-homogeneous boundary conditions on perforations.…”

Section: Introductionmentioning

confidence: 99%

Multiscale dimension reduction for flow and transport problems in thin domain with reactive boundaries

Vasilyeva,

Alekseev,

Chung

et al. 2020

Preprint

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In this paper, we consider flow and transport problems in thin domains. Modeling problems in thin domains occur in many applications, where thin domains lead to some type of reduced models.A typical example is one dimensional reduced-order model for flows in pipe-like geometries (e.g., blood vessels). In many reduced-order models, the model equations are described apriori by some analytical approaches. In this paper, we propose the use of multiscale methods, which are alternative to reduced-order models and can represent reduced-dimension modeling by using fewer basis functions (e.g., the use of one basis function corresponds to one dimensional approximation).The mathematical model considered in the paper is described by a system of equations for velocity, pressure, and concentration, where the flow is described by the Stokes equations, and the transport is described by an unsteady convection-diffusion equation with non-homogeneous boundary conditions on walls (reactive boundaries). We start with the finite element approximation of the problem on unstructured grids and use it as a reference solution for two and three-dimensional model problems. Fine grid approximation resolves complex geometries on the grid level and leads to a large discrete system of equations that is computationally expensive to solve. To reduce the size of the discrete systems, we develop a multiscale model reduction technique, where we construct local multiscale basis functions to generate a lower-dimensional model on a coarse grid. The proposed multiscale model reduction is based on the Discontinuous Galerkin Generalized Multiscale Finite Element Method (DG-GMsGEM). In DG-GMsFEM for flow problems, we start with constructing the snapshot space for each interface between coarse grid cells to capture possible flows. For the reduction of the snapshot space size, we perform a dimension reduction via a solution of the local spectral problem and use eigenvectors corresponding to the smallest eigenvalues as multiscale basis functions for the approximation on the coarse grid. For the transport problem, we construct multiscale basis functions for each interface between coarse grid cells and present additional basis functions to capture non-homogeneous boundary conditions on walls. Finally, we will present some numerical simulations for three test geometries for two and three-dimensional problems to demonstrate the method's performance.

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“…Multiscale methods are widely used to solve various problems in domains with complex heterogeneities. Some of the popular methods are the multiscale finite element method(MsFEM) [27,24,28], mixed multiscale finite element method(Mixed MsFEM) [29,30], generalize multiscale finite element method(GMsFEM) [25,31,32,33,34], heterogeneous multiscale methods [35,36,37], multiscale finite volume method (MsFVM) [38,39,40], constraint energy minimizing generalized multiscale finite element method (CEM-GMsFEM) [41,41] and etc. Recently, in [42,43,44], the authors present a special design of the multiscale basis functions to solve problems in fractured porous media, which obtains the basis functions based on the constrained energy minimization problems and nonlocal multicontinuum (NLMC) method.…”

Section: Introductionmentioning

confidence: 99%

Mixed Generalized Multiscale Finite Element Method for Darcy-Forchheimer Model

et al. 2019

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In this paper, the solution of the Darcy-Forchheimer model in high contrast heterogeneous media is studied. This problem is solved by a mixed finite element method (MFEM) on a fine grid (the reference solution), where the pressure is approximated by piecewise constant elements; meanwhile, the velocity is discretized by the lowest order Raviart-Thomas elements. The solution on a coarse grid is performed by using the mixed generalized multiscale finite element method (mixed GMsFEM). The nonlinear equation can be solved by the well known Picard iteration. Several numerical experiments are presented in a two-dimensional heterogeneous domain to show the good applicability of the proposed multiscale method.

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A Generalized Multiscale Finite Element Method (GMsFEM) for perforated domain flows with Robin boundary conditions

Cited by 27 publications

References 19 publications

A Multiscale Method for the Heterogeneous Signorini Problem

A Multiscale Method for the Heterogeneous Signorini Problem

Multiscale dimension reduction for flow and transport problems in thin domain with reactive boundaries

Mixed Generalized Multiscale Finite Element Method for Darcy-Forchheimer Model

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