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generalized multiscale finite element method for a strain-limiting nonlinear elasticity model

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2019年 359卷 153-165页

作者： Fu, Shubin Chung, Eric Mai, Tina Chinese Univ Hong Kong Dept Math Shatin Hong Kong Peoples R China Duy Tan Univ Inst Res & Dev Da Nang 550000 Vietnam

In this paper, we consider multiscale methods for nonlinear elasticity. In particular, we investigate the generalized multiscale finite element method (GMsFEM) for a strain-limiting elasticity problem. Being a special case of the naturally implicit constitutive theory of nonlinear elasticity, strain-limiting relation has presented an interesting class of material bodies, for which strains remain bounded (even infinitesimal) while stresses can become arbitrarily large. The nonlinearity and material heterogeneities can create multiscale features in the solution, and multiscale methods are therefore necessary. To handle the resulting nonlinear monotone quasilinear elliptic equation, we use linearization based on the Picard iteration. We consider two types of basis functions, offline and online basis functions, following the general framework of GMsFEM. The offline basis functions depend nonlinearly on the solution. Thus, we design an indicator function and we will recompute the offline basis functions when the indicator function predicts that the material property has significant change during the iterations. On the other hand, we will use the residual based online basis functions to reduce the error substantially when updating basis functions is necessary. Our numerical results show that the above combination of offline and online basis functions is able to give accurate solutions with only a few basis functions per each coarse region and adaptive updating basis functions in selected iterations. (C) 2019 Elsevier B.V. All rights reserved.

关键词： generalized multiscale finite element method Strain-limiting nonlinear elasticity Adaptivity Residual based online multiscale basis functions

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generalized multiscale finite element method for piezoelectric problem in heterogeneous media

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ENGINEERING ANALYSIS WITH BOUNDARY elementS 2022年 135卷 12-25页

作者： Ammosov, Dmitry Vasilyeva, Maria Nasedkin, Andrey Efendiev, Yalchin North Eastern Fed Univ Lab Computat Technol Modeling Multiphys & Multisc Yakutsk 677980 Republic Of Sak Russia Texas A&M Univ Dept Math & Stat 6300 Ocean Dr Corpus Christi TX 78412 USA Southern Fed Univ Inst Math Mech & Comp Sci Rostov Na Donu 344006 Russia Texas A&M Univ Dept Math College Stn TX 77843 USA North Eastern Fed Univ Multiscale Model Reduct Yakutsk 677980 Republic Of Sak Russia

In this paper, we study multiscale methods for piezocomposites. We consider a model of static piezoelectric problem that consists of deformation with respect to components of displacements and a function of electric potential. This problem includes the equilibrium equations, the quasi-electrostatic equation for dielectrics, and a system of coupled constitutive relations for mechanical and electric fields. We consider a model problem that consists of coupled differential equations. The first equation describes the deformations and is given by the elasticity equation and includes the effect of the electric field. The second equation is for the electric field and has a contribution from the elasticity equation. In previous findings, numerical homogenization methods are proposed and used for piezocomposites. We consider the generalized multiscale finite element method (GMsFEM), which is more general and is known to handle complex heterogeneities. The main idea of the GMsFEM is to develop additional degrees of freedom and can go beyond numerical homogenization. We consider both coupled and split basis functions. In the former, the multiscale basis functions are constructed by solving coupled local problems. In particular, coupled local problems are solved to generate snapshots. Furthermore, in the snapshot space, a local spectral decomposition is performed to identify multiscale basis functions. Our approaches share some common concepts with meshless methods as they solve the underlying problem on a coarse mesh, which does not conform heterogeneities and contrast. We discuss this issue in the paper. We show that with a few basis functions per coarse element, one can achieve a good approximation of the solution. Numerical results are presented.

关键词： generalized multiscale finite element method multiscale method Piezoelectric composite Heterogeneous media

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generalized multiscale finite element method for Multicontinuum Coupled Flow and Transport Model

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Lobachevskii Journal of Mathematics 2024年第11期45卷 5343-5356页

作者： Ammosov, D.A. Huang, J. Leung, W.T. Shan, B. Computational Technologies and Artificial Intelligence North-Eastern Federal University Yakutsk 677980 Russian Federation School of Mathematics and Computational Science Xiangtan University National Center for Applied Mathematics in Hunan and Hunan Key Laboratory for Computation and Simulation in Science and Engineering Hunan Xiangtan 411105 China Department of Mathematics City University of Hong Kong Hong Kong Department of Mathematics Texas A & M University College Station 77843 TX United States

Abstract: In this paper, we consider a coupled flow and transport process described by partial differential equations for pressure and concentration. We derive the multicontinuum coupled flow and transport model using the multicontinuum homogenization method. We formulate cell problems with constraints that take into account various homogenized effects. For the resulting multicontinuum model, we develop a multiscale solver based on the generalized multiscale finite element method. The combined approach of multicontinuum and multiscale modeling makes it possible to consider complex heterogeneous media with high contrast. We present numerical results for a model problem with two continua and heterogeneous coefficients separable into microscopic and macroscopic parts. The results show the high efficiency of the proposed multiscale approach. © Pleiades Publishing, Ltd. 2024.

关键词： coupled flow and transport generalized multiscale finite element method homogenization miscible displacement multicontinuum

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generalized multiscale finite element method for a nonlinear elastic strain-limiting Cosserat model

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JOURNAL OF COMPUTATIONAL PHYSICS 2024年 519卷

作者： Ammosov, Dmitry Mai, Tina Galvis, Juan North Eastern Fed Univ Computat Technol & Artificial Intelligence Yakutsk 677980 Republic Of Sak Russia Duy Tan Univ Inst Res & Dev Da Nang 550000 Vietnam Duy Tan Univ Fac Nat Sci Da Nang 550000 Vietnam Univ Nacl Colombia Dept Matemat Carrera 45 26-85Edificio Uriel Gutierrez Bogota Colombia

For nonlinear Cosserat elasticity, we consider multiscale methods in this paper. In particular, we explore the generalized multiscale finite element method (GMsFEM) to solve an isotropic Cosserat problem with strain-limiting property (ensuring bounded linearized strains even under high stresses). Such strain-limiting Cosserat model can find potential applications in solids and biological fibers. However, Cosserat media with naturally rotational degrees of freedom, nonlinear constitutive relations, high contrast, and heterogeneities may produce challenging multiscale characteristics in the solution, and upscaling by multiscale methods is necessary. Therefore, we utilize the offline and residual-based online (adaptive or uniform) GMsFEM in this context while handling the nonlinearity by Picard iteration. Through various two-dimensional experiments (for perforated, composite, and stochastically heterogeneous media with small and big strain-limiting parameters), our numerical results show the approaches' convergence, efficiency, and robustness. In addition, these results demonstrate that such approaches provide good accuracy, the online GMsFEM gives more accurate solutions than the offline one, and the online adaptive strategy has similar accuracy to the uniform one but with fewer degrees of freedom.

关键词： generalized multiscale finite element method Adaptivity Residual-based online multiscale basis functions Strain-limiting Nonlinear Cosserat elasticity Heterogeneous media

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generalized multiscale finite element method for language competition modeling I: Offline approach

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2024年 442卷

作者： Ammosov, D. A. Malysheva, N. V. Zamorshchikova, L. S. Ammosov NEFU Belinskogo Str 58 Yakutsk 677000 Russia

This paper develops a multiscale solver for the problem of two languages competing in a heterogeneous medium. The mathematical model contains terms for language group switching, diglossia, population growth, and diffusive -convective spatial spread. The fine grid approximation is based on the finite element method. We use the finite difference method with an implicit scheme for time approximation. We use the Picard method to linearize the nonlinear terms. We use the generalized multiscale finite element method for approximation on a coarse grid. Numerical results are presented for a model two -language interaction problem. The results show that the proposed multiscale approach can significantly reduce the computational cost of the problem while maintaining high accuracy.

关键词： Language competition finite element method generalized multiscale finite element method Heterogeneous media Picard method Nonlinear problem

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generalized multiscale finite element method for multicontinua unsaturated flow problems in fractured porous media

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2020年第0期370卷 112594-000页

作者： Spiridonov, Denis Vasilyeva, Maria Chung, Eric T. North Eastern Fed Univ Multiscale Model Reduct Lab Yakutsk 677980 Republic Of Sak Russia Texas A&M Univ Inst Sci Computat College Stn TX 77843 USA North Eastern Fed Univ Dept Computat Technol Yakutsk 677980 Republic Of Sak Russia CUHK Dept Math Hong Kong Peoples R China

In this paper, we present a multiscale method for simulations of the multicontinua unsaturated flow problems in heterogeneous fractured porous media. The mathematical model is described by the system of Richards equations for each continuum that is coupled by the specific transfer term. To illustrate the idea of our approach, we consider a dual continua background model with discrete fractures networks that is generalized as a multicontinua model for unsaturated fluid flow in the complex heterogeneous porous media. We present fine grid approximation based on the finite element method and Discrete Fracture Model (DFM) approach. In this model, we construct an unstructured fine grid that takes into account complex fracture geometries for two and three dimensional formulations. Due to construction of the unstructured grid, the fine grid approximation leads to the very large system of equations. For reduction of the discrete system size, we develop a multiscale method for coarse grid approximation of the coupled problem using generalized multiscale finite element method (GMsFEM). In this method, we construct coupled multiscale basis functions that are used to construct highly accurate coarse grid approximation. The multiscale method allowed us to capture detailed interactions between multiple continua. The adaptive approach is investigated, where we consider two approaches for multiscale basis functions construction: (1) based on the spectral characteristics of the local problems and (2) using simplified multiscale basis functions. We investigate accuracy of the proposed method for the several test problems in two and three dimensional formulations. We present a comparison of the relative error for different number of basis functions and for adaptive approach. Numerical results illustrate that the presented method provides accurate solution of the unsaturated multicontinua problem on the coarse grid with huge reduction of the discrete system size. (C) 2019 Elsevier B.V.

关键词： Unsaturated flow Fractured porous media Multicontinuum media generalized multiscale finite element method GMsFEM multiscale method

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A generalized multiscale finite element method (GMsFEM) for perforated domain flows with Robin boundary conditions

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2019年 357卷 319-328页

作者： Spiridonov, Denis Vasilyeva, Maria Leung, Wing Tat North Eastern Fed Univ Yakutsk 677980 Republic Of Sak Russia Texas A&M Univ Inst Sci Computat College Stn TX 77843 USA Univ Texas Austin Inst Computat Engn & Sci Austin TX 78712 USA

In this work, we develop a multiscale method for elliptic problems in perforated media with Robin boundary conditions. Perforated media are encountered in many applications, e.g., in material science applications, where the material properties are prescribed outside perforation. Such a perforated structure can significantly affect the solution. For the numerical solution of the equation, it is necessary to construct a computational grid with elements resolving the perforations. The computational grid calculated in this way is fine-scale grid and this can lead to a large number of unknowns. For the numerical solution of our problem, we construct an approximation of the equation on a coarse grid using the generalized multiscale finite element method (GMsFEM). The main idea of the GMsFEM is to construct multiscale basis functions on a coarse grid, which can reduce the computational cost. The multiscale basis function construction requires snapshots and local spectral problems, which are developed in the paper. In the paper, we study Robin boundary conditions, which arise in many applications. Previous works (Chung et al., 2016) considered Dirichlet and Neumann boundary conditions. The construction of multiscale basis functions differs from those in Chung et al. 2016 [1,2] and as we need to impose non-homogeneous Robin boundary conditions. We present several numerical examples. In these examples, perforated domains with many inclusions are considered. Our numerical results show a good agreement between the coarse and the fine-grid simulations. (C) 2019 Elsevier B.V. All rights reserved.

关键词： generalized multiscale finite element method GMsFEM Perforated domain Robin boundary conditions

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A generalized multiscale finite element method for poroelasticity problems II: Nonlinear coupling

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2016年 297卷 132-146页

作者： Brown, Donald L. Vasilyeva, Maria Univ Nottingham Sch Math Sci Univ Pk Nottingham NG7 2RD England North Eastern Fed Univ Inst Math & Informat Yakutsk 677980 Republic Sakha Russia Inst Computat Math & Math Geophys SB RAS Novosibirsk 630090 Russia

In this paper, we consider the numerical solution of some nonlinear poroelasticity problems that are of Biot type and develop a general algorithm for solving nonlinear coupled systems. We discuss the difficulties associated with flow and mechanics in heterogeneous media with nonlinear coupling. The central issue being how to handle the nonlinearities and the multiscale scale nature of the media. To compute an efficient numerical solution we develop and implement a generalized multiscale finite element method (GMsFEM) that solves nonlinear problems on a coarse grid by constructing local multiscale basis functions and treating part of the nonlinearity locally as a parametric value. After linearization with a Picard Iteration, the procedure begins with construction of multiscale bases for both displacement and pressure in each coarse block by treating the staggered nonlinearity as a parametric value. Using a snapshot space and local spectral problems, we construct an offline basis of reduced dimension. From-here an online, parametric dependent, space is constructed. Finally, after multiplying by a multiscale partitions of unity, the multiscale basis is constructed and the coarse grid problem then can be solved for arbitrary forcing and boundary conditions. We implement this algorithm on a geometry with a linear and nonlinear pressure dependent permeability field and compute error between the multiscale solution with the fine-scale solutions. (C) 2015 Elsevier B.V. All rights reserved.

关键词： generalized multiscale finite element method Nonlinear poroelasticity problems

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Least-squares mixed generalized multiscale finite element method

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COMPUTER methodS IN APPLIED MECHANICS AND ENGINEERING 2016年 311卷 764-787页

作者： Chen, Fuchen Chung, Eric Jiang, Lijian Hunan Univ Coll Math & Econometr Changsha 410082 Hunan Peoples R China Chinese Univ Hong Kong Dept Math Hong Kong Hong Kong Peoples R China Hunan Univ Inst Math Changsha 410082 Hunan Peoples R China

In this paper, we present an approximation of elliptic problems with multiscale and high-contrast diffusion coefficients. A mixed formulation is considered such that both pressure and velocity are approximated simultaneously. This formulation arises naturally in many applications such as flows in porous media. Due to the multiscale nature of the solutions, using model reduction is required to efficiently obtain approximate solutions. There are many multiscale approaches for elliptic problems in mixed formulation. These approaches include numerical homogenization and mixed multiscale finite element method, which aim to obtain a coarse accurate representation of the velocity without using an accurate representation for pressure. It has been a challenging task to construct a method giving accurate representation for both pressure and velocity. The goal in this paper is to construct multiscale basis functions for both pressure and velocity. We will apply the framework of generalized multiscale finite element method (GMsFEM), and design systematic strategies for the construction of basis. The construction involves snapshot spaces and dimension reduction via local spectral problems. The mixed formulation is minimized in the sense of least-squares. The compatibility condition for multiscale finite element spaces of the pressure and velocity is not required in the least-squares mixed form. This gives more flexibility for the construction of multiscale basis functions for velocity and pressure. Convergence analysis is carried out for the least-squares mixed GMsFEM. Several numerical examples for various permeability fields are presented to show the performance of the presented method. The numerical results show that the least-squares mixed GMsFEM can give accurate approximation for both pressure and velocity using only a few basis functions per coarse element. (C) 2016 Elsevier B.V. All rights reserved.

关键词： Least-squares mixed formulation generalized multiscale finite element method Flux correction High-contrast coefficients

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A generalized multiscale finite element method for high-contrast single-phase flow problems in anisotropic media

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2015年 277卷 202-214页

作者： Ren, Jun Presho, Michael Texas A&M Univ Dept Math College Stn TX 77843 USA Texas A&M Univ Inst Sci Computat College Stn TX 77843 USA Univ Texas Austin Inst Computat Engn & Sci Austin TX 78712 USA

In this paper, we describe a systematic framework called the generalized multiscale finite element method (GMsFEM) for performing multiscale simulations, and we investigate the application of this framework to the high-contrast flow problems in anisotropic media. We present a multiscale model reduction technique for constructing a reduced-dimension space based on localized spectral decompositions. We discuss the multiscale coarse space both in parameter-independent and parameter-dependent cases, Furthermore, we highlight the relation between the solution convergence and the spectral behavior of the eigenvalue problems. A variety of numerical examples are presented to illustrate the performance of the proposed methodology. (C) 2014 Elsevier B.V. All rights reserved.

关键词： generalized multiscale finite element method High-contrast permeability Anisotropic media Single-phase flow

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