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

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 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 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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Online Meshfree generalized multiscale finite element method for Flows in Fractured Media

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Lobachevskii Journal of Mathematics 2024年第11期45卷 5391-5401页

作者： Nikiforov, D.Y. Stepanov, S.P. Lazarev, N.P. North-Eastern Federal University Yakutsk 677000 Russian Federation North-Caucasus Federal University Stavropol Russian Federation

Abstract: Modeling fracture systems in porous media requires a specialized approach due to the high permeability of fractures, which significantly influences transport and flow processes. Fracture systems can exist at various scales and differ in their origins, complicating their modeling. This paper presents a generalized multiscale finite element method (GMsFEM) utilizing a meshfree approach for constructing the coarse grid. We develop a methodology for creating online multiscale basis functions computed during the solution of non-stationary problems, which account for changes in local subdomains. This approach reduces the number of offline basis functions required to achieve high accuracy. Comparative studies demonstrate the advantages of the online approach over traditional finite element methods on detailed grids. The online multiscale approach proves effective for problems in fractured porous media, albeit requiring additional computational resources. © Pleiades Publishing, Ltd. 2024.

关键词： discrete fractures generalized multiscale finite element method mathematical modeling meshfree method

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Numerical simulation of language interactions using online coupled generalized multiscale finite element method

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2023年 423卷

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

In this paper, we present a new mathematical model of the interaction of two languages. In the model, we distinguish percentages of people who speak a non-target language, a target language with low and high proficiency, and both languages considering the low and high proficiency levels in the target language. Therefore, the solution consists of five fields. Furthermore, we assume the diffusive and convective spread of the languages, considering the overflow between them. Thus, the mathematical model is defined by a coupled system of partial differential equations for the five *** the mathematical model is coupled and the medium is heterogeneous, we have implemented a multiscale method. The proposed multiscale method is based on the generalized multiscale finite element method (GMsFEM). In addition to offline multiscale basis functions, we also construct online multiscale basis functions. The online basis functions can account for changes in the heterogeneity of the medium caused by migration flows. Numerical results have shown that such online enrichment can significantly improve the accuracy of multiscale modeling.(c) 2022 Elsevier B.V. All rights reserved.

关键词： Language interaction Sociolinguistics Interdisciplinary research Mathematical modeling generalized multiscale finite element method Online basis functions

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

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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 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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multiscale model reduction for a thermoelastic model with phase change using a generalized multiscale finite-element method

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THEORETICAL AND MATHEMATICAL PHYSICS 2022年第2期211卷 595-610页

作者： Ammosov, D. A. Vasil'ev, V. I. Vasil'eva, M. V. Stepanov, S. P. MK Ammosov North Eastern Fed Univ Yakutsk Russia Texas A&M Univ Dept Math & Stat Corpus Christi TX USA

The development of the cryolithozone requires building and numerically implementing mathematical models of multiphysics thermoelastic processes involving with first-order phase transitions and occurring in the foundations of engineering structures and buildings. Numerical implementation of such models is associated with computational difficulties due to various types of heterogeneities in applied problems and the nonlinearity of governing equations, which require very fine grids, increasing computational costs. We develop a numerical method for solving a thermoelasticity problem with phase transitions based on the generalized multiscale finite-element method (GMsFEM). The main idea of the GMsFEM is to construct multiscale basis functions that take the medium heterogeneities into account. The approximation on a fine grid is carried out using the finite-element method with standard linear basis functions. To verify the accuracy of the proposed multiscale method, we solve two- and three-dimensional problems in heterogeneous media. Numerical results show that the multiscale method can provide a good approximation to the solution of the thermoelasticity problem with a phase transition on a fine grid with a significant reduction in the dimensionality of the discrete problem.

关键词： cryolithozone heterogeneous medium mathematical modeling thermoelasticity phase transition generalized multiscale finite element method

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Application of the generalized multiscale finite element method in an inverse random source problem

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JOURNAL OF COMPUTATIONAL PHYSICS 2021年 429卷 110032-110032页

作者： Fu, Shubin Zhang, Zhidong Univ Wisconsin Madison Dept Math Madison WI 53706 USA Sun Yat Sen Univ Sch Math Zhuhai Zhuhai 519082 Guangdong Peoples R China

In this work, an inverse random source problem in the fractional diffusion equation with heterogeneous medium is considered. The measurements used are the statistical moments of the realizations of the single point data u(x(0), t, omega). We build the representation of the solution u in integral sense, then prove that the unknowns can be bounded by the moments theoretically. For the numerical reconstruction, to handle the highly heterogeneous medium, the generalized multiscale finite element method (GMsFEM) will be employed in the forward problem solver. With the simulated data, we establish an iterative algorithm of regularized Levenberg-Marquardt type, and some numerical results generated from this algorithm are displayed. (C) 2020 Elsevier Inc. All rights reserved.

关键词： Inverse problem Fractional diffusion equation Random source generalized multiscale finite element method Heterogeneous medium Regularized iterative algorithm

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