Generalized Multiscale Finite Element Method Research Articles

Generalized multiscale finite element method for language competition modeling II: Online approach

In this paper, we consider language competition in a heterogeneous medium. The mathematical model considers bilingualism, population growth, and social pressure to use specific languages (diglossia). The transition from one language group to another is determined by probabilities that depend on status, group size, and spatial location. The model considers not only diffusive spread but also convective spread, which models seasonal migrations of speaker populations. We develop multiscale approaches for coarse grid modeling while maintaining reasonable accuracy. We present offline and online approaches based on the Generalized Multiscale Finite Element Method. We consider a model problem in a heterogeneous environment to test these approaches. The results show that both approaches can significantly reduce the discrete problem’s size. However, the online approach achieves better accuracy.

Meshfree multiscale method for the infiltration problem in permafrost

This study introduces a multiscale simulation approach, employing the Meshfree Generalized Multiscale Finite Element Method (GMsFEM), to numerically model fluid seepage in permafrost soil conditions. Combining Meshfree and Finite Element Methods, the coarse-grid system is addressed, while the standard Finite Element Method operates on the fine grid. For representing fractures at the fine grid level, the Discrete Fracture Model (DFM) is utilized. The findings demonstrate the method’s efficacy in accurately depicting soil heterogeneity and the impact of fractures on soil temperature.

Regularized coupling multiscale method for thermomechanical coupled problems

The coupling effects in multiphysics processes are often neglected in designing multiscale methods. The coupling may be described by a non-positive definite operator, which in turn brings significant challenges in multiscale simulations. In the paper, we develop a regularized coupling multiscale method based on the generalized multiscale finite element method (GMsFEM) to solve coupled thermomechanical problems, and it is referred to as the coupling generalized multiscale finite element method (CGMsFEM). The method consists of defining the coupling multiscale basis functions through local regularized coupling spectral problems in each coarse-grid block, which can be implemented by a novel design of two relaxation parameters. Compared to the standard GMsFEM, the proposed method can not only accurately capture the multiscale coupling correlation effects of multiphysics problems but also greatly improve computational efficiency with fewer multiscale basis functions. In addition, the convergence analysis is also established, and the optimal error estimates are derived, where the upper bound of errors is independent of the magnitude of the relaxation coefficient. Several numerical examples for periodic, random microstructure, and random material coefficients are presented to validate the theoretical analysis. The numerical results show that the CGMsFEM shows better robustness and efficiency than uncoupled GMsFEM.

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

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.

Convergence of the CEM-GMsFEM for compressible flow in highly heterogeneous media

This paper presents and analyses a Constraint Energy Minimization Generalized Multiscale Finite Element Method (CEM-GMsFEM) for solving single-phase non-linear compressible flows in highly heterogeneous media. The construction of CEM-GMsFEM hinges on two crucial steps: First, the auxiliary space is constructed by solving local spectral problems, where the basis functions corresponding to small eigenvalues are captured. Then the basis functions are obtained by solving local energy minimization problems over the oversampling domains using the auxiliary space. The basis functions have exponential decay outside the corresponding local oversampling regions. The convergence of the proposed method is provided, and we show that this convergence only depends on the coarse grid size and is independent of the heterogeneities. An online enrichment guided by a posteriori error estimator is developed to enhance computational efficiency. Several numerical experiments on a three-dimensional case to confirm the theoretical findings are presented, illustrating the performance of the method and giving efficient and accurate numerical solutions.

An Online Generalized Multiscale Finite Element Method for Dual-continuum Unsaturated Filtration Problem in Domains with Rough Boundaries

An Online Generalized Multiscale Finite Element Method for Dual-continuum Unsaturated Filtration Problem in Domains with Rough Boundaries

Online Coupled Generalized Multiscale Finite Element Method for the Poroelasticity Problem in Three-Dimensional Media

Online Coupled Generalized Multiscale Finite Element Method for the Poroelasticity Problem in Three-Dimensional Media

Deep ensemble learning for high-dimensional subsurface fluid flow modeling

The accuracy of Deep Learning (DL) algorithms can be improved by combining several deep learners into an ensemble. This avoids the continuous endeavor required to adjust the architecture of individual networks or the nature of the propagation. This study investigates prediction improvements possible using Deep Ensemble Learning (DEL) to determine four distinct multiscale basis functions in the mixed Generalized Multiscale Finite Element Method (GMsFEM), involving the permeability field as the only input. 376,250 samples were initially generated, filtered down to 367,811 after data pre-processing. A standard Convolutional Neural Network (CNN) named SkiplessCNN and three skip connection-based CNNs named FirstSkipCNN, MidSkipCNN, and DualSkipCNN were developed for the base learners. For each basis function, these four CNNs were combined into an ensemble model using linear regression and ridge regression, separately, as part of the stacking technique. A comparison of the coefficient of determination (R2) and Mean Squared Error (MSE) confirms the effectiveness of all three skip connections in enhancing the performance of the standard CNN, with DualSkip being the most effective among them. Additionally, as evaluated on the testing subset, the combined models meaningfully outperform the individual models for all basis functions. The case that applies linear regression delivers R2 ranging from 0.8456 to 0.9191 and MSE ranging from 0.0092 to 0.0369. The ridge regression case achieves marginally better predictions with R2 ranging from 0.8539 to 0.922, and MSE ranging from 0.009 to 0.0349 because its solution involves more evenly distributed weights.

A computational macroscopic model of piezomagnetoelectric materials using Generalized Multiscale Finite Element Method

Piezomagnetoelectric materials have many applications in various areas of science and engineering. For example, such materials can be used in sensors, transducers, energy harvesters, etc. The mathematical model of piezomagnetoelectricity consists of a coupled system of partial differential equations for the electric potential, magnetic potential, and mechanical displacements.We develop multiscale algorithms based on the Generalized Multiscale Finite Element Method for solving the piezomagnetoelectricity problem. We consider two ways of constructing multiscale basis functions: split and coupled. In the split approach, we construct the basis functions separately for each field. The coupled approach constructs multiscale basis functions by solving coupled local problems. The split way requires fewer computational resources than the coupled one, but it may give a less accurate solution in the case of a strongly coupled medium. The coupled approach considers the mutual influence of the fields, which is especially important when they are strongly coupled. We use the multiscale partition of unity functions to obtain conforming basis functions for both approaches.We consider two- and three-dimensional model problems to test the effectiveness of the proposed multiscale approaches. On average, the coupled basis functions show better results than the split ones. We also consider the application of the single-step hermitian and skew-hermitian splitting for solving the discrete problem on a coarse grid. The results have demonstrated that one can successfully apply this technique to solving the piezomagnetoelectricity problem on a coarse grid.

Fast online adaptive enrichment for poroelasticity with high contrast

In this work, we develop an online adaptive enrichment method within the framework of the Constraint Energy Minimizing Generalized Multiscale Finite Element Method (CEM-GMsFEM) for solving the linear heterogeneous poroelasticity models with coefficients of high contrast. The proposed method makes use of information of residual-driven error indicators to enrich the multiscale spaces for both the displacement and the pressure variables in the model. Additional online basis functions are constructed in oversampled regions accordingly and are adaptively chosen to reduce the error the most. A complete theoretical analysis of the online enrichment algorithm is provided and justified by thorough numerical experiments.

Online Multiscale Finite Element Simulation of Thermo-Mechanical Model with Phase Change

This paper presents a thermo-mechanical model with phase transition considering changes in the mechanical properties of the medium. The proposed thermo-mechanical model is described by a system of partial differential equations for temperature and displacements. In the model, soil deformations occur due to porosity growth caused by ice and water density differences. A finite-element approximation of this model on a fine grid is presented. The linearization from the previous time step is used to handle the nonlinearity of the problem. For reducing the size of the discrete problem, offline and online multiscale approaches based on the Generalized Multiscale Finite Element Method (GMsFEM) are proposed. A two-dimensional model problem simulating the heaving process of heterogeneous soil with a stiff inclusion was considered for testing the mathematical model and the multiscale approaches. Numerical solutions depict the process of soil heaving caused by changes in porosity due to the phase transition. The movement of the phase transition interface was observed. The change of medium properties, including the elastic modulus, was traced and corresponds to the phase transition interface. The proposed multiscale approaches significantly reduce the size of the discrete problem while maintaining reasonable accuracy. However, the online multiscale approach achieves better accuracy than the offline approach with fewer degrees of freedom.

An exponential integration generalized multiscale finite element method for parabolic problems

We consider linear and semilinear parabolic problems posed in high-contrast multiscale media in two dimensions. The presence of high-contrast multiscale media adversely affects the accuracy, stability, and overall efficiency of numerical approximations such as finite elements in space combined with some time integrator. In many cases, implementing time discretizations such as finite differences or exponential integrators may be impractical because each time iteration needs the computation of matrix operators involving very large and ill-conditioned sparse matrices. Here, we propose an efficient Generalized Multiscale Finite Element Method (GMsFEM) that is robust against the high-contrast diffusion coefficient. We combine GMsFEM with exponential integration in time to obtain a good approximation of the final time solution. Our approach is efficient and practical because it computes matrix functions of small matrices given by the GMsFEM method. We present representative numerical experiments that show the advantages of combining exponential integration and GMsFEM approximations. The constructions and methods developed here can be easily adapted to three-dimensional domains.

Multiscale solver for multi-component reaction–diffusion systems in heterogeneous media

Coupled nonlinear system of reaction–diffusion equations describing multi-component (species) interactions with heterogeneous coefficients is considered. Finite volume method based approximation for the space is used to construct semi-discrete form for the computation of numerical solutions. Two techniques for time approximations, namely, a fully implicit (FI) and a semi-implicit (SI) schemes are examined. The fully implicit scheme is constructed using Newton’s method and leads to the coupled system of equations on each nonlinear and time iterations which is computationally rather expensive. In order to minimize the latter hurdle, an efficient and fast multiscale solver is proposed for reaction–diffusion systems in heterogeneous media. To construct fast solver, we apply a semi-implicit scheme that leads to an uncoupled system for each individual component. Problems in heterogeneous domains require a very fine grid for accurate solutions of large systems of equations at each time step iteration. Here we present a multiscale model reduction technique to reduce the size of the discrete system. Multiscale solver is based on the uncoupled operator of the problem and constructed by the use of Generalized Multiscale Finite Element Method (GMsFEM). In GMsFEM we use a diffusion part of the operator and construct multiscale basis functions by solving spectral problems in each local domain associated with the coarse grid nodes. We collect multiscale basis functions to construct a projection/prolongation matrix and generate reduced order model on the coarse grid for fast solution. Moreover, the prolongation operator is used to reconstruct a fine-scale solution and accurate approximation of the reaction part of the problem which then leads to a very accurate and computationally effective multiscale solver. We provide numerical results for two species competition test problems in two-dimensional domain with heterogeneous inclusions. Our computed solutions predict strong competition between the two species and demonstrate that diffusion can dictate the dominance of one species over the other in heterogeneous environments. We investigate the influence of number of the multiscale basis functions to the method accuracy and ability to work with different values of the diffusion coefficients.

Generalized Multiscale Finite Element Method for Highly Heterogeneous Compressible Flow

In this paper, we study the generalized multiscale finite element method (GMsFEM) for single phase compressible flow in highly heterogeneous porous media. We follow the major steps of the GMsFEM to construct permeability dependent offline basis for fast coarse-grid simulation. The offline coarse space is efficiently constructed only once based on the initial permeability field with parallel computing. A rigorous convergence analysis is performed for two types of snapshot spaces. The analysis indicates that the convergence rates of the proposed multiscale method depend on the coarse meshsize and the eigenvalue decay of the local spectral problem. To further increase the accuracy of multiscale method, residual driven online multiscale basis is added to the offline space. The construction of online multiscale basis is based on a carefully design error indicator motivated by the analysis. We find that online basis is particularly important for the singular source. Rich numerical tests on typical 3D highly heterogeneous medias are presented to demonstrate the impressive computational advantages of the proposed multiscale method.

Multiscale model reduction technique for fluid flows with heterogeneous porous inclusions

Numerical treatment of the problem of two-dimensional viscous fluid flow in and around circular porous inclusions is considered. The mathematical model is described by Navier–Stokes equation in the free flow domain Ωf and nonlinear convective Darcy–Brinkman–Forchheimer equations in porous subdomains Ωp. It is well-known that numerical solutions of the problems in such heterogeneous domains require a very fine computational mesh that resolve inclusions on the grid level. The size alteration of the relevant system requires model reduction techniques. Here, we present a multiscale model reduction technique based on the Generalized Multiscale Finite Element Method (GMsFEM). We discuss construction of the multiscale basis functions for the velocity fields based on the solution of the local problems with and without oversampling strategy. Three test cases are considered for a given choice of the three key model parameters, namely, the Reynolds number (Re), the Forchheimer coefficient (C) and the Darcy number (Da). For the test runs, the Reynolds number values are taken to be Re=1,10,100 while the Forchheimer coefficient and Darcy number are chosen as C=1,10 and Da=10−5,10−4,10−3, respectively. We numerically study the convergence of the method as we increase the number of multiscale basis functions in each domain, and observe good performance of the multiscale method.

Generalized Multiscale Finite Element Method for scattering problem in heterogeneous media

In this paper, we consider the scattering problem in the heterogeneous domain. The mathematical model is described by the Helmholtz equation related to wave propagation with absorbing boundary conditions. For the solution of the problem using the finite element method, we construct an unstructured fine grid that resolves perforation on the grid level. Such approximations lead to a large system of equations. To reduce the size of the discrete system, we use a multiscale approximation on a coarse grid using the Generalized Multiscale Finite Element Method. We construct a multiscale space using the solution of local spectral problems on the snapshot space in each local domain. Two types of multiscale basis functions are presented and investigated. We present numerical results for the Helmholtz problem in a heterogeneous domain with heterogeneous properties on obstacles. The proposed method is studied for different wavenumbers and numbers of the multiscale basis functions.

Numerical simulation of language interactions using online coupled Generalized Multiscale Finite Element Method

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 fields.Since 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.

Meshfree Generalized Multiscale Finite Element Method

In this paper, we propose a new multiscale approach with a meshfree coarse scale. A coarse scale is constructed on the basis of an already existing computational grid on a fine scale, depending on the heterogeneous parameters of the problem. This approach is based on the Generalized Multiscale Finite Element Method (GMsFEM), where the heterogeneous parameters of the problem are taken into account on a coarse scale using multiscale basis functions. These multiscale basis functions are constructed at an offline stage using local spectral problems. To represent the fractures on a fine grid, the Discrete Fracture Model (DFM) is used. The results of a numerical solution for two-dimensional and three-dimensional problems are presented.

Multiscale model reduction for stochastic elasticity problems using ensemble variable-separated method

Elasticity is a fundamental model in mechanics and material sciences. In the article, we present an ensemble variable-separated multiscale method for elasticity problems in random media. A variable-separated method is utilized to provide a low-rank approximation for random fields. The variable-separated method is integrated with the Generalized Multiscale Finite Element Method (GMsFEM) to obtain an efficient reduced model. To this end, a few local problems are solved for snapshots to construct the multiscale basis functions in GMsFEM. An ensemble method is used to construct stochastic basis functions shared by a series of elasticity equations, and a residual decomposition is adopted to calculate the deterministic physical components for all ensemble members. The ensemble variable-separation is used to solve the local problems of the multiscale basis functions and achieves efficient online-computation for the basis. Then, under the Galerkin projection of the generalized finite element method, the solutions of stochastic elasticity problems are projected onto the low-dimensional space spanned by the online multiscale basis functions. This renders an effective coarse model. The ensemble variable-separated multiscale method is applied to steady-state elastic deformation and elastic wave propagation in random media. A few numerical results are carried out to demonstrate the accuracy and efficiency of the presented method.

A computational macroscale model for the time fractional poroelasticity problem in fractured and heterogeneous media

In this paper, we consider the poroelasticity problem with a time memory formalism that couples the pressure and displacement, and we assume this multiphysics process occurs in multicontinuum media. A coupled system of equations for pressures in each continuum and elasticity equations for displacements of the medium are included in the mathematical model.Based on the Caputo’s time fractional derivative, we derive an implicit finite difference approximation for time discretization. Also, a Discrete Fracture Model (DFM) is used to model fluid flow through fractures and treat the complex network of fractures. Further, we develop a coarse grid approximation based on the Generalized Multiscale Finite Element Method (GMsFEM), where we solve local spectral problems for construction of the multiscale basis functions. The main idea of the proposed method is to reduce the dimensionality of the problem because our model equation has multiple fractional powers, there multiple unknowns with memory effects. Consequently, the solution is on a coarse grid, which saves some computational time.We present numerical results for the two-dimensional model problems in fractured heterogeneous porous media. After, we investigate error analysis between reference (fine-scale) solution and multiscale solution with different numbers of multiscale basis functions. The results show that on a coarse grid, the proposed approach can achieve good accuracy.