Coarse-scale Equations Research Articles

On the upscaling of the diffusion equation in a heterogeneous medium by the two different methods

Upscaling the flow equations appears in many studies related to diffusion, heat conductivity, and flows in porous media. Variable coefficients of the original fine-scale process description are substituted by averaged constant values. The different upscaling procedures have been suggested in the literature. The question arises, whether they result in the same or different upscaled models; moreover, whether the solutions of the coarse-scale equations provide a reasonably accurate description of the fine scale. In this work, we consider three sample 1D diffusion problems with periodic coefficients: diffusion with or without the external source, as well as diffusion with advection. These problems are upscaled by the two methods. The method of direct upscaling selects the averaged coefficients to provide the minimum difference between the solutions of the coarse-scale and fine-scale models. The method of continuous upscaling, developed previously, consists of the continuous averaging transformation between the distant scales. New expressions for the upscaled diffusion coefficients were derived for this case. It turns out that the direct upscaling results in multiple optimum parameters of the upscaled model. Meanwhile, continuous upscaling points at one of them. The coarse-scale approximation of a fine-scale solution may be unsatisfactory, even for the best choice of the upscaled coefficient. A numerical study demonstrates a nontrivial behavior of the diffusion coefficient under continuous upscaling, while it converges to an asymptotic value.

Machine-learning-based data-driven discovery of nonlinear phase-field dynamics.

One of the main questions regarding complex systems at large scales concerns the effective interactions and driving forces that emerge from the detailed microscopic properties. Coarse-grained models aim to describe complex systems in terms of coarse-scale equationswith a reduced number of degrees of freedom. Recent developments in machine-learning algorithms have significantly empowered the discovery process of governing equationsdirectly from data. However, it remains difficult to discover partial differential equations(PDEs) with high-order derivatives. In this paper, we present data-driven architectures based on a multilayer perceptron, a convolutional neural network (CNN), and a combination of a CNN and long short-term memory structures for discovering the nonlinear equationsof motion for phase-field models with nonconserved and conserved order parameters. The well-known Allen-Cahn, Cahn-Hilliard, and phase-field crystal models were used as test cases. Two conceptually different types of implementations were used: (a) guided by physical intuition (such as the local dependence of the derivatives) and (b) in the absence of any physical assumptions (black-box model). We show that not only can we effectively learn the time derivatives of the field in both scenarios, but we can also use the data-driven PDEs to propagate the field in time and achieve results in good agreement with the original PDEs.

A deep learning based nonlinear upscaling method for transport equations

We will develop a nonlinear upscaling method for nonlinear transport equations. The proposed scheme gives a coarse scale equation for the cell average of the solution. In order to compute the parameters in the coarse scale equation, a local downscaling operator is constructed. This downscaling operation recovers fine scale properties using cell averages. This is achieved by solving the equation on an oversampling region with the given cell average as constraint. Due to the nonlinearity, one needs to compute these downscaling operations on the fly and cannot pre-compute these quantities. In order to give an efficient downscaling operation, we apply a deep learning approach. We will use a deep neural network to approximate the downscaling operation. Our numerical results show that the proposed scheme can achieve good accuracy and efficiency.

A variational multiscale immersed meshfree method for heterogeneous materials

We introduce an immersed meshfree formulation for modeling heterogeneous materials with flexible non-body-fitted discretizations, approximations, and quadrature rules. The interfacial compatibility condition is imposed by a volumetric constraint, which avoids a tedious contour integral for complex material geometry. The proposed immersed approach is formulated under a variational multiscale based formulation, termed the variational multiscale immersed method (VMIM). Under this framework, the solution approximation on either the foreground or the background can be decoupled into coarse-scale and fine-scale in the variational equations, where the fine-scale approximation represents a correction to the residual of the coarse-scale equations. The resulting fine-scale solution leads to a residual-based stabilization in the VMIM discrete equations. The employment of reproducing kernel (RK) approximation for the coarse- and fine-scale variables allows arbitrary order of continuity in the approximation, which is particularly advantageous for modeling heterogeneous materials. The effectiveness of VMIM is demonstrated with several numerical examples, showing accuracy, stability, and discretization efficiency of the proposed method.

K-Values-Based Upscaling of Compositional Simulation

Summary Enhanced-oil-recovery (EOR) processes involve complex flow, transport, and thermodynamic interactions; as a result, compositional simulation is necessary for accurate representation of the physics. Flow simulation of compositional systems with high-resolution reservoir models is computationally intensive because of the large number of unknowns and the strong nonlinear interactions. Thus, there is a great need for upscaling methods of compositional processes. The complex multiscale interactions between the phase behavior and the heterogeneities lie at the core of the difficulty in constructing consistent upscaling procedures. We use a mass-conservative formulation and introduce upscaled phase-molar-mobility functions for coarse-scale modeling of multiphase flow. These upscaled flow functions account for the subgrid effects caused by the absolute permeability and relative permeability variations, as well as the effects of compressibility. Upscaling of the phase behavior is performed as follows. We assume that instantaneous thermodynamic equilibrium is valid on the fine scale, and we derive coarse-scale equations in which the phase behavior may not necessarily be at equilibrium. The upscaled thermodynamic functions, which represent differences in the component fugacities, are used to account for the nonequilibrium effects on the coarse scale. We demonstrate that the upscaled phase-behavior functions transform the equilibrium phase space on the fine scale to a region of similar shape, but with tilted tie-lines on the coarse space. The numerical framework uses K-values that depend on the orientation of the tie-lines in the new nonequilibrium phase space and the sign of upscaled thermodynamic functions. The proposed methodology is applied to challenging gas-injection problems with large numbers of components and highly heterogeneous permeability fields. The K-value-based coarse-scale operator produces results that are in good agreement with the fine-scale solutions for the quantities of interest, including the component overall compositions and saturation distributions.

Spatial averaging of a dissipative particle dynamics model for active suspensions

Starting from a fine-scale dissipative particle dynamics (DPD) model of self-motile point particles, we derive meso-scale continuum equations by applying a spatial averaging version of the Irving--Kirkwood--Noll procedure. Since the method does not rely on kinetic theory, the derivation is valid for highly concentrated particle systems. Spatial averaging yields a stochastic continuum equations similar to those of Toner and Tu. However, our theory also involves a constitutive equation for the average fluctuation force. According to this equation, both the strength and the probability distribution vary with time and position through the effective mass density. The statistics of the fluctuation force also depend on the fine scale dissipative force equation, the physical temperature, and two additional parameters which characterize fluctuation strengths. Although the self-propulsion force entering our DPD model contains no explicit mechanism for aligning the velocities of neighboring particles, our averaged coarse-scale equations include the commonly encountered cubically nonlinear (internal) body force density.

Implementation of the Multiscale Stochastic Finite Element Method on Elliptic PDE Problems

In this study, a multi-scale finite element method was proposed to solve two linear scale-coupling stochastic elliptic PDE problems, a tightly stretched wire and flow through porous media. At microscopic level, the main idea was to form coarse-scale equations with a prescribed analytic form that may differ from the underlying fine-scale equations. The relevant stochastic homogenization theory was proposed to model the effective global material coefficient matrix. At the macroscopic level, the Karhunen–Loeve decomposition was coupled with a Polynomial Chaos expansion in conjunction with a Galerkin projection to achieve an efficient implementation of the randomness into the solution procedure. Various stochastic methods were used to plug the microscopic cell to the global system. Strategy and relevant algorithms were developed to boost computational efficiency and to break the curse of dimension. The results of numerical examples were shown consistent with ones from literature. It indicates that the proposed numerical method can act as a paradigm for general stochastic partial differential equations involving multi-scale stochastic data. After some modification, the proposed numerical method could be extended to diverse scientific disciplines such as geophysics, material science, biological systems, chemical physics, oceanography, and astrophysics, etc.

Dispersive computational continua

Abstract The two primary objectives of the present manuscript are: (i) to develop a variant of the computational continua formulation (C 2 ) with outstanding dispersive properties, and (ii) to conduct a rigorous dispersion analysis of it. The ability of the C 2 formulation to capture dispersive behavior stems from its underlying formulation, which does not explicitly assume scale separation and accounts for microstructures of finite size. The dispersion study in heterogeneous elastic media with periodic microstructure has been conducted using both analytical and numerical approaches. The so-called analytical dispersion analysis is based on the Floquet–Bloch wave solution, while the numerical dispersion analysis is based on the modal analysis of the discrete coupled fine–coarse-scale equations. The dispersive curves obtained from the dispersive C 2 formulations were compared with the classical exact Floquet–Bloch wave solution, hereafter referred to as the reference dispersive curve. It has been observed that in the case of the unit cell sizes being either half of the coarse-scale element size or equal to it, the dispersive curves obtained by the dispersive C 2 formulation are practically identical to the reference solution. For other cases, the dispersive C 2 solution is in good agreement with the reference solution. The dispersion analysis results have been further verified by the wave propagation problem in a periodic heterogeneous medium with a wavelength comparable to the microstructural size.

Variational Multiscale Modeling of Charge Transport in Li Metal Battery

Battery performance highly depends on charge transport phenomena. The length scale of battery charge transport modeling is from nanometers to millimeters, therefore it makes the computational modeling difficult to accomplish. Moreover, the charge transport is well known at macro-scale and is easy to examine but there is difficulty in understanding the phenomena at micro-scale. Hence there is need for a framework that not only predicts the performance at macro-scale but also couples micro-scale details (e.g., microstructure and electrochemical reactions). The variational multiscale method (VMM) can be utilized as a paradigm for deriving models and numerical methods capable of dealing with multiscale charge transport phenomena [1]. The VMM can be applied to a broad range of problems and applications like complex electrochemical systems (e.g., Lithium batteries). VMM modeling is one of the essential tools to ascertain varied features of a problem concerning multiscale charge transport phenomena. In addition, VMM can be utilized to investigate continuum-scale systems with tracking detailed, electrochemical dynamics at the micro-scale [2]. Thus developing a VMM modeling framework to investigate the aforementioned problem is of much importance for obtaining reliable computational simulation results. We will focus on building a link between battery micro-scale details and battery continuum level performance via VMM. The basic idea is to decompose the battery electrochemical modeling ODE/PDE solutions into two separate solutions including fine and coarse scale contributions. Fine scale equations will be solved with regard to coarse scale residuals and then the solution will be eliminated from coarse scale equations. Consequently, modified coarse scale equations which include the effects of fine scale can be obtained [3]. We will validate our model against Li metal batteries experimental studies on charge transport. Employing VMM, we can obtain charge transport at different levels and we expect the simulation and experimental data show better accuracy than traditional continue level simulations. Reference [1] T.J.R. Hughes, G.R. Feijόo, L. Mazzei, J. Quincy, Computer Methods in Applied Mechanics and Engineering, 3-24, 166 (1998) [2] S. Lee, The University of Michigan, 2011 [3] M.G. Larson, A. Målqvist, Computer Methods in Applied Mechanics and Engineering, 2313-2324, 196 (2007)

Adaptive heterogeneous multiscale methods for immiscible two-phase flow in porous media

In this contribution, we present the first formulation of a heterogeneous multiscale method for an incompressible immiscible two-phase flow system with degenerate permeabilities. The method is in a general formulation, which includes oversampling. We do not specify the discretization of the derived macroscopic equation, but we give two examples of possible realizations, suggesting a finite element solver for the fine scale and a vertex-centered finite volume method for the effective coarse scale equations. Assuming periodicity, we show that the method is equivalent to a discretization of the homogenized equation. We provide an a posteriori estimate for the error between the homogenized solutions of the pressure and saturation equations and the corresponding HMM approximations. The error estimate is based on the results recently achieved as reported by Cancès et al. (Math. Comp. 83(285):153–188, 2014). An a posteriori error estimate for vertex-centered finite volume discretizations of immiscible incompressible two-phase flow.

An Adaptive Discontinuous Galerkin Multiscale Method for Elliptic Problems

An adaptive discontinuous Galerkin multiscale method driven by an energy norm a posteriori error bound is proposed. The method is based on splitting the problem into a coarse and fine scale. Localized fine scale constituent problems are solved on patches of the domain and are used to obtain a modified coarse scale equation. The coarse scale equation has considerably less degrees of freedom than the original problem. The a posteriori error bound is used within an adaptive algorithm to tune the critical parameters, i.e., the refinement level and the size of the different patches on which the fine scale constituent problems are solved. The fine scale computations are completely parallelizable, since no communication between different processors is required for solving the constituent fine scale problems. The convergence of the method, the performance of the adaptive strategy, and the computational effort involved are investigated through a series of numerical experiments.

Geological Storage of CO2: Modeling Applications for Large‐Scale SimulationsGeofluids (2012) NordbottenJ.M., CeliaM.A., Wiley & Sons, Hoboken, New Jersey, USA241 pp. ISBN 978‐0‐470‐88946‐6

Geological Storage of <scp>CO</scp>₂: Modeling Applications for Large‐Scale Simulations<i>Geofluids</i> (2012) NordbottenJ.M., CeliaM.A., <i>Wiley &amp; Sons</i>, Hoboken, New Jersey, USA241<i> pp. ISBN 978‐0‐470‐88946‐6</i>

Operator-Based Multiscale Method for Compressible Flow

Summary Multiscale methods have been developed for accurate and efficient numerical solution of flow problems in large-scale heterogeneous reservoirs. A scalable and extendible Operator-Based Multiscale Method (OBMM) is described here. OBMM is cast as a general algebraic framework. It is natural and convenient to incorporate more physics in OBMM for multiscale computation. In OBMM, two operators are constructed: prolongation and restriction. The prolongation operator is constructed by assembling the multiscale basis functions. The specific form of the restriction operator depends on the coarse-scale discretization formulation (e.g., finite-volume or finite-element). The coarse-scale pressure equation is obtained algebraically by applying the prolongation and restriction operators to the fine-scale flow equations. Solving the coarse-scale equation results in a high-quality coarse-scale pressure. The fine-scale pressure can be reconstructed by applying the prolongation operator to the coarse-scale pressure. A conservative fine-scale velocity field is then reconstructed to solve the transport (saturation) equation. We describe the OBMM approach for multiscale modeling of compressible multiphase flow. We show that extension from incompressible to compressible flows is straightforward. No special treatment for compressibility is required. The efficiency of multi-scale formulations over standard fine-scale methods is retained by OBMM. The accuracy of OBMM is demonstrated using several numerical examples including a challenging depletion problem in a strongly heterogeneous permeability field (SPE 10).

A hierarchical multiscale framework for problems with multiscale source terms

This paper presents a hierarchical multiscale framework for problems that involve multiscale source terms. An assumption on the additive decomposition of the source function results in consistent decoupling of the fully coupled system and constitutes the new method. The structure of this decomposition is investigated and its mathematical implications are delineated. This method results in variational embedding of fine-scale information that is derived from the fine-scale equations, in the corresponding coarse-scale equations. It therefore provides a mathematically consistent way of bridging information between disparate spatial scales in the response function that are induced by multiscale forcing functions.

ON NUMERICAL UPSCALING FOR FLOWS IN HETEROGENEOUS POROUS MEDIA

AbstractA numerical upscaling approach, NU, for solving multiscale elliptic problems is discussed. The main components of this NU are: i) local solve of aux-iliary problems in grid blocks and formal upscaling of the obtained results to build a coarse scale equation; ii) global solve of the upscaled coarse scale equation; and iii) reconstruction of a fine scale solution by solving local block problems on a dual coarse grid. By its structure NU is similar to other methods for solving multiscale elliptic problems, such as the multiscale finite element method, the multiscale mixed finite element method, the numerical subgrid upscaling method, heterogeneous mul-tiscale method, and the multiscale finite volume method. The difference with those methods is in the way the coarse scale equation is build and solved, and in the way the fine scale solution is reconstructed. Essential components of the presented here NU approach are the formal homogenization in the coarse blocks and the usage of so called multipoint flux approximation method, MPFA. Unlike the usual usage of MPFA as a discretization method for single scale elliptic problems with tensor discontinuous coefficients, we consider its usage as a part of a numerical upscaling approach. An aim of this paper is to compare the performance of NU with the one of MsFEM for ceratin multiscale problems. In particular, it is shown that the resonance effect, which limits the application of the Multiscale FEM, does not appear, or it is significantly relaxed, when the presented here numerical upscaling approach is applied

Analysis Of Upwind Method For Piecewise Deterministic Markov Processes

AbstractA numerical upscaling approach, NU, for solving multiscale elliptic problems is discussed. The main components of this NU are: i) local solve of aux-iliary problems in grid blocks and formal upscaling of the obtained results to build a coarse scale equation; ii) global solve of the upscaled coarse scale equation; and iii) reconstruction of a fine scale solution by solving local block problems on a dual coarse grid. By its structure NU is similar to other methods for solving multiscale elliptic problems, such as the multiscale finite element method, the multiscale mixed finite element method, the numerical subgrid upscaling method, heterogeneous mul-tiscale method, and the multiscale finite volume method. The difference with those methods is in the way the coarse scale equation is build and solved, and in the way the fine scale solution is reconstructed. Essential components of the presented here NU approach are the formal homogenization in the coarse blocks and the usage of so called multipoint flux approximation method, MPFA. Unlike the usual usage of MPFA as a discretization method for single scale elliptic problems with tensor discontinuous coefficients, we consider its usage as a part of a numerical upscaling approach. An aim of this paper is to compare the performance of NU with the one of MsFEM for ceratin multiscale problems. In particular, it is shown that the resonance effect, which limits the application of the Multiscale FEM, does not appear, or it is significantly relaxed, when the presented here numerical upscaling approach is applied.

Generalized Mathematical Homogenization: From theory to practice

A Generalized Mathematical Homogenization (GMH) theory for coupling atomistic and continuum descriptions at finite temperature has been extended to account for many-body potentials. GMH gives rise to the constitutive law-free coarse scale equations where coupled continuum thermo-mechanical equations are directly derived from molecular dynamics equations. An integrated computational approach, which seamlessly integrates ABAQUS for coarse scale (finite element method) computations with a molecular dynamics code, has been developed. The accuracy and efficiency of the method has been studied for crystal silicon structures including failure predictions of nanowires.

A Multiresolution Strategy for Reduction of Elliptic PDEs and Eigenvalue Problems

In many practical problems coefficients of PDEs are changing across many spatial or temporal scales, whereas we might be interested in the behavior of the solution only on some relatively coarse scale. We approach the problem of capturing the influence of fine scales on the behavior of the solution on a coarse scale using the multiresolution strategy. Considering adjacent scales of a multiresolution analysis, we explicitly eliminate variables associated with the finer scale, which leaves us with a coarse-scale equation. We use the term reduction to designate a recursive application of this procedure over a finite number of scales.We present a multiresolution strategy for reduction of self-adjoint, strictly elliptic operators in one and two dimensions. It is known that the non-standard form for a wide class of operators has fast off-diagonal decay and the rate of decay is controlled by the number of vanishing moments of the wavelet. We prove that the reduction procedure preserves the rate of decay over any finite number of scales and therefore results in sparse matrices for computational purposes. Furthermore, the reduction procedure approximately preserves small eigenvalues of self-adjoint, strictly elliptic operators. We also introduce a modified reduction procedure which preserves the small eigenvalues with greater accuracy than the standard reduction procedure and obtain estimates for the perturbation of those eigenvalues. Finally, we discuss potential extensions of the reduction procedure to parabolic and hyperbolic problems.

Coarse scale models of two phase flow in heterogeneous reservoirs: volume averaged equations and their relationship to existing upscaling techniques

Coarse scale equations applicable to two phase flow in heterogeneous reservoirs are developed via a volume average of the fine scale saturation equation. The resulting coarse scale description, in the form of an average saturation equation and a sequence of moment equations, is then used to assess several methods for the upscaling of detailed reservoir characterizations. The approaches considered are the use of pseudo‐relative permeabilities, the nonuniform coarsening of the fine grid geostatistical model, and the use of higher moments of the fine scale variables. Using the volume averaged equations as a framework, the underlying assumptions, strengths and limitations of each of these methods are discussed. Previous findings demonstrating the accuracy of coarse scale relative permeability functions based on higher moments are readily explained in terms of the volume average saturation and moment equations. Finally, the volume averaged equations are simplified for the unit mobility ratio case and applied to the direct solution of a model coarse scale problem.