On pore-scale modeling and simulation of reactive transport in 3D geometries

Pore-scale modeling and simulation of reactive flow in porous media has a range of diverse applications, and poses a number of research challenges. It is known that the morphology of a porous medium has significant influence on the local flow rate, which can have a substantial impact on the rate of chemical reactions. While there are a large number of papers and software tools dedicated to simulating either fluid flow in 3D computerized tomography (CT) images or reactive flow using pore-network models, little attention to date has been focused on the pore-scale simulation of sorptive transport in 3D CT images, which is the specific focus of this paper. Here we first present an algorithm for the simulation of such reactive flows directly on images, which is implemented in a sophisticated software package. We then use this software to present numerical results in two resolved geometries, illustrating the importance of pore-scale simulation and the flexibility of our software package.Comment: 15 pages, 6 figure

Comparative Analysis of High Performance Solvers for 3D Elliptic Problems

Abstract. The presented comparative analysis concerns two iterative solvers for 3D linear boundary value problems of elliptic type. After applying the Finite Difference Method (FDM) or the Finite Element Method (FEM) discretization a system of linear algebraic equations has to be solved, where the stiffness matrix is large, sparse and symmetric positive definite. It is well known that the preconditioned conjugate gradient method is the best tool for efficient solution of large-scale symmetric systems with sparse positive definite matrices. Here, the performance of two preconditioners is studied, namely the Modified Incomplete Cholesky factorization MIC(0) and the Circulant Block-Factorization (CBF) preconditioning. Portable parallel codes are developed based on Message Passing Interface (MPI) standards. The comparative analysis is mostly based on the execution times to run the parallel codes. The number of iterations for both preconditioners are also discussed. The performed numerical tests on parallel computer systems demonstrate the level of efficiency of the developed algorithms. The obtained parallel speed-up and efficiency well illustrate the scope of efficient applications.

Comparative Analysis of High Performance Solvers for 3D Elasticity Problems

Abstract. We consider the numerical solution of 3D linear elasticity equations. The investigated problem is described by a coupled system of second order elliptic partial differential equations. This system is then discretized by conforming or nonconforming finite elements. After applying the Finite Element Method (FEM) based discretization, a system of linear algebraic equations has to be solved. In this system the stiffness matrix is large, sparse and symmetric positive definite. In the solution process we utilize a well-known fact that the preconditioned conjugate gradient method is the best tool for efficient solution of large-scale symmetric systems with sparse positive definite matrices. In this context, the displacement decomposition (DD) technique is applied at the first step to construct a preconditioner that is based on a decoupled block diagonal part of the original matrix. Then two preconditioners, namely the Modified Incomplete Cholesky factorization MIC(0) and the Circulant Block-Factorization (CBF) preconditioning, are used to precondition thus obtained block diagonal matrix. As far as the parallel implementation of the proposed solution methods is concerned, we utilize the Message Passing Interface (MPI) communication libraries. The aim of our work is to compare the performance of the two proposed preconditioners: the DD MIC(0) and the DD CBF. The presented comparative analysis is based on the execution times of actual codes run on modern parallel computers. Performed numerical tests demonstrate the level of parallel efficiency and robustness of the proposed algorithms. Furthermore, we discuss the number of iterations resulting from utilization of both preconditioners

High Performance Solver for 3D Elasticity Problems

Abstract. In this paper we consider numerical solution of 3D linear elasticity equations described by a coupled system of second order elliptic partial differential equations. This system is discretized by trilinear parallelepipedal finite elements. Preconditioned Conjugate Gradient iterative method is used for solving large-scale linear algebraic systems arising after the Finite Element Method (FEM) discretization of the problem. The displacement decomposition technique is applied at the first step to construct a preconditioner using the decoupled block diagonal part of the original matrix. Then circulant block factorization is used to precondition thus obtained block diagonal matrix. Since both preconditioning techniques, displacement decomposition and circulant block factorization, are highly parallelizable, a portable parallel FEM code utilizing MPI for communication is implemented. Results of numerical tests performed on a number of modern parallel computers using real life engineering problems from the geosciences (geomechanics in particular) are reported and discussed.

Reduced Multiplicative (BURA-MR) and Additive (BURA-AR) Best Uniform Rational Approximation Methods and Algorithms for Fractional Elliptic Equations

Numerical methods for spectral space-fractional elliptic equations are studied. The boundary value problem is defined in a bounded domain of general geometry, Ω⊂Rd, d∈{1,2,3}. Assuming that the finite difference method (FDM) or the finite element method (FEM) is applied for discretization in space, the approximate solution is described by the system of linear algebraic equations Aαu=f, α∈(0,1). Although matrix A∈RN×N is sparse, symmetric and positive definite (SPD), matrix Aα is dense. The recent achievements in the field are determined by methods that reduce the original non-local problem to solving k auxiliary linear systems with sparse SPD matrices that can be expressed as positive diagonal perturbations of A. The present study is in the spirit of the BURA method, based on the best uniform rational approximation rα,k(t) of degree k of tα in the interval [0,1]. The introduced additive BURA-AR and multiplicative BURA-MR methods follow the observation that the matrices of part of the auxiliary systems possess very different properties. As a result, solution methods with substantially improved computational complexity are developed. In this paper, we present new theoretical characterizations of the BURA parameters, which gives a theoretical justification for the new methods. The theoretical estimates are supported by a set of representative numerical tests. The new theoretical and experimental results raise the question of whether the almost optimal estimate of the computational complexity of the BURA method in the form O(Nlog2N) can be improved