Korea-Australia Rheology Journal, Vol.25, No.1, pp.55-64 (2013) www.springer.com/13367 DOI: 10.1007/s13367-013-0006-9 An efficient iterative scheme for the highly constrained augmented Stokes problem for the numerical simulation of flows in porous media Wook Ryol Hwang1,*, Oliver G. Harlen2,§ and Mark A. Walkley3 1School of Mechanical Engineering, Research Center for Aircraft Parts Technology (ReCAPT), Gyeongsang National University, Gajwa-dong 900, Jinju, 660-701, Korea 2Department of Applied Mathematics, University of Leeds, Leeds, LS2 9JT, United Kingdom 3School of Computing, University of Leeds, Leeds, LS2 9JT, United Kingdom (Received November 24, 2012; final revision received January 21, 2013; accepted January 28, 2013) In this work, we present a new efficient iterative solution technique for large sparse matrix systems that are necessary in the mixed finite-element formulation for flow simulations of porous media with complex 3D architectures in a representative volume element. Augmented Stokes flow problems with the periodic boundary condition and the immersed solid body as constraints have been investigated, which form a class of highly constrained saddle point problems mathematically. By solving the generalized eigenvalue problem based on block reduction of the discrete systems, we investigate structures of the solution space and its sub- spaces and propose the exact form of the block preconditioner. The exact Schur complement using the fun- damental solution has been proposed to implement the block-preconditioning problem with constraints. Additionally, the algebraic multigrid method and the diagonally scaled conjugate gradient method are applied to the preconditioning sub-block system and a Krylov subspace method (MINRES) is employed as an outer solver. We report the performance of the present solver through example problems in 2D and 3D, in comparison with the approximate Schur complement method. We show that the number of iterations to reach the convergence is independent of the problem size, which implies that the performance of the present iterative solver is close to O(N). Keywords: flow in porous media, representative volume element (RVE), iterative solver, block precondi- tioning, algebraic multigrid method 1. Introduction of complex porous microstructures for the application to the composite manufacturing. There are two necessary In this work, we consider a fast and efficient iterative ingredients in dealing with this class of porous media solution technique for the numerical simulation of flows flows: one is the treatment of the periodic boundary con- in porous media with complex micro-architectures to dition and the other is the introduction of the solid bodies investigate the flow behaviors such as the permeability, within the flow. As we will introduce later, mathematical the mobility of fluids with shear-dependent viscosity or treatments for these two lead to highly constrained flow the flow resistance of viscoelastic fluids, which has var- problem, the so-called augmented Stokes problem, par- ious industrial applications: e.g. liquid molding in com- ticularly with mixed formulation of the finite-element posite manufacturing, the packed-bed reactor in chemical method. A suitably preconditioned iterative scheme is engineering, the secondary-oil recovery in petroleum essential for successful flow simulations of large-scale industries and various filters in automobile industries. Due problems, as the use of a direct solver is impractical and to its repeated structure, it is necessary to introduce a rep- even impossible for large 3D problems. In this work, we resentative volume element containing a small number of aim to develop a new efficient iterative solution technique microstructures with periodic boundary conditions for for the highly constrained large sparse matrix system effective numerical simulations. Good examples are the using specific choices of block-preconditioning, which is works of one of the authors’ group (Wang and Hwang, specifically tailored for flow simulations of porous media 2008; Liu and Hwang, 2009; Hwang and Advani, 2010; within a unit cell. Liu and Hwang, 2012) in which the authors modeled 2D To introduce the problem, let us first consider the stan- and 3D structures in bi-periodic or tri-periodic unit cells dard Stokes problem in a domain Ω. Since fluid inertia is containing fibers or fiber tows to predict the permeability often neglected on scale of interest with flows in porous media, the Stokes flow is usually assumed as follows: * Corresponding author: [email protected] § Co-corresponding author: [email protected] ∇⋅ σ ==0, ∇⋅ u 0 (1) © 2013 The Korean Society of Rheology and Springer 55 Wook Ryol Hwang, Oliver G. Harlen and Mark A. Walkley subjected to the following boundary conditions: It is worthwhile to briefly introduce their Krylov subspace/ multigrid method, as we will further extend their method in uu==, on ∂Ω and t t, on ∂Ω .(2) u t this work for much more complex highly constrained system The stress is σ = –2pI + ηD with the pressure p, the with the presence of the immersed solid bodies or the peri- identity tensor I, the viscosity η and the rate-of-the defor- odic boundary condition. For the discrete Stokes problem, mation tensor D = ()12⁄ ()∇uu+()∇ T . Suppose that the Silverster and Wathen (1994) introduced a finite element domain boundary Γ∂Ω()= be composed of the Dirichlet- mass matrix as a block preconditioner for the pressure ∂Ω ∂Ω type boundary u and the Neumann-type boundary t unknowns and the discrete Laplacian as a preconditioning ∂Ω ∂Ω ∂Ω and = u∪ t . From the standard Galerkin approx- block for the velocity unknowns. The mass matrix has been imation with the velocity and the pressure as the primitive shown to be spectrally equivalent to the exact Schur com- –1 variables, one can obtain the following weak form for this plement SGK= GT and it can be solved by a diagonally problem: Find ()u,p such that scaled CG method within a fixed number of iterations (Elman et al., 2005). Also, the discrete Laplacian can be ()Ω∇⋅v ηDu[] []Ωv tv⋅Γ –∫Ω p d +∫Ω 2 :D d = ∫∂Ω d , (3) optimally inverted by the multigrid method for the precon- t ditioning problem of the velocity unknowns. They showed ()Ω∇⋅u ∫Ω q d = 0 (4) that this combination of the approximate preconditioners clusters the eigenvalue spectrum independent of the mesh for all the admissible weighting functions ()v,q . The dis- size and thereby the convergence of the Krylov subspace crete finite element matrix system for Eqs. (3) and (4) can outer iterative scheme (MINRES) can be guaranteed within be written in a block-matrix form with a suitable com- a fixed number of iterations. The CPU time as well as the bination of discretized spaces for the velocity and the pres- memory usage is observed to scale linearly with the number sure as of degree of freedom. That is, the ultimate ON() perfor- ˜ mance of the solution scheme has been established. KG⎛⎞u ⎛⎞˜f KG ⎜⎟==, A .(5)In this work, we aim to develop a new efficient iter- T ⎝⎠ Stokes T G 0 ⎝⎠p˜ 0 G 0 ative solution technique for the highly constrained large The variable u˜ is a collective unknown for the discrete sparse matrix system using specific choices of block-pre- velocity variables, p˜ for the discrete pressure variables conditioning. Augmented Stokes flow problems with the and ˜f is the work equivalent nodal force due to the trac- periodic boundary condition and the immersed solid tion boundary condition. (The symbols with tilde indicate body as additional constraints have been investigated. By the discretized variables throughout this work.) solving the generalized eigenvalue problem in block The linear system shown in Eq.(5) is already highly con- matrix form, we propose an exact form of the block pre- strained and can be classified as the saddle point problem, conditioner to achieve the ON() performance of the iter- which means that the matrix AStokes is indefinite though ative solution technique. The exact Schur complement symmetric and it will have positive and also negative using the fundamental solution has been proposed to eigenvalues and pivots while elimination (Strang, 2007). implement the block preconditioner with the constraints. (We will consider additional constraints to the discrete The paper is organized as follows. In Sec. 2, we intro- Stokes problem in Eq. (5) in this work and therefore the duce the mathematical framework for the two highly class of the flow problem of interest in the present work constrained Stokes problem by presenting the weak form may be best called highly constrained augmented Stokes and the structure of the block matrices in their dis- problem.) This complication has originated from the pres- cretized form. In Sec. 3, we investigate the solution ence of the incompressibility constraint. Although the space of the discretized weak form by solving the gen- basic methods for solving large sparse indefinite problems eralized eigenvalue problem with block reduction and are the minimum residual (MINRES) and the generalized then propose an exact form of the block preconditioner minimum residual (GMRES), the iterative methods do not that guarantees fixed number of iteration for conver- behave satisfactory or even do not converge without a gence. In Sec. 4, we introduce the implementation tech- suitable choice of the preconditioner. An extremely effi- niques for this preconditioned iterative method, cient, indeed ON(), iterative scheme for the discrete particularly the exact Schur complement method using Stokes problem of Eq. (5) has been proposed by Silverster the fundamental solution. Finally, the performance of the and Wathen (1994). This approach formulates a block present solver will be presented through example prob- structured preconditioner and uses appropriate techniques lems in 2D and 3D, in comparison with the approximate for each block system such that it can be solved with opti- Schur complement method. We show that the number of mal efficiency.