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Download ##Common.Downloadpdf DOI: 10.24178/ijsms.2017.2.1.01 IJSMS Vol 2(1) Mar 2017 Treatment of Block-Based Sparse Matrices in Domain Decomposition Method Abul Mukid Mohammad Mukaddes Ryuji Shioya Department of Industrial and Production Engineering Faculty of Information Science and Arts Shahjalal University of Science and Technology Toyo University Sylhet, Bangladesh Kawagoe, Japan [email protected] [email protected] Masao Ogino Hiroshi Kanayama Information Technology Center, Nagoya University, Faculty of Mathematical and Physical Sciences Nagoya, Japan Japan Women’s University, Japan [email protected]–u.ac.jp [email protected] Abstract— The domain decomposition method involves the finite considered in this paper. The hierarchical domain element solution of problems in the parallel computer. The finite decomposition method (HDDM) has been adopted for the element discretization leads to the solution of large systems of parallel solver in those modules. Simplified diagonal scaling linear equation whose matrix is naturally sparse. The use of and balancing domain decomposition (BDD) are used to proper storing techniques for sparse matrix is fundamental accelerate the CG iterations. especially when dealing with large scale problems typical of industrial applications. The aim of this research is to review the There are several ways to speed-up the domain sparsity pattern of the matrices originating from the decomposition method; one-reducing CG iterations using a discretization of the elasto-plastic and thermal-convection suitable preconditioner [4]; two-storing the matrix efficiently problems. Some practical strategies dealing with sparsity pattern [5]; three-constructing the program more parallel. The finite in the finite element code of adventure system are recalled. element discretization of the partial differential equations Several efficient storage schemes to store the matrix originating leads to sparse linear system with different sparsity patterns from elasto-plastic and thermal-convection problems have been based on the selected problems. An efficient storing technique proposed. In the proposed technique, inherent block pattern of can reduce the computational time of matrix-vector the matrix is exploited to locate the matrix element. The multiplication and the used memory. computation in the high performance computer shows better performance compared to the conventional skyline storage The sparsity pattern is described by the distribution of non- method used by the most of the researchers. zero elements of the sparse matrix. The distribution of non- zero elements depends on the topology of the adopted Keywords: sparse matrix, compressed sparse, block size, domain computational grid, on the kind of finite element method decompose, adventure system; chosen and on the kind of problem discretized. Thus the sparsity pattern is completely known before the construction I. INTRODUCTION of the matrix from its nodal connectivity. Therefore the matrix With the advent of high performance computer, can be stored efficiently by excluding the unnecessary a priori researchers are showing interest to analyze large scale three known zero-elements. The use of efficient storing techniques dimensional problems using the finite element method. To get for sparse matrices is computationally important especially the accurate simulation results, the finite elementIJSMS grids should when dealing with large scale problems in the high also be finer. The degrees of freedom (dof) grow rapidly as the performance computer. Moreover, the matrix results in the grid gets finer and nowadays it is customary to deal with discretization of different problems have its own sparsity trillions of dof. At the same time, domain decomposition pattern. The matrix originating from the elasto-plastic and method has become much popular to treat those large scale thermal problems is symmetric while from the thermal- problems in the parallel cluster computer. This method convection is asymmetric. Again structural and thermal- decomposes the whole problem into subproblems and divides convection problem have some inherent block shape. Suppose, the dof into two groups. One group belongs to the dof within thermal problem has one dof per node, elasto-plastic problem the subdomains and the other belongs to the shared dof. The has three dof per node and thermal-convection problem has shared dof is solved by the conjugate gradient (CG) method five dof per node. This dof per node shape the matrices as which involves finite element solution for each subproblem block matrices. The built-in block shape is exploited in storing and matrix-vector multiplication in each CG iteration. of the sparse matrices. In this paper, several sparse matrix We have been developing the Adventure system [1] for the storage schemes have been reviewed and some efficient solution of large scale engineering problems. Among the techniques are proposed. The proposed techniques are modules of the adventure system, Adventure_Solid [2], problem sensitive. The proposed sparse matrix storing Adventure_Thermal [3, 4] and Adventure_sFlow have been International Journal of System Modeling and Simulation (ISSN Online: 2518-0959) 1 DOI: 10.24178/ijsms.2017.2.1.01 IJSMS Vol 2(1) Mar 2017 techniques have been implemented in Adventure_Solid and 0 uI Adventure_sFlow and good performances have been achieved. (1a) Find: u (0) K K K u 0 f I , II IB IT B I II. DOMAIN DECOMPOSTIION 0 uT Discretization of partial differential equations in a domain 0 using the finite element method leads to large and sparse uI linear systems like (0) (0) 0 (1b)Calculate: g w K BI K BB K BT uB f B Ku f . (1) u The domain decomposition method decomposes the T domain into N non-overlapping subdomains, (i) . Thus Step 1: (iteration) the stiffness matrix K of equation (1) could be generated by k subassembly: (2a) Find (k ) : K K K wk 0 N II IB IT (i) (i) (i)T K R K R (2) i1 0 k (i)T where R is the 0-1 matrix which translates the global indices of the nodes into local (subdomain) numbering. (2b) Find: Sw(k) K K K wk BI BB BT Denoting u (i) as the vector corresponding to the elements 0 in (i) , it can be expressed as u (i) R (i)T u . Each u (i) is split (k ) (k ) (k) (i) into degrees of freedom u , which correspond to (i) , 2(c) Update for each k iteration: u B , g and w B (i) k k called interface degrees of freedom, u I for interior degrees g g k (i) k k of freedom and uT for essential boundary conditions (for the w Sw subdomains which has these boundary conditions). The k1 k k k ub ub w (i) (i) subdomain matrix K , vector u are then split accordingly: g k 1 g k k Swk k1 k1 (i) (i) (i) (i) g g K II K IB K IT u I If /* convergence * / 0 0 (i) (i) (i) (i) (i) g g K K BI K BB K BT , u B (3). k1 k1 (i) (i) (i) (i) g g K K K u k TI TB TT T k k g g Similarly equation (1) can be written as k 1 k k k 1 w w g K II K IB K IT u I f I A. Scope of Sparse Storage Schemes K K K u f (4). BI BB BT B B We remark that the Schur complement matrix S is not KTI KTB KTT uT fT constructed explicitly, though the multiplication of a vector with the Schur complement is required in the CG method. After eliminating the interior degrees of freedom, problem The action of the matrix S on a typical vector x can be (4) is reduced to a problem on interface IJSMSimplemented by using the block elements of the matrix K i . Again the Su B g (5) y S x N i i iT N (i) (i) (i)T i1 RB S RB x where the Schur complement S i1 RB S RB is assumed N i i i (6) i1 RB S x to be positive definite, uB is the vector of the unknown N i i (i) variables on the interface, g is a known vector and S are i1 RB y the local Schur complements of subdomain i 1,..., N , where, assumed to be positive semi-definite. The problem (5) is i i i i iT i 1 i i y S x K BB K IB K II K IB x . (7) solved by the conjugate gradient method. The flow of (i) algorithm of the basic domain decomposition method is shown Action of S on a vector can be implemented by solving next. i a Dirichlet problem on and it needs subdomain wise (0) matrix-vector multiplication twice in each iteration as shown Step 0: Initialize: u B in equation (7). The contribution of the matrix in equation (7) International Journal of System Modeling and Simulation (ISSN Online: 2518-0959) 2 DOI: 10.24178/ijsms.2017.2.1.01 IJSMS Vol 2(1) Mar 2017 comes from subdomains matrix which is stored using non- zero-only sparse matrix storage schemes. III. ADVENTURE SYSTEM We have been developing the adventure system [1] which is a group module to solve the large scale engineering problems. Initially, its target was to solve 100 million degrees of freedom. But some of its module is now able to solve more than 1000 million degrees of freedom. Hierarchical Domain Decomposition Method (HDDM), a parallel finite element technique is used to solve the engineering problems in the Fig 2. Numerical Values of the Matrix parallel computer. The HDDM system uses the conjugate gradient method to solve the interface problem. Matrix-vector IV. SPARSE MATRICES STORING TECHNIQUES multiplication is the most time consuming part in the Efficient storage of sparse matrix effects on both conjugate gradient method. The HDDM system can be computational time and memory especially in the matrix- accelerated either by using a preconditioner or by reducing vector multiplication part.
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