GPU-based matrix-free finite element solver exploiting symmetry of elemental matrices
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GPU-based matrix-free finite element solver exploiting symmetry of elemental matrices Utpal Kiran1 · Sachin Singh Gautam1 · Deepak Sharma1 Received: 4 January 2020 / Accepted: 10 June 2020 © Springer-Verlag GmbH Austria, part of Springer Nature 2020
Abstract Matrix-free solvers for finite element method (FEM) avoid assembly of elemental matrices and replace sparse matrix-vector multiplication required in iterative solution method by an element level dense matrix-vector product. In this paper, a novel matrixfree strategy for FEM is proposed which computes element level matrix-vector product by using only the symmetric part of the elemental matrices. The proposed strategy is developed to take advantage of the massive parallelism of Graphics Processing Unit (GPU). A unique data structure is also introduced which ensures localized and coalesced memory access suitable for a GPU while storing only the symmetric part of the elemental matrices. In addition, the proposed strategy emphasizes the efficient use of register cache, uniform workload distribution, reducing thread synchronization, and maintaining sufficient granularity to make the best use of GPU resources. The performance of the proposed strategy is evaluated by solving elasticity and heat conduction problems using 4-noded quadrilateral element with two degrees of freedom (DOFs) and one DOF per node, respectively. The performance is compared with the matrixfree solver strategies on GPU from the literature. It is found that a maximum speedup of 4.9 × is obtained for the elasticity problem and a maximum of 3.2 × speedup for the heat conduction problem. Further, the proposed strategy takes the least amount of GPU memory as compared to the existing strategies. Keywords Matrix-free solver · Finite element method · GPU · CUDA · Parallel computing
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Deepak Sharma [email protected] Utpal Kiran [email protected] Sachin Singh Gautam [email protected]
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Department of Mechanical Engineering, Indian Institute of Technology, Guwahati, Assam 781039, India
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U. Kiran et al.
Mathematics Subject Classification 74S05 · 65Y05
1 Introduction Finite element method (FEM) is one of the most extensively used numerical methods to solve real-world problems governed by ordinary/partial differential equations. The popularity of FEM is primarily due to its ability to handle complex geometries, high accuracy, and applicability to a wide range of problems. However, FEM can be computationally expensive for complex real-life problems [3,19,22] that require a large number of degrees of freedom (DOFs) to obtain desired solution. Although there has been an exponential increase in computational resources, the computational cost of FEM is still the main bottleneck for many large-scale problems. In the literature, the high performance computing (HPC) techniques have been used to handle expensive computation required in FEM [41,45]. Recently, Graphics Processing Unit (GPU)-based computation has become immensely popular for HPC implementation. The importance of GPU for HPC applications can be un
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