Research Papers

Large Scale Finite Element Analysis Via Assembly-Free Deflated Conjugate Gradient

[+] Author and Article Information
Praveen Yadav

Department of Mechanical Engineering,
Madison, WI 53706

Krishnan Suresh

Department of Mechanical Engineering,
Madison, WI 53706
e-mail: suresh@engr.wisc.edu

Contributed by the Computers and Information Division of ASME for publication in the JOURNAL OF COMPUTING AND INFORMATION SCIENCE IN ENGINEERING. Manuscript received June 27, 2014; final manuscript received September 10, 2014; published online October 7, 2014. Editor: Bahram Ravani.

J. Comput. Inf. Sci. Eng 14(4), 041008 (Oct 07, 2014) (9 pages) Paper No: JCISE-14-1225; doi: 10.1115/1.4028591 History: Received June 27, 2014; Revised September 10, 2014

Large-scale finite element analysis (FEA) with millions of degrees of freedom (DOF) is becoming commonplace in solid mechanics. The primary computational bottleneck in such problems is the solution of large linear systems of equations. In this paper, we propose an assembly-free version of the deflated conjugate gradient (DCG) for solving such equations, where neither the stiffness matrix nor the deflation matrix is assembled. While assembly-free FEA is a well-known concept, the novelty pursued in this paper is the use of assembly-free deflation. The resulting implementation is particularly well suited for large-scale problems and can be easily ported to multicore central processing unit (CPU) and graphics-programmable unit (GPU) architectures. For demonstration, we show that one can solve a 50 × 106 degree of freedom system on a single GPU card, equipped with 3 GB of memory. The second contribution is an extension of the “rigid-body agglomeration” concept used in DCG to a “curvature-sensitive agglomeration.” The latter exploits classic plate and beam theories for efficient deflation of highly ill-conditioned problems arising from thin structures.

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Fig. 1

A two-level geometric multigrid

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Fig. 2

(a) Finite element mesh and (b) agglomeration of mesh nodes into four groups

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Fig. 3

Example of “thick” solids

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Fig. 4

Examples of “thin” solids

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Fig. 5

Curvature effects in thin structures

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Fig. 6

Congruency in a finite element mesh

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Fig. 7

Most of the distinct elements are localized

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Fig. 8

Partitioning mesh-nodes into groups. (a) Finite element mesh. (b) Partitioning into 32 groups. (c) Partitioning into 64 groups.

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Fig. 9

SpMV implementation in GPU

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Fig. 10

GPU implementation of prolongation

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Fig. 11

GPU implementation for restriction

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Fig. 12

A beam geometry and its mesh

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Fig. 13

Assembly-free SpMV on the CPU with and without exploiting element-congruency

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Fig. 14

(a) Knuckle geometry and loading. (b) Voxel mesh with 3.16 × 106 DOF.

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Fig. 15

Static displacement and stress for knuckle

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Fig. 16

Visual representation of 100 and 1000 agglomeration groups

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Fig. 17

Convergence of DCG versus Jacobi-PCG

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Fig. 18

Loading on a thin plate

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Fig. 19

Convergence of DCG versus Jacobi-PCG for thin plate

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Fig. 20

CUDA profile for RBM deflation

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Fig. 21

Structural problem over a Thomas engine

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Fig. 22

Deflection from a 50 × 106 DOF system




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