Proceedings of Computational Engineering Conference

JSCESVol.13, 2008 May

Evaluation of Parallel Version of Nonlinear
FEM Solver FINAS*Chuanrong JIN, Takahisa CHIBA, Yutaka HASEGAWA, Isao MATSUMOTO and Hitoshi NAKAMURA1)
1) Science & Engineering Systems Division, ITOCHU Techno-Solutions Corporation（3-2-5, Kasumigaseki, Chiyoda-ku, Tokyo 100-6080, Japan）
FINAS* is an in-house nonlinear FEM solver, which has been developed as a core solver ofmulti-physics analysis systems in ITOCHU techno-solutions corporation. In this paper, speed andaccuracy of the parallel version of FINAS* are evaluated.
Key Words: Parallel computing, nonlinear FEM

1.

INTRODUCTION

BDD improves the convergence by coupling local problems

FINAS* is a core solver of multi-physics systems, whichon subdomains with a coarse problem which propagates thehas been developed in ITOCHU techno-solutions corporation,

error globally, but needs to solve the local problems on allconsisting of linear and nonlinear FEM components ofsubdomains in every iteration step that is typically singular. In

structural analysis, heat transfer analysis, electromagnetic fieldorder to avoid the singularity and reducing the computing cost,analysis, civil analysis and seismic analysis. FINAS* is

BDD_DIAG simplified the local problem into a diagonal

parallelized based on MPI with several parallel algorithms ofmatrix.

linear equation solver implemented.
The objective of this paper is to evaluate speed and accuracy

MBDD, a modified version of BDD, will be presented inother paper by the authors.

of the parallel version of FINAS*. An elasto-plastic problem
with large deformation is calculated and comparison is
performed among different parallel algorithms and a directsolver.

3.

CONVERGENCE OF BDD
As known, the convergence rate of PCG and DDM isdependent on the matrix property, fast for well conditioned
models such as blocky structures but slow for ill conditioned

2.

PARALLEL ALGORITHMS IN FINAS*
The following parallel algorithms of linear equation solver

are implemented in FINAS*.
・Conjugate gradient method with Jacobi preconditioning,
conveniently called as PCG in this paper.
・ Domain decomposition method with diagonal scaling
preconditioning (DDM).

ones for example thin-walled structures. BDD is proposed toimprovethematrix

Neumann-Neumanncondition

preconditioning

byand

thebalancing

reducethedependency on the matrix property [1].
In order to verify the BDD’s convergence, three models arecalculated. One is a cube, a most typical blocky structure, asshown in Figure 1. The second is a pipe, a thin-walled

・Balancing domain decomposition method (BDD) [1].

structure, as shown in Figure 2. Another is a wheel, a usual

・Balancing domain decomposition method with diagonal

FEM model in practice, as shown in Figure 3.

scaling (BDD_DIAG) [2].
・Modified balancing decomposition method (MBDD).
In all these methods, the analysis domain is divided intosome subdomains calculated on different CPUs in parallel.
PCG solves the linear system of all the free degrees of freedom,
while DDM and BDD solve only that on the boundary ofsubdomains and involve calculating the Schur complement ofeach subdomain. For that reason, the computing cost of DDM
or BDD in each iteration step is more than that of PCG,
whereas the number of iteration steps can be much less.

Figure 1 A cube model of 24,000 DOFs

elements. Five parallel algorithms described in Section 2 areapplied decomposing the whole analysis elements into 4096
subdomains as shown in Figure 7 and parallelizing as 4processes. The results of vonMises stress and displacement arecompared to that of a direct solver for all these algorithms, andFigure 2 A pipe model of 101,520 DOFs

that the differences are in the range of error tolerances isconfirmed. The quadratic convergence of the Newton-Raphson
iteration is achieved by all these parallel algorithms as same asthat by the direct solver.
The specification of the computer used for this evaluation isas following:
・OS: SUSE linux enterprise server 9・CPU: Opteron 8468(2.0GHz)×4

Figure 3 A wheel model of 76,371 DOFs

・Memory: 16GB.

It can be found from Figure 4-6 that the BDD’s convergenceis much faster than that of DMM for all three models.
Relative res idual norm

1.E+01
1.E+00
1.E-01 0102030

40

50

60

1.E-02
1.E-03
1.E-04BDD

1.E-05DDM

1.E-06
1.E-07

Number of iteration

Figure 7 4096 decomposed subdomains of the wheel model

Relative res idual norm

Figure 4 BDD’s Convergence of the cube model
1.E+02
1.E+01
1.E+00
1.E-01 0
1.E-02
1.E-03
1.E-04
1.E-05
1.E-06
1.E-07

The ratios of the elapsed time of the above algorithms to2000

4000

6000

8000

that of PCG are listed in Table 1. In this analysis, MBDD isthe fastest.
Table 1 Ratios of elapsed time to that of PCG

BDD

DDM

Number of itera tions

Figure 5 BDD’s Convergence of the pipe model

Rela tive res idua l norm

1.E+0
1.E-1

0200

400

600

800

1000

PCG

1DDM

2.73

BDD

0.77

BDD_DIAG

0.87

MBDD

0.53

1.E-2

5.

1.E-3
1.E-4

BDD

The convergence of BDD implemented in FINAS* is

1.E-5

DDM

1.E-6

CONCLUSIONS

1.E-7
Number of iteration

Figure 6 BDD’s Convergence of the wheel model

verified by three typical models. The calculation accuracy ofthe parallel version of FINAS* is confirmed by comparing to a
direct solver. The speed of five parallel algorithms is tested
and MBDD is the most effective for the wheel model of 1.5M
elements of elasto-plastic analysis with large deformation.

4.

EVALUATION REULTSREFFERENCES

A rotation bending problem of a wheel is calculated as anelasto-plastic analysis considering the effect of large1)J.

MANDEL:Balancing

domaindecomposition,deformation. The elasto-plastic model is given as a J2 model

Communications on Numerical Methods in Engineering,with isotropic hardening. The logarithmic strain and theVol. 9, pp.233-241, 1993.

updated Lagrange algorithm are adopted for large deformation
analysis.
The FEM mesh consists of 299,569 nodes and 1,572,842

2)M. OGINO: Massively parallel computing using domain
decomposition finite element method, Ph.D. thesis, Kyushu
University, 2003.