Domain Decomposition Approach Applied for Two- and Three-dimensional Problems via Direct Solution Methodology

- Journal title : International Journal of Aeronautical and Space Sciences
- Volume 16, Issue 2, 2015, pp.177-189
- Publisher : The Korean Society for Aeronautical & Space Sciences
- DOI : 10.5139/IJASS.2015.16.2.177

Title & Authors

Domain Decomposition Approach Applied for Two- and Three-dimensional Problems via Direct Solution Methodology

Kwak, Jun Young; Cho, Haeseong; Chun, Tae Young; Shin, SangJoon; Bauchau, Olivier A.;

Kwak, Jun Young; Cho, Haeseong; Chun, Tae Young; Shin, SangJoon; Bauchau, Olivier A.;

Abstract

This paper presents an all-direct domain decomposition approach for large-scale structural analysis. The proposed approach achieves computational robustness and efficiency by enforcing the compatibility of the displacement field across the sub-domain boundaries via local Lagrange multipliers and augmented Lagrangian formulation (ALF). The proposed domain decomposition approach was compared to the existing FETI approach in terms of the computational time and memory usage. The parallel implementation of the proposed algorithm was described in detail. Finally, a preliminary validation was attempted for the proposed approach, and the numerical results of two- and three-dimensional problems were compared to those obtained through a dual-primal FETI approach. The results indicate an improvement in the performance as a result of the implementing the proposed approach.

Keywords

Finite Element Tearing and Interconnecting (FETI);Augmented Lagrangian Formulation;Domain Decomposition Method;Lagrange multipliers;Parallel computing;

Language

English

References

1.

Bathe, K. J., Finite Element Procedures, Prentice Hall, Inc., Englewood Cliffs, New Jersey, 1996..

2.

Irons, B. M., "A frontal solution program for finite element analysis." International Journal for Numerical Methods in Engineering, Vol. 2, 1970, pp. 5-32.

3.

Duff, I. S. and Reid, J. K., "The multifrontal solution of indefinite sparse symmetric linear equations." ACM Transactions on Mathematical Software, Vol. 9, No. 3, 1973, pp. 302-325.

4.

Kim, J. H. and Kim, S. J., "Multifrontal solver combined with graph partitioners." AIAA Journal, Vol. 37, No. 8, 1999, pp. 964-970.

5.

Gould, N. I. H. and Scott, J. A., "A Numerical Evaluation of Sparse Direct Solvers for the Solution of Large Sparse Symmetric Linear Systems of Equations." ACM Transactions on Mathematical, Vol. 33, No. 2, 2007, pp. 10.

6.

O. Schenk, and K. Gartner. PARDISO User Guide Verson 5.0.0., 2014.

7.

Schenk , O., Gartner, K. and Fichtner, W., "Efficient Sparse LU Factorization with Left-Rghit Looking Strategy on Shared Memory Multiprocessors." BIT Numerical Mathematics, Vol. 40, No. 1, 1999, pp. 158-176.

8.

Saad, Y. and Schultz, M. H., "GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems" SIAM Journal on Scientific and Statistical Computing, Vol. 7, No. 3, 1986, pp. 856-869.

9.

Farhat, C. and Roux, F.-X., "A method of finite element tearing and interconnecting and its parallel solution algorithm." International Journal for Numerical Methods in Engineering, Vol. 32, 1991, pp. 1205-1227.

10.

Farhat, C., Chen, P. S. Mandel, J. and Roux, F.-X., "The two-level FETI method for static and dynamic plate problems. Part I an optimal iterative solver for biharmonic systems." Computer Methods and Applied Mechanics and Engineering, Vol. 155, 1998, pp. 129-151.

11.

Farhat, C., Chen, P. S. Mandel, J. and Roux, F.-X., "The two-level FETI method. Part II extension to shell problems, parallel implementation and performance results." Computer Methods and Applied Mechanics and Engineering, Vol. 155, 1998, pp. 153-179.

12.

Farhat, C., Chen, P. S. and Roux, F.-X., "The dual schur complement method with well-posed local neumann problems: Regulariztion with a perturbed lagrangian formulation." SIAM Journal on Scientific Computing, Vol. 14, No. 3, 1993, pp. 752-759.

13.

Dostal, R. Z., Friedlander, A. and Santos, S. A., "Solution of coercive and semicoercive contact problems by FETI domain decomposition." Contemporary Mathematics, Vol. 218, 1998.

14.

Farhat, C., Lesoinne, M., and Pierson, K., "A scalable dual-primal domain decomposition method." Numerical Linear Algebra with Applications, Vol. 7, 2000, pp. 687-714.

15.

Farhat, C., M. Lesoinne, P. Tallec, L., Pierson, K. and Rixen, D., "FETI-DP: a dual-primal unified FETI method - Part I: A faster alternative to the two-level FETI method." International Journal for Numerical Methods in Engineering, Vol. 50, 2001, pp. 1523-1544.

16.

Hackbusch, R. W., Khoromskij, B. N. and Kriemann, R., "Direct Schur complement method by domain decomposition based on H-matrix approximation." Computing and Visualization in Science, Vol. 8, 2005, pp. 179-188.

17.

Li, R. K., "Fast and highly scalable parallel computations for fundamental matrix problems on distributed memory systems." The Journal of Supercomputing, Vol. 54, 2010, pp. 271-297.

18.

Gueye, I., Arem S. E., Feyel F., Roux R.X. and Gailletaud G., "A new parallel sparse direct solver: Presentation and numerical experiments in large-scale structural mechanics parallel computing." International Journal for Numerical Methods in Engineering, Vol. 88, 2011, pp. 370-384.

19.

Tak M. and Park T., "High scalable non-overlapping domain decomposition method using a direct method for finite element analysis." Computer Methods in Applied Mechanics and Engineering, Vol. 264, 2013, pp. 108-128.

20.

O.A. Bauchau, A. Epple, and C.L. Bottasso. Scaling of constraints and augmented Lagrangian formulations in multibody dynamics simulations. Journal of Computational and Nonlinear Dynamics, Vol. 4, No. 2, 2009, pp. 1-9.

21.

O.A. Bauchau. "Parallel computation approaches for flexible multibody dynamics simulations", Journal of the Franklin Institute, Vol. 347, No. 1, February 2010, pp. 53-68.

22.

Kwak J. Y., Cho H. S., Shin S. J. and Bauchau O. A., "Development of finite element domain decomposition method using local and mixed Lagrange multipliers." Journal of the Computational Structural Engineering Institute of Korea, Vol. 25, No. 6, 2012, pp. 469-476.

23.

Kwak , J. Y., Chun, T. Y., Shin, S. J. and Bauchau, O.A., "Domain decomposition approach to flexible multibody dynamics simulation." Computational Mechanics, Vol. 53, 2014, pp. 147-158.

24.

Tong P. and Pian T. H. H., "A hybid-element approach to crack problems in plane elasticity." International Journal for Numerical Methods in Engineering, Vol. 7, 1973, pp. 297-308.

25.

Park K. C. and Felippa C. A., "A variational framework for solution method developments in structural mechanics." Journal of Applied Mechanics, Vol. 65, 1998, pp. 242-249.

26.

Park K. C., Felippa C. A. and Gumaste U.A., "A localized version of the method of Lagrange multipliers and its applications." Computational Mechanics, Vol. 24, 2000, pp. 476-490.

27.

Fortin M. and Glowinski R., "Augmented Lagrangian Methods: Applications to the Numerical Solution of Boundary- Value Problems." North-Holland, Amsterdam, the Netherlands, 1983.

28.

Gill P. E., Murray W., Saunders M. A. and Wright M. H., "Sequential quadratic programming methods for nonlinear programming." In E.J. Haug, editor, Computer-Aided Analysis and Optimization of Mechanical System Dynamics, Springer-Verlag, Berlin, Heidelberg, 1984, pp. 679-697.

29.

Bayo E., Garcia de Jalon J. and Serna M. A., "A modified Lagrangian formulation for the dynamic analysis of constrained mechanical systems." Computer Methods in Applied Mechanics and Engineering, Vol. 71, 1998, pp. 183-195.

30.

Bayo E., Garcia de Jalon J., Avello A. and Cuadrado J., "An efficient computational method for real time multibody dynamic simulation in fully Cartesian coordinates." Computer Methods in Applied Mechanics and Engineering, Vol. 92, 1991, pp. 377-395.

31.

Bayo E. and Ledesma R., "Augmented Lagrangian and mass-orthogonal projection methods for constrained multibody dynamics." Nonlinear Dynamics, Vol. 9, 1996, pp. 113-130.

32.

Bottasso C.L., Bauchau O.A. and Cardona A., "Timestep-size-independent conditioning and sensitivity to perturbations in the numerical solution of index three differential algebraic equations." SIAM Journal on Scientific Computing, Vol. 29, No. 1, 2007, pp. 397-414.

33.

Linear Algebra PACKage (LAPACK). http://www.netlib.org/lapack, 2014.

34.

Scalable Linear Algebra PACKage (ScaLAPACK). http://www.netlib.org/scalapack, 2014.

35.

Intel Math Kernel Library (Intel MKL) 11.0. http://software.intel.com/en-us/intel-mkl, 2014.

36.

Ahmad S., Irons B. H. and Zienkiewicz, O. C., "Analysis of thick and thin shell structures by curved finite elements." International Journal for Numerical Methods in Engineering, Vol. 2, 1970, pp. 419-451.

37.

Ashwell D.G. and Gallagher R.H., Finite Elements for Thin Shells and Curved Members. JohnWiley & Sons, New York, 1976.

38.

Ugural A.C., Stresses in Plates and Shells. McGraw-Hill Book Company, New-York, second edition, 1999.