Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-30T16:52:44.819Z Has data issue: false hasContentIssue false
  • English
  • Français

Algebraic Multigrid Preconditioning for Finite Element Solution of Inhomogeneous Elastic Inclusion Problems in Articular Cartilage

Published online by Cambridge University Press:  03 June 2015

Zhengzheng Hu  and
Mansoor A Haider
Zhengzheng Hu*
Affiliation:
Department of Mathematics, North Carolina State University, Box 8205, Raleigh, NC 27695, USA
Mansoor A Haider*
Affiliation:
Department of Mathematics, North Carolina State University, Box 8205, Raleigh, NC 27695, USA
*
Corresponding author. URL: http://www4.ncsu.edu/∼mahaider/ Email: zhu4@ncsu.edu
Email: m haider@ncsu.edu
Get access

Abstract

In studying biomechanical deformation in articular cartilage, the presence of cells (chondrocytes) necessitates the consideration of inhomogeneous elasticity problems in which cells are idealized as soft inclusions within a stiff extracellular matrix. An analytical solution of a soft inclusion problem is derived and used to evaluate iterative numerical solutions of the associated linear algebraic system based on discretization via the finite element method, and use of an iterative conjugate gradient method with algebraic multigrid preconditioning (AMG-PCG). Accuracy and efficiency of the AMG-PCG algorithm is compared to two other conjugate gradient algorithms with diagonal preconditioning (DS-PCG) or a modified incomplete LU decomposition (Euclid-PCG) based on comparison to the analytical solution. While all three algorithms are shown to be accurate, the AMG-PCG algorithm is demonstrated to provide significant savings in CPU time as the number of nodal unknowns is increased. In contrast to the other two algorithms, the AMG-PCG algorithm also exhibits little sensitivity of CPU time and number of iterations to variations in material properties that are known to significantly affect model variables. Results demonstrate the benefits of algebraic multigrid preconditioners for the iterative solution of assembled linear systems based on finite element modeling of soft elastic inclusion problems and may be particularly advantageous for large scale problems with many nodal unknowns.

Keywords

92C1074B0565M55Biomechanicsinterface problemconjugate gradient method
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 3 , Issue 6 , December 2011 , pp. 729 - 744
Copyright
Copyright © Global-Science Press 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Bower, A. F., Applied Mechanics of Solids, CRC Press, 2009, Boca Raton.Google Scholar
[2] Briggs, W. L., Henson, V. E. and Mccormick, S. F., A Multigrid Tutorial, 2nd Edition, SIAM, Philadelphia, 2000.CrossRefGoogle Scholar
[3] Chen, Z. and Zou, J., Finite element methods and their convergence for elliptic and parabolic interface problems, Numer. Math., 79 (1998), pp. 175–202.Google Scholar
[4] Cleary, A. J., Falgout, R. D., Henson, V. E. and Johns, J. E., Coarse grid selection for parallel algebraic multigrid, in Ferriera, A., Rollin, J., Simon, H. and Teng, S.-H., editors, Proceedings of the Fifth International Symposium on Solving Irregularly Structured Problems in Parallel, Lecture Notes in Computer Science, 1457 (1998), Springer-Verlag.Google Scholar
[5] Eshelby, J. D., The determination of the elastic field of an ellipsoidal inclusion, and related problems, Proc. Roy. Soc. A, 241 (1957), pp. 376–396.Google Scholar
[6] Gong, Y., Li, B. and Li, Z., Immersed-interface finite-element methods for elliptic interface problems with non-homogeneous jump conditions, SIAM J. Numer. Anal., 46(1) (2008), pp. 472–495.Google Scholar
[7] Gong, Y. and Li, Z., Immersed interface finite element methods for elasticity interface problems with non-homogeneous jump conditions, Numer. Math. Theo. Meth. Appl., 3 (2010), pp. 23–39.Google Scholar
[8] Griebel, M., Oeltz, D. and Schweitzer, M. A., An algebraic multigrid method for linear elasticity, SIAM J. Sci. Comput., 25 (2003), pp. 385–407.Google Scholar
[9] Hackbusch, W., Multi-Grid Methods and Applications, Springer-Verlag, New York, 1985.Google Scholar
[10] Henson, V. E. and Yang, U. M., BoomerAMG: a parallel algebraic multigrid solver and pre-conditioner, Appl. Numer. Math., 41 (2002), pp. 155–177.CrossRefGoogle Scholar
[11] HYPRE User’s Manual, Software Version: 2.4.0b, 2008.Google Scholar
[12] Hysome, D. and Pothen, A., A scalable parallel algorithm for incomplete factor preconditioning, SIAM J. Sci. Comput., 22 (2001), pp. 2194–2215.Google Scholar
[13] Jones, M. T. and Plassman, P. E., A parallel graph coloring heuristic, SIAM J. Sci. Com-put., 14 (1983), pp. 654–669.CrossRefGoogle Scholar
[14] Luby, M., A simple parallel algorithm for the maximal independent set problem, SIAM J. Sci. Comput., 15 (1986), pp. 1036–1053.Google Scholar
[15] Kim, E., Guilak, F. and Haider, M. A., An axisymmetric boundary element model for determination of articular cartilage pericellular matrix properties in situ via inverse analysis of chondron deformation, J. Biomech. Eng., 132 (2010), 031011.CrossRefGoogle ScholarPubMed
[16] Mow, V. C., Holmes, M. H. and Lai, W. M., Fluid transport and mechanical properties of articular cartilage: a review, J. Biomech., 17 (1984), pp. 377–394.Google Scholar
[17] Xiao, Y., Zhang, P. and Shu, S., Algebraic multigrid methods for elastic structures with highly discontinuous coefficients, Math. Comput. Simul., 76 (2007), pp. 249–262.Google Scholar
[18] Ruge, J. W. and Stüben, K., Algebraic multigrid (AMG), in McCormick, S. F., editor, Multi-grid Methods, volume 3 of Frontiers in Applied Mathematics, 3 (1987), pp. 73–130, SIAM, Philadelphia.Google Scholar
[19] Trefethen, L. D. and Bau, D. III, Numerical Linear Algebra, SIAM, Philadelphia, 1997.CrossRefGoogle Scholar
[20] Trottenberg, U., Oosterlee, C. W. and Schuller, A., Multigrid, Academic Press, London, 2000.Google Scholar
[21] Yang, U. M., Parallel algebraic multigrid methods-high performance preconditioners, in Bruaset, A. M. and Tveito, A., editors, Numerical Solution of Partial Differential Equations on Parallel Computers, pp. 209–236, Springer-Verlag, 2006.Google Scholar
[22] Yang, X., Li, B. and Li, Z., The immersed interface method for elasticity problems with interfaces, Dynam. Cont. Dis. Ser. A, 10 (2003), pp. 783–808.Google Scholar