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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
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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

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