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.