Performance of Parallel Conjugate Gradient Solvers in Meshfree Analysis

Meshfree analysis methods, on a per degree of freedom basis, are typically more computationally expensive and yet more accurate than finite element methods. For very large models, whether meshfree or finite element, the memory and computational effort associated with direct equation solvers makes them prohibitively expensive. In this work, the performance of different linear equation solvers with meshfree analysis methods is explored. In particular, parallel conjugate gradient solvers with both Jacobi diagonal preconditioning and incomplete Cholesky factorization preconditioning are tested on a number of different meshfree analysis applications and compared against the performance of a fast direct sparse equation solver. It is found in these exploratory computations that as the support size of meshfree shape functions increases, the condition number of the associated stiffness matrices increases, and the relative efficiency of iterative solvers suffers somewhat. Nevertheless, for normalized support sizes between one and two, the performance of both conjugate gradient solvers compares very favorably with that of the sparse direct solver for intermediate ( ) 10 ( 5 O N ≈ ) and large problems.