The solution of elliptic diffusion operators is the computational bottleneck in many simulations in a wide range of engineering and scientific disciplines. We present a truly scalable-ultrascalable-algebraic multigrid (AMG) linear solver for the diffusion operator in unstructured elasticity problems. Scalability is demonstrated with speedup studies of a non-linear micro-finite element analyses of a human vertebral body with over a half of a billion degrees of freedom on up to 4088 processors on the ACSI White machine. This work is significant because in the domain of unstructured implicit finite element analysis in solid mechanics with complex geometry, this is the first demonstration of a highly parallel and efficient application of a mathematically optimal linear solution method on a common large scale computing platform — the IBM SP Power3.

Accurate micro-finite element analyses of whole bones require the solution of large sets of algebraic equations. Multigrid has proven to be an effective approach to the design of highly scalable linear solvers for solid mechanics problems. We present some of the first applications of scalable linear solvers, on massively parallel computers, to whole vertebral body structural analysis. We analyze the performance of our algebraic multigrid (AMG) methods on problems with over 237 million degrees of freedom on IBM SP parallel computers. We demonstrate excellent parallel scalability, both in the algorithms and the implementations, and analyze the nodal performance of the important AMG kernels on the IBM Power3 and Power4 architectures.