high performance computing on graphics processing units: hgpu.org

Popular eigen-solvers such as block-Lanczos require repeated inversion of an eigen-matrix. This is a bottleneck in large-scale modal problems with millions of degrees of freedom. On the other hand, the classic Rayleigh-Ritz conjugate gradient method only requires a matrix-vector multiplication, and is therefore potentially scalable to such problems. However, as is well-known, the Rayleigh-Ritz has serious numerical deficiencies, and has largely been abandoned by the finite element community. In this paper, we address these deficiencies through subspace augmentation, and consider a subspace augmented Rayleigh-Ritz conjugate gradient method (SaRCG). SaRCG is numerically stable and does not entail explicit inversion. As a specific application, we consider the modal analysis of geometrically complex structures discretized via non-conforming voxels. The resulting large-scale eigen-problems are then solved via SaRCG. The voxelization structure is also exploited to render the underlying matrix-vector multiplication assembly-free. The implementation of SaRCG on multi-core CPUs, and graphics-programmable-units (GPUs) is discussed, followed by numerical experiments and case-studies.