Assembly Free Large Scale Modal Analysis on the Graphics Programmable Unit

Abstract

Popular eigensolvers such as blockLanczos require repeated inversion of an eigenmatrix. This is a bottleneck in largescale modal problems with millions of degrees of freedom. On the other hand, the classic Rayleigh–Ritz conjugate gradient method only requires a matrixvector multiplication, and is therefore potentially scalable to such problems. However, as is wellknown, the Rayleigh–Ritz has serious numerical deficiencies, and has largely been abandoned by the finiteelement 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 nonconforming voxels. The resulting largescale eigenproblems are then solved via SaRCG. The voxelization structure is also exploited to render the underlying matrixvector multiplication assemblyfree. The implementation of SaRCG on multicore central processing units (CPUs) and graphicsprogrammable units (GPUs) is discussed, followed by numerical experiments and casestudies.