Preconditioned gradient methods for sparse linear systems for very `large structural` problems

This paper deals with background and practical experience with preconditioned gradient methods for sparse linear systems for `very large` structural problems. The conjugate gradient method with diagonal preconditioning (CG/D) is demonstrated to substantially increase the size of structural problems that can be analyzed, significantly reduce computer storage requirements, and cut computing cost; thus allowing for much more detailed modeling and increased engineering efficiency. For one case for a structural system with 396,087 unknowns, the conjugate gradient method with diagonal preconditioning is demonstrated to be a factor of sixty faster than the direct method. For certain problems, however, the number of iterations required by the CG/D method is excessive and improved methods are needed. A stand-alone iterative solver research computer program was developed to evaluate the merits of various matrix preconditioners. A matrix preconditoner based on a shifted incomplete Cholesky factorization algorithm was demonstrated to be superior to other choices. The stand-alone program incorporates an effective data management strategy which utilizes disk and solid state auxiliary computer storage devices to make it possible to efficiently solve excessively large structural problems on state-of-the-art vector and parallel computers. The background of gradient methods, algorithms for their implementation, and practical experience in their applications to structural problems are presented