On the application of an element-by-element lanczos solver to large offshore structural engineering problems

Abstract

Iterative methods for solving large sets of linear equations have been used as an alternative to direct methods of solution since the early beginning of numerical analysis. The conjugate gradient method (CCM), one of the most widely used, seeks a solution that minimizes the potential energy of the finite element assemblage. Recently, the use of Lanczos algorithm for the solution of large sets of linear equations has been re-examined. Lanczos biorthogonalization procedure is an oblique projection method that provides a solution approximation whose residual is orthogonal to a Krylov subspace. It has been shown that Lanczos and CGM share several properties but the former has the advantage of not being necessary to compute the approximated solution at each iteration. Jacobi preconditioning can also be employed in order to accelerate convergence. The Lanczos procedure was implemented using an element-by-element (EBE) scheme. The applications spread over typical offshore engineering problems encompassing regular and irregular meshes. These problems are normally ill-conditioned when compared with continuum problems. For all the analyses addressed the element-by-element Lanczos procedures presented outstanding computational efficiency.