Improved Quasi‐Newton Methods for Large Nonlinear Problems

Abstract

In this work, schemes based on limited‐memory quasi‐Newton methods are investigated, as applied to solving large systems of nonlinear equations with sparse symmetric Jacobian matrices. Problems in mechanics typically give rise to such systems when the method of finite elements is employed to solve them. An attempt is made to develop algorithms that take advantage of sparsity and can effectively use a variable amount of storage according to the availability. The use of preconditioning matrices as initial approximations to the tangent stiffness matrix is suggested in order to accelerate convergence when the available high‐speed storage exceeds the needs of purely vectorial methods but is not sufficient to house a full factorization of the tangent stiffness. The limited‐memory quasi‐Newton methods are also combined with the concept of truncation, based on a preconditioned conjugate gradient iterative solver of the linearized equations, to produce quite efficient algorithms.