Composite Newton-PCG and quasi-Newton iterations for nonlinear consolidation

Abstract

Newton's method is theoretically attractive for solving a system of nonlinear equations, but it is difficult to use in practice because it requires the solution of a linear system at each iteration. For large-scale problems where the dimension of the system may be several thousand, the solution of such linear system may be extremely expensive. The burden of equation-solving is more evident in two- and three-dimensional consolidation problems due to the coupling of soil skeletal displacements and pore fluid pressure which tends to increase the size of the matrix equation significantly. Two iterative techniques for solving such system of nonlinear equations are thus investigated in this paper and compared with the performance of the standard Newton's method in which the linear system is solved by direct triangular factorization. The first of these two techniques is a combination of Newton iteration for the nonlinear system and preconditioned conjugate gradients (PCG) with a global elastic tangent operator preconditioner for the linearized system. The PCG iterative linear equation solver is shown to dramatically improve the efficiency of Newton's method, particularly when the system of linear equations is large. The second technique involves a secant-Newton iteration in which the rank-two BFGS inverse update is employed. For large symmetric systems it is shown that this technique can also be competitive with the standard Newton's method, although it appears to be incapable of handling nonsymmetric systems well. A simple line search algorithm based on the majorization principle is employed to force all of these iterative techniques to produce a norm-reducing solution. Numerical results employing realistic constitutive models for soils are then presented to compare the performance of each of these iterative techniques in the analysis of typical nonlinear consolidation problems.