Multilevel algorithms for 3D simulation of nonlinear elasticity problems

Abstract

This study is devoted to the numerical solution of 3D elasticity problems in multilayer media. The problem is described by a coupled system of second-order nonlinear elliptic partial differential equations with strongly varying coefficients. The boundary value problem is discretized by trilinear finite elements. The goal of the paper is to analyze the performance of three hierarchical algorithms for the arisen discrete problems. The secant method is applied as a general outer nonlinear iterative procedure. The two-level block-size reduction block-incomplete LU (BSR BILU) algorithm, and the algebraic multilevel iteration (AMLI) algorithm are implemented as preconditioners to the linearized problems into the framework of the conjugate gradient method. The BSR BILU is a special case two-level algorithm while the AMLI preconditioner is based on a regular multilevel mesh refinement. A simple patched local refinement in combination with the BSR BILU algorithm is considered at the end, as an efficient approach for problems with localized zones of active interactions. The developed FEM codes are applied for 3D simulation of pile systems in weak multilayer soil media. The benchmark problem is taken from the real-life bridge engineering practice. The presented test data illustrate the abilities of the proposed methods and algorithms as well as the robustness of the codes.