Compact Combination of the Finite Element, Linear Iteration and Finite Difference Methods

Abstract

The Galerkin Finite Element Method (FEM), Newton Linear Iteration Method (LIM)¹ and Newmark Finite Difference Method (FDM) form a powerful trio of numerical techniques for obtaining approximate solutions to Non-Linear Initial Boundary Value Problems (NLIBVP) governed by Non-Linear Partial Differential Equations (NLPDE). The FEM performs the spatial discretization, the LIM the successive linearizations and the FDM the time step integration. These three methods can be applied in any order to get from the exact continuous problem to its approximate discrete counterpart. However, tradition and common sense suggest the FEM be applied first, the LIM next and the FDM last. The following diagram clarifies this sequence 231-1 where “NLODE” stands for Non-Linear—Ordinary Differential Equation (in time), and the resulting system of linear Algebraic Equations “ALGEQ” is directly solvable by a variety of standard methods of linear algebra. The use of numerical quadrature formulas during the FE phase and the use of the backward Euler finite difference method with projection for the time integration of the flow rule in plasticity must also be mentioned in order to complete this list of numerical techniques used in a program.