ABSTRACT

In the finite element analysis of linear systems, we obtain the system of equilibrium equations in the form of

KU P= (3.1)

where: K is the system stiffness matrix P is the global load vector U is the unknown displacement

The stiffness matrix is usually symmetric, and positive definite. In nonlinear finite element analysis, at the nth load increment, we have the following incremental form of system of equilibrium equations:

K U P It n n 1∆ = − − (3.2)

In this equation, Kt is the tangential stiffness matrix, P is the load vector, and In−1 is the internal resisting force vector at the end of the previous increment. These systems of linear equations are usually written in the following generic form:

Ax b= (3.3)

The displacement (solution) vector x is computed by solving the system of linear equations. In general, there are three classes of methods for solving the system of linear equations:

1. Direct methods 2. Iterative methods 3. Semi-iterative methods

A direct method solves the system of equations to arrive at the solution vector in a finite number of computational operations. An iterative method approaches the solution vector of the system in a number of iterations. If the iterative method is convergent, then as the number of iterations increases, the solution vector will converge to the exact solution. However, the number of iterations to reach convergence cannot be determined in advance. A semi-iterative method is a direct method in exact mathematical operations, but it is implemented as an iterative method in practice. Because of their unique numerical characteristics, iterative and semi-iterative methods have been increasingly used in solving very large systems, often on massively parallel, distributed or vector computers. Formation of the actual structural system stiffness matrix for very large systems may not be efficient and practical.