Solution of progressively changing equilibrium equations for nonlinear structures

Abstract In the analysis of nonlinear structures by tangent stiffness methods, the equilibrium equations change progressively during the analysis. When direct methods of solving these equations are used, it may be possible to re-use a substantial part of the previously reduced coefficient matrix, and hence substantially reduce the equation solving effort. This paper examines procedures for re-solving equations when only selected parts of the reduced matrix need to be modified. The Crout and Cholesky algorithms are first reviewed for initial complete reduction and subsequent selective reduction of the coefficient matrix, and it is shown that the Cholesky algorithm is superior for selective reduction. A general procedure is then presented for identifying those parts of the coefficient matrix which remain unchanged as the structure changes. Finally, a general purpose in-core equation solver is presented, in which those parts of the previously reduced matrix which need to be modified are determined automatically during the solution process, and only these portions are changed. The equation solver is based on the Cholesky algorithm, and is applicable to both positive-definite and well conditioned non-positive-definite symmetrical systems of equations.