A quadratically convergent algorithm for the computation of stability points: The application of the determinant of the tangent stiffness matrix Original Research Article

This paper presents a quadratically convergent algorithm for the computation of stability points (limit and bifurcation points) in the finite element formulation of the problems of the nonlinear structural mechanics. One such approach was given by Wriggers and coworkers [Comput. Methods Appl. Mech. Engrg. 70 (1988) 329–347 and Int. J. Numer. Methods Engrg. 30 (1990) 155–176]. Their approach employs the eigenvector equation KTψ = 0 as the characterization of the stability point. In the present paper, an alternative approach is proposed, which uses the equation det KT = 0 as the stability point condition. In combination with Newtonʹs iteration scheme, this condition has so far been considered unsuitable for the implementation in the finite element analysis because it has been believed that it requires the full assembly of the derivative of the global tangent stiffness matrix, which is a very costly operation. Objectives of our paper are: (i) to derive a new quadratically convergent algorithm for the computation of stability points, which uses the condition det KT = 0 for the characterization of the stability point; (ii) to show that the full assembly of the global tangent stiffness matrix derivative is not required, and that the linearization of the determinant of the global tangent stiffness matrix can be done in an element-by-element fashion; and (iii) to indicate that, in terms of the number of algebraic operations, memory requirements and the computer programming effort, our algorithm practically equals the one based on an eigenvector equation, but has the advantage of a larger radius of convergence. An additional benefit of the proposed algorithm is that it can also be used without any modification as a comprehensive path-following procedure to calculate regular and singular (stability) points. The effectiveness of our algorithm is illustrated by numerical examples. Its convergence characteristics are compared with those of the algorithm based on the eigenvector equation. We show stability analyses of elastic-plastic planar frames. The algorithm is, of course, valid generally and is not limited to this particular kind of structures.