Unconditionally stable element-by-element algorithms for dynamic problems
A collection of results is presented regarding the consistency, stability and accuracy of operator split methods and product formula algorithms for general nonlinear equations of evolution. These results are then applied to the structural dynamics problem. The basic idea is to exploit an element-by-element additive decomposition of a particular form of the discrete dynamic equations resulting from a finite element discretization. It is shown that such a particular form of the discrete dynamic equations is obtained when velocity and stress are taken as unknowns. By applying the general product formula technique to the element-by-element decomposition, unconditionally stable algorithms are obtained that involve only element coefficient matrices. The storage requirements and operation counts are comparable to those of explicit methods. The method places no restriction on the topology of the finite element mesh.
© 1983 Elsevier. Received 8 February 1982. Revised manuscript received 2 July 1982. The authors would like to thank Prof. T.J.R. Hughes for bringing some of the ideas contained in this paper to our attention and for supplying us with a preprint of . Grants for the partial support of this work from the Lawrence Livermore National Laboratory and Genera1 Motors Research Laboratories are gratefully acknowledged.