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.