A FETI‐DP method for the parallel iterative solution of indefinite and complex‐valued solid and shell vibration problems

The dual-primal finite element tearing and interconnecting (FETI-DP) domain decomposition method (DDM) is extended to address the iterative solution of a class of indefinite problems of the form (K-σ 2 M)u = f, and a class of complex problems of the form (K-σ 2 M+iσD)u = f, where K, M, and D are three real symmetric matrices arising from the finite element discretization of solid and shell dynamic problems, i is the imaginary complex number, and a is a real positive number. A key component of this extension is a new coarse problem based on the free-space solutions of Navier's equations of motion. These solutions are waves, and therefore the resulting DDM is reminiscent of the FETI-H method. For this reason, it is named here the FETI-DPH method. For a practically large a range, FETI-DPH is shown numerically to be scalable with respect to all of the problem size, substructure size, and number of substructures. The CPU performance of this iterative solver is illustrated on a 40-processor computing system with the parallel solution, for various a ranges, of several large-scale, indefinite, or complex-valued systems of equations associated with shifted eigenvalue and forced frequency response structural dynamics problems.