A waveform relaxation Newmark method for structural dynamics problems

Abstract

In the conventional Newmark family for time integration of hyperbolic problems, both explicit and implicit methods are inherently sequential in the time domain and not well suited for parallel implementations due to unavoidable processor communication at every time step. In this work we propose a Waveform Relaxation Newmark (WRN β ) algorithm for the solution of linear second-order hyperbolic systems of ODEs in time, which retains the unconditional stability of the implicit Newmark scheme with the advantage of the lower computational cost of explicit time integration schemes. This method is unstructured in the time domain and is well suited for parallel implementation. We consider a Jacobi and Gauss-Seidel type splitting and study their convergence and stability. The performance of the WRN β algorithm is compared to a standard implicit Newmark method and the obtained results confirm the effectiveness of the Waveform Relaxation Newmark algorithm as a new class of more efficient time integrators, which is applicable, as shown in the numerical examples, to both the finite element method and meshfree methods (e.g. the reproducing kernel particle method).

A Newmark space-time formulation in structural dynamics

“…The parameters β and γ are chosen as 1 4 and 1 2 in this paper. These settings result in constant acceleration within each interval [t i−1 , t i ] and unconditional stability with respect to the size of the chosen time step length Δt [1,68,69].…”

“…visualizes the iterative enrichment procedure as a step-wise adaption of the solution in the whole time domain.The solution ansatz (83) as well as the decomposition of the space-time excitation term (82) are inserted into the space-time equation(69):…”