A velocity/stress mixed stabilized nodal finite element for elastodynamics: Analysis and computations with strongly and weakly enforced boundary conditions

• Published in: Computer methods in applied mechanics and engineering Vol. 325; no. C; pp. 532 - 576
• Main Authors: Scovazzi, G., Song, T., Zeng, X.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01-10-2017; Elsevier BV; Elsevier
• Subjects: Boundary conditions; Computational mathematics; Computer simulation; Data structures; Elastodynamics; ENGINEERING; Finite element analysis; Finite element method; Nodal stress or nodal strain; Stability analysis; Stabilized methods; Strain; Stresses; Studies; Time domain analysis; Variational multiscale analysis; Wave equation; Wave propagation; Weak boundary conditions; Wave equation; Nodal stress or nodal strain; Elastodynamics; Variational multiscale analysis; Weak boundary conditions; Stabilized methods
• ISSN: 0045-7825; 1879-2138
• DOI: 10.1016/j.cma.2017.07.018

A new nodal mixed finite element is proposed for the simulation of linear elastodynamics and wave propagation problems in time domain. Our method is based on equal-order interpolation discrete spaces for both the velocity (or displacement) and stress (or strain) tensor variables. The mixed form is derived using either the velocity/stress or velocity/strain pair of unknowns, the latter being instrumental in extensions of the method to nonlinear mechanics. The proposed approach works equally well on hexahedral or tetrahedral grids and, for this reason, it is suitable for time-domain engineering applications in complex geometry. The peculiarity of the proposed approach is the use of the rate form of the stress update equation, which yields a set of governing equations with the structure of a non-dissipative space/time Friedrichs’ system. We complement standard traction boundary conditions for the stress with strongly and weakly enforced boundary conditions for the velocity (or displacement). Weakly enforced boundary conditions are particularly suitable when considering complex geometrical shapes, because they do not require dedicated data structures for the imposition of the boundary degrees of freedom, but, rather, they utilize the structure of the variational formulation. We also show how the framework of weakly enforced boundary conditions can be used to develop variational forms for multi-domain simulations of heterogeneous media. A complete analysis including stability and convergence proofs is included, in the case of a space–time variational approach. A series of computational tests are used to demonstrate and verify the performance of the proposed approach.