Multi-field spacetime discontinuous Galerkin methods for linearized elastodynamics

• Published in: Computer methods in applied mechanics and engineering 2009-12, Vol.199 (1), p.34-47
• Main Authors: Miller, S.T., Kraczek, B., Haber, R.B., Johnson, D.D.
• Format: Article
• Language: English
• Subjects: Computational techniques; Discontinuous Galerkin; Elastodynamics; Exact sciences and technology; Fundamental areas of phenomenology (including applications); Mathematical methods in physics; Physics; Solid mechanics; Spacetime finite element; Structural and continuum mechanics; Vibration, mechanical wave, dynamic stability (aeroelasticity, vibration control...)
• ISSN: 0045-7825; 1879-2138
• DOI: 10.1016/j.cma.2009.09.012

We extend the single-field spacetime discontinuous Galerkin (SDG) method for linearized elastodynamics of Abedi et al. [1] to multi-field versions. A three-field method, in displacement, velocity and strain, is derived by invoking a Bubnov–Galerkin weighted residuals procedure on the system of spacetime field equations and the corresponding jump conditions. A two-field formulation, in displacement and velocity, and the one-field displacement formulation of [1] are obtained from the three-field model through strong enforcement of kinematic compatibility relations. All of these formulations balance linear and angular momentum at the element level, and we prove that they are energy-dissipative and unconditionally stable. As in [1], we implement the SDG models using a causal, advancing-front meshing procedure that enables a patch-by-patch solution procedure with linear complexity in the number of spacetime elements. Numerical results show that the three-field formulation is most efficient, wherein all interpolated fields converge at the optimal, O ( h p + 1 ) , rate. For a given mesh size, the three-field model delivers error values that are more than an order of magnitude smaller than those of the one- and two-field models. The three-field formulation’s efficiency is also superior, independent of whether the comparison is based on matching polynomial orders or matching convergence rates.