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Multi-field spacetime discontinuous Galerkin methods for linearized elastodynamics. (English) Zbl 1231.74430
Summary: We extend the single-field spacetime discontinuous Galerkin (SDG) method for linearized elastodynamics of R. Abedi [Comput. Methods Appl. Mech. Eng. 195, No. 25–28, 3247–3273 (2006; Zbl 1130.74044)] 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 [loc. cit.] 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 [loc. cit.], 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, $O\left({h}^{p+1}\right)$, 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- andtwo-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.
##### MSC:
 74S05 Finite element methods in solid mechanics 74B15 Equations linearized about a deformed state (small deformations superposed on large) 74H15 Numerical approximation of solutions for dynamical problems in solid mechanics