Adaptive finite element procedures for linear and non‐linear dynamics

Abstract

SUMMARYThis paper discusses implementation and adaptivity of the Discontinuous Galerkin (DG) "nite element method as applied to linear and non-linear structural dynamic problems. By the DG method, both displacements and velocities are approximated as piecewise bilinear functions in space and time and may be discontinuous at the discrete time levels. Both implicit and explicit iterative algorithms for solving the resulted system of coupled equations are derived. They are third-order accurate and, while the implicit procedure is unconditionally stable, the explicit one is conditionally stable. An h-adaptive procedure based on the Zienkiewicz}Zhu error estimate using the SPR technique is applied. Numerical examples are presented to show the suitability of the DG method for both linear and non-linear structural dynamic analysis.

Error Assessment in Structural Transient Dynamics

“…For instance, the Newmark method is not using any time variational form. However, there are some a posteriori error estimation techniques using the full variational formulation both for the problem approximation and for the error assessment strategy, see [35,50,29,30].…”

“…Note that here u O is not given as part of the data λ O and λ O N characterizing (30). The function u O has to be computed as the solution of (31).…”

“…In the following, recovery type error estimates in the framework of elastodynamics, see [29,30,31], are applied to the DG formulations. That is, they assess the error with respect to the solution U of the discrete problem (16).…”

“…The local error indicators η t n ≈ ||E t (t n )|| 2 m+a are computed in references [29,30] using the time-discontinuities of the DG approximation U, namely…”

“…In this context, different estimates provide also indicators driving mesh adaptive procedures, either using energy-like measures [29,30,31,32,33,34] or QoI [35,36,37,38]. Estimates providing error bounds are also available both for energy-like error measures [39,40,41] and goal-oriented ones [42,43,44,45,46,47,31].…”