Adaptive superposition of finite element meshes in elastodynamic problems

Abstract

SUMMARYAn adaptive finite element procedure is developed for modelling transient phenomena in elastic solids, including both wave propagation and structural dynamics. Although both temporal and spatial adaptivity are addressed, the novel feature of the formulation is the use of mesh superposition to produce spatial refinement (referred to as s-adaptivity) in transient problems. Spatial error estimation is based on superconvergent patch recovery of higher-order accurate stresses and is used to guide mesh adaptivity, while the temporal error estimation is based on the assumption of linearly varying third-order time derivatives of the displacement field and is used to adjust the time step size for the HHT-variant of the Newmark direct numerical integration method. Spatial adaptivity of the mesh is performed using a hierarchical h-refinement scheme that is efficiently implemented using a structured version of finite element mesh superposition. This particular spatial adaptivity scheme is extremely fast and consequently makes it feasible to repeatedly update both the mesh and the time increment as required in an adaptive transient analysis. This work represents the initial effort in applying this type of spatial adaptivity to transient problems. Three example problems are given to demonstrate the performance characteristics of the s-adaptive procedure.

Adaptive superposition of finite element meshes in non‐linear transient solid mechanics problems

“…These error estimators are defined exactly the same as in the linear transient analysis of Yue and Robbins [17]. It should be emphasized that the present choice of error estimators is perfectly suitable for the present case of material non-linearity caused by progressive damage.…”

“…However, the inherent speed and efficiency of mesh superposition warrants research efforts to overcome these difficulties for computationally demanding non-linear transient problems. Therefore, in the present study, the authors extend the proven s-adaptive technique described in [17] to materially non-linear transient problems caused by progressive damage.…”

“…Babuska et al [1] and Yue and Robbins [17] showed that the use of relative variables in a hierarchical finite element formulation produces a better-conditioned system of equations than nonhierarchical forms. Thus, s-refinement inherently produces better conditioned systems of equations than h-refinement, and the relative superiority of s-refinement increases with the size of the problem being solved.…”

“…While this technique had been previously applied to static problems, it proved to be particularly advantageous for transient problems [17] where speed and efficiency of the adaptive procedures are of paramount importance.…”

“…Yue and Robbins [17] developed a structured s-adaptive finite element procedure for linear elastodynamic problems that simultaneously controlled both the spatial and temporal errors. The spatial error estimation was based on the superconvergent patch recovery technique of Zienkiewicz and Zhu [18,19], while the temporal error estimation was based on the assumption of linearly varying third-order time derivatives of the displacement field as discussed by Wiberg and Li [20].…”