Speaker
Description
It is well known that in Finite Element (FE) simulations, the selected mesh has a strong influence on the quality of the results. Especially in the case of highly distorted meshes, large discrepancies between the numerical and the analytical solution can be observed. To address this issue, various elements based on the Petrov-Galerkin FE method have been developed in recent years (see, e.g., [1, 3]). In contrast to the Bubnov-Galerkin method, which is commonly used in most FE formulations, the Petrov-Galerkin method employs different ansatz spaces for the test and trial functions. While this approach typically improves accuracy in elastostatic simulations, its extension to elastodynamic problems is not straightforward. In [2], it is shown that special discretization techniques are necessary to achieve conservation of energy in the case of unsymmetric mass and stiffness matrices, as they occur in the Petrov-Galerkin FE method. Otherwise, unbounded energy growth may occur over time. Wang and Hillman [2] proposed a modification to the Newmark method to address this issue. In this contribution, we present an alternative approach. The core idea is to introduce the velocity field as an independent variable and enforce its relationship with the displacement field through a special constraint. We demonstrate that, under certain conditions, this approach enables energy-conserving simulations.
References
[1] Pfefferkorn R, Betsch P. Mesh distortion insensitive and locking-free Petrov-Galerkin low-order EAS elements for linear elasticity, Int J Numer Methods Eng. 2021; 122(23):6924-6954.
https://doi.org/10.1002/nme.6817
[2] Wang J, Hillman MC. Temporal stability of collocation, Petrov-Galerkin, and other non-symmetric methods in elastodynamics and an energy conserving time integration, Comput. Methods Appl. Mech. Engrg. 2022; 393:114738.
https://doi.org/10.1016/j.cma.2022.114738
[3] Xie Q, Sze KY, Zhou YX. Modified and Trefftz unsymmetric finite element models, Int J Mech Mater Des. 2016; 12:53-70.
https://doi.org/10.1007/s10999-014-9289-3
