Novel quadratic Bézier triangular and tetrahedral elements using existing mesh generators: Extension to nearly incompressible implicit and explicit elastodynamics in finite strains

Abstract

We present a novel unified finite element framework for performing computationally efficient large strain implicit and explicit elastodynamic simulations using triangular and tetrahedral meshes that can be generated using the existing mesh generators. For the development of a unified framework, we use Bézier triangular and tetrahedral elements that are directly amenable for explicit schemes using lumped mass matrices and employ a mixed displacement-pressure formulation for dealing with the numerical issues arising due to volumetric and shear locking. We demonstrate the accuracy of the proposed scheme by studying several challenging benchmark problems in finite strain elastostatics and nonlinear elastodynamics modelled with nearly incompressible hyperelastic and von Mises elastoplastic material models. We show that Bézier elements, in combination with the mixed formulation, help in developing a simple unified finite element formulation that is accurate, robust, and computationally very efficient for performing a wide variety of challenging nonlinear elastostatic and implicit and explicit elastodynamic simulations.

A linearized consistent mixed displacement-pressure formulation for hyperelasticity

“…Remark I: The formulation can also be written in terms of the symmetric part of the strain and stress measures by following the Voigt notation, see [2][3][4][5]74]. Such an approach, however, leads to complications in transforming the fourth-and third-order tensors resulting from the electrical, magnetic and coupled energy functions in electro-and magnetomechanics problems.…”

“…For this purpose, we consider the widely-used Q1/P0 finite element discretisation, in which it is possible to condense the pressure DOFs out. Other discretisations, for example, P2/P0 [1] or the recently proposed BT2/BT0 [5], are equally applicable.…”

“…The performance of the proposed two-field formulation against the other two two-field formulations is assessed by using the Ladyzhenskaya-Babuska-Brezzi (LBB)-stable BT2/BT1 element [5] that employs a quadratic Bézier element for the displacement field and a linear Bézier element for the pressure field. We refer the reader to Kadapa [5,74] for the comprehensive details on Bézier elements. The tetrahedral finite element mesh used for the analysis, as shown in Fig.…”

“…In the literature, there exist various techniques for addressing the issue of incompressibility. For the truly incompressible case, the mixed displacementpressure formulation is the only computationally-efficient option, which, however, results in saddlepoint problems [1][2][3][4][5]. To overcome the numerical issues in solving saddle-point problems, the widely followed approach in computational solid mechanics is to impose the incompressibility constraint weakly using quasi-incompressible approximations.…”

“…This deterioration of accuracy is due to the absence of the relation between the pressure and the volumetric energy function in the perturbed Lagrangian approach. Although such an issue can be overcome by employing the weak statement for the two-field mixed formulation [5,20,22], this approach results in unsymmetric matrix systems for the majority of volumetric energy functions. Note that any solver for unsymmetric matrices is expensive when compared to the one for symmetric matrices.…”