A stabilised Petrov–Galerkin formulation for linear tetrahedral elements in compressible, nearly incompressible and truly incompressible fast dynamics

Abstract

A mixed second order stabilised Petrov-Galerkin finite element framework was recently introduced by the authors (C.H.Lee, A.J.Gil and J.Bonet. "Development of a stabilised Petrov-Galerkin formulation for conservation laws in Lagrangian fast solid dynamics", CMAME, 268:40-64, 2014). The new mixed formulation, written as a system of conservation laws for the linear momentum and the deformation gradient, performs extremely well in bending dominated scenarios (even when linear tetrahedral elements are used) yielding equal order of convergence for displacements and stresses. In this paper, this formulation is further enhanced for nearly and truly incompressible deformations with three key novelties. First, a new conservation law for the Jacobian of the deformation is added into the system providing extra flexibility to the scheme. Second, a variationally consistent PetrovGalerkin stabilisation methodology is derived. Third, an adapted fractional step method is presented for both incompressible and nearly incompressible materials in the context of nonlinear elastodynamics. For completeness and ease of understanding, these three improvements are presented both in small and large strain regimes, studying the eigenstructure of the resulting systems. A series of numerical examples are presented in order to demonstrate the robustness of the enhanced methodology with respect to the work previously published by the authors.

An upwind vertex centred Finite Volume solver for Lagrangian solid dynamics

“…More recently, the p-F formulation was improved in [5] for the case of nearly incompressible materials, by means of an additional conservation law for the Jacobian of the deformation J (widely known as volume map conservation law [42,[54][55][56][57]), providing extra flexibility for the calculation of the volumetric stress. This innovative idea extended the range of use of the formulation to nearly and fully incompressible media.…”

Explicit mixed strain-displacement finite element for dynamic geometrically non-linear solid mechanics

“…We also observe that this approach is not appealing for the implicit case since it results in a non-symmetric matrix. Let us consider the one-dimensional problem in figure (2). This corresponds to a bar with unit cross section, discretized using two elements of length h. Constant material properties Young's modulus E and density ρ are assumed.…”

“…A vast literature exists proposing cures for such problems, with early proposals based on reduced integration techniques or on the use of corrected assumed-strains, see for instance, the F-bar method [1]. Mixed displacement-pressure approaches started to be employed in the 90s for the solution of truly incompressible problems [1,2,3,4,5]. Such techniques were also shown to provide an accuracy advantage when used in application to compressible problems (see e.g.…”

Explicit mixed strain–displacement finite elements for compressible and quasi-incompressible elasticity and plasticity

“…All these mixed schemes are stabilized via the Variational Multi-Scale method (VMS) [21,31,32,33,41], and, specifically, the Orthogonal Subgrid Scales method, in order to gain control of all the variables while circumventing the restrictiveness of the inf-sup compatibility conditions on the choice of the interpolation spaces. Similar mixed approaches have been proposed using alternative stabilization techniques, like SUPG [4,5,29,36] or FIC [41].…”

“…Using displacements u as primary variables, the first proposals were based on reduced integration [30,38,39], and extended to the use of assumed deformations [47] and the B-bar method [31]. Mixed pressuredisplacement u − p approaches were introduced in the 90's and used thenceforth to address the incompressible limit [18,29,40,49]. The reason for using the pressure as independent variable is to gain control on it and ensure stability; this results in an overall satisfactory behavior of strains and stresses in quasi-incompressible situations.…”