Stabilized Mixed Tetrahedrals with Volume and Area Bubble Functions at Large Deformations

Abstract

In this contribution, stabilized mixed finite tetrahedral elements are presented in order to avoid volume locking and stress oscillations. Geometrically non-linear elastic problems are addressed. The mixed method of incompatible modes is considered. As a key idea, volume and area bubble functions are used for the method of incompatible modes [1], thus giving it the interpretation of a mixed finite element method with stabilization terms. Concerning non-linear problems these are nonlinearly dependent on the current deformation state, however, linearly dependent stabilization terms are used. The approach becomes most attractive for the numerical implementation, since the use of quantities related to the previous Newton iteration step is completely avoided. The variational formulation for the standard two-field method, the method of incompatible modes in finite deformation problems is derived for a hyper elastic Neo-Hookean material. In the representative examples Cook's membrane problem and a block under central pressure illustrate the good performance of the presented approaches compared to existing finite element formulations.We introduce the space of compatible displacements and pressures as U = {u ∈ H 1 (B)} and P = {p ∈ L 2 (B)}. Then the standard two-field variational formulation for the geometrically non-linear problem readswith the bulk modulus K and the variational terms G int (δu, u) = B δE : S iso dV, b(p, δu) = B δE : pJC −1 dV b(δp, u) = B δp(J − 1)dV, c(δp, p) = B δppdV.(2)Here, E denotes the right Cauchy-Green tensor and the isochoric part of second Piola-Kirchhoff stress tensor S iso is derived from the energy function describing the hyperelastic material. The discretization with linear displacement and pressure fields is known to violate the Babuška-Brezzi condition and causes stress oscillation in numerical results. Now, bilinear stabilization terms are introduced, extending the nonlinear formulation in Eqn. 1 into a three-field formulation.1Here, the additional stabilization terms for the method of incompatible modes (IM) are introduced in the same way as in [1] with the space of incompatible displacements introduced as V = {v ∈ H 1 (B)}. 2 Finite element matrix formulation For the discretization, standard shape functions of the isoparametric concept are considered for the compatible part of the displacements and pressures. For the incompatible part of the displacements, the volume bubble function [2] and area bubble functions [3] are considered. In the discretized FE formulation of Eqn. 3, the stabilization terms result in constant matricesk e and do not depend on quantities from previous Newton-iterations, see [1] for details.R u R p = ne A e=1 Be B T u (τ iso(k) + p (k) 1)dv Be N T (J (k) − 1) − p (k) K dv − ne A e=1k e û (k) p (k) = ne A e=1 F ext 0 ,k e = 0 k e pv [k e pv ] −1 0 k e vp (4)

An error indicator for parameter identification with stabilized mixed tetrahedrals

“…with compatible u and incompatible v parts of the displacements, the pressure p and material parameters κ ∈ R npar , see [4,6] for more details. Advantages of this formulation is the elimination of volume locking and damping of stress-oscillation in contrast to linear finite elements with (nearly) incompressible material behavior.…”

“…The goal is to apply the error indicators to the finite element method for tetrahedral elements of low order which are preferable for adaptive mesh refinements and in addition reduce computational effort. Additional stabilization terms in the element formulation [4,6] reduce volume locking effects making the elements suitable for (nearly) incompressible material behavior. Numerical examples illustrate the progress on this work.…”