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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)

SUMMARYGoal-oriented error estimation allows to refine meshes in space and time with respect to arbitrary quantities. The required dual problems that need to be solved usually require weak formulations and the Galerkin method in space and time to be established. Unfortunately, this does not obviously leads to structures of standard finite element implementations for solid mechanics. These are characterized by a combination of variables at nodes (e.g. displacements) and at integration points (e.g. internal variables) and are solved with a two-level Newton method because of local uncoupled and global coupled equations. Therefore, we propose an approach to approximate the dual problem while maintaining these structures. The primal and the dual problems are derived from a multifield formulation. Discretization in time and space with appropriate shape functions and rearrangement yields the desired result. Details on practical implementation as well as applications to elasto-plasticity are given. Numerical examples demonstrate the effectiveness of the procedure.

In numerical simulations with the finite element method the dependency on the mesh -and for time-dependent problems on the time discretization -arises. Adaptive refinements in space (and time) based on goal-oriented error estimation [1] become more and more popular for finite element analyses to balance computational effort and accuracy of the solution. The introduction of a goal quantity of interest defines a dual problem which has to be solved to estimate the error with respect to it. Often such procedures are based on a space-time Galerkin framework for instationary problems [2]. Discretization results in systems of equations in which the unknowns are nodal values. Contrary, in current finite element implementations for pathdependent problems some quantities storing information about the path-dependence are located at the integration points of the finite elements [3], e.g. plastic strains etc. In this contribution we propose an approach -similar to [4] for sensitivity analysis -for the approximation of the dual problem which mainly maintains the structure of current finite element implementations for path-dependent problems. Here, the dual problem is introduced after discretization. A numerical example illustrates the approach.

The identification of parameters in constitutive laws considering inhomogeneous states of stress and strain is realized by iteratively minimizing a least squares functional. In each iterative step of this optimization problem a finite element analysis is carried out which results in a significant higher numerical cost than a single finite element analysis. Consequently, an efficient discretization is required to keep the numerical cost low. To address this problem an adaptive mesh refinement is considered which is based on a posteriori error indicators [1] leading to refinements appropriate to the parameter identification problem. 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.