Approximation and reconstruction of stresses in the deformed configuration for hyperelastic material models

这个项目的目标是通过有希望的有限元方法来更好地理解有限应变下的弹性行为，到目前为止，有限元方法主要是在线弹性的背景下进行研究的。具体地说，这包括三角形和四面体上的非协调P2元，它们在局部动量守恒和超稳定方面具有有利的性质。我们的计划是调查这些有利的性质中有多少会延续到超弹性的情况。动量守恒应力可以利用局部面片上的计算从位移-压力近似重建。在超弹性材料模型的情况下，由于直接由位移-压力近似产生的输入应力不再是分段线性的，所以重建稍微复杂一些。然而，更严重的是与使用这些应力重建来提供后验误差估计器相关的困难。问题的非线性使得问题变得更加复杂，我们试图尽可能地扩大适用范围。还将从数学和力学两个方面研究在H(Div)协调的有限元空间中直接计算应力近似的方法。为此，我们将在应力对称性的处理和单元间连续性条件的实施方面对最小二乘有限元方法进行修改。最后，我们将重点关注用于直接计算动量守恒的应力近似的Hellinger-Reissner原理。对于所有这些方法，参数Raviart-Thomas有限元空间适合于使用完全设置在材料构型中的公式来近似柯西应力。

The goal of this project is to provide an improved understanding of elastic behavior at finite strains by promising finite element approaches which have so far mostly been studied in the context of linear elasticity. In particular, these include nonconforming P2 elements on triangles and tetrahedra which have advantageous properties with respect to local momentum conservation and inf-sup stability. Our plan is to investigate how much of these favourable properties carry over to the hyperelastic situation. Momentum-conservative stresses can be reconstructed from displacement-pressure approximations using computations on local patches. In the case of hyperelastic material models, the reconstruction is somewhat more involved since the input stress arising directly from the displacement-pressure approximation is not piecewise linear anymore. Much more severe, however, are the difficulties associated with the use of these stress reconstructions to provide an a posteriori error estimator. The nonlinearity of the problem makes the situation much more complicated and we attempt to widen the range of applicability as much aspossible.Approaches which compute stress approximations directly in H (div)-conforming finite element spaces will also be studied from the mathematical as well as from the mechanical side. To this end, least-squares finite element methods will be modified concerning the treatment of stress symmetry and the enforcement of inter-element continuity conditions.Finally, we will focus our attention on the Hellinger-Reissner principle for the direct computation of stress approximations which are momentum-conservative. For all these approaches, parametric Raviart-Thomas finite element spaces lend themselves for the approximation of the Cauchy stresses using a formulation that is completely set in the material configuration.