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

该项目的目标是通过有前途的有限元方法提供对有限应变下弹性行为的更好理解，这些方法迄今为止主要是在线性弹性的背景下研究的。特别地，这些包括三角形和四面体上的非相容P2元素，它们在局部动量守恒和中支持稳定性方面具有有利的性质。我们的计划是调查有多少这些有利的性质延续到超弹性的情况。动量守恒应力可以通过局部块的计算从位移-压力近似中重建。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，因此重建稍微复杂一些。然而，更严重的是，使用这些应力重建来提供后验误差估计的困难。问题的非线性使情况更加复杂，我们试图尽可能地扩大适用范围。在符合H （div）的有限元空间中直接计算应力近似的方法也将从数学和力学方面进行研究。为此，将对最小二乘有限元方法在处理应力对称性和执行单元间连续性条件方面进行修改。最后，我们将把注意力集中在Hellinger-Reissner原理上，用于直接计算动量保守的应力近似。对于所有这些方法，参数Raviart-Thomas有限元空间可以使用完全在材料构型中设置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.