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

超弹性材料模型变形构型中应力的近似和重建

• 批准号: 392587488
• 负责人: Professorin Dr. Fleurianne Bertrand
• 金额: --
• 依托单位: Institut für Mechanik
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2017
• 资助国家: 德国
• 起止时间: 2016-12-31 至 2021-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/392587488?language=en
• 关键词: Approximation; reconstruction; stresses; deformed; configuration

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.

该项目的目标是提供一个更好的理解有限应变下的弹性行为的有前途的有限元方法，迄今为止主要是在线性弹性的背景下研究。特别地，这些包括在三角形和四面体上的具有相对于局部动量守恒和inf-sup稳定性的有利性质的P2元素。我们的计划是调查这些有利的性质有多少延续到超弹性的情况。动量守恒应力可以用局部面片上的计算从位移-压力近似中重建出来。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，因此重建在某种程度上更加复杂。然而，更严重的是与使用这些应力重建提供后验误差估计相关的困难。问题的非线性使情况更加复杂，我们试图扩大适用范围尽可能多的aspossibility.Approaches直接计算应力近似在H（div）-协调有限元空间也将研究从数学以及从机械方面。最后，我们将着重讨论直接计算动量守恒应力近似的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配置中的公式来近似柯西应力。