In this work, we generalize the mass-conserving mixed stress (MCS) finite element method for Stokes equations (Gopalakrishnan et al., 2019), involving normal velocity and tangential-normal stress continuous fields, to incompressible finite elasticity. By means of the three-field Hu–Washizu principle, introducing the displacement gradient and 1st Piola–Kirchhoff stress tensor as additional fields, we circumvent the inversion of the constitutive law. We lift the arising distributional derivatives of the displacement gradient to a regular auxiliary displacement gradient field. Static condensation can be applied at the element level, providing a global pure displacement problem to be solved. We present a stabilization motivated by Hybrid Discontinuous Galerkin methods. A solving algorithm is discussed, which asserts the solvability of the arising linearized subproblems for problems with physically positive eigenvalues. The excellent performance of the proposed method is corroborated by several numerical experiments.

中文翻译：

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不可压缩有限弹性的四场混合公式

在这项工作中，我们将斯托克斯方程的质量守恒混合应力 （MCS） 有限元方法（Gopalakrishnan et al.， 2019）推广到不可压缩的有限弹性，涉及法向速度和切向-法向应力连续场。通过三场 胡-Washizu 原理，引入位移梯度和第一 Piola-Kirchhoff 应力张量作为附加场，我们规避了本构定律的反转。我们将位移梯度产生的分布导数提升到一个规则的辅助位移梯度场。静态冷凝可以在单元级别应用，从而提供要求解的全局纯位移问题。我们提出了一种由混合间断 Galerkin 方法驱动的稳定化。讨论了一种求解算法，该算法断言对于具有物理正特征值的问题，产生的线性化子问题的可求解性。所提方法的优异性能得到了多次数值实验的证实。