标题： 一种不可压缩有限弹性四场混合公式 显示英文标题

标题： A four-field mixed formulation for incompressible finite elasticity

摘要： 在本工作中，我们将质量守恒混合应力（MCS）有限元方法推广到不可压缩有限弹性问题，该方法用于斯托克斯方程 [Gopalakrishnan J., Lederer P., and Schöberl J., 一种质量守恒的混合应力公式用于斯托克斯方程，IMA数值分析杂志 40(3), 1838-1874 (2019)]，涉及法向速度和切向-法向应力连续场。通过三场Hu-Washizu变分原理，引入位移梯度和第一皮奥拉-基尔霍夫应力张量作为额外场，我们避免了本构定律的求逆。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中。可以在单元层面应用静态凝聚，提供一个全局纯位移问题进行求解。我们提出了一种基于混合不连续伽辽金方法的稳定化策略。讨论了一种求解算法，该算法证明了对于具有物理正特征值的问题，所得到的线性化子问题的可解性。所提出方法的优异性能通过多个数值实验得到了验证。 显示英文摘要

摘要： In this work, we generalize the mass-conserving mixed stress (MCS) finite element method for Stokes equations [Gopalakrishnan J., Lederer P., and Sch\"oberl J., A mass conserving mixed stress formulation for the Stokes equations, IMA Journal of Numerical Analysis 40(3), 1838-1874 (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.