An important requirement in the standard finite element method (FEM) is that all elements in the underlying mesh must be tangle-free i.e., the Jacobian must be positive throughout each element. To relax this requirement, an isoparametric tangled finite element method (i-TFEM) was recently proposed for linear elasticity problems. It was demonstrated that i-TFEM leads to optimal convergence even for severely tangled meshes. In this paper, i-TFEM is generalized to nonlinear elasticity. Specifically, a variational formulation is proposed that leads to local modification in the tangent stiffness matrix associated with tangled elements, and an additional piece-wise compatibility constraint. i-TFEM reduces to standard FEM for tangle-free meshes. The effectiveness and convergence characteristics of i-TFEM are demonstrated through a series of numerical experiments, involving both compressible and in-compressible problems.

翻译：在标准有限元方法中，一个重要的要求是底层网格中的所有元素必须是不缠结的，即雅可比矩阵在每个元素中必须始终为正。为了放宽这个要求，最近提出了一个用于线性弹性问题的等参异形定形有限元法(i-TFEM)。证明了即使在严重缠结的网格中，i-TFEM也能够导致最优收敛。在本文中，将i-TFEM推广到非线性弹性。具体而言，提出了一个变分形式，它导致了与缠结元素相关的切线刚度矩阵中的局部修改和一个额外的分段兼容性约束。对于无缠结网格，i-TFEM还原为标准FEM。通过一系列数值实验，包括可压缩和不可压缩问题，证明了i-TFEM的有效性和收敛特性。