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：标准有限元方法(FEM)的一个重要要求是底层网格中的所有单元必须无缠绕，即每个单元内的雅可比行列式必须始终为正。为放宽这一要求，近期针对线弹性问题提出了等参缠绕有限元方法(i-TFEM)。研究表明，即使对于严重缠绕的网格，i-TFEM仍能实现最优收敛。本文将i-TFEM推广至非线性弹性问题。具体而言，提出了一种变分公式，该方法通过引入与缠绕单元相关的切线刚度矩阵局部修正，以及附加的分段相容性约束。当网格无缠绕时，i-TFEM可退化为标准有限元方法。通过一系列涉及可压缩与不可压缩问题的数值实验，验证了i-TFEM的有效性与收敛特性。