An isoparametric tangled finite element method for handling higher-order elements with negative Jacobian

This paper presents an isoparametric tangled finite element method (i-TFEM) method for handling tangled high order/curvilinear meshes. Tangled elements, i.e. elements with negative Jacobian determinant, frequently occur during various stages of analysis and optimization, leading to erroneous results in standard finite element method (FEM). The proposed i-TFEM is an extension of standard FEM to allow for tangled elements. Specifically, a novel variational formulation is proposed that leads to a simple modification of the standard FEM stiffness matrix and additional piece-wise compatibility constraints. Moreover, i-TFEM reduces to the standard FEM for non-tangled (regular) meshes. The accuracy of the proposed i-TFEM is demonstrated for tangled 9-node quadrilateral (Q9) and 6-node triangular (T6) elements. Numerical experiments involving linear and nonlinear elasticity and Poisson problems illustrate that the accuracy and convergence rate of the proposed i-TFEM over a tangled mesh is comparable to that of the standard FEM over a non-tangled mesh.