A computationally efficient isoparametric tangled finite element method for handling inverted quadrilateral and hexahedral elements

The finite element method (FEM) requires elements to be tangle-free i.e. the Jacobian must be positive throughout the element. In particular, quadrilateral and hexahedral elements are required to be convex. However, generating high quality tangle-free meshes, especially 3D hexahedral meshes, remains an open challenge. Recently, the tangled finite element method (TFEM) was proposed to handle concave (tangled) quadrilateral elements. However, even in 2D, it was found to be computationally expensive and programmatically complex. Here, we present a computationally efficient isoparametric-TFEM (i-TFEM) framework for inverted 2D quadrilateral and 3D hexahedral elements. i-TFEM employs the properties of isoparametric elements to make the formulation computationally much more efficient. In i-TFEM, the constraint on full invertibility (convexity) is replaced by partial invertibility by modifying the elemental stiffness matrices of the concave elements, and by incorporating certain piecewise-compatibility conditions. The proposed i-TFEM is simple, efficient, and provides accurate solutions with optimal convergence rate even in the presence of severely inverted elements. Moreover, i-TFEM requires minimal changes to the standard FEM framework and reduces to standard FEM for meshes without tangled elements. The accuracy and efficiency of i-TFEM are demonstrated by solving elastostatics problems on several 2D and 3D tangled meshes.