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

Abstract

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.

Appendix A: Identification Identification of the positive and negative Jacobian regions

While the proposed i-TFEM implementation does not require explicit identification of the $J^+$ and $J^-$ regions, for completeness, we provide here a procedure for identifying these two regions.

Recall that the determinant of the Jacobian associated with isoparametric mapping is given by:

$$\begin{aligned} |\varvec{J}| = \left| {\begin{array}{*{20}c} x (\xi , \eta ),_{\xi } &{} y (\xi , \eta ),_{\xi } \\ x (\xi , \eta ),_{\eta } &{} y (\xi , \eta ),_{\eta } \\ \end{array}} \right| \end{aligned}$$

(A.1)

When an element becomes tangled, the parametric space can be decomposed into a positive $|\varvec{J}|$ region and a negative $|\varvec{J}|$ region, denoted as $J^+$ and $J^-$ respectively. The two regions are divided by the $|\varvec{J}| = 0$ curve, i.e., we can identify the $J^+$ and $J^-$ regions by computing the $\varvec{|J|} = 0$ curve.

To illustrate, we consider a 4-noded bilinear quadrilateral (Q4) element for simplicity. Let the element be defined by the vertices $\left[ (x_1, y_1), (x_2, y_2), (x_3, y_3), (x_4, y_4)\right]$. Given the bilinear mapping, the Jacobian is given by

$$\begin{aligned} \varvec{J} = \frac{1}{4} \left[ {\begin{array}{*{20}c} \eta -1 &{} 1 - \eta &{} 1 + \eta &{} -(1 + \eta )\\ \xi -1 &{} -(1+ \xi ) &{} 1 + \xi &{} 1 - \xi \\ \end{array}} \right] \left[ {\begin{array}{*{20}c} x_1 &{} x_2 &{} x_3 &{} x_4\\ y_1 &{} y_2 &{} y_3 &{} y_4\\ \end{array}} \right] ^T \end{aligned}$$

The Jacobian determinant can be expressed as a straight line:

$$\begin{aligned} \left| \varvec{J} \right| = \text {det} (\varvec{J}) = c_0 + c_1\xi + c_2\eta \end{aligned}$$

(A.2)

where,

$$\begin{aligned} c_0&={\left[ \left( x_1 -x_3\right) \left( y_2 -y_4\right) - \left( x_2 -x_4\right) \left( y_1 -y_3\right) \right] }/{8},\\ c_1&= \left[ \left( x_3 - x_4 \right) \left( y_1 - y_2\right) - \left( x_1 - x_2 \right) \left( y_3 - y_4\right) \right] /8,\\ c_2&= \left[ \left( x_2 -x_3\right) \left( y_1 -y_4\right) - \left( x_1 -x_4\right) \left( y_2 -y_3\right) \right] /8. \end{aligned}$$

Thus, the two regions can be computed easily.

For a 9-node quadrilateral (Q9) element, the $|\varvec{J}| = 0$ is no longer a straight line. However, it is possible to derive an expression for this curve. For example, one can show that, for the tangled Q9 element in Fig. 2b, the $\varvec{|J|} = 0$ curve is given by

$$\begin{aligned}{} & {} 0.0605 \xi ^2 \eta ^2 - 0.0055 \left( \xi ^2 \eta + \xi \eta ^ 2 \right) - 0.0129 \left( \xi ^2 + \eta ^ 2 \right) \\{} & {} \quad - 3.1762 \left( \xi + \eta \right) + 1.2898 \xi \eta + 1.2127 = 0. \end{aligned}$$

Consequently, one can identify the two regions $J^+$ and $J^-$.