Physics-Based Simulation

Kinematics

Similar to 2D triangle elements, we use $\beta ,\gamma ,\tau \in [0,1]$ with $\beta +\gamma +\tau =1$ to express the material space coordinates of an arbitrary point $\mathbf{X}$ in the tetrahedral element defined by vertices ${\mathbf{X}}_1,{\mathbf{X}}_2,{\mathbf{X}}_3,$ and ${\mathbf{X}}_4$ as follows: $$\begin{array}{rl}\mathbf{X}(\beta ,\gamma ,\tau )& ={\mathbf{X}}_1+\beta ({\mathbf{X}}_2-{\mathbf{X}}_1)+\gamma ({\mathbf{X}}_3-{\mathbf{X}}_1)+\tau ({\mathbf{X}}_4-{\mathbf{X}}_1)\\ & =(1-\beta -\gamma -\tau ){\mathbf{X}}_1+\beta {\mathbf{X}}_2+\gamma {\mathbf{X}}_3+\tau {\mathbf{X}}_4.\end{array}$$ Here, $\mathbf{X}$ is a linear function of $(\beta ,\gamma ,\tau )$. Using linear shape functions, the approximate world-space coordinate $\hat{\mathbf{x}}$ is also a linear function of $(\beta ,\gamma ,\tau )$: $$\begin{array}{r}\mathbf{x}(\beta ,\gamma ,\tau )\approx \hat{\mathbf{x}}(\beta ,\gamma ,\tau )=(1-\beta -\gamma -\tau ){\mathbf{x}}_1+\beta {\mathbf{x}}_2+\gamma {\mathbf{x}}_3+\tau {\mathbf{x}}_4,\end{array}$$ where $\mathbf{x}({\mathbf{X}}_i)$ is denoted as ${\mathbf{x}}_i$. This implies that the shape functions are: $$\begin{array}{rl}{N}_1(\beta ,\gamma ,\tau )& =1-\beta -\gamma -\tau ,\\ N_2(\beta ,\gamma ,\tau )& =\beta ,\\ N_3(\beta ,\gamma ,\tau )& =\gamma ,\\ N_4(\beta ,\gamma ,\tau )& =\tau .\end{array}$$