Linear Finite Elements

From the governing equations in the continuous setting, we derived the discretized weak form system (nd equations) using the backward Euler time integration rule:

$$\begin{array}{rl}& M_{\hat{a}b}\frac{x_{b|\hat{i}}^n-(x_{b|\hat{i}}^{n-1}+h{V}_{b|\hat{i}}^{n-1})}{\Delta t^2}\\ & ={\int }_{\partial {\Omega }^0}{N}_{\hat{a}}(\mathbf{X})T_{\hat{i}}(\mathbf{X},t^n)ds(\mathbf{X})-{\int }_{{\Omega }^0}{N}_{\hat{a},j}(\mathbf{X})P_{\hat{i}j}(\mathbf{X},t^n)d\mathbf{X}.\end{array}\text{\hspace{1em}(19.1)}$$ In this chapter, we'll start by discussing the shape function $N_{\hat{a}}$ in the context of linear finite elements. This exploration will help us understand the underlying implementation detailed in Inversion-Free Elasticity.

We'll focus specifically on simplex finite elements. In 2D, the 2-simplex is a triangle, and we've consistently used triangle meshes throughout this book to discretize the solid domain into a disjoint set of triangular elements.

Definition 19.1 (Simplex).
An n-simplex is a geometric object with $n+1$ vertices that exists in an $n$-dimensional space. It cannot fit in any space of smaller dimension.