Skip to content

Shape Functions and Finite Element Approximation

To handle the weak form of the principle of virtual work computationally, the body domain is divided into a finite number of elements, and the material-point coordinates, displacement, and test function within each element are interpolated from nodal values and shape functions. Spatial derivatives of the shape functions are covered in Spatial Derivatives of Shape Functions, discretization of the weak form in Discretization of Internal Virtual Work, and the specific forms of shape functions for each element type in Element Numbering System and Shape Function Library and subsequent sections.

Domain Partitioning and Element-wise Sums of Integrals

The domain \(\Omega_0\) in the reference configuration and the domain \(\Omega\) in the current configuration are approximated by the unions of elements \(\Omega^e_0\) and \(\Omega^e\), respectively:

\[ \Omega_0 \approx \Omega_0^h = \bigcup_e \Omega^e_0, \qquad \Omega \approx \Omega^h = \bigcup_e \Omega^e \]

(\(e\) is the element number, and element boundaries are shared between adjacent elements.) This decomposes the volume and surface integrals in the principle of virtual work into sums of integrals over individual elements:

\[ \int_{\Omega_0} (\cdot)\, dV \approx \sum_e \int_{\Omega^e_0} (\cdot)\, dV, \qquad \int_{\Gamma_{0t}} (\cdot)\, d\Gamma \approx \sum_e \int_{\Gamma^e_{0t}} (\cdot)\, d\Gamma \]

(The same applies to the current configuration by replacing \(dV \to dv\), \(\Omega^e_0 \to \Omega^e\), and \(\Gamma^e_{0t} \to \Gamma^e_t\).) Hereafter, evaluation of the weak form reduces to constructing integrals on an element-by-element basis.

Interpolation Using Nodal Values and Shape Functions (Isoparametric Elements)

Each element \(\Omega^e_0\) is assigned \(n_e\) nodes. Let the reference-configuration coordinates and nodal displacements of element node \(\alpha = 1, \ldots, n_e\) be \(\boldsymbol{X}^e_\alpha, \boldsymbol{u}^e_\alpha\). The element nodal vectors \(\boldsymbol{X}^e = (\boldsymbol{X}^{eT}_1, \ldots, \boldsymbol{X}^{eT}_{n_e})^T\) and \(\boldsymbol{u}^e = (\boldsymbol{u}^{eT}_1, \ldots, \boldsymbol{u}^{eT}_{n_e})^T\) are formed by arranging these values; they contain only the components of the nodes constituting the element, extracted from the global nodal vectors \(\boldsymbol{X}^n, \boldsymbol{u}^n\) (\(n_g\) is the total number of nodes), for element \(e\).

Using the natural coordinates \(\boldsymbol{r}\), which are local coordinates within the element, as parameters, the shape functions \(N_\alpha^e(\boldsymbol{r})\) interpolate the material coordinates, displacement, and test function within the element using the same shape functions (isoparametric element and Galerkin method):

\[ \boldsymbol{X} = \sum_{\alpha=1}^{n_e} N_\alpha^e(\boldsymbol{r})\, \boldsymbol{X}^e_\alpha, \qquad \boldsymbol{u} = \sum_{\alpha=1}^{n_e} N_\alpha^e(\boldsymbol{r})\, \boldsymbol{u}^e_\alpha, \qquad \delta\boldsymbol{u} = \sum_{\alpha=1}^{n_e} N_\alpha^e(\boldsymbol{r})\, \delta\boldsymbol{u}^e_\alpha. \]

The shape functions are constructed to satisfy the following two properties, and the element geometry is chosen so that the mapping \(\boldsymbol{r}\mapsto\boldsymbol{X}\) from natural coordinates to material coordinates is one-to-one within the element:

\[ \sum_{\alpha=1}^{n_e} N_\alpha^e(\boldsymbol{r}) = 1, \qquad N_\beta^e(\boldsymbol{r}_\alpha) = \delta_{\alpha\beta} \]

(\(\boldsymbol{r}_\alpha\) is the point in natural coordinates corresponding to node \(\alpha\), and \(\delta_{\alpha\beta}\) is the Kronecker delta.) The first equation guarantees reproduction of rigid-body translation, and the second guarantees that the interpolated value coincides with the nodal value at each node. The specific forms of \(n_e\) and \(N_\alpha^e\) for each element type are given in Element Numbering System and Shape Function Library and subsequent sections. To avoid cumbersome notation, dependence on the element type is represented by the element-wise superscript \(e\).

With the interpolation rules above, the integrand of the weak form can be expressed solely in terms of the element nodal values \(\boldsymbol{u}^e, \delta\boldsymbol{u}^e\) and \(N_\alpha^e\). Strain, on the other hand, is derived from the interpolated displacement and the strain-displacement relation, while stress is derived from that strain and the material constitutive law; these quantities are not directly interpolated from nodal values. They are evaluated at integration points within the element (Numerical Integration).

Ordering Rule for Global Nodal Vectors

Physical quantities assigned to nodes are arranged in the global nodal vector in ascending order of node number → degree of freedom. If, at node \(\alpha\), degree-of-freedom component \(i\) is denoted by \(u_{i\alpha}\), then in three dimensions (\(i=1,2,3\)) and two dimensions (\(i=1,2\)), respectively,

\[ \boldsymbol{u}^n = (u_{11}, u_{21}, u_{31},\ u_{12}, u_{22}, u_{32},\ \ldots,\ u_{1 n_g}, u_{2 n_g}, u_{3 n_g})^T, \]
\[ \boldsymbol{u}^n = (u_{11}, u_{21},\ u_{12}, u_{22},\ \ldots,\ u_{1 n_g}, u_{2 n_g})^T \]

The coordinates \(\boldsymbol{X}^n\) and the test function \(\delta\boldsymbol{u}^n\) follow the same ordering. Hereafter, matrix- and vector-form derivations are written using the three-dimensional case as representative.