Isogeometric finite element method
Formal definitions of the reference one- and two-dimensional isogeometric finite elements differ from classical elements only in the shape functions and selection of nodes. Shape functions for isogeometric elements are defined in such a way that it is possible to sum them into global B-spline basis functions. All other mathematical formalism described in this chapter remains valid if we replace the classical elements with isogeometric elements. For isogeometric elements of degree \( p \), it is necessary to define an appropriate number of nodes, specifically \( (p+1) \) in one dimension and \( (p+1)^2 \) in two dimensions, and providing formulas for appropriate shape functions. For isogeometric analysis, there is also a specific mapping recipe that defines the composition of local shape functions into global basis functions over the computational grid. The global Lagrange basis function is made up of local shape functions differently, and the global B-spline basis function is made up of local shape functions differently. Below, as an example, we give the definitions of one and two-dimensional reference isogeometric second degree finite elements, for which the given shape functions can be summed into second order B-spline functions of continuity \( C^1 \). Classical finite elements have local shape functions which, for the sake of distinction, can be grouped into global Lagrange basis functions. For example, for the second order \( p=2 \), the local shape functions consist of the second order \( C^1 \) over the insides of individual finite elements, however on the interfaces between elements (over vertices in one dimension or over edges in two dimensions) they have continuity \( C^0 \).
We call a one-dimensional reference isogeometric finite element of the second degree
\( \left( \hat{K}, X\left(\hat{K}\right), \Pi_p \right) \)
defined by the following four steps
- Geometry: \( \hat{K}=[0,1] \subset {\cal R} \)
- Selection of nodes: \( \hat{a}_1, \hat{a}_2 \) nodes associated with vertices 0 and 1 of the element, and \( \hat{a}_3 \) node associated with the interior (0,1) of the element
- Definition of element shape function \( X \left( \hat{K}\right)=span \{ \hat{\chi}_j \in {\cal P}^p\left(\hat{K}\right),j=1,...,3 \} \) where \( {\cal P}^p\left(\hat{K}\right) \) are degree polynomials \( p \) specified on the interval \( \hat{K} =(0,1) \) and \( \hat{\chi}_1(\xi)=\frac{1}{2}(1-\xi)^2 \), \( \hat{\chi}_2(\xi)=\frac{1}{2}\xi^2 \), \( \hat{\chi}_3(\xi)=-\xi^2+x+\frac{1}{2} \).
- Definition of interpolation by projection operator \( \Pi_p:H^1\left( \hat{K} \right) \rightarrow X\left( \hat{K}\right) \). For a given function \( u \in H^1\left(\hat{K} \right) \), its projection-based interpolant is \( \Pi_pu\in X\left( \hat{K}\right) \) is defined by the following conditions:
\( \Pi_p u(\hat{a}_1)=u(\hat{a}_1) \)
\( \Pi_p u(\hat{a}_2)=u(\hat{a}_2) \)
\( \| \left( \Pi_p u \right)' -u' \|_{H^1(0,1)}\rightarrow min \)
where \( \| \left( \Pi_p u \right)' -u' \|_{H^1(0,1)} = \int_0^1 \left( \left( \Pi_p u \right)' -u' \right)^2 d\xi \) is a norm in Sobolew’s space \( H^1(0,1) \).
A two-dimensional reference isogeometric finite element of the second degree is a four
\( \left( \hat{K}, X\left(\hat{K}\right), \Pi_p \right) \)
defined by the following four steps
- Geometry: \( \hat{K}=[0,1]^2 \subset {\cal R}^2 \)
- Selection of nodes: \( \hat{a}_1, \hat{a}_2, \hat{a}_3, \hat{a}_4, \) nodes associated with element vertices (0,0), (1,0), (1,1), (0,1), \( \hat{a}_5, \hat{a}_6, \hat{a}_7, \hat{a}_8 \) the nodes associated with the element edges, and \( \hat{a}_9 \) the node associated with the interior of the element.
- Definition of element shape function \( X \left( \hat{K}\right)=span \{ \hat{\chi}_j \in {\cal S}^{(2,2)}\left(\hat{K}\right),j=1,...,9 \} \) where \( {\cal S}^{(2,2)}\left(\hat{K}\right) \) are second order polynomials with respect to the variable \( \xi_1 \) and relative to the variable \( \xi_2 \), defined at \( \hat{K}=[0,1]^2 \). We define shape functions \( \hat{\phi}_1(\xi_1,\xi_2)=\hat{\chi}_1(\xi_1)\hat{\chi}_1(\xi_2) \), \( \hat{\phi}_2(\xi_1,\xi_2)=\hat{\chi}_2(\xi_1)\hat{\chi}_1(\xi_2) \), \( \hat{\phi}_3(\xi_1,\xi_2)=\hat{\chi}_2(\xi_1)\hat{\chi}_2(\xi_2) \), \( \hat{\phi}_3(\xi_1,\xi_2)=\hat{\chi}_1(\xi_1)\hat{\chi}_2(\xi_2) \), \( \hat{\phi}_{5,2}(\xi_1,\xi_2)=\hat{\chi}_{3}(\xi_1)\hat{\chi}_1(\xi_2) \), \( \hat{\phi}_{6,2}(\xi_1,\xi_2)=\hat{\chi}_2(\xi_1)\hat{\chi}_{3}(\xi_2) \), \( \hat{\phi}_{7}(\xi_1,\xi_2)=\hat{\chi}_{3}(\xi_1)\hat{\chi}_2(\xi_2) \), \( \hat{\phi}_{8}(\xi_1,\xi_2)=\hat{\chi}_1(\xi_1)\hat{\chi}_{3}(\xi_2) \), \( \hat{\phi}_{9}(\xi_1,\xi_2)=\hat{\chi}_{3}(\xi_1)\hat{\chi}_{3}(\xi_2) \).
- Definition of the projection-based interpolation operator \( \Pi_p:H^1\left( \hat{K} \right) \rightarrow X\left( \hat{K}\right) \). For a given function \( u \in H^1\left(\hat{K} \right) \), its projection based interpolant is \( \Pi_pu\in X\left( \hat{K}\right) \) is defined by the following conditions:
\( \Pi_p u(0,0)=u(0,0) \)
\( \Pi_p u(1,0)=u(1,0) \)
\( \Pi_p u(0,1)=u(0,1) \)
\( \Pi_p u(1,1)=u(1,1) \)
\( \| \left( \Pi_p u \right)' -u' \|_{H^1(\hat{K})}\rightarrow min \)
where \( \| \left( \Pi_p u \right)' -u' \|_{H^1(\hat{K})} = \int_{\hat{K}} \left( \left( \Pi_p u \right)' -u' \right)^2 d\xi_1d\xi_2 \) it is a semorma in the Sobolev space \( H^1(\hat{K}) \).