Appendix A. Some basic computations between reference and real elements

Volume integral

One has

\[\int_T f(x)\ dx = \int_{\widehat{T}} \widehat{f}(\widehat{x}) |\mbox{vol}\left( \frac{\partial\tau(\widehat{x})}{\partial \widehat{x}_0}; \frac{\partial\tau(\widehat{x})}{\partial \widehat{x}_1}; \ldots; \frac{\partial\tau(\widehat{x})}{\partial \widehat{x}_{P-1}} \right)|\ d\widehat{x}.\]

Denoting \(J_{\tau}(\widehat{x})\) the jacobian

\[\fbox{$ J_{\tau}(\widehat{x}) := |\mbox{vol}\left( \frac{\partial\tau(\widehat{x})}{\partial \widehat{x}_0}; \frac{\partial\tau(\widehat{x})}{\partial \widehat{x}_1}; \ldots; \frac{\partial\tau(\widehat{x})}{\partial \widehat{x}_{P-1}} \right)| = (\mbox{det}(K(\widehat{x})^T K(\widehat{x})))^{1/2}$,}\]

one finally has

\[\fbox{$\int_T f(x)\ dx = \int_{\widehat{T}} \widehat{f}(\widehat{x}) J_{\tau}(\widehat{x})\ d\widehat{x}$.}\]

When \(P = N\), the expression of the jacobian reduces to \(J_{\tau}(\widehat{x}) = |\mbox{det}(K(\widehat{x}))|\).

Surface integral

With \(\Gamma\) a part of the boundary of \(T\) a real element and \(\widehat{\Gamma}\) the corresponding boundary on the reference element \(\widehat{T}\), one has

\[\fbox{$\int_{\Gamma} f(x)\ d\sigma = \int_{\widehat{\Gamma}}\widehat{f}(\widehat{x}) \|B(\widehat{x})\widehat{n}\| J_{\tau}(\widehat{x})\ d\widehat{\sigma}$,}\]

where \(\widehat{n}\) is the unit normal to \(\widehat{T}\) on \(\widehat{\Gamma}\). In a same way

\[\fbox{$\int_{\Gamma} F(x)\cdot n\ d\sigma = \int_{\widehat{\Gamma}} \widehat{F}(\widehat{x})\cdot(B(\widehat{x})\cdot\widehat{n}) J_{\tau}(\widehat{x})\ d\widehat{\sigma}$,}\]

For \(n\) the unit normal to \(T\) on \(\Gamma\).

Derivative computation

One has

\[\nabla f(x) = B(\widehat{x})\widehat{\nabla} \widehat{f}(\widehat{x}).\]

Second derivative computation


\[\nabla^2 f = \left[\frac{\partial^2 f}{\partial x_i \partial x_j}\right]_{ij},\]

the \(N \times N\) matrix and

\[\widehat{X}(\widehat{x}) = \sum_{k = 0}^{N-1}\widehat{\nabla}^2\tau_k(\widehat{x})\frac{\partial f}{\partial x_k}(x) = \sum_{k = 0}^{N-1}\sum_{i = 0}^{P-1} \widehat{\nabla}^2\tau_k(\widehat{x})B_{ki}\frac{\partial \widehat{f}}{\partial \widehat{x}_i}(\widehat{x}),\]

the \(P \times P\) matrix, then

\[\widehat{\nabla}^2 \widehat{f}(\widehat{x}) = \widehat{X}(\widehat{x}) + K(\widehat{x})^T \nabla^2 f(x) K(\widehat{x}),\]

and thus

\[\nabla^2 f(x) = B(\widehat{x})(\widehat{\nabla}^2 \widehat{f}(\widehat{x}) - \widehat{X}(\widehat{x})) B(\widehat{x})^T.\]

In order to have uniform methods for the computation of elementary matrices, the Hessian is computed as a column vector \(H f\) whose components are \(\frac{\partial^2 f}{\partial x^2_0}, \frac{\partial^2 f}{\partial x_1\partial x_0},\ldots, \frac{\partial^2 f}{\partial x^2_{N-1}}\). Then, with \(B_2\) the \(P^2 \times P\) matrix defined as

\[\left[B_2(\widehat{x})\right]_{ij} = \sum_{k = 0}^{N-1} \frac{\partial^2 \tau_k(\widehat{x})}{\partial \widehat{x}_{i / P} \partial \widehat{x}_{i\mbox{ mod }P}} B_{kj}(\widehat{x}),\]

and \(B_3\) the \(N^2 \times P^2\) matrix defined as

\[\left[B_3(\widehat{x})\right]_{ij} = B_{i / N, j / P}(\widehat{x}) B_{i\mbox{ mod }N, j\mbox{ mod }P}(\widehat{x}),\]

one has

\[\fbox{$H f(x) = B_3(\widehat{x}) \left(\widehat{H}\ \widehat{f}(\widehat{x}) - B_2(\widehat{x})\widehat{\nabla} \widehat{f}(\widehat{x})\right)$.}\]

Example of elementary matrix

Assume one needs to compute the elementary “matrix”:

\[t(i_0, i_1, \ldots, i_7) = \int_{T}\varphi_{i_1}^{i_0} \partial_{i_4}\varphi_{i_3}^{i_2} \partial^2_{i_7/ P, i_7\mbox{ mod } P}\varphi_{i_6}^{i_5}\ dx,\]

The computations to be made on the reference elements are

\[\widehat{t}_0(i_0, i_1, \ldots,i_7) = \int_{\widehat{T}}(\widehat{\varphi})_{i_1}^{i_0} \partial_{i_4}(\widehat{\varphi})_{i_3}^{i_2} \partial^2_{i_7 / P, i_7\mbox{ mod } P}(\widehat{\varphi})_{i_6}^{i_5} J(\widehat{x})\ d\widehat{x},\]


\[\widehat{t}_1(i_0, i_1, \ldots, i_7) = \int_{\widehat{T}}(\widehat{\varphi})_{i_1}^{i_0} \partial_{i_4}(\widehat{\varphi})_{i_3}^{i_2} \partial_{i_7}(\widehat{\varphi})_{i_6}^{i_5} J(\widehat{x})\ d\widehat{x},\]

Those two tensor can be computed once on the whole reference element if the geometric transformation is linear (because \(J(\widehat{x})\) is constant). If the geometric transformation is non-linear, what has to be stored is the value on each integration point. To compute the integral on the real element a certain number of reductions have to be made:

  • Concerning the first term (\(\varphi_{i_1}^{i_0}\)) nothing.
  • Concerning the second term (\(\partial_{i_4}\varphi_{i_3}^{i_2}\)) a reduction with respect to \(i_4\) with the matrix \(B\).
  • Concerning the third term (\(\partial^2_{i_7 / P, i_7\mbox{ mod }P} \varphi_{i_6}^{i_5}\)) a reduction of \(\widehat{t}_0\) with respect to \(i_7\) with the matrix \(B_3\) and a reduction of \(\widehat{t}_1\) with respect also to \(i_7\) with the matrix \(B_3 B_2\)

The reductions are to be made on each integration point if the geometric transformation is non-linear. Once those reductions are done, an addition of all the tensor resulting of those reductions is made (with a factor equal to the load of each integration point if the geometric transformation is non-linear).

If the finite element is non-\(\tau\)-equivalent, a supplementary reduction of the resulting tensor with the matrix \(M\) has to be made.