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¶

Denoting

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

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

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

and

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

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.

Appendix A. Some basic computations between reference and real elements¶

Volume integral¶

Surface integral¶

Derivative computation¶

Second derivative computation¶

Example of elementary matrix¶

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Appendix A. Some basic computations between reference and real elements¶

Volume integral¶

Surface integral¶

Derivative computation¶

Second derivative computation¶

Example of elementary matrix¶

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