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.
