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

## Volume integral¶

One has Denoting the jacobian one finally has When , the expression of the jacobian reduces to .

## Surface integral¶

With a part of the boundary of a real element and the corresponding boundary on the reference element , one has where is the unit normal to on . In a same way For the unit normal to on .

## Derivative computation¶

One has ## Second derivative computation¶

Denoting the matrix and the matrix, then and thus In order to have uniform methods for the computation of elementary matrices, the Hessian is computed as a column vector whose components are . Then, with the matrix defined as and the matrix defined as one has ## Example of elementary matrix¶

Assume one needs to compute the elementary “matrix”: The computations to be made on the reference elements are and Those two tensor can be computed once on the whole reference element if the geometric transformation is linear (because 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 ( ) nothing.
• Concerning the second term ( ) a reduction with respect to with the matrix .
• Concerning the third term ( )` a reduction of with respect to with the matrix and a reduction of with respect also to with the matrix 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- -equivalent, a supplementary reduction of the resulting tensor with the matrix has to be made.

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