Piola Transformation

The Piola transformation maps vectors between Eulerian and Lagrangian coordinates in continuum mechanics. It is named after Gabrio Piola.

Definition

Let $F: \mathbb^d \rightarrow \mathbb^d$ with $F( \hat) = B \hat +b, ~ B \in \mathbb^, ~ b \in \mathbb^$ an affine transformation. Let $K=F(\hat)$ with $\hat$ a domain with Lipschitz boundary. The mapping

$p: L^2( \hat )^d \rightarrow L^2(K)^d, \quad \hat \mapsto p(\hat)(x) := \frac \cdot B \hat (\hat)$
is called Piola transformation. The usual definition takes the absolute value of the determinant, although some authors make it just the determinant.

Note: for a more general definition in the context of tensors and elasticity, as well as a proof of the property that the Piola transform conserves the flux of tensor fields across boundaries, see Ciarlet's book.

See also

* Piola–Kirchhoff stress tensor
*Raviart–Thomas basis functions
* Raviart–Thomas Element

References

Continuum mechanics

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