Piola transformation

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

## Definition

Let [math]\displaystyle{ F: \mathbb{R}^d \rightarrow \mathbb{R}^d }[/math] with [math]\displaystyle{ F( \hat{x}) = B \hat{x} +b, ~ B \in \mathbb{R}^{d,d}, ~ b \in \mathbb{R}^{d} }[/math] an affine transformation. Let [math]\displaystyle{ K=F(\hat{K}) }[/math] with [math]\displaystyle{ \hat{K} }[/math] a domain with Lipschitz boundary. The mapping

[math]\displaystyle{ p: L^2( \hat{K} )^d \rightarrow L^2(K)^d, \quad \hat{q} \mapsto p(\hat{q})(x) := \frac{1}{|\det(B)|} \cdot B \hat{q} (\hat{x}) }[/math] is called Piola transformation. The usual definition takes the absolute value of the determinant, although some authors make it just the determinant.[1]

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.[2]

## See also

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

## References

1. ↑ Rognes, Marie E.; Kirby, Robert C.; Logg, Anders (2010). "Efficient Assembly of [math]\displaystyle{ H(\mathrm{div}) }[/math] and [math]\displaystyle{ H(\mathrm{curl}) }[/math] Conforming Finite Elements". SIAM Journal on Scientific Computing 31 (6): 4130–4151. doi:10.1137/08073901X.
2. ↑ Ciarlet, P. G. (1994). Three-dimensional elasticity. 1. Elsevier Science. ISBN 9780444817761.

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