Piola transformation

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

## Definition

Let F:ℝd->ℝd with F(x^)=Bx^+b,B∈ℝd,d,b∈ℝd an affine transformation. Let K=F(K^) with K^ a domain with Lipschitz boundary. The mapping

p:L2(K^)d->L2(K)d,q^↦p(q^)(x):=1|det⁡(B)|⋅Bq^(x^) 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 H(div) and H(curl) 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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