Piola transformation

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

Definition

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

$${\displaystyle p:L^{2}({\hat {K}})^{d}\rightarrow L^{2}(K)^{d},\quad {\hat {q}}\mapsto p({\hat {q}})(x):={\frac {1}{|\det(B)|}}\cdot B\cdot {\hat {q}}({\hat {x}})}$$

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 ${\displaystyle H(\mathrm {div} )}$ and ${\displaystyle H(\mathrm {curl} )}$ Conforming Finite Elements". SIAM Journal on Scientific Computing. 31 (6): 4130–4151. arXiv:1205.3085. doi:10.1137/08073901X.
2. Ciarlet, P. G. (1994). Three-dimensional elasticity. 1. Elsevier Science. ISBN 9780444817761.