Physics:Piola transformation

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

Definition

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

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

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