Piola transformation

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

Let ${\displaystyle F:{\mathbb{ℝ}}^d\to {\mathbb{ℝ}}^d}$ 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

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

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

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