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

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