Piola transformation

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

Definition

Let $F: \mathbb{R}^d \rightarrow \mathbb{R}^d$ with $F( \hat{x}) = B \hat{x} +b, ~ B \in \mathbb{R}^{d,d}, ~ b \in \mathbb{R}^{d}$ an affin linear Transformation. Let $K=F(\hat{K})$ with $\hat{K}$ a area with lipschitz Boundary. The Mapping

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

is called Piola Transformation. The absolute value of the Determinant is the usual Definition, some Authors make it just the Determinant.^[1]

See also

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

References

1. http://arxiv.org/pdf/1205.3085.pdf