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 affine transformation. Let $K=F({\hat {K}})$ with ${\hat {K}}$ a domain 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}})

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]

References

↑ https://arxiv.org/pdf/1205.3085.pdf
↑ Ciarlet, P. G. (1994). Three-dimensional elasticity. 1. Elsevier Science. ISBN 9780444817761.

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[1] ttps://arxiv.org/pdf/1205.3085.pdf

[Ciarlet-2] Ciarlet, P. G. (1994). Three-dimensional elasticity. 1. Elsevier Science. ISBN 9780444817761.

Piola transformation

Definition

See also

References