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Piola transformation

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

Definition

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

p : L 2 ( K ^ ) d → L 2 ( K ) d , q ^ ↦ p ( q ^ ) ( x ) := 1 | det ( B ) | ⋅ B q ^ ( x ^ ) {\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}})} 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

References

  1. Rognes, Marie E.; Kirby, Robert C.; Logg, Anders (2010). "Efficient Assembly of H ( d i v ) {\displaystyle H(\mathrm {div} )} and H ( c u r l ) {\displaystyle H(\mathrm {curl} )} Conforming Finite Elements". SIAM Journal on Scientific Computing. 31 (6): 4130–4151. arXiv:1205.3085. doi:10.1137/08073901X.
  2. Ciarlet, P. G. (1994). Three-dimensional elasticity. Vol. 1. Elsevier Science. ISBN 9780444817761.


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