Piola transformation explained

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

$$p:\; L^2(\backslash hat)^d\; \backslash rightarrow\; L^2(K)^d,\; \backslash quad\; \backslash hat\; \backslash mapsto\; p(\backslash hat)(x)\; :=\; \backslash frac$$

\cdot B \hat (\hat) 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

Notes and References

1. 1205.3085. 10.1137/08073901X. Efficient Assembly of H( div ) and H( curl ) Conforming Finite Elements. SIAM Journal on Scientific Computing. 31. 6. 4130–4151. 2010. Rognes. Marie E.. Marie Rognes . Kirby. Robert C.. Logg. Anders.
2. Book: Ciarlet, P. G.. Three-dimensional elasticity. Elsevier Science. 1994. 9780444817761. 1.