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Piola transformation

From Wikipedia, the free encyclopedia

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

Definition

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Let with an affine transformation. Let with a domain with Lipschitz boundary. The mapping

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

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References

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  1. ^ Rognes, Marie E.; Kirby, Robert C.; Logg, Anders (2010). "Efficient Assembly of and 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.