Piola transformation

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

$$p:\; L^2(\; \backslash hat\{K\}\; )^d\; \backslash rightarrow\; L^2(K)^d,\; \backslash quad\; \backslash hat\{q\}\; \backslash mapsto\; p(\backslash hat\{q\})(x)\; :=\; \backslash frac\{1\}$$

is called Piola transformation. The usual definition takes the absolute value of the determinant, although some authors make it just the determinant.{{Cite journal |arxiv = 1205.3085|doi = 10.1137/08073901X|title = Efficient Assembly of $H(\backslash mathrm\{div\})$ and $H(\backslash mathrm\{curl\})$ Conforming Finite Elements|journal = SIAM Journal on Scientific Computing|volume = 31|issue = 6|pages = 4130–4151|year = 2010|last1 = Rognes|first1 = Marie E.|author1-link= Marie Rognes |last2 = Kirby|first2 = Robert C.|last3 = Logg|first3 = Anders}}

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.{{cite book|title=Three-dimensional elasticity|last=Ciarlet|first=P. G.|publisher=Elsevier Science|year=1994|isbn=9780444817761|volume=1}}

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

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Category:Continuum mechanics

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