]> https://gitweb.dealii.org/ - dealii-svn.git/commitdiff
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authorbangerth <bangerth@0785d39b-7218-0410-832d-ea1e28bc413d>
Fri, 18 May 2012 14:03:19 +0000 (14:03 +0000)
committerbangerth <bangerth@0785d39b-7218-0410-832d-ea1e28bc413d>
Fri, 18 May 2012 14:03:19 +0000 (14:03 +0000)
git-svn-id: https://svn.dealii.org/branches/branch_higher_derivatives@25518 0785d39b-7218-0410-832d-ea1e28bc413d

deal.II/higher_derivatives.tex [new file with mode: 0644]

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+\documentclass{article}
+\usepackage{amsmath}
+\begin{document}
+
+For ``normal'' elements, we have that
+\begin{gather*}
+  \varphi(\mathbf x)
+  =
+  \hat\varphi(\mathbf{\hat x})
+\end{gather*}
+where $\mathbf x = \Phi(\mathbf{\hat x})$. Thus, the gradient of a shape
+function in real coordinates is given by
+\begin{gather*}
+  \nabla\varphi(\mathbf x)
+  =
+  \frac{\partial \mathbf{\hat x}}{\partial \mathbf{x}}
+  \hat\nabla\hat\varphi(\mathbf{\hat x})
+\end{gather*}
+where the matrix $\frac{\partial \mathbf{\hat x}}{\partial \mathbf{x}}$ is the
+gradient of the inverse of the mapping for which it can be shown that
+\begin{gather*}
+  \frac{\partial \mathbf{\hat x}}{\partial \mathbf{x}}
+  =
+  \frac{\partial \Phi^{-1}(\mathbf{x})}{\partial \mathbf{x}}
+  =
+  \left(\frac{\partial \Phi(\mathbf{\hat x})}{\partial \mathbf{\hat x}}\right)^{-1}
+  =
+  J^{-1}.
+\end{gather*}
+Note that $\Phi$ is a polynomial mapping and so $J=\frac{\partial
+  \Phi(\mathbf{\hat x})}{\partial \mathbf{\hat x}}$ is easy to
+compute. Computing $J^{-1}$ then involves inverting a $d\times d$ matrix.
+
+As a consequence, we have just shown that we can write
+\begin{gather*}
+  \nabla\phi(\mathbf x) =
+  J^{-1} \hat\nabla \hat\varphi(\mathbf{\hat x})
+\end{gather*}
+and we can generalize this formula for second derivatives:
+\begin{gather*}
+  \nabla^2\phi(\mathbf x) =
+  [J^{-1} \hat\nabla][J^{-1} \hat\nabla] \hat\varphi(\mathbf{\hat x}).
+\end{gather*}
+Note, however, that for non-affine mappings, $J$ is not a constant matrix and
+so we need to apply the chain rule to obtain
+\begin{align*}
+  [\nabla^2\phi(\mathbf x)]_{ij}
+  &=
+  [J^{-1} \hat\nabla]_{i}[J^{-1} \hat\nabla]_j \hat\varphi(\mathbf{\hat x})
+  \\
+  &=
+  [J^{-1}_{ik} \hat\partial_k][J^{-1}_{jl} \hat\partial_l] \hat\varphi(\mathbf{\hat x})
+  \\
+  &=
+  J^{-1}_{ik} J^{-1}_{jl} \hat\partial_k \hat\partial_l
+  \hat\varphi(\mathbf{\hat x})
+  +
+  J^{-1}_{ik} (\hat\partial_k J^{-1}_{jl}) \hat\partial_l \hat\varphi(\mathbf{\hat x}).
+\end{align*}
+The difficulty is the derivative of the inverse of the Jacobian,
+$J^{-1}$. Since $J$ is a polynomial itself, the entries of $J^{-1}$ are
+rational functions and computing their derivatives is, at least,
+awkward. However, we can use that
+\begin{gather*}
+  0 = \hat\partial_k \mathbf I_{pl} = \hat\partial_k (J_{pm}J^{-1}_{ml})
+  =
+  (\hat\partial_k J_{pm})J^{-1}_{ml}
+  +
+  J_{pm}(\hat\partial_k J^{-1}_{ml}).
+\end{gather*}
+Multiplying this equation with $J^{-1}_{jp}$ from the left and summing over
+$p$ yields the formula
+\begin{gather*}
+  \delta_{jm}
+  \hat\partial_k J^{-1}_{ml}
+  =
+  -
+  J^{-1}_{jp}(\hat\partial_k J_{pm})J^{-1}_{ml},
+\end{gather*}
+that is
+\begin{gather*}
+  \hat\partial_k J^{-1}_{jl}
+  =
+  -
+  J^{-1}_{jp}(\hat\partial_k J_{pm})J^{-1}_{ml}.
+\end{gather*}
+Inserting this into the formula for the second derivative yields
+\begin{align*}
+  [\nabla^2\phi(\mathbf x)]_{ij}
+  &=
+  J^{-1}_{ik} J^{-1}_{jl} \hat\partial_k \hat\partial_l
+  \hat\varphi(\mathbf{\hat x})
+  -
+  J^{-1}_{ik}
+  J^{-1}_{jp}(\hat\partial_k J_{pm})J^{-1}_{ml}
+  \hat\partial_l \hat\varphi(\mathbf{\hat x}).
+\end{align*}
+All of these terms are easy to compute since they only involve the already
+existing inverse Jacobian matrices $J^{-1}$ and the second derivatives
+$\hat\partial_k J_{pm} = \hat\partial_k \hat\partial_p \Phi_{m}$
+\marginpar{Is indeed $J_{pm} = \hat\partial_p \Phi_{m}$ or is it $J_{pm} = \hat\partial_m \Phi_{p}$?}
+of the polynomial forward mapping $\Phi$. Note that as expected, if the
+mapping is affine then $\hat\partial_k J_{pm}=0$ and the formula above reduces to
+the one expected.
+
+\end{document}

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