From 4d69e491715094cb1733c80d566aa1fa7e2f5fa6 Mon Sep 17 00:00:00 2001 From: Maien Hamed Date: Thu, 10 Sep 2015 12:16:38 +0200 Subject: [PATCH] Added a subsection about mapping higher-order derivatives in Mapping documentation --- doc/news/changes.h | 6 ++++++ include/deal.II/fe/fe_poly.h | 14 ++++++++---- include/deal.II/fe/mapping.h | 42 ++++++++++++++++++++++++++++++++++++ 3 files changed, 58 insertions(+), 4 deletions(-) diff --git a/doc/news/changes.h b/doc/news/changes.h index c57b6101cb..64c86b4ec8 100644 --- a/doc/news/changes.h +++ b/doc/news/changes.h @@ -278,6 +278,12 @@ inconvenience this causes. (Timo Heister, Lei Qiao, 2015/09/09) +
  • New: Introduced third-order derivatives of the shape functions, which + can now be accessed through FEValues and FEValuesViews using similar interfaces + as shape_values, shape_derivatives and shape_hessians. + (Maien Hamed, 2015/09/08) +
    • +
  • Cleanup: TableIndices can now be used (constructed and accessed) with N > 7.
    diff --git a/include/deal.II/fe/fe_poly.h b/include/deal.II/fe/fe_poly.h index 0b03e783df..79331d4e85 100644 --- a/include/deal.II/fe/fe_poly.h +++ b/include/deal.II/fe/fe_poly.h @@ -445,12 +445,18 @@ protected: * to the Jacobian pushed forward gradient and second derivative. * * Before the correction, the third derivatives would be given by + * @f[ * D_{ijkl} = \frac{d^3\phi_i}{d \hat x_J d \hat x_K d \hat x_L} (J_{jJ})^{-1} (J_{kK})^{-1} (J_{lL})^{-1}, - * where J_{iI}=\frac{d x_i}{d \hat x_I}. After the correction, the correct + * @f] + * where $J_{iI}=\frac{d x_i}{d \hat x_I}$. After the correction, the correct * third derivative would be given by - * \frac{d^3\phi_i}{d x_j d x_k d x_l} = D_{ijkl} - H_{mjl} \frac{d^2 \phi_i}{d x_k d x_m} - H_{mkl} \frac{d^2 \phi_i}{d x_j d x_m} - H_{mjk} \frac{d^2 \phi_i}{d x_l d x_m} - K_{mjkl} \frac{d \phi_i}{d x_m}, - * where H_{ijk} = \frac{d^2 x_i}{d \hat x_J d \hat x_K} (J_{jJ})^{-1} (J_{kK})^{-1}, - * and K_{ijkl} = \frac{d^3 x_i}{d \hat x_J d \hat x_K d \hat x_L} (J_{jJ})^{-1} (J_{kK})^{-1} (J_{lL})^{-1} + * @f[ + * \frac{d^3\phi_i}{d x_j d x_k d x_l} = D_{ijkl} - H_{mjl} \frac{d^2 \phi_i}{d x_k d x_m} + * - H_{mkl} \frac{d^2 \phi_i}{d x_j d x_m} - H_{mjk} \frac{d^2 \phi_i}{d x_l d x_m} + * - K_{mjkl} \frac{d \phi_i}{d x_m}, + * @f] + * where $H_{ijk}$ is the Jacobian pushed-forward derivative and $K_{ijkl}$ is + * the Jacobian pushed-forward second derivative. */ void correct_third_derivatives (internal::FEValues::FiniteElementRelatedData &output_data, diff --git a/include/deal.II/fe/mapping.h b/include/deal.II/fe/mapping.h index 18b33925cc..1d178f9bf6 100644 --- a/include/deal.II/fe/mapping.h +++ b/include/deal.II/fe/mapping.h @@ -215,6 +215,48 @@ enum MappingType * transformed is specified by their MappingType argument. See the documentation there * for possible choices. * + *

    Derivatives of the mapping

    + * + * Some applications require the derivatives of the mapping, of which the first order + * derivative is the mapping Jacobian, $J_{iJ}(\hat{\mathbf x})=\frac{\partial x_i}{\partial \hat x_J}$, + * described above. Higher order derivatives of the mapping are similarly + * defined, for example the Jacobian derivative, + * $\hat H_{iJK}(\hat{\mathbf x}) = \frac{\partial^2 x_i}{\partial \hat x_J \partial \hat x_K}$, + * and the Jacobian second derivative, + * $\hat K_{iJKL}(\hat{\mathbf x}) = \frac{\partial^3 x_i}{\partial \hat x_J \partial + * \hat x_K \partial \hat x_L}$. + * It is also useful to define the "pushed-forward" versions of the higher order derivatives: + * the Jacobian pushed-forward + * derivative, $H_{ijk}(\hat{\mathbf x}) = \frac{\partial^2 x_i}{\partial \hat x_J \partial + * \hat x_K}(J_{jJ})^{-1}(J_{kK})^{-1}$, + * and the Jacobian pushed-forward second derivative, + * $K_{ijkl}(\hat{\mathbf x}) = \frac{\partial^3 x_i}{\partial \hat x_J \partial \hat x_K \partial + * \hat x_L}(J_{jJ})^{-1}(J_{kK})^{-1}(J_{lL})^{-1}$. + * These pushed-forward versions can be used to compute the higher order derivatives of functions + * defined on the reference cell with respect to the + * real cell coordinates. for instance, the Jacobian derivative with respect to the real cell coordinates is + * given by: + * + * @f[ + * \frac{\partial}{\partial x_j}\left[J_{iJ}(\hat{\mathbf x})\right] = + * H_{ikn}(\hat{\mathbf x})J_{nJ}(\hat{\mathbf x}), + * @f] + * and the derivative of the Jacobian inverse with respect to the real cell coordinates is similarly given by: + * @f[ + * \frac{\partial}{\partial x_j}\left[\left(J_{iJ}(\hat{\mathbf x})\right)^{-1}\right] + * = -H_{nik}(\hat{\mathbf x})\left(J_{nJ}(\hat{\mathbf x})\right)^{-1}. + * @f] + * + * In a similar fashion, higher order derivatives, with respect to the real cell coordinates, of functions + * defined on the reference cell can + * be defined using the Jacobian pushed-forward higher-order derivatives. + * For example, the derivative, with respect to the real cell coordinates, of the Jacobian pushed-forward + * derivative is given by: + * + * @f[ + * \frac{\partial}{\partial x_l}\left[H_{ijk}(\hat{\mathbf x})\right] = K_{ijkl}(\hat{\mathbf x}) + * -H_{mjl}(\hat{\mathbf x})H_{imk}(\hat{\mathbf x})-H_{mkl}(\hat{\mathbf x})H_{imk}(\hat{\mathbf x}). + * @f] * *

    References

    * -- 2.39.5