From: Wolfgang Bangerth Date: Fri, 14 Aug 2015 16:42:23 +0000 (-0500) Subject: General doc updates. X-Git-Tag: v8.4.0-rc2~599^2 X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=refs%2Fpull%2F1362%2Fhead;p=dealii.git General doc updates. This provides a good number of general doc updates for the Mapping base class. No functional changes of any sort. --- diff --git a/include/deal.II/fe/mapping.h b/include/deal.II/fe/mapping.h index fcd23e9d69..1778ed3022 100644 --- a/include/deal.II/fe/mapping.h +++ b/include/deal.II/fe/mapping.h @@ -111,79 +111,88 @@ enum MappingType /** - * Abstract base class for mapping classes. + * @short Abstract base class for mapping classes. * - * The interface for filling the tables of FEValues is provided. Everything - * else has to happen in derived classes. + * This class declares the interface for the functionality to describe mappings + * from the reference (unit) cell to a cell in real space, as well as for filling + * the information necessary to use the FEValues, FEFaceValues, and FESubfaceValues + * classes. Concrete implementations of these interfaces are provided in + * derived classes. * *

Mathematics of the mapping

* - * The mapping is a transformation $\mathbf x = \Phi(\mathbf{\hat x})$ which - * maps the reference cell [0,1]dim to the actual grid cell in - * Rspacedim. In order to describe the application of the mapping - * to different objects, we introduce the notation for the Jacobian - * $J(\mathbf{\hat x}) = \nabla\Phi(\mathbf{\hat x})$. For instance, if + * The mapping is a transformation $\mathbf x = \mathbf F_K(\hat{\mathbf x})$ which + * maps points $\hat{\mathbf x}$ in the reference cell $[0,1]^\text{dim}$ to points + * $\mathbf x$ in the actual grid cell + * $K\subset{\mathbb R}^\text{spacedim}$. Many of the applications of such mappings + * require the Jacobian of this mapping, + * $J(\hat{\mathbf x}) = \hat\nabla {\mathbf F}_K(\hat{\mathbf x})$. For instance, if * dim=spacedim=2, we have * @f[ - * J(\mathbf{\hat x}) = \left(\begin{matrix} + * J(\hat{\mathbf x}) = \left(\begin{matrix} * \frac{\partial x}{\partial \hat x} & \frac{\partial x}{\partial \hat y} * \\ * \frac{\partial y}{\partial \hat x} & \frac{\partial y}{\partial \hat y} * \end{matrix}\right) * @f] * - *

Mapping of functions

+ *

%Mapping of scalar functions

* - * Functions are simply mapped such that + * The shape functions of scalar finite elements are typically defined on a reference + * cell and are then simply mapped according to the rule * @f[ - * u(\mathbf x) = u\bigl(\Phi(\mathbf{\hat x})\bigr) - * = \hat u(\mathbf{\hat x}). + * \varphi(\mathbf x) = \varphi\bigl(\mathbf F_K(\hat{\mathbf x})\bigr) + * = \hat \varphi(\hat{\mathbf x}). * @f] - * Since finite element shape functions are usually defined on the reference - * cell, nothing needs to be done for them. For a function defined on the - * computational domain, the quadrature points need to be mapped, which is - * done in fill_fe_values() if @p update_quadrature_points is set in the - * update flags. The mapped quadrature points are then accessed through - * FEValuesBase::quadrature_point(). * - * @todo Add a function transform_quadrature_points for this. * - *

Mapping of integrals

+ *

%Mapping of integrals

* - * The volume form $d\hat x$ is mapped such that for a grid cell Z + * Using simply a change of variables, integrals of scalar functions over a cell + * $K$ can be expressed as an integral over the reference cell $\hat K$. + * Specifically, The volume form $d\hat x$ is transformed so that * @f[ - * \int_Z u(\mathbf x)\,d\mathbf x = \int_{\hat Z} \hat - * u(\mathbf{\hat x}) \left|\text{det}J(\mathbf{\hat x})\right| - * \,d\mathbf{\hat x}. + * \int_K u(\mathbf x)\,dx = \int_{\hat K} \hat + * u(\hat{\mathbf x}) \left|\text{det}J(\hat{\mathbf x})\right| + * \,d\hat x. * @f] * - * The transformed quadrature weights $\left|\text{det}J(\mathbf{\hat - * x})\right|$ are accessed through FEValuesBase::JxW() and computed in - * fill_fe_values(), if @p update_JxW_values is set in the update flags. + * In expressions where such integrals are approximated by quadrature, + * this then leads to terms of the form + * @f[ + * \int_K u(\mathbf x)\,dx + * \approx + * \sum_{q} + * \hat u(\hat{\mathbf x}_q) + * \underbrace{\left|\text{det}J(\hat{\mathbf x}_q)\right| w_q}_{=: \text{JxW}_q}. + * @f] + * Here, the weights $\text{JxW}_q$ of each quadrature point (where JxW + * mnemonically stands for Jacobian times Quadrature Weights) take the + * role of the $dx$ in the original integral. Consequently, they appear in all + * code that computes integrals approximated by quadrature, and are accessed + * by FEValues::JxW(). * - * @todo Add a function transform_quadrature_weights for this. + * @todo Document what happens in the codimension-1 case. * - * @todo Add documentation on the codimension-one case * - *

Mapping of vector fields, differential forms and gradients of vector + *

%Mapping of vector fields, differential forms and gradients of vector * fields

* - * The transformation of vector fields, differential forms - * (gradients/jacobians) and gradients of vector fields between the reference - * cell and the actual grid cell follows the general form + * The transformation of vector fields or differential forms + * (gradients of scalar functions) $\mathbf v$, and gradients of vector fields $\mathbf T$ + * follows the general form * * @f[ - * \mathbf v(\mathbf x) = \mathbf A(\mathbf{\hat x}) - * \mathbf{\hat v}(\mathbf{\hat x}), + * \mathbf v(\mathbf x) = \mathbf A(\hat{\mathbf x}) + * \hat{\mathbf v}(\hat{\mathbf x}), * \qquad - * \mathbf T(\mathbf x) = \mathbf A(\mathbf{\hat x}) - * \mathbf{\hat T}(\mathbf{\hat x}) \mathbf B(\mathbf{\hat x}), + * \mathbf T(\mathbf x) = \mathbf A(\hat{\mathbf x}) + * \hat{\mathbf T}(\hat{\mathbf x}) \mathbf B(\hat{\mathbf x}). * @f] - * - * where v is a vector field or a differential form and and T a - * tensor field of gradients. The differential forms A and B are - * determined by the MappingType enumerator. These transformations are - * performed through the functions transform(). See the documentation there + * The differential forms A and B are + * determined by the kind of object being transformed. These transformations are + * performed through the transform() functions, and the type of object being + * transformed is specified by their MappingType argument. See the documentation there * for possible choices. * * @@ -237,6 +246,18 @@ public: std_cxx11::array, GeometryInfo::vertices_per_cell> get_vertices (const typename Triangulation::cell_iterator &cell) const; + /** + * Returns whether the mapping preserves vertex locations, i.e., whether the + * mapped location of the reference cell vertices (given by + * GeometryInfo::unit_cell_vertex()) equals the result of + * cell-@>vertex(). + * + * For example, implementations in derived classes return @p true for + * MappingQ, MappingQ1, MappingCartesian, but @p false for MappingQEulerian, + * MappingQ1Eulerian, and MappingFEField. + */ + virtual + bool preserves_vertex_locations () const = 0; /** * @name Mapping points between reference and real cells @@ -296,16 +317,15 @@ public: * Transforms the point @p p on the real @p cell to the corresponding point * on the unit cell, and then projects it to a dim-1 point on the face with * the given face number @p face_no. Ideally the point @p p is near the face - * @ face_no, but any point in the cell can technically be projected. + * @p face_no, but any point in the cell can technically be projected. * * This function does not make physical sense when dim=1, * so it throws an exception in this case. */ Point - project_real_point_to_unit_point_on_face ( - const typename Triangulation::cell_iterator &cell, - const unsigned int &face_no, - const Point &p) const; + project_real_point_to_unit_point_on_face (const typename Triangulation::cell_iterator &cell, + const unsigned int &face_no, + const Point &p) const; /** * @} @@ -313,17 +333,9 @@ public: /** - * Returns whether the mapping preserves vertex locations, i.e., whether the - * mapped location of the reference cell vertices (given by - * GeometryInfo::unit_cell_vertex()) equals the result of - * cell-@>vertex(). - * - * For example, implementations in derived classes return @p true for - * MappingQ, MappingQ1, MappingCartesian, but @p false for MappingQEulerian, - * MappingQ1Eulerian. + * @name Exceptions + * @{ */ - virtual - bool preserves_vertex_locations () const = 0; /** * Exception @@ -358,6 +370,9 @@ public: << "fraction of " << arg2 << " in quadrature point " << arg3 << "."); + /** + * @} + */ /** * @name Interface with FEValues @@ -722,33 +737,36 @@ public: *
  • @p mapping_contravariant: maps a vector field on the reference cell * is to the physical cell through the Jacobian: * @f[ - * \mathbf u(\mathbf x) = J(\mathbf{\hat x})\mathbf{\hat u}(\mathbf{\hat x}). + * \mathbf u(\mathbf x) = J(\hat{\mathbf x})\hat{\mathbf u}(\hat{\mathbf x}). * @f] * In physics, this is usually referred to as the contravariant * transformation. Mathematically, it is the push forward of a vector field. * *
  • @p mapping_covariant: maps a field of one-forms on the reference - * cell to a field of one-forms on the physical cell. (theoretically this - * would refer to a DerivativeForm<1, dim, 1> but it canonically identified - * with a Tensor<1,dim>). Mathematically, it is the pull back of the + * cell to a field of one-forms on the physical cell. (Theoretically this + * would refer to a DerivativeForm<1,dim,1> but we canonically identify + * this type with a Tensor<1,dim>). Mathematically, it is the pull back of the * differential form * @f[ - * \mathbf u(\mathbf x) = J(J^{T} J)^{-1}(\mathbf{\hat x})\mathbf{\hat - * u}(\mathbf{\hat x}). + * \mathbf u(\mathbf x) = J(\hat{\mathbf x})(J(\hat{\mathbf x})^{T} J(\hat{\mathbf x}))^{-1}\hat{\mathbf + * u}(\hat{\mathbf x}). * @f] + * Gradients of scalar differentiable functions are transformed this way. + * * In the case when dim=spacedim the previous formula reduces to * @f[ - * \mathbf u(\mathbf x) = J^{-T}(\mathbf{\hat x})\mathbf{\hat - * u}(\mathbf{\hat x}). + * \mathbf u(\mathbf x) = J(\hat{\mathbf x})^{-T}\hat{\mathbf + * u}(\hat{\mathbf x}) * @f] - * Gradients of scalar differentiable functions are transformed this way. + * because we assume that the mapping $\mathbf F_K$ is always invertible, + * and consequently its Jacobian $J$ is an invertible matrix. * *
  • @p mapping_piola: A field of dim-1-forms on the reference cell * is also represented by a vector field, but again transforms differently, * namely by the Piola transform * @f[ - * \mathbf u(\mathbf x) = \frac{1}{\text{det}J(\mathbf x)} - * J(\mathbf x) \mathbf{\hat u}(\mathbf x). + * \mathbf u(\mathbf x) = \frac{1}{\text{det}\;J(\mathbf x)} + * J(\mathbf x) \hat{\mathbf u}(\mathbf x). * @f] * */ @@ -763,31 +781,32 @@ public: /** * Transform a field of differential forms from the reference cell to the - * physical cell. It is useful to think of $\mathbf{T} = D \mathbf u$ and - * $\mathbf{\hat T} = \hat D \mathbf{\hat u}$, with $\mathbf u$ a vector + * physical cell. It is useful to think of $\mathbf{T} = \nabla \mathbf u$ and + * $\hat{\mathbf T} = \hat \nabla \hat{\mathbf u}$, with $\mathbf u$ a vector * field. The mapping types currently implemented by derived classes are: *
      *
    • @p mapping_covariant: maps a field of forms on the reference cell to * a field of forms on the physical cell. Mathematically, it is the pull * back of the differential form * @f[ - * \mathbf T(\mathbf x) = \mathbf{\hat T}(\mathbf{\hat x}) - * J*(J^{T} J)^{-1}(\mathbf{\hat x}). - * @f] - * n the case when dim=spacedim the previous formula reduces to - * @f[ - * \mathbf T(\mathbf x) = \mathbf{\hat u}(\mathbf{\hat x}) - * J^{-1}(\mathbf{\hat x}). + * \mathbf T(\mathbf x) = \hat{\mathbf T}(\hat{\mathbf x}) + * J(\hat{\mathbf x})(J(\hat{\mathbf x})^{T} J(\hat{\mathbf x}))^{-1}. * @f] * Jacobians of spacedim-vector valued differentiable functions are * transformed this way. + * + * In the case when dim=spacedim the previous formula reduces to + * @f[ + * \mathbf T(\mathbf x) = \hat{\mathbf u}(\hat{\mathbf x}) + * J(\hat{\mathbf x})^{-1}. + * @f] *
    * @note It would have been more reasonable to make this transform a * template function with the rank in DerivativeForm@<1, dim, * rank@>. Unfortunately C++ does not allow templatized virtual * functions. This is why we identify DerivativeForm@<1, dim, * 1@> with a Tensor@<1,dim@> when using - * mapping_covariant() in the function transform above this one. + * mapping_covariant() in the function transform() above this one. */ virtual void @@ -799,39 +818,39 @@ public: /** * Transform a tensor field from the reference cell to the physical cell. - * This tensors are usually the Jacobians in the reference cell of + * These tensors are usually the Jacobians in the reference cell of * vector fields that have been pulled back from the physical cell. The * mapping types currently implemented by derived classes are: *
      - *
    • @p mapping_contravariant_gradient, it assumes $\mathbf u(\mathbf x) - * = J \mathbf{\hat u}$ so that + *
    • @p mapping_contravariant_gradient: it assumes $\mathbf u(\mathbf x) + * = J \hat{\mathbf u}$ so that * @f[ * \mathbf T(\mathbf x) = - * J(\mathbf{\hat x}) \mathbf{\hat T}(\mathbf{\hat x}) - * J^{-1}(\mathbf{\hat x}). + * J(\hat{\mathbf x}) \hat{\mathbf T}(\hat{\mathbf x}) + * J(\hat{\mathbf x})^{-1}. * @f] - *
    • @p mapping_covariant_gradient, it assumes $\mathbf u(\mathbf x) = - * J^{-T} \mathbf{\hat u}$ so that + *
    • @p mapping_covariant_gradient: it assumes $\mathbf u(\mathbf x) = + * J^{-T} \hat{\mathbf u}$ so that * @f[ * \mathbf T(\mathbf x) = - * J^{-T}(\mathbf{\hat x}) \mathbf{\hat T}(\mathbf{\hat x}) - * J^{-1}(\mathbf{\hat x}). + * J(\hat{\mathbf x})^{-T} \hat{\mathbf T}(\hat{\mathbf x}) + * J(\hat{\mathbf x})^{-1}. * @f] - *
    • @p mapping_piola_gradient, it assumes $\mathbf u(\mathbf x) = - * \frac{1}{\text{det}J(\mathbf x)} J(\mathbf x) \mathbf{\hat u}(\mathbf x)$ + *
    • @p mapping_piola_gradient: it assumes $\mathbf u(\mathbf x) = + * \frac{1}{\text{det}\;J(\mathbf x)} J(\mathbf x) \hat{\mathbf u}(\mathbf x)$ * so that * @f[ * \mathbf T(\mathbf x) = - * \frac{1}{\text{det}J(\mathbf x)} - * J(\mathbf{\hat x}) \mathbf{\hat T}(\mathbf{\hat x}) - * J^{-1}(\mathbf{\hat x}). + * \frac{1}{\text{det}\;J(\mathbf x)} + * J(\hat{\mathbf x}) \hat{\mathbf T}(\hat{\mathbf x}) + * J(\hat{\mathbf x})^{-1}. * @f] *
    - * @todo The formulas for mapping_covariant_gradient(), - * mapping_contravariant_gradient() and mapping_piola_gradient() are only + * @todo The formulas for mapping_covariant_gradient, + * mapping_contravariant_gradient and mapping_piola_gradient are only * true as stated for linear mappings. If, for example, the mapping is - * bilinear then there is a missing term associated with the derivative of - * J. + * bilinear (or has a higher order polynomial degree) then there is a + * missing term associated with the derivative of $J$. */ virtual void