Mapping with Lagrange basis functions. More...
#include <t8_geometry_lagrange.hxx>
Public Member Functions | |
| t8_geometry_lagrange (int dim) | |
| Constructor of the Lagrange geometry with a given dimension. More... | |
| t8_geometry_type_t | t8_geom_get_type () const |
| Get the type of this geometry. More... | |
| void | t8_geom_evaluate (t8_cmesh_t cmesh, t8_gloidx_t gtreeid, const double *ref_coords, const size_t num_points, double *out_coords) const |
| Maps points from the reference space to the physical space \( \mathbb{R}^3 \). More... | |
| virtual void | t8_geom_evaluate_jacobian (t8_cmesh_t cmesh, t8_gloidx_t gtreeid, const double *ref_coords, const size_t num_points, double *jacobian) const |
| Compute the Jacobian of the t8_geom_evaluate map at a point in the reference space. More... | |
| virtual void | t8_geom_load_tree_data (t8_cmesh_t cmesh, t8_gloidx_t gtreeid) |
| Update a possible internal data buffer for per tree data. More... | |
 Public Member Functions inherited from t8_geometry_with_vertices | |
| t8_geometry_with_vertices (int dimension, std::string name) | |
| virtual | ~t8_geometry_with_vertices () |
| The destructor. More... | |
| virtual bool | t8_geom_tree_negative_volume () const |
| Check if the currently active tree has a negative volume. More... | |
| t8_geometry_type_t | t8_geom_get_type () const |
| Get the type of this geometry. More... | |
| Public Member Functions inherited from t8_geometry | |
| t8_geometry (int dim, std::string name) | |
| virtual | ~t8_geometry () |
| The destructor. More... | |
| virtual void | t8_geom_point_batch_inside_element (t8_forest_t forest, t8_locidx_t ltreeid, const t8_element_t *element, const double *points, const int num_points, int *is_inside, const double tolerance) const |
| Query whether a batch of points lies inside an element. More... | |
| int | t8_geom_get_dimension () const |
| Get the dimension of this geometry. More... | |
| const std::string | t8_geom_get_name () const |
| Get the name of this geometry. More... | |
| const size_t | t8_geom_get_hash () const |
Additional Inherited Members | |
| Protected Attributes inherited from t8_geometry_with_vertices | |
| t8_gloidx_t | active_tree |
| t8_eclass_t | active_tree_class |
| const double * | active_tree_vertices |
| Protected Attributes inherited from t8_geometry | |
| int | dimension |
| The dimension of reference space for which this is a geometry. | |
| std::string | name |
| The name of this geometry. | |
| size_t | hash |
| The hash of the name of this geometry. | |
Detailed Description
Mapping with Lagrange basis functions.
The enumeration of the nodal basis functions depends on the node numbering scheme. While this is a convention, we came up with the following rules to guide us when constructing new mappings:
- Be compatible with lower degree elements. It means that when we read the first n nodes, it gives a valid lower-degree element. This rule is made attainable by the fact that the nodes spanning the Lagrange basis are equidistant. As an example, consider a degree four segment element whose five nodes in increasing coordinate direction are numbered as 0-3-2-4-1. Reading the first two nodes gives us a valid linear segment, while reading two first three nodes provides a valid quadratic segment. In general, we need to find for an element of degree d which nodes coincide with the nodes of elements with degree lower than d. Those nodes must be numbered first.
- Faces are oriented outwards the volume (3D) or the plane (2D). This is just a convention to ensure that the node numbering be consistent.
- The mapping by the Lagrange geometry falls back to the linear geometry for degree one. It means that the starting point for the node number assignment is the numbering defined in the t8_element.c file.
- The node numbering is performed in increasing spatial dimension, which results in a hierarchical construction of elements. The node numbers are assigned based on increasing face IDs, then increasing edge IDs, then increasing vertex IDs. For volume elements, the volume nodes are assigned lastly. This hierarchical construction satisfies rule 1 at the mesh level.
You can verify these rules by checking the documentation of the t8_geom_xxx_basis methods of this class (e.g. t8_geometry_lagrange::t8_geom_t6_basis).
Constructor & Destructor Documentation
◆ t8_geometry_lagrange()
| t8_geometry_lagrange::t8_geometry_lagrange | ( | int | dim | ) |
Constructor of the Lagrange geometry with a given dimension.
The geometry is compatible with all tree types and uses as many vertices as the number of Lagrange basis functions used for the mapping. The vertices are saved via the t8_cmesh_set_tree_vertices function.
- Parameters
-
[in] dim 0 <= dim <= 3. Element dimension in the parametric space. E.g. dim = 2 for a T8_ECLASS_QUAD element.
Member Function Documentation
◆ t8_geom_evaluate()
|
virtual |
Maps points from the reference space to the physical space \( \mathbb{R}^3 \).
For linear elements, it gives the same result as t8_geom_compute_linear_geometry.
The mapping is performed via Lagrange interpolation according to
\( \mathbf{x}(\vec{\xi}) = \sum\limits_{i=1}^{N_{\mathrm{vertex}}} \psi_i(\vec{\xi}) \mathbf{x}_i \)
where \( \vec{\xi} \) is the point in the reference space to be mapped, \( \mathbf{x} \) is the mapped point we search, \( \psi_i(\vec{\xi}) \) are the basis functions associated with the vertices, and \( \mathbf{x}_i \) are the vertices of the current tree in the physical space. The basis functions are specific to the tree type, see e.g. t8_geom_t6_basis. The vertices of the current tree were set with t8_cmesh_set_tree_vertices.
- Parameters
-
[in] cmesh The cmesh in which the point lies. [in] gtreeid The global tree (of the cmesh) in which the reference point is. [in] ref_coords Array of dimension x num_points entries, specifying points in the reference space. [in] num_points Number of points to map. Currently, only one point is supported. [out] out_coords Coordinates of the mapped points in physical space of ref_coords. The length is num_points * 3.
Implements t8_geometry.
◆ t8_geom_evaluate_jacobian()
|
virtual |
Compute the Jacobian of the t8_geom_evaluate map at a point in the reference space.
- Parameters
-
[in] cmesh The cmesh in which the point lies. [in] gtreeid The global tree (of the cmesh) in which the reference point is. [in] ref_coords Array of dimension x num_points entries, specifying points in the reference space. [in] num_points Number of points to map. [out] jacobian The Jacobian at ref_coords. Array of size num_points x dimension x 3. Indices \( 3 \cdot i\) , \( 3 \cdot i+1 \) , \( 3 \cdot i+2 \) correspond to the \( i \)-th column of the Jacobian (Entry \( 3 \cdot i + j \) is \( \frac{\partial f_j}{\partial x_i} \)).
Implements t8_geometry.
◆ t8_geom_get_type()
|
inlinevirtual |
◆ t8_geom_load_tree_data()
|
inlinevirtual |
Update a possible internal data buffer for per tree data.
This function is called before the first coordinates in a new tree are evaluated. You can use it for example to load the polynomial degree of the Lagrange basis into an internal buffer (as is done in the Lagrange geometry).
- Parameters
-
[in] cmesh The cmesh. [in] gtreeid The global tree.
Reimplemented from t8_geometry_with_vertices.
The documentation for this struct was generated from the following files:
- src/t8_geometry/t8_geometry_implementations/t8_geometry_lagrange.hxx
- src/t8_geometry/t8_geometry_implementations/t8_geometry_lagrange.cxx