Struct t8_geometry_lagrange

Inheritance Relationships

Base Type

Struct Documentation

struct t8_geometry_lagrange : public t8_geometry_with_vertices

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:

  1. 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.

  2. Faces are oriented outwards the volume (3D) or the plane (2D). This is just a convention to ensure that the node numbering be consistent.

  3. 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.cxx file.

  4. 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).

Public Functions

t8_geometry_lagrange()

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.

virtual ~t8_geometry_lagrange()
inline virtual t8_geometry_type_t t8_geom_get_type() const

Get the type of this geometry.

Returns:

The type.

virtual 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 \( \mathbb{R}^3 \).

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:

  • cmesh – [in] The cmesh in which the point lies.

  • gtreeid – [in] The global tree (of the cmesh) in which the reference point is.

  • ref_coords – [in] Array of tree dimension x num_points entries, specifying points in the reference space.

  • num_points – [in] Number of points to map. Currently, only one point is supported.

  • out_coords – [out] Coordinates of the mapped points in physical space of ref_coords. The length is num_points * 3.

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.

Parameters:

  • cmesh – [in] The cmesh in which the point lies.

  • gtreeid – [in] The global tree (of the cmesh) in which the reference point is.

  • ref_coords – [in] Array of tree dimension x num_points entries, specifying points in the reference space.

  • num_points – [in] Number of points to map.

  • jacobian – [out] 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 \( \frac{\partial f_j}{\partial x_i} \)).

inline 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. 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:

  • cmesh – [in] The cmesh.

  • gtreeid – [in] The global tree.

virtual bool t8_geom_check_tree_compatibility() const

Check for compatibility of the currently loaded tree with the geometry. This geometry supports lines, triangles, quadrilaterals and hexahedra up to degree 2.

Returns:

True if the geometry is compatible with the tree.