discretize.TreeMesh

class discretize.TreeMesh(h=None, origin=None, **kwargs)

Bases: discretize._extensions.tree_ext._TreeMesh, discretize.base.BaseTensorMesh, discretize.operators.InnerProducts, discretize.operators.DiffOperators, discretize.mixins.mesh_io.TreeMeshIO, discretize.mixins.InterfaceMixins

TreeMesh is a class for adaptive QuadTree (2D) and OcTree (3D) meshes.

Attributes
area

area has been deprecated. See face_areas for documentation

areaFx

areaFx has been deprecated. See face_x_areas for documentation

areaFy

areaFy has been deprecated. See face_y_areas for documentation

areaFz

areaFz has been deprecated. See face_z_areas for documentation

average_cell_to_edge
average_cell_to_face

Construct the averaging operator on cell centers to cell faces.

average_cell_to_face_x

Construct the averaging operator on cell centers to cell x-faces.

average_cell_to_face_y

Construct the averaging operator on cell centers to cell y-faces.

average_cell_to_face_z

Construct the averaging operator on cell centers to cell z-faces.

average_cell_vector_to_face

Construct the averaging operator on cell centers to cell faces.

average_edge_to_cell

Construct the averaging operator on cell edges to cell centers.

average_edge_to_cell_vector

Construct the averaging operator on cell edges to cell centers.

average_edge_to_face_vector

Construct the averaging operator on cell edges in the x direction to cell faces.

average_edge_x_to_cell

Construct the averaging operator on cell edges in the x direction to cell centers.

average_edge_y_to_cell

Construct the averaging operator on cell edges in the y direction to cell centers.

average_edge_z_to_cell

Construct the averaging operator on cell edges in the z direction to cell centers.

average_face_to_cell

Construct the averaging operator on cell faces to cell centers.

average_face_to_cell_vector

Construct the averaging operator on cell faces to cell centers.

average_face_x_to_cell

Construct the averaging operator on cell faces in the x direction to cell centers.

average_face_y_to_cell

Construct the averaging operator on cell faces in the y direction to cell centers.

average_face_z_to_cell

Construct the averaging operator on cell faces in the z direction to cell centers.

average_node_to_cell

Construct the averaging operator on cell nodes to cell centers.

average_node_to_edge

Construct the averaging operator on cell nodes to cell edges, keeping each dimension separate.

average_node_to_edge_x

Averaging operator on cell nodes to x-edges

average_node_to_edge_y

Averaging operator on cell nodes to y-edges

average_node_to_edge_z

Averaging operator on cell nodes to z-edges

average_node_to_face

Construct the averaging operator on cell nodes to cell edges, keeping each dimension separate.

average_node_to_face_x

Averaging operator on cell nodes to x-faces

average_node_to_face_y

Averaging operator on cell nodes to y-faces

average_node_to_face_z

Averaging operator on cell nodes to z-faces

axis_u

Deprecated since version 0.7.0.

axis_v

Deprecated since version 0.7.0.

axis_w

Deprecated since version 0.7.0.

boundary_edge_vector_integral

Represents the operation of integrating a vector function on the boundary

boundary_edges
boundary_face_outward_normals
boundary_face_scalar_integral

Represents the operation of integrating a scalar function on the boundary

boundary_faces
boundary_node_vector_integral

Represents the operation of integrating a vector function dotted with the boundary normal

boundary_nodes
cellBoundaryInd

cellBoundaryInd has been deprecated. See cell_boundary_indices for documentation

cellGrad

cellGrad has been deprecated. See cell_gradient for documentation

cellGradBC

cellGradBC has been deprecated. See cell_gradient_BC for documentation

cellGradStencil

cellGradStencil has been deprecated. See cell_gradient_stencil for documentation

cellGradx

cellGradx has been deprecated. See cell_gradient_x for documentation

cellGrady

cellGrady has been deprecated. See cell_gradient_y for documentation

cellGradz

cellGradz has been deprecated. See cell_gradient_z for documentation

cell_boundary_indices

Returns a tuple of arrays of indexes for boundary cells in each direction

cell_centers

Returns a numpy arrayof shape (n_cells, dim) with the center locations of all cells in order.

cell_centers_x

Cell-centered grid vector (1D) in the x direction.

cell_centers_y

Cell-centered grid vector (1D) in the y direction.

cell_centers_z

Cell-centered grid vector (1D) in the z direction.

cell_gradient

Cell centered Gradient operator built off of the faceDiv operator.

cell_gradient_BC

The cell centered Gradient boundary condition matrix

cell_gradient_x

Cell centered Gradient operator in x-direction (Gradx)

cell_gradient_y

Cell centered Gradient operator in y-direction (Grady)

cell_gradient_z

Cell centered Gradient operator in z-direction (Gradz)

cell_nodes

The index of nodes for each cell.

cell_state
cell_volumes

Returns a numpy array of length n_cells with the volumes (areas in 2D) of all cells in order.

dim

The dimension of the mesh (1, 2, or 3).

edge

edge has been deprecated. See edge_lengths for documentation

edgeCurl

edgeCurl has been deprecated. See edge_curl for documentation

edgeEx

edgeEx has been deprecated. See edge_x_lengths for documentation

edgeEy

edgeEy has been deprecated. See edge_y_lengths for documentation

edgeEz

edgeEz has been deprecated. See edge_z_lengths for documentation

edge_curl

Construct the 3D curl operator.

edge_lengths

Returns a numpy array of length n_edges with the length of all edges ordered by x, then y, then z.

edge_nodes

The index of nodes for every edge.

edge_tangents

Edge Tangents

edges

Edge grid

edges_x

Returns a numpy array of shape (n_edges_x, dim) with the centers of all non-hanging edges along the first dimension in order.

edges_y

Returns a numpy array of shape (n_edges_y, dim) with the centers of all non-hanging edges along the second dimension in order.

edges_z

Returns a numpy array of shape (n_edges_z, dim) with the centers of all non-hanging edges along the third dimension in order.

faceBoundaryInd

faceBoundaryInd has been deprecated. See face_boundary_indices for documentation

faceDiv

faceDiv has been deprecated. See face_divergence for documentation

faceDivx

faceDivx has been deprecated. See face_x_divergence for documentation

faceDivy

faceDivy has been deprecated. See face_y_divergence for documentation

faceDivz

faceDivz has been deprecated. See face_z_divergence for documentation

face_areas

Returns a numpy array of length n_faces with the area (length in 2D) of all faces ordered by x, then y, then z.

face_boundary_indices

Returns a tuple of arrays of indexes for boundary faces in each direction

face_divergence

Construct divergence operator (face-stg to cell-centres).

face_normals

Face Normals

face_x_divergence

Construct divergence operator in the x component (face-stg to cell-centres).

face_y_divergence
face_z_divergence

Construct divergence operator in the z component (face-stg to cell-centers).

faces

Face grid

faces_x

Returns a numpy array of shape (n_faces_x, dim) with the centers of all non-hanging faces along the first dimension in order.

faces_y

Returns a numpy array of shape (n_faces_y, dim) with the centers of all non-hanging faces along the second dimension in order.

faces_z

Returns a numpy array of shape (n_faces_z, dim) with the centers of all non-hanging faces along the third dimension in order.

fill

How filled is the mesh compared to a TensorMesh? As a fraction: [0, 1].

finalized
h
h_gridded

Returns an (n_cells, dim) numpy array with the widths of all cells in order

hanging_edges_x

Returns a numpy array of shape (n_hanging_edges_x, dim) with the centers of all hanging edges along the first dimension in order.

hanging_edges_y

Returns a numpy array of shape (n_hanging_edges_y, dim) with the centers of all hanging edges along the second dimension in order.

hanging_edges_z

Returns a numpy array of shape (n_hanging_edges_z, dim) with the centers of all hanging edges along the third dimension in order.

hanging_faces_x

Returns a numpy array of shape (n_hanging_faces_x, dim) with the centers of all hanging faces along the first dimension in order.

hanging_faces_y

Returns a numpy array of shape (n_hanging_faces_y, dim) with the centers of all hanging faces along the second dimension in order.

hanging_faces_z

Returns a numpy array of shape (n_hanging_faces_z, dim) with the centers of all hanging faces along the third dimension in order.

hanging_nodes

Returns a numpy array of shape (n_nodes, dim) with the locations of all hanging nodes in order.

hx

Width of cells in the x direction

hy

Width of cells in the y direction

hz

Width of cells in the z direction

maxLevel

maxLevel has been deprecated. See max_used_level for documentation

max_level

The maximum possible level for a cell on this mesh

max_used_level

The maximum level used, which may be less than max_level.

n_cells

Number of cells

n_edges

Total number of non-hanging edges amongst all dimensions

n_edges_per_direction

The number of edges in each direction

n_edges_x

Number of non-hanging edges oriented along the first dimension

n_edges_y

Number of non-hanging edges oriented along the second dimension

n_edges_z

Number of non-hanging edges oriented along the third dimension

n_faces

Total number of non-hanging faces amongst all dimensions

n_faces_per_direction

The number of faces in each direction

n_faces_x

Number of non-hanging faces oriented along the first dimension

n_faces_y

Number of non-hanging faces oriented along the second dimension

n_faces_z

Number of non-hanging faces oriented along the third dimension

n_hanging_edges

Total number of hanging edges amongst all dimensions

n_hanging_edges_x

Number of hanging edges oriented along the first dimension

n_hanging_edges_y

Number of hanging edges oriented along the second dimension

n_hanging_edges_z

Number of hanging edges oriented along the third dimension

n_hanging_faces

Total number of hanging faces amongst all dimensions

n_hanging_faces_x

Number of hanging faces oriented along the first dimension

n_hanging_faces_y

Number of hanging faces oriented along the second dimension

n_hanging_faces_z

Number of hanging faces oriented along the third dimension

n_hanging_nodes

Number of hanging nodes

n_nodes

Number of non-hanging nodes

n_total_edges

Total number of non-hanging and hanging edges amongst all dimensions

n_total_edges_x

Number of non-hanging and hanging edges oriented along the first dimension

n_total_edges_y

Number of non-hanging and hanging edges oriented along the second dimension

n_total_edges_z

Number of non-hanging and hanging edges oriented along the third dimension

n_total_faces

Total number of hanging and non-hanging faces amongst all dimensions

n_total_faces_x

Number of non-hanging and hanging faces oriented along the first dimension

n_total_faces_y

Number of non-hanging and hanging faces oriented along the second dimension

n_total_faces_z

Number of non-hanging and hanging faces oriented along the third dimension

n_total_nodes

Number of non-hanging and hanging nodes

nodalGrad

nodalGrad has been deprecated. See nodal_gradient for documentation

nodalLaplacian

nodalLaplacian has been deprecated. See nodal_laplacian for documentation

nodal_gradient

Construct gradient operator (nodes to edges).

nodal_laplacian
nodes

Returns a numpy array of shape (n_nodes, dim) with the locations of all non-hanging nodes in order.

nodes_x

Nodal grid vector (1D) in the x direction.

nodes_y

Nodal grid vector (1D) in the y direction.

nodes_z

Nodal grid vector (1D) in the z direction.

normals

normals has been deprecated. See face_normals for documentation

orientation
origin

Origin of the mesh

permuteCC

permuteCC has been deprecated. See permute_cells for documentation

permuteE

permuteE has been deprecated. See permute_edges for documentation

permuteF

permuteF has been deprecated. See permute_faces for documentation

permute_cells

Permutation matrix re-ordering of cells sorted by x, then y, then z

permute_edges

Permutation matrix re-ordering of edges sorted by x, then y, then z

permute_faces

Permutation matrix re-ordering of faces sorted by x, then y, then z

project_edge_to_boundary_edge
project_face_to_boundary_face
project_node_to_boundary_node
reference_is_rotated

True if the axes are rotated from the traditional <X,Y,Z> system

reference_system

The type of coordinate reference frame.

rotation_matrix

Builds a rotation matrix to transform coordinates from their coordinate system into a conventional cartesian system.

shape_cells

The number of cells in each direction

stencil_cell_gradient
stencil_cell_gradient_x

Cell gradient stencil matrix to total (including hanging) x faces

stencil_cell_gradient_y

Cell gradient stencil matrix to total (including hanging) y faces

stencil_cell_gradient_z

Cell gradient stencil matrix to total (including hanging) z faces

tangents

tangents has been deprecated. See edge_tangents for documentation

vectorCCx

vectorCCx has been deprecated. See cell_centers_x for documentation

vectorCCy

vectorCCy has been deprecated. See cell_centers_y for documentation

vectorCCz

vectorCCz has been deprecated. See cell_centers_z for documentation

vectorNx

vectorNx has been deprecated. See nodes_x for documentation

vectorNy

vectorNy has been deprecated. See nodes_y for documentation

vectorNz

vectorNz has been deprecated. See nodes_z for documentation

vntE

Total number of hanging and non-hanging edges in a [nx,ny,nz] form

vntF

Total number of hanging and non-hanging faces in a [nx,ny,nz] form

vol

vol has been deprecated. See cell_volumes for documentation

x0

Methods

average_cell_to_total_face_x(self)

Average matrix for cell center to total (including hanging) x faces

average_cell_to_total_face_y(self)

Average matrix for cell center to total (including hanging) y faces

average_cell_to_total_face_z(self)

Average matrix for cell center to total (including hanging) z faces

cell_gradient_weak_form_robin([alpha, beta, …])

Robin boundary condition for the weak formulation of the cell gradient

cell_levels_by_index(indices)

Fast function to return a list of levels for the given cell indices

copy()

Make a copy of the current mesh

edge_divergence_weak_form_robin([alpha, …])

Robin boundary condition for the weak formulation of the edge divergence

finalize(self)

Finalize the TreeMesh Called after finished cronstruction of the mesh.

from_omf(element)

Convert an OMF element to it’s proper discretize type.

getBCProjWF(*args, **kwargs)

getBCProjWF has been deprecated.

getBCProjWF_simple(*args, **kwargs)

getBCProjWF_simple has been deprecated.

getEdgeInnerProduct(*args, **kwargs)

getEdgeInnerProduct has been deprecated.

getEdgeInnerProductDeriv(*args, **kwargs)

getEdgeInnerProductDeriv has been deprecated.

getFaceInnerProduct(*args, **kwargs)

getFaceInnerProduct has been deprecated.

getFaceInnerProductDeriv(*args, **kwargs)

getFaceInnerProductDeriv has been deprecated.

getInterpolationMat(*args, **kwargs)

getInterpolationMat has been deprecated.

getTensor(*args, **kwargs)

getTensor has been deprecated.

get_BC_projections(BC[, discretization])

The weak form boundary condition projection matrices.

get_BC_projections_simple([discretization])

The weak form boundary condition projection matrices when mixed boundary condition is used

get_boundary_cells(self[, active_ind, direction])

Returns the indices of boundary cells in a given direction given an active index array.

get_cells_along_line(self, x0, x1)

Finds the cells along a line segment defined by two points

get_edge_inner_product([model, …])

Generate the edge inner product matrix

get_edge_inner_product_deriv(model[, …])

Parameters

get_face_inner_product([model, …])

Generate the face inner product matrix

get_face_inner_product_deriv(model[, …])

Parameters

get_interpolation_matrix(locs[, …])

Produces interpolation matrix

get_overlapping_cells(self, rectangle)

get_tensor(key)

Returns a tensor list.

insert_cells(self, points, levels[, finalize])

Insert cells into the TreeMesh that contain given points

isInside(*args, **kwargs)

isInside has been deprecated.

is_inside(pts[, location_type])

Determines if a set of points are inside a mesh.

number(self)

Number the cells, nodes, faces, and edges of the TreeMesh

plotGrid(*args, **kwargs)

plotGrid has been deprecated.

plotImage(*args, **kwargs)

plotImage has been deprecated.

plotSlice(*args, **kwargs)

plotSlice has been deprecated.

plot_3d_slicer(v[, xslice, yslice, zslice, …])

Plot slices of a 3D volume, interactively (scroll wheel).

plot_grid([ax, nodes, faces, centers, …])

Plot the nodal, cell-centered and staggered grids.

plot_image(v[, v_type, grid, view, ax, …])

Plots fields on the given mesh.

plot_slice(v[, v_type, normal, ind, …])

Plots slice of fields on the given 3D mesh.

point2index(locs)

Finds cells that contain the given points.

projectEdgeVector(*args, **kwargs)

projectEdgeVector has been deprecated.

projectFaceVector(*args, **kwargs)

projectFaceVector has been deprecated.

project_edge_vector(edge_vector)

Project vectors onto the edges of the mesh

project_face_vector(face_vector)

Project vectors onto the faces of the mesh.

readModelUBC(*args, **kwargs)

readModelUBC has been deprecated.

readUBC(file_name[, directory])

read_UBC(meshFile[, directory])

Read UBC 3D OcTree mesh file Input: :param str meshFile: path to the UBC GIF OcTree mesh file to read :rtype: discretize.TreeMesh :return: The octree mesh

read_model_UBC(file_name)

Read UBC OcTree model and get vector :param string file_name: path to the UBC GIF model file to read :rtype: numpy.ndarray :return: OcTree model

refine(self, function[, finalize])

Refine a TreeMesh using a user supplied function.

refine_ball(self, points, radii, levels[, …])

Refines the TreeMesh around points with the given radii

refine_box(self, x0s, x1s, levels[, finalize])

Refines the TreeMesh within the axis aligned boxes to the desired level

save([file_name, verbose])

Save the mesh to json :param str file: file_name for saving the casing properties :param str directory: working directory for saving the file

setCellGradBC(*args, **kwargs)

setCellGradBC has been deprecated.

set_cell_gradient_BC(BC)

Function that sets the boundary conditions for cell-centred derivative operators.

toVTK([models])

Convert this mesh object to it’s proper VTK or pyvista data object with the given model dictionary as the cell data of that dataset.

to_omf([models])

Convert this mesh object to it’s proper omf data object with the given model dictionary as the cell data of that dataset.

to_vtk([models])

Convert this mesh object to it’s proper VTK or pyvista data object with the given model dictionary as the cell data of that dataset.

validate()

Every object will be valid upon initialization

writeModelUBC(*args, **kwargs)

writeModelUBC has been deprecated.

writeUBC(*args, **kwargs)

writeUBC has been deprecated.

writeVTK(file_name[, models, directory])

Makes and saves a VTK object from this mesh and given models

write_UBC(file_name[, models, directory])

Write UBC ocTree mesh and model files from a octree mesh and model.

write_model_UBC(file_name, model[, directory])

Writes a model associated with a TreeMesh to a UBC-GIF format model file.

write_vtk(file_name[, models, directory])

Makes and saves a VTK object from this mesh and given models

deserialize

equals

serialize

to_dict

Attributes

TreeMesh.area

area has been deprecated. See face_areas for documentation

TreeMesh.areaFx

areaFx has been deprecated. See face_x_areas for documentation

TreeMesh.areaFy

areaFy has been deprecated. See face_y_areas for documentation

TreeMesh.areaFz

areaFz has been deprecated. See face_z_areas for documentation

TreeMesh.average_cell_to_edge
TreeMesh.average_cell_to_face

Construct the averaging operator on cell centers to cell faces.

TreeMesh.average_cell_to_face_x

Construct the averaging operator on cell centers to cell x-faces.

TreeMesh.average_cell_to_face_y

Construct the averaging operator on cell centers to cell y-faces.

TreeMesh.average_cell_to_face_z

Construct the averaging operator on cell centers to cell z-faces.

TreeMesh.average_cell_vector_to_face

Construct the averaging operator on cell centers to cell faces.

TreeMesh.average_edge_to_cell

Construct the averaging operator on cell edges to cell centers.

TreeMesh.average_edge_to_cell_vector

Construct the averaging operator on cell edges to cell centers.

TreeMesh.average_edge_to_face_vector

Construct the averaging operator on cell edges in the x direction to cell faces.

TreeMesh.average_edge_x_to_cell

Construct the averaging operator on cell edges in the x direction to cell centers.

TreeMesh.average_edge_y_to_cell

Construct the averaging operator on cell edges in the y direction to cell centers.

TreeMesh.average_edge_z_to_cell

Construct the averaging operator on cell edges in the z direction to cell centers.

TreeMesh.average_face_to_cell

Construct the averaging operator on cell faces to cell centers.

TreeMesh.average_face_to_cell_vector

Construct the averaging operator on cell faces to cell centers.

TreeMesh.average_face_x_to_cell

Construct the averaging operator on cell faces in the x direction to cell centers.

TreeMesh.average_face_y_to_cell

Construct the averaging operator on cell faces in the y direction to cell centers.

TreeMesh.average_face_z_to_cell

Construct the averaging operator on cell faces in the z direction to cell centers.

TreeMesh.average_node_to_cell

Construct the averaging operator on cell nodes to cell centers.

TreeMesh.average_node_to_edge

Construct the averaging operator on cell nodes to cell edges, keeping each dimension separate.

TreeMesh.average_node_to_edge_x

Averaging operator on cell nodes to x-edges

TreeMesh.average_node_to_edge_y

Averaging operator on cell nodes to y-edges

TreeMesh.average_node_to_edge_z

Averaging operator on cell nodes to z-edges

TreeMesh.average_node_to_face

Construct the averaging operator on cell nodes to cell edges, keeping each dimension separate.

TreeMesh.average_node_to_face_x

Averaging operator on cell nodes to x-faces

TreeMesh.average_node_to_face_y

Averaging operator on cell nodes to y-faces

TreeMesh.average_node_to_face_z

Averaging operator on cell nodes to z-faces

TreeMesh.axis_u

Deprecated since version 0.7.0: axis_u will be removed in discretize 1.0.0, it is replaced by mesh.orientation for better mesh orientation validation.

See also

orientation
TreeMesh.axis_v

Deprecated since version 0.7.0: axis_v will be removed in discretize 1.0.0, it is replaced by mesh.orientation for better mesh orientation validation.

See also

orientation
TreeMesh.axis_w

Deprecated since version 0.7.0: axis_w will be removed in discretize 1.0.0, it is replaced by mesh.orientation for better mesh orientation validation.

See also

orientation
TreeMesh.boundary_edge_vector_integral

Represents the operation of integrating a vector function on the boundary

This matrix represents the boundary surface integral of a vector function multiplied with a finite volume test function on the mesh.

In 1D and 2D, the operation assumes that the right array contains only a single component of the vector u. In 3D, however, we must assume that u will contain each of the three vector components, and it must be ordered as, [edges_1_x, ... ,edge_N_x, edge_1_y, ..., edge_N_y, edge_1_z, ..., edge_N_z] , where N is the number of boundary edges.

Returns
scipy.sparse.csr_matrix

Sparse matrix of shape (n_edges, n_boundary_edges) for 1D or 2D mesh, (n_edges, 3*n_boundary_edges) for a 3D mesh.

Notes

The integral we are representing on the boundary of the mesh is

\[\int_{\Omega} \vec{w} \cdot (\vec{u} \times \hat{n}) \partial \Omega\]

In discrete form this is:

\[w^T * P * u_b\]

where w is defined on all edges, and u_b is all three components defined on boundary edges.

TreeMesh.boundary_edges
TreeMesh.boundary_face_outward_normals
TreeMesh.boundary_face_scalar_integral

Represents the operation of integrating a scalar function on the boundary

This matrix represents the boundary surface integral of a scalar function multiplied with a finite volume test function on the mesh.

Returns
scipy.sparse.csr_matrix

Sparse matrix of shape (n_faces, n_boundary_faces)

Notes

The integral we are representing on the boundary of the mesh is

\[\int_{\Omega} u\vec{w} \cdot \hat{n} \partial \Omega\]

In discrete form this is:

\[w^T * P * u_b\]

where w is defined on all faces, and u_b is defined on boundary faces.

TreeMesh.boundary_faces
TreeMesh.boundary_node_vector_integral

Represents the operation of integrating a vector function dotted with the boundary normal

This matrix represents the boundary surface integral of a vector function dotted with the boundary normal and multiplied with a scalar finite volume test function on the mesh.

Returns
scipy.sparse.csr_matrix

Sparse matrix of shape (n_nodes, ndim * n_boundary_nodes).

Notes

The integral we are representing on the boundary of the mesh is

\[\int_{\Omega} (w \vec{u}) \cdot \hat{n} \partial \Omega\]

In discrete form this is:

\[w^T * P * u_b\]

where w is defined on all nodes, and u_b is all three components defined on boundary nodes.

TreeMesh.boundary_nodes
TreeMesh.cellBoundaryInd

cellBoundaryInd has been deprecated. See cell_boundary_indices for documentation

TreeMesh.cellGrad

cellGrad has been deprecated. See cell_gradient for documentation

TreeMesh.cellGradBC

cellGradBC has been deprecated. See cell_gradient_BC for documentation

TreeMesh.cellGradStencil

cellGradStencil has been deprecated. See cell_gradient_stencil for documentation

TreeMesh.cellGradx

cellGradx has been deprecated. See cell_gradient_x for documentation

TreeMesh.cellGrady

cellGrady has been deprecated. See cell_gradient_y for documentation

TreeMesh.cellGradz

cellGradz has been deprecated. See cell_gradient_z for documentation

TreeMesh.cell_boundary_indices

Returns a tuple of arrays of indexes for boundary cells in each direction xdown, xup, ydown, yup, zdown, zup

TreeMesh.cell_centers

Returns a numpy arrayof shape (n_cells, dim) with the center locations of all cells in order.

TreeMesh.cell_centers_x

Cell-centered grid vector (1D) in the x direction.

TreeMesh.cell_centers_y

Cell-centered grid vector (1D) in the y direction.

TreeMesh.cell_centers_z

Cell-centered grid vector (1D) in the z direction.

TreeMesh.cell_gradient

Cell centered Gradient operator built off of the faceDiv operator. Grad = - (Mf)^{-1} * Div * diag (volume)

TreeMesh.cell_gradient_BC

The cell centered Gradient boundary condition matrix

TreeMesh.cell_gradient_x

Cell centered Gradient operator in x-direction (Gradx) Grad = sp.vstack((Gradx, Grady, Gradz))

TreeMesh.cell_gradient_y

Cell centered Gradient operator in y-direction (Grady) Grad = sp.vstack((Gradx, Grady, Gradz))

TreeMesh.cell_gradient_z

Cell centered Gradient operator in z-direction (Gradz) Grad = sp.vstack((Gradx, Grady, Gradz))

TreeMesh.cell_nodes

The index of nodes for each cell.

Returns
numpy.ndarray of ints

Index array of shape (n_cells, 4) if 2D, or (n_cells, 6) if 3D

Notes

These indices will also point to hanging nodes.

TreeMesh.cell_state
TreeMesh.cell_volumes

Returns a numpy array of length n_cells with the volumes (areas in 2D) of all cells in order.

TreeMesh.dim

The dimension of the mesh (1, 2, or 3).

Returns
int

dimension of the mesh

TreeMesh.edge

edge has been deprecated. See edge_lengths for documentation

TreeMesh.edgeCurl

edgeCurl has been deprecated. See edge_curl for documentation

TreeMesh.edgeEx

edgeEx has been deprecated. See edge_x_lengths for documentation

TreeMesh.edgeEy

edgeEy has been deprecated. See edge_y_lengths for documentation

TreeMesh.edgeEz

edgeEz has been deprecated. See edge_z_lengths for documentation

TreeMesh.edge_curl

Construct the 3D curl operator.

TreeMesh.edge_lengths

Returns a numpy array of length n_edges with the length of all edges ordered by x, then y, then z.

TreeMesh.edge_nodes

The index of nodes for every edge.

The index of the nodes at each end of every (including hanging) edge.

Returns
tuple of numpy.ndarray of ints

One numpy array for each edge type (x, y, (z)) for this mesh.

Notes

These arrays will also index into the hanging nodes.

TreeMesh.edge_tangents

Edge Tangents

Returns
numpy.ndarray

normals, (n_edges, dim)

TreeMesh.edges

Edge grid

TreeMesh.edges_x

Returns a numpy array of shape (n_edges_x, dim) with the centers of all non-hanging edges along the first dimension in order.

TreeMesh.edges_y

Returns a numpy array of shape (n_edges_y, dim) with the centers of all non-hanging edges along the second dimension in order.

TreeMesh.edges_z

Returns a numpy array of shape (n_edges_z, dim) with the centers of all non-hanging edges along the third dimension in order.

TreeMesh.faceBoundaryInd

faceBoundaryInd has been deprecated. See face_boundary_indices for documentation

TreeMesh.faceDiv

faceDiv has been deprecated. See face_divergence for documentation

TreeMesh.faceDivx

faceDivx has been deprecated. See face_x_divergence for documentation

TreeMesh.faceDivy

faceDivy has been deprecated. See face_y_divergence for documentation

TreeMesh.faceDivz

faceDivz has been deprecated. See face_z_divergence for documentation

TreeMesh.face_areas

Returns a numpy array of length n_faces with the area (length in 2D) of all faces ordered by x, then y, then z.

TreeMesh.face_boundary_indices

Returns a tuple of arrays of indexes for boundary faces in each direction xdown, xup, ydown, yup, zdown, zup

TreeMesh.face_divergence

Construct divergence operator (face-stg to cell-centres).

TreeMesh.face_normals

Face Normals

Returns
numpy.ndarray

normals, (n_faces, dim)

TreeMesh.face_x_divergence
TreeMesh.face_y_divergence
TreeMesh.face_z_divergence
TreeMesh.faces

Face grid

TreeMesh.faces_x

Returns a numpy array of shape (n_faces_x, dim) with the centers of all non-hanging faces along the first dimension in order.

TreeMesh.faces_y

Returns a numpy array of shape (n_faces_y, dim) with the centers of all non-hanging faces along the second dimension in order.

TreeMesh.faces_z

Returns a numpy array of shape (n_faces_z, dim) with the centers of all non-hanging faces along the third dimension in order.

TreeMesh.fill

How filled is the mesh compared to a TensorMesh? As a fraction: [0, 1].

TreeMesh.finalized
TreeMesh.h
TreeMesh.h_gridded

Returns an (n_cells, dim) numpy array with the widths of all cells in order

TreeMesh.hanging_edges_x

Returns a numpy array of shape (n_hanging_edges_x, dim) with the centers of all hanging edges along the first dimension in order.

TreeMesh.hanging_edges_y

Returns a numpy array of shape (n_hanging_edges_y, dim) with the centers of all hanging edges along the second dimension in order.

TreeMesh.hanging_edges_z

Returns a numpy array of shape (n_hanging_edges_z, dim) with the centers of all hanging edges along the third dimension in order.

TreeMesh.hanging_faces_x

Returns a numpy array of shape (n_hanging_faces_x, dim) with the centers of all hanging faces along the first dimension in order.

TreeMesh.hanging_faces_y

Returns a numpy array of shape (n_hanging_faces_y, dim) with the centers of all hanging faces along the second dimension in order.

TreeMesh.hanging_faces_z

Returns a numpy array of shape (n_hanging_faces_z, dim) with the centers of all hanging faces along the third dimension in order.

TreeMesh.hanging_nodes

Returns a numpy array of shape (n_nodes, dim) with the locations of all hanging nodes in order.

TreeMesh.hx

Width of cells in the x direction

Returns
numpy.ndarray

Deprecated since version 0.5.0: hx will be removed in discretize 1.0.0 to reduce namespace clutter, please use mesh.h[0].

TreeMesh.hy

Width of cells in the y direction

Returns
numpy.ndarray or None

Deprecated since version 0.5.0: hy will be removed in discretize 1.0.0 to reduce namespace clutter, please use mesh.h[1].

TreeMesh.hz

Width of cells in the z direction

Returns
numpy.ndarray or None

Deprecated since version 0.5.0: hz will be removed in discretize 1.0.0 to reduce namespace clutter, please use mesh.h[2].

TreeMesh.maxLevel

maxLevel has been deprecated. See max_used_level for documentation

TreeMesh.max_level

The maximum possible level for a cell on this mesh

TreeMesh.max_used_level

The maximum level used, which may be less than max_level.

TreeMesh.n_cells

Number of cells

TreeMesh.n_edges

Total number of non-hanging edges amongst all dimensions

TreeMesh.n_edges_per_direction

The number of edges in each direction

Returns
n_edges_per_directiontuple

[n_edges_x, n_edges_y, n_edges_z], (dim, )

Notes

Also accessible as vnE.

Examples

>>> import discretize
>>> import matplotlib.pyplot as plt
>>> import numpy as np
>>> M = discretize.TensorMesh([np.ones(n) for n in [2,3]])
>>> M.plot_grid(edges=True, show_it=True)

(Source code, png, pdf)

../../_images/discretize-TreeMesh-1.png
TreeMesh.n_edges_x

Number of non-hanging edges oriented along the first dimension

TreeMesh.n_edges_y

Number of non-hanging edges oriented along the second dimension

TreeMesh.n_edges_z

Number of non-hanging edges oriented along the third dimension

TreeMesh.n_faces

Total number of non-hanging faces amongst all dimensions

TreeMesh.n_faces_per_direction

The number of faces in each direction

Returns
n_faces_per_directiontuple

[n_faces_x, n_faces_y, n_faces_z], (dim, )

Notes

Also accessible as vnF.

Examples

>>> import discretize
>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> M = discretize.TensorMesh([np.ones(n) for n in [2,3]])
>>> M.plot_grid(faces=True, show_it=True)

(Source code, png, pdf)

../../_images/discretize-TreeMesh-2.png
TreeMesh.n_faces_x

Number of non-hanging faces oriented along the first dimension

TreeMesh.n_faces_y

Number of non-hanging faces oriented along the second dimension

TreeMesh.n_faces_z

Number of non-hanging faces oriented along the third dimension

TreeMesh.n_hanging_edges

Total number of hanging edges amongst all dimensions

TreeMesh.n_hanging_edges_x

Number of hanging edges oriented along the first dimension

TreeMesh.n_hanging_edges_y

Number of hanging edges oriented along the second dimension

TreeMesh.n_hanging_edges_z

Number of hanging edges oriented along the third dimension

TreeMesh.n_hanging_faces

Total number of hanging faces amongst all dimensions

TreeMesh.n_hanging_faces_x

Number of hanging faces oriented along the first dimension

TreeMesh.n_hanging_faces_y

Number of hanging faces oriented along the second dimension

TreeMesh.n_hanging_faces_z

Number of hanging faces oriented along the third dimension

TreeMesh.n_hanging_nodes

Number of hanging nodes

TreeMesh.n_nodes

Number of non-hanging nodes

TreeMesh.n_total_edges

Total number of non-hanging and hanging edges amongst all dimensions

TreeMesh.n_total_edges_x

Number of non-hanging and hanging edges oriented along the first dimension

TreeMesh.n_total_edges_y

Number of non-hanging and hanging edges oriented along the second dimension

TreeMesh.n_total_edges_z

Number of non-hanging and hanging edges oriented along the third dimension

TreeMesh.n_total_faces

Total number of hanging and non-hanging faces amongst all dimensions

TreeMesh.n_total_faces_x

Number of non-hanging and hanging faces oriented along the first dimension

TreeMesh.n_total_faces_y

Number of non-hanging and hanging faces oriented along the second dimension

TreeMesh.n_total_faces_z

Number of non-hanging and hanging faces oriented along the third dimension

TreeMesh.n_total_nodes

Number of non-hanging and hanging nodes

TreeMesh.nodalGrad

nodalGrad has been deprecated. See nodal_gradient for documentation

TreeMesh.nodalLaplacian

nodalLaplacian has been deprecated. See nodal_laplacian for documentation

TreeMesh.nodal_gradient

Construct gradient operator (nodes to edges).

TreeMesh.nodal_laplacian
TreeMesh.nodes

Returns a numpy array of shape (n_nodes, dim) with the locations of all non-hanging nodes in order.

TreeMesh.nodes_x

Nodal grid vector (1D) in the x direction.

TreeMesh.nodes_y

Nodal grid vector (1D) in the y direction.

TreeMesh.nodes_z

Nodal grid vector (1D) in the z direction.

TreeMesh.normals

normals has been deprecated. See face_normals for documentation

TreeMesh.orientation
TreeMesh.origin

Origin of the mesh

TreeMesh.permuteCC

permuteCC has been deprecated. See permute_cells for documentation

TreeMesh.permuteE

permuteE has been deprecated. See permute_edges for documentation

TreeMesh.permuteF

permuteF has been deprecated. See permute_faces for documentation

TreeMesh.permute_cells

Permutation matrix re-ordering of cells sorted by x, then y, then z

TreeMesh.permute_edges

Permutation matrix re-ordering of edges sorted by x, then y, then z

TreeMesh.permute_faces

Permutation matrix re-ordering of faces sorted by x, then y, then z

TreeMesh.project_edge_to_boundary_edge
TreeMesh.project_face_to_boundary_face
TreeMesh.project_node_to_boundary_node
TreeMesh.reference_is_rotated

True if the axes are rotated from the traditional <X,Y,Z> system with vectors of \((1,0,0)\), \((0,1,0)\), and \((0,0,1)\)

TreeMesh.reference_system

The type of coordinate reference frame. Can take on the values

TreeMesh.rotation_matrix

Builds a rotation matrix to transform coordinates from their coordinate system into a conventional cartesian system. This is built off of the three axis_u, axis_v, and axis_w properties; these mapping coordinates use the letters U, V, and W (the three letters preceding X, Y, and Z in the alphabet) to define the projection of the X, Y, and Z durections. These UVW vectors describe the placement and transformation of the mesh’s coordinate sytem assuming at most 3 directions.

Why would you want to use these UVW mapping vectors the this rotation_matrix property? They allow us to define the relationship between local and global coordinate systems and provide a tool for switching between the two while still maintaing the connectivity of the mesh’s cells. For a visual example of this, please see the figure in the docs for the InterfaceVTK.

TreeMesh.shape_cells

The number of cells in each direction

Returns
tuple of ints

Notes

Also accessible as vnC.

TreeMesh.stencil_cell_gradient
TreeMesh.stencil_cell_gradient_x

Cell gradient stencil matrix to total (including hanging) x faces

TreeMesh.stencil_cell_gradient_y

Cell gradient stencil matrix to total (including hanging) y faces

TreeMesh.stencil_cell_gradient_z

Cell gradient stencil matrix to total (including hanging) z faces

TreeMesh.tangents

tangents has been deprecated. See edge_tangents for documentation

TreeMesh.vectorCCx

vectorCCx has been deprecated. See cell_centers_x for documentation

TreeMesh.vectorCCy

vectorCCy has been deprecated. See cell_centers_y for documentation

TreeMesh.vectorCCz

vectorCCz has been deprecated. See cell_centers_z for documentation

TreeMesh.vectorNx

vectorNx has been deprecated. See nodes_x for documentation

TreeMesh.vectorNy

vectorNy has been deprecated. See nodes_y for documentation

TreeMesh.vectorNz

vectorNz has been deprecated. See nodes_z for documentation

TreeMesh.vntE

Total number of hanging and non-hanging edges in a [nx,ny,nz] form

TreeMesh.vntF

Total number of hanging and non-hanging faces in a [nx,ny,nz] form

TreeMesh.vol

vol has been deprecated. See cell_volumes for documentation

TreeMesh.x0

Methods

TreeMesh.average_cell_to_total_face_x(self)

Average matrix for cell center to total (including hanging) x faces

TreeMesh.average_cell_to_total_face_y(self)

Average matrix for cell center to total (including hanging) y faces

TreeMesh.average_cell_to_total_face_z(self)

Average matrix for cell center to total (including hanging) z faces

TreeMesh.cell_gradient_weak_form_robin(alpha=1.0, beta=0.0, gamma=0.0)

Robin boundary condition for the weak formulation of the cell gradient

This function returns the necessary parts for the weak form of the cell gradient operator to represent the Robin boundary conditions.

The implementation assumes a ghost cell that mirrors the boundary cells across the boundary faces, with a piecewise linear approximation to the values at the ghost cell centers.

The parameters can either be defined as a constant applied to the entire boundary, or as arrays that represent those values on the discretize.base.BaseTensorMesh.boundary_faces().

The returned arrays represent the proper boundary conditions on a solution u such that the inner product of the gradient of u with a test function y would be <y, gradient*u> = y.dot((-face_divergence.T*cell_volumes + A)*u + y.dot(b).

The default values will produce a zero-dirichlet boundary condition.

Parameters
alpha, betascalar or array_like

Parameters for the Robin boundary condition. array_like must be defined on each boundary face.

gamma: scalar or array_like

right hand side boundary conditions. If this parameter is array like, it can be fed either a (n_boundary_faces,) shape array or an (n_boundary_faces, n_rhs) shape array if multiple systems have the same alpha and beta parameters.

Returns
Ascipy.sparse.csr_matrix

Matrix to add to (-face_divergence.T * cell_volumes)

bnumpy.ndarray

Array to add to the result of the (-face_divergence.T * cell_volumes + A) @ u.

Notes

The weak form is obtained by multiplying the gradient by a (piecewise-constant) test function, and integrating over the cell, i.e.

(1)\[\int_V \vec{y} \cdot \nabla u \partial V\]

This equation can be transformed to reduce the differentiability requirement on u to be,

(2)\[-\int_V u (\nabla \cdot \vec{y}) \partial V + \int_{dV} u \vec{y} \partial V.\]

The first term in equation :eq:transformed is constructed using the matrix operators defined on this mesh as D = discretize.operators.DiffOperators.face_divergence() and V, a diagonal matrix of discretize.base.BaseMesh.cell_volumes(), as

\[-(D*y)^T*V*u.\]

This function returns the necessary matrices to complete the transformation of equation :eq:transformed. The second part of equation :eq:transformed becomes,

(3)\[\int_V \nabla \cdot (\phi u) \partial V = \int_{\partial\Omega} \phi\vec{u}\cdot\hat{n} \partial a\]

which is then approximated with the matrices returned here such that the full form of the weak formulation in a discrete form would be.

\[y^T(-D^T V + B)u + y^Tb\]

Examples

We first create a very simple 2D tensor mesh on the [0, 1] boundary:

>>> import matplotlib.pyplot as plt
>>> import scipy.sparse as sp
>>> import discretize
>>> mesh = discretize.TensorMesh([32, 32])

Define the alpha, beta, and gamma parameters for a zero - Dirichlet condition on the boundary, this corresponds to setting:

>>> alpha = 1.0
>>> beta = 0.0
>>> gamma = 0.0
>>> A, b = mesh.cell_gradient_weak_form_robin(alpha, beta, gamma)

We can then represent the operation of taking the weak form of the gradient of a function defined on cell centers with appropriate robin boundary conditions as:

>>> V = sp.diags(mesh.cell_volumes)
>>> D = mesh.face_divergence
>>> phi = np.sin(np.pi * mesh.cell_centers[:, 0]) * np.sin(np.pi * mesh.cell_centers[:, 1])
>>> phi_grad = (-D.T @ V + A) @ phi + b

(Source code)

TreeMesh.cell_levels_by_index(indices)[source]

Fast function to return a list of levels for the given cell indices

Parameters
index: array_like of length (N)

Cell indexes to query

Returns
numpy.array of length (N)

Levels for the cells.

TreeMesh.copy()

Make a copy of the current mesh

classmethod TreeMesh.deserialize(items, **kwargs)
TreeMesh.edge_divergence_weak_form_robin(alpha=0.0, beta=1.0, gamma=0.0)

Robin boundary condition for the weak formulation of the edge divergence

This function returns the necessary parts to form the full weak form of the edge divergence using the nodal gradient with appropriate boundary conditions.

The alpha, beta, and gamma parameters can be scalars, or arrays. If they are arrays, they can either be the same length as the number of boundary faces, or boundary nodes. If multiple parameters are arrays, they must all be the same length.

beta can not be 0.

It is assumed here that quantity that is approximated on the boundary is the gradient of another quantity. See the Notes section for explicit details.

Parameters
alpha, betascalar or array_like

Parameters for the Robin boundary condition. array_like must be defined on either boundary faces or boundary nodes.

gamma: scalar or array_like

right hand side boundary conditions. If this parameter is array like, it can be fed either a (n_boundary_XXX,) shape array or an (n_boundary_XXX, n_rhs) shape array if multiple systems have the same alpha and beta parameters.

Notes

For these returned operators, it is assumed that the quantity on the boundary is related to the gradient of some other quantity.

The weak form is obtained by multiplying the divergence by a (piecewise-constant) test function, and integrating over the cell, i.e.

(4)\[\int_V y \nabla \cdot \vec{u} \partial V\]

This equation can be transformed to reduce the differentiability requirement on \(\vec{u}\) to be,

(5)\[-\int_V \vec{u} \cdot (\nabla y) \partial V + \int_{dV} y \vec{u} \cdot \hat{n} \partial S.\]

Furthermore, when applying these types of transformations, the unknown vector \(\vec{u}\) is usually related to some scalar potential as:

(6)\[\vec{u} = \nabla \phi\]

Thus the robin conditions returned by these matrices apply to the quantity of \(\phi\).

\[ \begin{align}\begin{aligned}\alpha \phi + \beta \nabla \phi \cdot \hat{n} = \gamma\\\alpha \phi + \beta \vec{u} \cdot \hat{n} = \gamma\end{aligned}\end{align} \]

The returned operators cannot be used to impose a Dirichlet condition on \(\phi\).

TreeMesh.equals(other)[source]
TreeMesh.finalize(self)

Finalize the TreeMesh Called after finished cronstruction of the mesh. Can only be called once. After finalize is called, all other attributes and functions are valid.

static TreeMesh.from_omf(element)

Convert an OMF element to it’s proper discretize type. Automatically determines the output type. Returns both the mesh and a dictionary of model arrays.

TreeMesh.getBCProjWF(*args, **kwargs)

getBCProjWF has been deprecated. See get_BC_projections for documentation

TreeMesh.getBCProjWF_simple(*args, **kwargs)

getBCProjWF_simple has been deprecated. See get_BC_projections_simple for documentation

TreeMesh.getEdgeInnerProduct(*args, **kwargs)

getEdgeInnerProduct has been deprecated. See get_edge_inner_product for documentation

TreeMesh.getEdgeInnerProductDeriv(*args, **kwargs)

getEdgeInnerProductDeriv has been deprecated. See get_edge_inner_product_deriv for documentation

TreeMesh.getFaceInnerProduct(*args, **kwargs)

getFaceInnerProduct has been deprecated. See get_face_inner_product for documentation

TreeMesh.getFaceInnerProductDeriv(*args, **kwargs)

getFaceInnerProductDeriv has been deprecated. See get_face_inner_product_deriv for documentation

TreeMesh.getInterpolationMat(*args, **kwargs)

getInterpolationMat has been deprecated. See get_interpolation_matrix for documentation

TreeMesh.getTensor(*args, **kwargs)

getTensor has been deprecated. See get_tensor for documentation

TreeMesh.get_BC_projections(BC, discretization='CC')

The weak form boundary condition projection matrices.

Examples

# Neumann in all directions
BC = 'neumann'

# 3D, Dirichlet in y Neumann else
BC = ['neumann', 'dirichlet', 'neumann']

# 3D, Neumann in x on bottom of domain, Dirichlet else
BC = [['neumann', 'dirichlet'], 'dirichlet', 'dirichlet']
TreeMesh.get_BC_projections_simple(discretization='CC')

The weak form boundary condition projection matrices when mixed boundary condition is used

TreeMesh.get_boundary_cells(self, active_ind=None, direction='zu')

Returns the indices of boundary cells in a given direction given an active index array.

Parameters
active_indarray_like of bool, optional

If not None, then this must show which cells are active

direction: str, optional

must be one of (‘zu’, ‘zd’, ‘xu’, ‘xd’, ‘yu’, ‘yd’)

Returns
numpy.array

Array of indices for the boundary cells in the requested direction

TreeMesh.get_cells_along_line(self, x0, x1)

Finds the cells along a line segment defined by two points

Parameters
x0,x1array_like of length (dim)

Begining and ending point of the line segment.

Returns
list of ints

Indexes for cells that contain the a line defined by the two input points, ordered in the direction of the line.

TreeMesh.get_edge_inner_product(model=None, invert_model=False, invert_matrix=False, do_fast=True, **kwargs)

Generate the edge inner product matrix

Parameters
modelnumpy.ndarray

material property (tensor properties are possible) at each cell center (nC, (1, 3, or 6))

invert_modelbool

inverts the material property

invert_matrixbool

inverts the matrix

do_fastbool

do a faster implementation if available.

Returns
scipy.sparse.csr_matrix

M, the inner product matrix (nE, nE)

TreeMesh.get_edge_inner_product_deriv(model, do_fast=True, invert_model=False, invert_matrix=False, **kwargs)
Parameters
modelnumpy.ndarray

material property (tensor properties are possible) at each cell center (nC, (1, 3, or 6))

do_fastbool

do a faster implementation if available.

invert_modelbool

inverts the material property

invert_matrixbool

inverts the matrix

Returns
scipy.sparse.csr_matrix

dMdm, the derivative of the inner product matrix (nE, nC*nA)

TreeMesh.get_face_inner_product(model=None, invert_model=False, invert_matrix=False, do_fast=True, **kwargs)

Generate the face inner product matrix

Parameters
modelnumpy.ndarray

material property (tensor properties are possible) at each cell center (nC, (1, 3, or 6))

invert_modelbool

inverts the material property

invert_matrixbool

inverts the matrix

do_fastbool

do a faster implementation if available.

Returns
scipy.sparse.csr_matrix

M, the inner product matrix (nF, nF)

TreeMesh.get_face_inner_product_deriv(model, do_fast=True, invert_model=False, invert_matrix=False, **kwargs)
Parameters
modelnumpy.ndarray

material property (tensor properties are possible) at each cell center (nC, (1, 3, or 6))

do_fast :

bool do a faster implementation if available.

invert_modelbool

inverts the material property

invert_matrixbool

inverts the matrix

Returns
scipy.sparse.csr_matrix

dMdmu(u), the derivative of the inner product matrix for a certain u

TreeMesh.get_interpolation_matrix(locs, location_type='CC', zeros_outside=False, **kwargs)[source]

Produces interpolation matrix

Parameters
locnumpy.ndarray

Location of points to interpolate to

location_type: str

What to interpolate

location_type can be:

'Ex'    -> x-component of field defined on edges
'Ey'    -> y-component of field defined on edges
'Ez'    -> z-component of field defined on edges
'Fx'    -> x-component of field defined on faces
'Fy'    -> y-component of field defined on faces
'Fz'    -> z-component of field defined on faces
'N'     -> scalar field defined on nodes
'CC'    -> scalar field defined on cell centers
Returns
scipy.sparse.csr_matrix

M, the interpolation matrix

TreeMesh.get_overlapping_cells(self, rectangle)
TreeMesh.get_tensor(key)

Returns a tensor list.

Parameters
keystr

Which tensor (see below)

key can be:

'CC', 'cell_centers' -> location of cell centers
'N', 'nodes'         -> location of nodes
'Fx', 'faces_x'      -> location of faces with an x normal
'Fy', 'faces_y'      -> location of faces with an y normal
'Fz', 'faces_z'      -> location of faces with an z normal
'Ex', 'edges_x'      -> location of edges with an x tangent
'Ey', 'edges_y'      -> location of edges with an y tangent
'Ez', 'edges_z'      -> location of edges with an z tangent
Returns
list

list of the tensors that make up the mesh.

TreeMesh.insert_cells(self, points, levels, finalize=True)

Insert cells into the TreeMesh that contain given points

Insert cell(s) into the TreeMesh that contain the given point(s) at the assigned level(s).

Parameters
pointsarray_like with shape (N, dim)
levelsarray_like of integers with shape (N)
finalizebool, optional

Whether to finalize after inserting point(s)

Examples

>>> from discretize import TreeMesh
>>> mesh = TreeMesh([32,32])
>>> mesh.insert_cells([0.5, 0.5], mesh.max_level)
>>> print(mesh)
---- QuadTreeMesh ----
 origin: 0.00, 0.00
 hx: 32*0.03,
 hy: 32*0.03,
n_cells: 40
Fill: 3.91%
TreeMesh.isInside(*args, **kwargs)

isInside has been deprecated. See is_inside for documentation

TreeMesh.is_inside(pts, location_type='nodes', **kwargs)

Determines if a set of points are inside a mesh.

Parameters

pts (numpy.ndarray) – Location of points to test

Return type

numpy.ndarray

Returns

inside, numpy array of booleans

TreeMesh.number(self)

Number the cells, nodes, faces, and edges of the TreeMesh

TreeMesh.plotGrid(*args, **kwargs)

plotGrid has been deprecated. See plot_grid for documentation

TreeMesh.plotImage(*args, **kwargs)

plotImage has been deprecated. See plot_image for documentation

TreeMesh.plotSlice(*args, **kwargs)

plotSlice has been deprecated. See plot_slice for documentation

TreeMesh.plot_3d_slicer(v, xslice=None, yslice=None, zslice=None, v_type='CC', view='real', axis='xy', transparent=None, clim=None, xlim=None, ylim=None, zlim=None, aspect='auto', grid=[2, 2, 1], pcolor_opts=None, fig=None, **kwargs)

Plot slices of a 3D volume, interactively (scroll wheel).

If called from a notebook, make sure to set

%matplotlib notebook

See the class discretize.View.Slicer for more information.

It returns nothing. However, if you need the different figure handles you can get it via

fig = plt.gcf()

and subsequently its children via

fig.get_children()

and recursively deeper, e.g.,

fig.get_children()[0].get_children().

One can also provide an existing figure instance, which can be useful for interactive widgets in Notebooks. The provided figure is cleared first.

TreeMesh.plot_grid(ax=None, nodes=False, faces=False, centers=False, edges=False, lines=True, show_it=False, **kwargs)

Plot the nodal, cell-centered and staggered grids.

Parameters
axmatplotlib.axes.Axes or None, optional

The axes to draw on. None produces a new Axes.

nodes, faces, centers, edges, linesbool, optional

Whether to plot the corresponding item

show_itbool, optional

whether to call plt.show()

colorColor or str, optional

If lines=True, the color of the lines, defaults to first color.

linewidthfloat, optional

If lines=True, the linewidth for the lines.

Returns
matplotlib.axes.Axes

Axes handle for the plot

Other Parameters
edges_x, edges_y, edges_z, faces_x, faces_y, faces_zbool, optional

When plotting a TreeMesh, these are also options to plot the individual component items.

cell_linebool, optional

When plotting a TreeMesh, you can also plot a line through the cell centers in order.

slice{‘both’, ‘theta’, ‘z’}

When plotting a CylindricalMesh, which dimension to slice over.

Notes

Excess arguments are passed on to plot

Examples

Plotting a 2D TensorMesh grid

>>> from matplotlib import pyplot as plt
>>> import discretize
>>> import numpy as np
>>> h1 = np.linspace(.1, .5, 3)
>>> h2 = np.linspace(.1, .5, 5)
>>> mesh = discretize.TensorMesh([h1, h2])
>>> mesh.plot_grid(nodes=True, faces=True, centers=True, lines=True)
>>> plt.show()

(Source code, png, pdf)

../../_images/discretize-TreeMesh-4_00_00.png

Plotting a 3D TensorMesh grid

>>> from matplotlib import pyplot as plt
>>> import discretize
>>> import numpy as np
>>> h1 = np.linspace(.1, .5, 3)
>>> h2 = np.linspace(.1, .5, 5)
>>> h3 = np.linspace(.1, .5, 3)
>>> mesh = discretize.TensorMesh([h1, h2, h3])
>>> mesh.plot_grid(nodes=True, faces=True, centers=True, lines=True)
>>> plt.show()

(png, pdf)

../../_images/discretize-TreeMesh-4_01_00.png

Plotting a 2D CurvilinearMesh

>>> from matplotlib import pyplot as plt
>>> import discretize
>>> X, Y = discretize.utils.exampleLrmGrid([10, 10], 'rotate')
>>> M = discretize.CurvilinearMesh([X, Y])
>>> M.plot_grid()
>>> plt.show()

(png, pdf)

../../_images/discretize-TreeMesh-4_02_00.png

Plotting a 3D CurvilinearMesh

>>> from matplotlib import pyplot as plt
>>> import discretize
>>> X, Y, Z = discretize.utils.exampleLrmGrid([5, 5, 5], 'rotate')
>>> M = discretize.CurvilinearMesh([X, Y, Z])
>>> M.plot_grid()
>>> plt.show()

(png, pdf)

../../_images/discretize-TreeMesh-4_03_00.png

Plotting a 2D TreeMesh

>>> from matplotlib import pyplot as plt
>>> import discretize
>>> M = discretize.TreeMesh([32, 32])
>>> M.insert_cells([[0.25, 0.25]], [4])
>>> M.plot_grid()
>>> plt.show()

(png, pdf)

../../_images/discretize-TreeMesh-4_04_00.png

Plotting a 3D TreeMesh

>>> from matplotlib import pyplot as plt
>>> import discretize
>>> M = discretize.TreeMesh([32, 32, 32])
>>> M.insert_cells([[0.3, 0.75, 0.22]], [4])
>>> M.plot_grid()
>>> plt.show()

(png, pdf)

../../_images/discretize-TreeMesh-4_05_00.png
TreeMesh.plot_image(v, v_type='CC', grid=False, view='real', ax=None, clim=None, show_it=False, pcolor_opts=None, stream_opts=None, grid_opts=None, range_x=None, range_y=None, sample_grid=None, stream_thickness=None, stream_threshold=None, **kwargs)

Plots fields on the given mesh.

Parameters
vnumpy.ndarray

values to plot

v_type{‘CC’,’CCV’, ‘N’, ‘F’, ‘Fx’, ‘Fy’, ‘Fz’, ‘E’, ‘Ex’, ‘Ey’, ‘Ez’}

Where the values of v are defined.

view{‘real’, ‘imag’, ‘abs’, ‘vec’}

How to view the array.

axmatplotlib.axes.Axes, optional

The axes to draw on. None produces a new Axes.

climtuple of float, optional

length 2 tuple of (vmin, vmax) for the color limits

range_x, range_ytuple of float, optional

length 2 tuple of (min, max) for the bounds of the plot axes.

pcolor_optsdict, optional

Arguments passed on to pcolormesh

gridbool, optional

Whether to plot the edges of the mesh cells.

grid_optsdict, optional

If grid is true, arguments passed on to plot for grid

sample_gridtuple of numpy.ndarray, optional

If view == ‘vec’, mesh cell widths (hx, hy) to interpolate onto for vector plotting

stream_optsdict, optional

If view == ‘vec’, arguments passed on to streamplot

stream_thicknessfloat, optional

If view == ‘vec’, linewidth for streamplot

stream_thresholdfloat, optional

If view == ‘vec’, only plots vectors with magnitude above this threshold

show_itbool, optional

Whether to call plt.show()

numberingbool, optional

For 3D TensorMesh only, show the numbering of the slices

annotation_colorColor or str, optional

For 3D TensorMesh only, color of the annotation

Examples

2D TensorMesh plotting

>>> from matplotlib import pyplot as plt
>>> import discretize
>>> import numpy as np
>>> M = discretize.TensorMesh([20, 20])
>>> v = np.sin(M.gridCC[:, 0]*2*np.pi)*np.sin(M.gridCC[:, 1]*2*np.pi)
>>> M.plot_image(v)
>>> plt.show()

(Source code, png, pdf)

../../_images/discretize-TreeMesh-5_00_00.png

3D TensorMesh plotting

>>> import discretize
>>> import numpy as np
>>> M = discretize.TensorMesh([20, 20, 20])
>>> v = np.sin(M.gridCC[:, 0]*2*np.pi)*np.sin(M.gridCC[:, 1]*2*np.pi)*np.sin(M.gridCC[:, 2]*2*np.pi)
>>> M.plot_image(v, annotation_color='k')
>>> plt.show()

(png, pdf)

../../_images/discretize-TreeMesh-5_01_00.png
TreeMesh.plot_slice(v, v_type='CC', normal='Z', ind=None, slice_loc=None, grid=False, view='real', ax=None, clim=None, show_it=False, pcolor_opts=None, stream_opts=None, grid_opts=None, range_x=None, range_y=None, sample_grid=None, stream_threshold=None, stream_thickness=None, **kwargs)

Plots slice of fields on the given 3D mesh.

Parameters
vnumpy.ndarray

values to plot

v_type{‘CC’,’CCV’, ‘N’, ‘F’, ‘Fx’, ‘Fy’, ‘Fz’, ‘E’, ‘Ex’, ‘Ey’, ‘Ez’}, or tuple of these options

Where the values of v are defined.

normal{‘Z’, ‘X’, ‘Y’}

Normal direction of slicing plane.

indNone, optional

index along dimension of slice. Defaults to the center index.

slice_locNone, optional

Value along dimension of slice. Defaults to the center of the mesh.

view{‘real’, ‘imag’, ‘abs’, ‘vec’}

How to view the array.

axmatplotlib.axes.Axes, optional

The axes to draw on. None produces a new Axes. Must be None if v_type is a tuple.

climtuple of float, optional

length 2 tuple of (vmin, vmax) for the color limits

range_x, range_ytuple of float, optional

length 2 tuple of (min, max) for the bounds of the plot axes.

pcolor_optsdict, optional

Arguments passed on to pcolormesh

gridbool, optional

Whether to plot the edges of the mesh cells.

grid_optsdict, optional

If grid is true, arguments passed on to plot for the edges

sample_gridtuple of numpy.ndarray, optional

If view == ‘vec’, mesh cell widths (hx, hy) to interpolate onto for vector plotting

stream_optsdict, optional

If view == ‘vec’, arguments passed on to streamplot

stream_thicknessfloat, optional

If view == ‘vec’, linewidth for streamplot

stream_thresholdfloat, optional

If view == ‘vec’, only plots vectors with magnitude above this threshold

show_itbool, optional

Whether to call plt.show()

Examples

Plot a slice of a 3D TensorMesh solution to a Laplace’s equaiton.

First build the mesh:

>>> from matplotlib import pyplot as plt
>>> import discretize
>>> from pymatsolver import Solver
>>> hx = [(5, 2, -1.3), (2, 4), (5, 2, 1.3)]
>>> hy = [(2, 2, -1.3), (2, 6), (2, 2, 1.3)]
>>> hz = [(2, 2, -1.3), (2, 6), (2, 2, 1.3)]
>>> M = discretize.TensorMesh([hx, hy, hz])

then build the necessary parts of the PDE:

>>> q = np.zeros(M.vnC)
>>> q[[4, 4], [4, 4], [2, 6]]=[-1, 1]
>>> q = discretize.utils.mkvc(q)
>>> A = M.face_divergence * M.cell_gradient
>>> b = Solver(A) * (q)

and finaly, plot the vector values of the result, which are defined on faces

>>> M.plot_slice(M.cell_gradient*b, 'F', view='vec', grid=True, pcolor_opts={'alpha':0.8})
>>> plt.show()

(Source code, png, pdf)

../../_images/discretize-TreeMesh-6_00_00.png

We can use the slice_loc kwarg to tell `plot_slice where to slice the mesh. Let’s create a mesh with a random model and plot slice of it. The slice_loc kwarg automatically determines the indices for slicing the mesh along a plane with the given normal.

>>> M = discretize.TensorMesh([32, 32, 32])
>>> v = discretize.utils.random_model(M.vnC, seed=789).reshape(-1, order='F')
>>> x_slice, y_slice, z_slice = 0.75, 0.25, 0.9
>>> plt.figure(figsize=(7.5, 3))
>>> ax = plt.subplot(131)
>>> M.plot_slice(v, normal='X', slice_loc=x_slice, ax=ax)
>>> ax = plt.subplot(132)
>>> M.plot_slice(v, normal='Y', slice_loc=y_slice, ax=ax)
>>> ax = plt.subplot(133)
>>> M.plot_slice(v, normal='Z', slice_loc=z_slice, ax=ax)
>>> plt.tight_layout()
>>> plt.show()

(png, pdf)

../../_images/discretize-TreeMesh-6_01_00.png

This also works for TreeMesh. We create a mesh here that is refined within three boxes, along with a base level of refinement.

>>> TM = discretize.TreeMesh([32, 32, 32])
>>> TM.refine(3, finalize=False)
>>> BSW = [[0.25, 0.25, 0.25], [0.15, 0.15, 0.15], [0.1, 0.1, 0.1]]
>>> TNE = [[0.75, 0.75, 0.75], [0.85, 0.85, 0.85], [0.9, 0.9, 0.9]]
>>> levels = [6, 5, 4]
>>> TM.refine_box(BSW, TNE, levels)
>>> v_TM = discretize.utils.volume_average(M, TM, v)
>>> plt.figure(figsize=(7.5, 3))
>>> ax = plt.subplot(131)
>>> TM.plot_slice(v_TM, normal='X', slice_loc=x_slice, ax=ax)
>>> ax = plt.subplot(132)
>>> TM.plot_slice(v_TM, normal='Y', slice_loc=y_slice, ax=ax)
>>> ax = plt.subplot(133)
>>> TM.plot_slice(v_TM, normal='Z', slice_loc=z_slice, ax=ax)
>>> plt.tight_layout()
>>> plt.show()

(png, pdf)

../../_images/discretize-TreeMesh-6_02_00.png
TreeMesh.point2index(locs)[source]

Finds cells that contain the given points. Returns an array of index values of the cells that contain the given points

Parameters
locs: array_like of shape (N, dim)

points to search for the location of

Returns
numpy.array of integers of length(N)

Cell indices that contain the points

TreeMesh.projectEdgeVector(*args, **kwargs)

projectEdgeVector has been deprecated. See project_edge_vector for documentation

TreeMesh.projectFaceVector(*args, **kwargs)

projectFaceVector has been deprecated. See project_face_vector for documentation

TreeMesh.project_edge_vector(edge_vector)

Project vectors onto the edges of the mesh

Given a vector, edge_vector, in cartesian coordinates, this will project it onto the mesh using the tangents

Parameters
edge_vectornumpy.ndarray

edge vector with shape (n_edges, dim)

Returns
numpy.ndarray

projected edge vector, (n_edges, )

TreeMesh.project_face_vector(face_vector)

Project vectors onto the faces of the mesh.

Given a vector, face_vector, in cartesian coordinates, this will project it onto the mesh using the normals

Parameters
face_vectornumpy.ndarray

face vector with shape (n_faces, dim)

Returns
numpy.ndarray

projected face vector, (n_faces, )

TreeMesh.readModelUBC(*args, **kwargs)

readModelUBC has been deprecated. See read_model_UBC for documentation

classmethod TreeMesh.readUBC(file_name, directory='')
classmethod TreeMesh.read_UBC(meshFile, directory='')

Read UBC 3D OcTree mesh file Input: :param str meshFile: path to the UBC GIF OcTree mesh file to read :rtype: discretize.TreeMesh :return: The octree mesh

TreeMesh.read_model_UBC(file_name)

Read UBC OcTree model and get vector :param string file_name: path to the UBC GIF model file to read :rtype: numpy.ndarray :return: OcTree model

TreeMesh.refine(self, function, finalize=True)

Refine a TreeMesh using a user supplied function.

Refines the TreeMesh using a function that is recursively called on each cell of the mesh. It must accept an object of type discretize.TreeMesh.Cell and return an integer like object defining the desired level. The function can also simply be an integer, which will then cause all cells to be at least that level.

Parameters
functioncallable | int

a function describing the desired level, or an integer to refine all cells to at least that level.

finalizebool, optional

Whether to finalize the mesh

See also

discretize.TreeMesh.TreeCell

a description of the TreeCell object

Examples

>>> from discretize import TreeMesh
>>> mesh = TreeMesh([32,32])
>>> def func(cell):
>>>     r = np.linalg.norm(cell.center-0.5)
>>>     return mesh.max_level if r<0.2 else mesh.max_level-1
>>> mesh.refine(func)
>>> mesh
---- QuadTreeMesh ----
 origin: 0.00, 0.00
 hx: 32*0.03,
 hy: 32*0.03,
n_cells: 352
Fill: 34.38%
TreeMesh.refine_ball(self, points, radii, levels, finalize=True)

Refines the TreeMesh around points with the given radii

Refines the TreeMesh by determining if a cell intersects the given ball(s) to the prescribed level(s).

Parameters
pointsarray_like with shape (N, dim)

The centers of the balls

radiiarray_like with shape (N)

The radii of the balls

levelsarray_like of integers with shape (N)

The level to refine intersecting cells to

finalizebool, optional

Whether to finalize after refining

Examples

We create a simple mesh and refine the TreeMesh such that all cells that intersect the spherical balls are at the given levels.

>>> import discretize
>>> import matplotlib.pyplot as plt
>>> import matplotlib.patches as patches
>>> tree_mesh = discretize.TreeMesh([32, 32])
>>> tree_mesh.max_level
5

Next we define the center and radius of the two spheres, as well as the level we want to refine them to, and refine the mesh.

>>> centers = [[0.1, 0.3], [0.6, 0.8]]
>>> radii = [0.2, 0.3]
>>> levels = [4, 5]
>>> tree_mesh.refine_ball(centers, radii, levels)

Now lets look at the mesh, and overlay the balls on it to ensure it refined where we wanted it to.

>>> ax = tree_mesh.plot_grid()
>>> circ = patches.Circle(centers[0], radii[0], facecolor='none', edgecolor='r', linewidth=3)
>>> ax.add_patch(circ)
>>> circ = patches.Circle(centers[1], radii[1], facecolor='none', edgecolor='k', linewidth=3)
>>> ax.add_patch(circ)
>>> plt.show()

(Source code, png, pdf)

../../_images/discretize-TreeMesh-7.png
TreeMesh.refine_box(self, x0s, x1s, levels, finalize=True)

Refines the TreeMesh within the axis aligned boxes to the desired level

Refines the TreeMesh by determining if a cell intersects the given axis aligned box(es) to the prescribed level(s).

Parameters
x0sarray_like with shape (N, dim)

The minimum location of the boxes

x1sarray_like with shape (N, dim)

The maximum location of the boxes

levelsarray_like of integers with shape (N)

The level to refine intersecting cells to

finalizebool, optional

Whether to finalize after refining

Examples

We create a simple mesh and refine the TreeMesh such that all cells that intersect the boxes are at the given levels.

>>> import discretize
>>> import matplotlib.pyplot as plt
>>> import matplotlib.patches as patches
>>> tree_mesh = discretize.TreeMesh([32, 32])
>>> tree_mesh.max_level
5

Next we define the origins and furthest corners of the two rectangles, as well as the level we want to refine them to, and refine the mesh.

>>> x0s = [[0.1, 0.1], [0.8, 0.8]]
>>> x1s = [[0.3, 0.2], [0.9, 1.0]]
>>> levels = [4, 5]
>>> tree_mesh.refine_box(x0s, x1s, levels)

Now lets look at the mesh, and overlay the boxes on it to ensure it refined where we wanted it to.

>>> ax = tree_mesh.plot_grid()
>>> rect = patches.Rectangle([0.1, 0.1], 0.2, 0.1, facecolor='none', edgecolor='r', linewidth=3)
>>> ax.add_patch(rect)
>>> rect = patches.Rectangle([0.8, 0.8], 0.1, 0.2, facecolor='none', edgecolor='k', linewidth=3)
>>> ax.add_patch(rect)
>>> plt.show()

(Source code, png, pdf)

../../_images/discretize-TreeMesh-8.png
TreeMesh.save(file_name='mesh.json', verbose=False, **kwargs)

Save the mesh to json :param str file: file_name for saving the casing properties :param str directory: working directory for saving the file

TreeMesh.serialize()
TreeMesh.setCellGradBC(*args, **kwargs)

setCellGradBC has been deprecated. See set_cell_gradient_BC for documentation

TreeMesh.set_cell_gradient_BC(BC)

Function that sets the boundary conditions for cell-centred derivative operators.

Examples

..code:: python

# Neumann in all directions BC = ‘neumann’

# 3D, Dirichlet in y Neumann else BC = [‘neumann’, ‘dirichlet’, ‘neumann’]

# 3D, Neumann in x on bottom of domain, Dirichlet else BC = [[‘neumann’, ‘dirichlet’], ‘dirichlet’, ‘dirichlet’]

TreeMesh.toVTK(models=None)

Convert this mesh object to it’s proper VTK or pyvista data object with the given model dictionary as the cell data of that dataset.

Parameters
modelsdict(numpy.ndarray)

Name(‘s) and array(‘s). Match number of cells

TreeMesh.to_dict()
TreeMesh.to_omf(models=None)

Convert this mesh object to it’s proper omf data object with the given model dictionary as the cell data of that dataset.

Parameters
modelsdict(numpy.ndarray)

Name(‘s) and array(‘s). Match number of cells

TreeMesh.to_vtk(models=None)

Convert this mesh object to it’s proper VTK or pyvista data object with the given model dictionary as the cell data of that dataset.

Parameters
modelsdict(numpy.ndarray)

Name(‘s) and array(‘s). Match number of cells

TreeMesh.validate()[source]

Every object will be valid upon initialization

TreeMesh.writeModelUBC(*args, **kwargs)

writeModelUBC has been deprecated. See write_model_UBC for documentation

TreeMesh.writeUBC(*args, **kwargs)

writeUBC has been deprecated. See write_UBC for documentation

TreeMesh.writeVTK(file_name, models=None, directory='')

Makes and saves a VTK object from this mesh and given models

Parameters
file_namestr

path to the output vtk file or just its name if directory is specified

modelsdict

dictionary of numpy.array - Name(‘s) and array(‘s). Match number of cells

directorystr

directory where the UBC GIF file lives

TreeMesh.write_UBC(file_name, models=None, directory='')

Write UBC ocTree mesh and model files from a octree mesh and model. :param string file_name: File to write to :param dict models: Models in a dict, where each key is the file_name :param str directory: directory where to save model(s)

TreeMesh.write_model_UBC(file_name, model, directory='')

Writes a model associated with a TreeMesh to a UBC-GIF format model file.

Input: :param str file_name: File to write to or just its name if directory is specified :param str directory: directory where the UBC GIF file lives :param numpy.ndarray model: The model

TreeMesh.write_vtk(file_name, models=None, directory='')

Makes and saves a VTK object from this mesh and given models

Parameters
file_namestr

path to the output vtk file or just its name if directory is specified

modelsdict

dictionary of numpy.array - Name(‘s) and array(‘s). Match number of cells

directorystr

directory where the UBC GIF file lives