sfepy.discrete.iga.iga module

Isogeometric analysis utilities.

Notes

The functions compute_bezier_extraction_1d() and eval_nurbs_basis_tp() implement the algorithms described in [1].

[1] Michael J. Borden, Michael A. Scott, John A. Evans, Thomas J. R. Hughes:
Isogeometric finite element data structures based on Bezier extraction of NURBS, Institute for Computational Engineering and Sciences, The University of Texas at Austin, Austin, Texas, March 2010.
sfepy.discrete.iga.iga.combine_bezier_extraction(cs)[source]

For a nD B-spline parametric domain, combine the 1D element extraction operators in each parametric dimension into a single operator for each nD element.

Parameters:
cs : list of lists of 2D arrays

The element extraction operators in each parametric dimension.

Returns:
ccs : list of 2D arrays

The combined element extraction operators.

sfepy.discrete.iga.iga.compute_bezier_control(control_points, weights, ccs, conn, bconn)[source]

Compute the control points and weights of the Bezier mesh.

Parameters:
control_points : array

The NURBS control points.

weights : array

The NURBS weights.

ccs : list of 2D arrays

The combined element extraction operators.

conn : array

The connectivity of the global NURBS basis.

bconn : array

The connectivity of the Bezier basis.

Returns:
bezier_control_points : array

The control points of the Bezier mesh.

bezier_weights : array

The weights of the Bezier mesh.

sfepy.discrete.iga.iga.compute_bezier_extraction(knots, degrees)[source]

Compute local (element) Bezier extraction operators for a nD B-spline parametric domain.

Parameters:
knots : sequence of array or array

The knot vectors.

degrees : sequence of ints or int

Polynomial degrees in each parametric dimension.

Returns:
cs : list of lists of 2D arrays

The element extraction operators in each parametric dimension.

sfepy.discrete.iga.iga.compute_bezier_extraction_1d(knots, degree)[source]

Compute local (element) Bezier extraction operators for a 1D B-spline parametric domain.

Parameters:
knots : array

The knot vector.

degree : int

The curve degree.

Returns:
cs : array of 2D arrays (3D array)

The element extraction operators.

sfepy.discrete.iga.iga.create_boundary_qp(coors, dim)[source]

Create boundary quadrature points from the surface quadrature points.

Uses the Bezier element tensor product structure.

Parameters:
coors : array, shape (n_qp, d)

The coordinates of the surface quadrature points.

dim : int

The topological dimension.

Returns:
bcoors : array, shape (n_qp, d + 1)

The coordinates of the boundary quadrature points.

sfepy.discrete.iga.iga.create_connectivity(n_els, knots, degrees)[source]

Create connectivity arrays of nD Bezier elements.

Parameters:
n_els : sequence of ints

The number of elements in each parametric dimension.

knots : sequence of array or array

The knot vectors.

degrees : sequence of ints or int

The basis degrees in each parametric dimension.

Returns:
conn : array

The connectivity of the global NURBS basis.

bconn : array

The connectivity of the Bezier basis.

sfepy.discrete.iga.iga.create_connectivity_1d(n_el, knots, degree)[source]

Create connectivity arrays of 1D Bezier elements.

Parameters:
n_el : int

The number of elements.

knots : array

The knot vector.

degree : int

The basis degree.

Returns:
conn : array

The connectivity of the global NURBS basis.

bconn : array

The connectivity of the Bezier basis.

sfepy.discrete.iga.iga.eval_bernstein_basis(x, degree)[source]

Evaluate the Bernstein polynomial basis of the given degree, and its derivatives, in a point x in [0, 1].

Parameters:
x : float

The point in [0, 1].

degree : int

The basis degree.

Returns:
funs : array

The degree + 1 values of the Bernstein polynomial basis.

ders : array

The degree + 1 values of the Bernstein polynomial basis derivatives.

sfepy.discrete.iga.iga.eval_mapping_data_in_qp(qps, control_points, weights, degrees, cs, conn, cells=None)[source]

Evaluate data required for the isogeometric domain reference mapping in the given quadrature points. The quadrature points are the same for all Bezier elements and should correspond to the Bernstein basis degree.

Parameters:
qps : array

The quadrature points coordinates with components in [0, 1] reference element domain.

control_points : array

The NURBS control points.

weights : array

The NURBS weights.

degrees : sequence of ints or int

The basis degrees in each parametric dimension.

cs : list of lists of 2D arrays

The element extraction operators in each parametric dimension.

conn : array

The connectivity of the global NURBS basis.

cells : array, optional

If given, use only the given Bezier elements.

Returns:
bfs : array

The NURBS shape functions in the physical quadrature points of all elements.

bfgs : array

The NURBS shape functions derivatives w.r.t. the physical coordinates in the physical quadrature points of all elements.

dets : array

The Jacobians of the mapping to the unit reference element in the physical quadrature points of all elements.

sfepy.discrete.iga.iga.eval_nurbs_basis_tp(qp, ie, control_points, weights, degrees, cs, conn)[source]

Evaluate the tensor-product NURBS shape functions in a quadrature point for a given Bezier element.

Parameters:
qp : array

The quadrature point coordinates with components in [0, 1] reference element domain.

ie : int

The Bezier element index.

control_points : array

The NURBS control points.

weights : array

The NURBS weights.

degrees : sequence of ints or int

The basis degrees in each parametric dimension.

cs : list of lists of 2D arrays

The element extraction operators in each parametric dimension.

conn : array

The connectivity of the global NURBS basis.

Returns:
R : array

The NURBS shape functions.

dR_dx : array

The NURBS shape functions derivatives w.r.t. the physical coordinates.

det : array

The Jacobian of the mapping to the unit reference element.

sfepy.discrete.iga.iga.eval_variable_in_qp(variable, qps, control_points, weights, degrees, cs, conn, cells=None)[source]

Evaluate a field variable in the given quadrature points. The quadrature points are the same for all Bezier elements and should correspond to the Bernstein basis degree. The field variable is defined by its DOFs - the coefficients of the NURBS basis.

Parameters:
variable : array

The DOF values of the variable with n_c components, shape (:, n_c).

qps : array

The quadrature points coordinates with components in [0, 1] reference element domain.

control_points : array

The NURBS control points.

weights : array

The NURBS weights.

degrees : sequence of ints or int

The basis degrees in each parametric dimension.

cs : list of lists of 2D arrays

The element extraction operators in each parametric dimension.

conn : array

The connectivity of the global NURBS basis.

cells : array, optional

If given, use only the given Bezier elements.

Returns:
coors : array

The physical coordinates of the quadrature points of all elements.

vals : array

The field variable values in the physical quadrature points.

dets : array

The Jacobians of the mapping to the unit reference element in the physical quadrature points.

sfepy.discrete.iga.iga.get_bezier_element_entities(degrees)[source]

Get faces and edges of a Bezier mesh element in terms of indices into the element’s connectivity (reference Bezier element entities).

Parameters:
degrees : sequence of ints or int

Polynomial degrees in each parametric dimension.

Returns:
faces : list of arrays

The indices for each face or None if not 3D.

edges : list of arrays

The indices for each edge or None if not at least 2D.

vertices : list of arrays

The indices for each vertex.

Notes

The ordering of faces and edges has to be the same as in sfepy.discrete.fem.geometry_element.geometry_data.

sfepy.discrete.iga.iga.get_bezier_topology(bconn, degrees)[source]

Get a topology connectivity corresponding to the Bezier mesh connectivity.

In the referenced Bezier control points the Bezier mesh is interpolatory.

Parameters:
bconn : array

The connectivity of the Bezier basis.

degrees : sequence of ints or int

The basis degrees in each parametric dimension.

Returns:
tconn : array

The topology connectivity (corner nodes, or vertices, of Bezier elements) with vertex ordering suitable for a FE mesh.

sfepy.discrete.iga.iga.get_facet_axes(dim)[source]

For each reference Bezier element facet return the facet axes followed by the remaining (perpendicular) axis, as well as the remaining axis coordinate of the facet.

Parameters:
dim : int

The topological dimension.

Returns:
axes : array

The axes of the reference element facets.

coors : array

The remaining coordinate of the reference element facets.

sfepy.discrete.iga.iga.get_patch_box_regions(n_els, degrees)[source]

Get box regions of Bezier topological mesh in terms of element corner vertices of Bezier mesh.

Parameters:
n_els : sequence of ints

The number of elements in each parametric dimension.

degrees : sequence of ints or int

Polynomial degrees in each parametric dimension.

Returns:
regions : dict

The Bezier mesh vertices of box regions.

sfepy.discrete.iga.iga.get_raveled_index(indices, shape)[source]

Get a global raveled index corresponding to nD indices into an array of the given shape.

sfepy.discrete.iga.iga.get_surface_degrees(degrees)[source]

Get degrees of the NURBS patch surfaces.

Parameters:
degrees : sequence of ints or int

Polynomial degrees in each parametric dimension.

Returns:
sdegrees : list of arrays

The degrees of the patch surfaces, in the order of the reference Bezier element facets.

sfepy.discrete.iga.iga.get_unraveled_indices(index, shape)[source]

Get nD indices into an array of the given shape corresponding to a global raveled index.

sfepy.discrete.iga.iga.tensor_product(a, b)[source]

Compute tensor product of two 2D arrays with possibly different shapes. The result has the form:

c = [[a00 b, a01 b, ...],
     [a10 b, a11 b, ...],
      ...
      ...               ]