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:

cslist of lists of 2D arrays: The element extraction operators in each parametric dimension.

Returns:

ccslist 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_pointsarray: The NURBS control points.
weightsarray: The NURBS weights.
ccslist of 2D arrays: The combined element extraction operators.
connarray: The connectivity of the global NURBS basis.
bconnarray: The connectivity of the Bezier basis.

Returns:

bezier_control_pointsarray: The control points of the Bezier mesh.
bezier_weightsarray: 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:

knotssequence of array or array: The knot vectors.
degreessequence of ints or int: Polynomial degrees in each parametric dimension.

Returns:

cslist 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:

knotsarray: The knot vector.
degreeint: The curve degree.

Returns:

csarray 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:

coorsarray, shape (n_qp, d): The coordinates of the surface quadrature points.
dimint: The topological dimension.

Returns:

bcoorsarray, 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_elssequence of ints: The number of elements in each parametric dimension.
knotssequence of array or array: The knot vectors.
degreessequence of ints or int: The basis degrees in each parametric dimension.

Returns:

connarray: The connectivity of the global NURBS basis.
bconnarray: 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_elint: The number of elements.
knotsarray: The knot vector.
degreeint: The basis degree.

Returns:

connarray: The connectivity of the global NURBS basis.
bconnarray: 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:

xfloat: The point in [0, 1].
degreeint: The basis degree.

Returns:

funsarray: The degree + 1 values of the Bernstein polynomial basis.
dersarray: 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:

qpsarray: The quadrature points coordinates with components in [0, 1] reference element domain.
control_pointsarray: The NURBS control points.
weightsarray: The NURBS weights.
degreessequence of ints or int: The basis degrees in each parametric dimension.
cslist of lists of 2D arrays: The element extraction operators in each parametric dimension.
connarray: The connectivity of the global NURBS basis.
cellsarray, optional: If given, use only the given Bezier elements.

Returns:

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

qparray: The quadrature point coordinates with components in [0, 1] reference element domain.
ieint: The Bezier element index.
control_pointsarray: The NURBS control points.
weightsarray: The NURBS weights.
degreessequence of ints or int: The basis degrees in each parametric dimension.
cslist of lists of 2D arrays: The element extraction operators in each parametric dimension.
connarray: The connectivity of the global NURBS basis.

Returns:

Rarray: The NURBS shape functions.
dR_dxarray: The NURBS shape functions derivatives w.r.t. the physical coordinates.
detarray: 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:

variablearray: The DOF values of the variable with n_c components, shape (:, n_c).
qpsarray: The quadrature points coordinates with components in [0, 1] reference element domain.
control_pointsarray: The NURBS control points.
weightsarray: The NURBS weights.
degreessequence of ints or int: The basis degrees in each parametric dimension.
cslist of lists of 2D arrays: The element extraction operators in each parametric dimension.
connarray: The connectivity of the global NURBS basis.
cellsarray, optional: If given, use only the given Bezier elements.

Returns:

coorsarray: The physical coordinates of the quadrature points of all elements.
valsarray: The field variable values in the physical quadrature points.
detsarray: 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:

degreessequence of ints or int: Polynomial degrees in each parametric dimension.

Returns:

faceslist of arrays: The indices for each face or None if not 3D.
edgeslist of arrays: The indices for each edge or None if not at least 2D.
verticeslist 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:

bconnarray: The connectivity of the Bezier basis.
degreessequence of ints or int: The basis degrees in each parametric dimension.

Returns:

tconnarray: 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:

dimint: The topological dimension.

Returns:

axesarray: The axes of the reference element facets.
coorsarray: 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_elssequence of ints: The number of elements in each parametric dimension.
degreessequence of ints or int: Polynomial degrees in each parametric dimension.

Returns:

regionsdict: 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:

degreessequence of ints or int: Polynomial degrees in each parametric dimension.

Returns:

sdegreeslist 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, ...],
      ...
      ...               ]