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 Bspline 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 Bspline 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 Bspline 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 tensorproduct 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.