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.