"""
Module focused on the implementation of the Radial Basis Functions interpolation
technique. This technique is still based on the use of a set of parameters, the
so-called control points, as for FFD, but RBF is interpolatory. Another
important key point of RBF strategy relies in the way we can locate the control
points: in fact, instead of FFD where control points need to be placed inside a
regular lattice, with RBF we hano no more limitations. So we have the
possibility to perform localized control points refiniments.
The module is analogous to the freeform one.
:Theoretical Insight:
As reference please consult M.D. Buhmann, Radial Basis Functions, volume 12
of Cambridge monographs on applied and computational mathematics. Cambridge
University Press, UK, 2003. This implementation follows D. Forti and G.
Rozza, Efficient geometrical parametrization techniques of interfaces for
reduced order modelling: application to fluid-structure interaction coupling
problems, International Journal of Computational Fluid Dynamics.
RBF shape parametrization technique is based on the definition of a map,
:math:`\\mathcal{M}(\\boldsymbol{x}) : \\mathbb{R}^n \\rightarrow
\\mathbb{R}^n`, that allows the possibility of transferring data across
non-matching grids and facing the dynamic mesh handling. The map introduced
is defines as follows
.. math::
\\mathcal{M}(\\boldsymbol{x}) = p(\\boldsymbol{x}) +
\\sum_{i=1}^{\\mathcal{N}_C} \\gamma_i
\\varphi(\\| \\boldsymbol{x} - \\boldsymbol{x_{C_i}} \\|)
where :math:`p(\\boldsymbol{x})` is a low_degree polynomial term,
:math:`\\gamma_i` is the weight, corresponding to the a-priori selected
:math:`\\mathcal{N}_C` control points, associated to the :math:`i`-th basis
function, and :math:`\\varphi(\\| \\boldsymbol{x} - \\boldsymbol{x_{C_i}}
\\|)` a radial function based on the Euclidean distance between the control
points position :math:`\\boldsymbol{x_{C_i}}` and :math:`\\boldsymbol{x}`.
A radial basis function, generally, is a real-valued function whose value
depends only on the distance from the origin, so that
:math:`\\varphi(\\boldsymbol{x}) = \\tilde{\\varphi}(\\| \\boldsymbol{x}
\\|)`.
The matrix version of the formula above is:
.. math::
\\mathcal{M}(\\boldsymbol{x}) = \\boldsymbol{c} +
\\boldsymbol{Q}\\boldsymbol{x} +
\\boldsymbol{W^T}\\boldsymbol{d}(\\boldsymbol{x})
The idea is that after the computation of the weights and the polynomial
terms from the coordinates of the control points before and after the
deformation, we can deform all the points of the mesh accordingly. Among
the most common used radial basis functions for modelling 2D and 3D shapes,
we consider Gaussian splines, Multi-quadratic biharmonic splines, Inverted
multi-quadratic biharmonic splines, Thin-plate splines, Beckert and
Wendland :math:`C^2` basis and Polyharmonic splines all defined and
implemented below.
"""
import numpy as np
from pygem import RBF as OriginalRBF
from .cad_deformation import CADDeformation
[docs]class RBF(CADDeformation, OriginalRBF):
"""
Class that handles the Radial Basis Functions interpolation on CAD
geometries.
:param numpy.ndarray original_control_points: it is an
(*n_control_points*, *3*) array with the coordinates of the original
interpolation control points before the deformation. The default is the
vertices of the unit cube.
:param numpy.ndarray deformed_control_points: it is an
(*n_control_points*, *3*) array with the coordinates of the
interpolation control points after the deformation. The default is the
vertices of the unit cube.
:param func: the basis function to use in the transformation. Several basis
function are already implemented and they are available through the
:class:`~pygem.rbf_factory.RBFFactory` by passing the name of the right
function (see class documentation for the updated list of basis
function). A callable object can be passed as basis function. Default
is 'gaussian_spline'.
:param float radius: the scaling parameter r that affects the shape of the
basis functions. For details see the class
:class:`~pygem.radialbasis.RBF`. The default value is 0.5.
:param dict extra_parameter: the additional parameters that may be passed to
the kernel function. Default is None.
:param int u_knots_to_add: the number of knots to add to the NURBS surfaces
along `u` direction before the deformation. This parameter is useful
whenever the gradient of the imposed deformation present spatial scales
that are smaller than the local distance among the original poles of
the surface/curve. Enriching the poles will allow for a more accurate
application of the deformation, and might also reduce possible
mismatches between bordering faces. On the orther hand, it might result
in higher computational cost and bigger output files. Default is 0.
:param int v_knots_to_add: the number of knots to add to the NURBS surfaces
along `v` direction before the deformation. This parameter is useful
whenever the gradient of the imposed deformation present spatial scales
that are smaller than the local distance among the original poles of
the surface/curve. Enriching the poles will allow for a more accurate
application of the deformation, and might also reduce possible
mismatches between bordering faces. On the orther hand, it might result
in higher computational cost and bigger output files. Default is 0.
:param int t_knots_to_add: the number of knots to add to the NURBS curves
before the deformation. This parameter is useful whenever the gradient
of the imposed deformation present spatial scales that are smaller than
the local distance among the original poles of the surface/curve.
Enriching the poles will allow for a more accurate application of the
deformation, and might also reduce possible mismatches between
bordering faces. On the orther hand, it might result in higher
computational cost and bigger output files. Default is 0.
:param float tolerance: the tolerance involved in several internal
operations of the procedure (joining wires in a single curve before
deformation and placing new poles on curves and surfaces). Change the
default value only if the input file scale is significantly different
form mm, making some of the aforementioned operations fail. Default is
0.0001.
:cvar numpy.ndarray weights: the matrix formed by the weights corresponding
to the a-priori selected N control points, associated to the basis
functions and c and Q terms that describe the polynomial of order one
p(x) = c + Qx. The shape is (n_control_points+1+3)-by-3. It is computed
internally.
:cvar numpy.ndarray original_control_points: it is an
(*n_control_points*, *3*) array with the coordinates of the original
interpolation control points before the deformation.
:cvar numpy.ndarray deformed_control_points: it is an
(*n_control_points*, *3*) array with the coordinates of the
interpolation control points after the deformation.
:cvar callable basis: the basis functions to use in the
transformation.
:cvar float radius: the scaling parameter that affects the shape of the
basis functions.
:cvar dict extra_parameter: the additional parameters that may be passed to
the kernel function.
:cvar int u_knots_to_add: the number of knots to add to the NURBS surfaces
along `u` direction before the deformation.
:cvar int v_knots_to_add: the number of knots to add to the NURBS surfaces
along `v` direction before the deformation.
:cvar int t_knots_to_add: the number of knots to add to the NURBS curves
before the deformation.
:cvar float tolerance: the tolerance involved in several internal
operations of the procedure (joining wires in a single curve before
deformation and placing new poles on curves and surfaces).
:Example:
>>> from pygem.cad import RBF
>>> rbf = RBF()
>>> rbf.read_parameters(
>>> 'tests/test_datasets/parameters_test_ffd_iges.prm')
>>> input_cad_file_name = "input.iges"
>>> modified_cad_file_name = "output.iges"
>>> rbf(input_cad_file_name, modified_cad_file_name)
"""
def __init__(self,
original_control_points=None,
deformed_control_points=None,
func='gaussian_spline',
radius=0.5,
extra_parameter=None,
u_knots_to_add=0,
v_knots_to_add=0,
t_knots_to_add=0,
tolerance=1e-4):
OriginalRBF.__init__(self,
original_control_points=original_control_points,
deformed_control_points=deformed_control_points,
func=func,
radius=radius,
extra_parameter=extra_parameter)
CADDeformation.__init__(self,
u_knots_to_add=u_knots_to_add,
v_knots_to_add=v_knots_to_add,
t_knots_to_add=t_knots_to_add,
tolerance=tolerance)