Source code for felupe.mechanics._multipoint
# -*- coding: utf-8 -*-
"""
This file is part of FElupe.
FElupe is free software: you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation, either version 3 of the License, or
(at your option) any later version.
FElupe is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
GNU General Public License for more details.
You should have received a copy of the GNU General Public License
along with FElupe. If not, see <http://www.gnu.org/licenses/>.
"""
import numpy as np
from scipy.sparse import eye, lil_matrix
from ._helpers import Assemble, Results
[docs]
class MultiPointConstraint:
"""A Multi-point-constraint which connects a center-point to a list of points.
Parameters
----------
field : FieldContainer
A field container with the displacement field as first field.
points : (n,) ndarray
An array with indices of points to be connected to the center-point.
centerpoint : int
The index of the centerpoint.
skip : 3-tuple of bool, optional
A tuple with boolean values for each axis to skip. If True, the respective axis
is not connected. Default is (False, False, False).
multiplier : float, optional
A multiplier to penalize the relative displacements between the center-point and
the points. Default is 1e-3.
Notes
-----
A :class:`~felupe.MultiPointConstraint` is supported as an item in a
:class:`~felupe.Step`. It provides the assemble-methods
:meth:`MultiPointConstraint.assemble.vector() <felupe.MultiPointConstraint.assemble.vector>`
and :meth:`MultiPointConstraint.assemble.matrix() <felupe.MultiPointConstraint.assemble.matrix>`.
.. note::
Rotational degrees-of-freedom of the center-point are not connected to the
points.
Examples
--------
This example shows how to use a :class:`~felupe.MultiPointConstraint`.
An additional center-point is added to a mesh. By default, all *hanging* points are
collected in the mesh-attribute
:attr:`Mesh.points_without_cells <felupe.Mesh.points_without_cells>`. The degrees of
freedom of these points are considered as fixed, i.e. they are ignored. The center-
point is not connected to any cell and is added to the points-without-cells array
on :meth:`Mesh.update <felupe.Mesh.update>`. Hence, center-point has to be removed
manually.
.. pyvista-plot::
:context:
>>> import numpy as np
>>> import felupe as fem
>>>
>>> mesh = fem.Cube(n=3)
>>> mesh.update(points=np.vstack([mesh.points, [2.0, 0.5, 0.5]]))
>>>
>>> # prevent the field-values at the center-point to be treated as dof0
>>> mesh.points_without_cells = mesh.points_without_cells[:-1]
>>>
>>> region = fem.RegionHexahedron(mesh)
>>> displacement = fem.Field(region, dim=3)
>>> field = fem.FieldContainer([displacement])
>>>
>>> umat = fem.NeoHooke(mu=1.0, bulk=2.0)
>>> solid = fem.SolidBody(umat=umat, field=field)
A :class:`~felupe.MultiPointConstraint` defines the multi-point constraint which
connects the displacement degrees of freedom of the center-point with the dofs of
points located at :math:`x=1`.
.. pyvista-plot::
:context:
:force_static:
>>> import pyvista as pv
>>>
>>> mpc = fem.MultiPointConstraint(
... field=field,
... points=np.arange(mesh.npoints)[mesh.x == 1],
... centerpoint=-1,
... )
>>>
>>> plotter = pv.Plotter()
>>> actor_1 = plotter.add_points(
... mesh.points[mpc.points],
... point_size=16,
... color="red",
... )
>>> actor_2 = plotter.add_points(
... mesh.points[[mpc.centerpoint]],
... point_size=16,
... color="green",
... )
>>> mesh.plot(plotter=mpc.plot(plotter=plotter)).show()
The mesh is fixed on the left end face and a ramped :class:`~felupe.PointLoad` is
applied on the center-point of the :class:`~felupe.MultiPointConstraint`. All items
are added to a :class:`~felupe.Step` and a :class:`~felupe.Job` is evaluated.
.. pyvista-plot::
:context:
>>> boundaries = {"fixed": fem.Boundary(displacement, fx=0)}
>>> load = fem.PointLoad(field, points=[-1])
>>> table = fem.math.linsteps([0, 1], num=5, axis=0, axes=3)
>>>
>>> step = fem.Step(
... [solid, mpc, load], boundaries=boundaries, ramp={load: table}
... )
>>> job = fem.Job([step]).evaluate()
A view on the deformed mesh including the :class:`~felupe.MultiPointConstraint` is
plotted.
.. pyvista-plot::
:context:
:force_static:
>>> plotter = pv.Plotter()
>>>
>>> actor_1 = plotter.add_points(
... mesh.points[mpc.points] + displacement.values[mpc.points],
... point_size=16,
... color="red",
... )
>>> actor_2 = plotter.add_points(
... mesh.points[[mpc.centerpoint]] + displacement.values[[mpc.centerpoint]],
... point_size=16,
... color="green",
... )
>>> field.plot(
... "Displacement", component=None, plotter=mpc.plot(plotter=plotter)
... ).show()
See Also
--------
felupe.MultiPointContact : A frictionless point-to-rigid (wall) contact.
"""
def __init__(
self, field, points, centerpoint, skip=(False, False, False), multiplier=1e3
):
self.field = field
self.mesh = field.region.mesh
self.points = np.asarray(points)
self.centerpoint = centerpoint
self.mask = ~np.array(skip, dtype=bool)[: self.mesh.dim]
self.axes = np.arange(self.mesh.dim)[self.mask]
self.multiplier = multiplier
self.results = Results(stress=False, elasticity=False)
self.assemble = Assemble(vector=self._vector, matrix=self._matrix)
[docs]
def plot(self, plotter=None, color="black", **kwargs):
import pyvista as pv
if plotter is None:
plotter = pv.Plotter()
# get deformed points
x = self.mesh.points + self.field[0].values
x = np.pad(x, ((0, 0), (0, 3 - x.shape[1])))
pointa = x[self.centerpoint]
for pointb in x[self.points]:
plotter.add_mesh(pv.Line(pointa, pointb), color=color, **kwargs)
return plotter
def _vector(self, field=None, parallel=False):
"Calculate vector of residuals with RBE2 contributions."
if field is not None:
self.field = field
u = self.field.fields[0].values
N = self.multiplier * (-u[self.points] + u[self.centerpoint])
N[:, ~self.mask] = 0
r = lil_matrix(u.shape)
r[self.points] = -N
r[self.centerpoint] = N.sum(axis=0)
self.results.force = r.reshape(-1, 1).tocsr()
return self.results.force
def _matrix(self, field=None, parallel=False):
"Calculate stiffness with RBE2 contributions."
if field is not None:
self.field = field
indices = np.arange(self.mesh.ndof).reshape(self.mesh.points.shape)
td = [indices[self.points.reshape(-1, 1), ax].ravel() for ax in self.axes]
cd = [indices[self.centerpoint, ax].ravel() for ax in self.axes]
L = lil_matrix((self.mesh.ndof, self.mesh.ndof))
for t, c in zip(td, cd):
L[t.reshape(-1, 1), t] = eye(len(t)) * self.multiplier
L[t.reshape(-1, 1), c] = -self.multiplier
L[c.reshape(-1, 1), t] = -self.multiplier
L[c.reshape(-1, 1), c] = eye(len(c)) * self.multiplier * len(self.points)
self.results.stiffness = L.tocsr()
return self.results.stiffness
[docs]
class MultiPointContact:
"""A frictionless point-to-rigid (wall) contact which connects a center-point to a
list of points.
Parameters
----------
field : FieldContainer
A field container with the displacement field as first field.
points : (n,) ndarray
An array with indices of points to be connected to the center-point.
centerpoint : int
The index of the centerpoint.
skip : 3-tuple of bool, optional
A tuple with boolean values for each axis to skip. If True, the respective axis
is not connected. Default is (False, False, False).
multiplier : float, optional
A multiplier to penalize the relative displacements between the center-point and
the points in contact. Default is 1e-6.
Notes
-----
A :class:`~felupe.MultiPointContact` is supported as an item in a
:class:`~felupe.Step`. It provides the assemble-methods
:meth:`MultiPointContact.assemble.vector() <felupe.MultiPointContact.assemble.vector>`
and :meth:`MultiPointContact.assemble.matrix() <felupe.MultiPointContact.assemble.matrix>`.
Examples
--------
This example shows how to use a :class:`~felupe.MultiPointContact`.
An additional center-point is added to a mesh. By default, all *hanging* points are
collected in the mesh-attribute
:attr:`Mesh.points_without_cells <felupe.Mesh.points_without_cells>`. The degrees of
freedom of these points are considered as fixed, i.e. they are ignored. The center-
point is not connected to any cell and is added to the points-without-cells array
on :meth:`Mesh.update <felupe.Mesh.update>`. Hence, center-point has to be removed
manually.
.. pyvista-plot::
:context:
>>> import numpy as np
>>> import felupe as fem
>>>
>>> mesh = fem.Cube(n=3)
>>> mesh.update(points=np.vstack([mesh.points, [2.0, 0.5, 0.5]]))
>>>
>>> # prevent the field-values at the center-point to be treated as dof0
>>> mesh.points_without_cells = mesh.points_without_cells[:-1]
>>>
>>> region = fem.RegionHexahedron(mesh)
>>> displacement = fem.Field(region, dim=3)
>>> field = fem.FieldContainer([displacement])
>>>
>>> umat = fem.NeoHooke(mu=1.0, bulk=2.0)
>>> solid = fem.SolidBody(umat=umat, field=field)
A :class:`~felupe.MultiPointContact` defines the multi-point contact which
connects the displacement degrees of freedom of the center-point with the dofs of
points located at :math:`x=1` if they are in contact. Only the :math:`x`-component
is considered in this example.
.. pyvista-plot::
:context:
:force_static:
>>> import pyvista as pv
>>>
>>> contact = fem.MultiPointContact(
... field=field,
... points=np.arange(mesh.npoints)[mesh.x == 1],
... centerpoint=-1,
... skip=(False, True, True)
... )
>>>
>>> plotter = pv.Plotter()
>>> actor_1 = plotter.add_points(
... mesh.points[mpc.points],
... point_size=16,
... color="red",
... )
>>> actor_2 = plotter.add_points(
... mesh.points[[mpc.centerpoint]],
... point_size=16,
... color="green",
... )
>>> mesh.plot(plotter=contact.plot(plotter=plotter)).show()
The mesh is fixed on the left end face and a ramped :class:`~felupe.Boundary` is
applied on the center-point of the :class:`~felupe.MultiPointContact`. All items
are added to a :class:`~felupe.Step` and a :class:`~felupe.Job` is evaluated.
.. pyvista-plot::
:context:
>>> boundaries = {
... "fixed": fem.Boundary(displacement, fx=0),
... "control": fem.Boundary(displacement, fx=2, skip=(1, 0, 0)),
... "move": fem.Boundary(displacement, fx=2, skip=(0, 1, 1)),
... }
>>> table = fem.math.linsteps([0, -1, -1.5], num=5)
>>> step = fem.Step(
... [solid, contact],
... boundaries=boundaries,
... ramp={boundaries["move"]: table},
... )
>>> job = fem.Job([step]).evaluate()
A view on the deformed mesh including the :class:`~felupe.MultiPointContact` is
plotted.
.. pyvista-plot::
:context:
:force_static:
>>> plotter = pv.Plotter()
>>>
>>> actor_1 = plotter.add_points(
... mesh.points[contact.points] + displacement.values[contact.points],
... point_size=16,
... color="red",
... )
>>> actor_2 = plotter.add_points(
... mesh.points[[contact.centerpoint]] + displacement.values[[contact.centerpoint]],
... point_size=16,
... color="green",
... )
>>> field.plot(
... "Displacement", component=None, plotter=contact.plot(plotter=plotter)
... ).show()
See Also
--------
felupe.MultiPointConstraint : A Multi-point-constraint which connects a center-point
to a list of points.
"""
def __init__(
self, field, points, centerpoint, skip=(False, False, False), multiplier=1e6
):
self.field = field
self.mesh = field.region.mesh
self.points = np.asarray(points)
self.centerpoint = centerpoint
self.mask = ~np.array(skip, dtype=bool)[: self.mesh.dim]
self.axes = np.arange(self.mesh.dim)[self.mask]
self.multiplier = multiplier
self.results = Results(stress=False, elasticity=False)
self.assemble = Assemble(vector=self._vector, matrix=self._matrix)
[docs]
def plot(
self,
plotter=None,
offset=0,
show_edges=True,
color="black",
opacity=0.5,
**kwargs,
):
import pyvista as pv
if plotter is None:
plotter = pv.Plotter()
# get edge lengths of deformed enclosing box
x = self.mesh.points + self.field[0].values
edges = np.diag((x.max(axis=0) - x.min(axis=0))) + x.min(axis=0)
# plot a line or a rectangle for each active contact plane
for ax in self.axes:
# fill the point values of the normal axis with the centerpoint values
points = edges.copy()
points[:, ax] = x[self.centerpoint, ax] + offset
# scale the line or rectangle at the origin
origin = points.mean(axis=0)
points = (points - origin) * 1.05 + origin
# plot a line or a rectangle
if len(points) == 3:
plotter.add_mesh(
pv.Rectangle(points), color=color, opacity=opacity, **kwargs
)
else:
points = np.pad(points, ((0, 0), (0, 3 - points.shape[1])))
plotter.add_mesh(
pv.Line(*points), color=color, opacity=opacity, **kwargs
)
return plotter
def _vector(self, field=None, parallel=False):
"Calculate vector of residuals with RBE2 contributions."
if field is not None:
self.field = field
u = self.field.fields[0].values
Xc = self.mesh.points[self.centerpoint]
Xt = self.mesh.points[self.points]
xc = u[self.centerpoint] + Xc
xt = u[self.points] + Xt
mask = np.sign(-Xt + Xc) == np.sign(-xt + xc)
mask[:, ~self.mask] = True
n = -xt + xc
n[mask] = 0
r = lil_matrix(u.shape)
r[self.points] = -self.multiplier * n
r[self.centerpoint] = self.multiplier * n.sum(axis=0)
self.results.force = r.reshape(-1, 1).tocsr()
return self.results.force
def _matrix(self, field=None, parallel=False):
"Calculate stiffness with RBE2 contributions."
if field is not None:
self.field = field
u = self.field.fields[0].values
Xc = self.mesh.points[self.centerpoint]
Xt = self.mesh.points[self.points]
xc = u[self.centerpoint] + Xc
xt = u[self.points] + Xt
mask = np.sign(-Xt + Xc) != np.sign(-xt + xc)
masks = [mask[:, ax] for ax in self.axes]
indices = np.arange(self.mesh.ndof).reshape(self.mesh.points.shape)
td = [indices[self.points.reshape(-1, 1), ax].ravel() for ax in self.axes]
cd = [indices[self.centerpoint, ax].ravel() for ax in self.axes]
L = lil_matrix((self.mesh.ndof, self.mesh.ndof))
for t, c, m in zip(td, cd, masks):
L[t[m].reshape(-1, 1), t[m]] = eye(len(t[m])) * self.multiplier
L[t[m].reshape(-1, 1), c] = -self.multiplier
L[c.reshape(-1, 1), t[m]] = -self.multiplier
L[c.reshape(-1, 1), c] = eye(len(c)) * self.multiplier * len(self.points[m])
self.results.stiffness = L.tocsr()
return self.results.stiffness