Source code for felupe.mechanics._item
# -*- 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/>.
"""
from ._helpers import Assemble, Results
[docs]
class FormItem:
r"""An item to be used in a :class:`felupe.Step` with bilinear and optional linear
form objects based on weak-forms with methods for integration and assembly of
vectors / sparse matrices.
Parameters
----------
bilinearform : Form or None, optional
A bilinear form object (default is None). If None, the resulting matrix will be
filled with zeros.
linearform : Form or None, optional
A linear form object (default is None). If None, the resulting vector will be
filled with zeros.
sym : bool, optional
Flag to active symmetric integration/assembly for bilinear forms (default is
False).
kwargs : dict or None, optional
Dictionary with initial optional weakform-keyword-arguments (default is None).
ramp_item : str or int, optional
The item of the dict of keyword arguments which will be updated in a
:class:`~felupe.Step` (default is 0). By default, the first item of the dict
is selected. Optionally, also the key of the item to be updated may be
given.
Notes
-----
.. warning::
The computational cost of weak-forms defined by :func:`~felupe.Form` is much
higher compared to :class:`~felupe.IntegralForm`. Try to re-formulate the weak
form and use :class:`~felupe.IntegralForm` instead if performance is relevant.
Examples
--------
A :class:`~felupe.FormItem` is very flexible, e.g. it is possible to define linear-
elastic solid bodies, boundary conditions or hyperelastic mixed-field solid bodies
by their weak forms.
**Linear-Elasticity**
A :class:`~felupe.FormItem` is used to create a linear-elastic solid body.
>>> import felupe as fem
>>> from felupe.math import ddot, sym, trace, grad
>>>
>>> mesh = fem.Cube(n=11)
>>> region = fem.RegionHexahedron(mesh)
>>> field = fem.FieldContainer([fem.Field(region, dim=3)])
>>> boundaries, loadcase = fem.dof.uniaxial(field, clamped=True)
>>>
>>> @fem.Form(v=field, u=field)
... def bilinearform():
... def a(v, u, μ=1.0, λ=2.0):
... δε, ε = sym(grad(v)), sym(grad(u))
... return 2 * μ * ddot(δε, ε) + λ * trace(δε) * trace(ε)
... return [a]
>>>
>>> item = fem.FormItem(bilinearform, linearform=None, sym=True)
>>> step = fem.Step(items=[item], boundaries=boundaries)
>>> job = fem.Job(steps=[step]).evaluate()
**Boundary Condition**
A :class:`~felupe.FormItem` is used to create a boundary condition with externally
applied displacements which is used as a ramped-boundary in a :class:`Step`.
>>> import felupe as fem
>>> import numpy as np
>>> from felupe.math import ddot
>>>
>>> mesh = fem.Cube(n=4)
>>> region = fem.RegionHexahedron(mesh)
>>> field = fem.FieldContainer([fem.Field(region, dim=3)])
>>> boundaries = fem.dof.symmetry(field[0])
>>> solid = fem.SolidBody(umat=fem.NeoHookeCompressible(mu=1, lmbda=2), field=field)
>>>
>>> face = fem.Field(fem.RegionHexahedronBoundary(mesh, mask=mesh.x == 1), dim=3)
>>> right = fem.FieldContainer([face])
>>>
>>> @fem.Form(v=right)
... def linearform():
... def L(v, value, multiplier=100):
... u = right.extract(grad=False)[0]
... uext = value * np.array([1, 0, 0]).reshape(3, 1, 1)
... return multiplier * ddot(v, u - uext)
...
... return [L]
>>>
>>>
>>> @fem.Form(v=right, u=right)
... def bilinearform():
... return [lambda v, u, value, multiplier=100: multiplier * ddot(v, u)]
>>>
>>> move = fem.FormItem(bilinearform, linearform, kwargs={"value": 0}, ramp_item=0)
>>> values = fem.math.linsteps([0, 1], num=5)
>>> step = fem.Step(items=[solid, move], boundaries=boundaries, ramp={move: values})
>>> job = fem.Job(steps=[step]).evaluate()
**Hu-Washizu (Mixed) Three-Field Formulation**
A :class:`~felupe.FormItem` is used to create a mixed three-field variation suitable
for nearly-incompressible hyperelastic solid bodies.
>>> import felupe as fem
>>> from felupe.math import ddot, grad
>>>
>>> mesh = fem.Cube(n=3)
>>> region = fem.RegionHexahedron(mesh)
>>> field = fem.FieldsMixed(region, n=3)
>>>
>>> @fem.Form(v=field)
... def linearform():
... def L1(du, **kwargs):
... dW = linearform.dW = kwargs["umat"].gradient(kwargs["field"].extract())
... return fem.math.ddot(fem.math.grad(du), dW[0])
...
... def L2(dp, **kwargs):
... return dp[0] * linearform.dW[1]
...
... def L3(dJ, **kwargs):
... return dJ[0] * linearform.dW[2]
...
... return [L1, L2, L3]
>>>
>>> @fem.Form(v=field, u=field)
... def bilinearform():
... def a11(du, Du, **kwargs):
... d2W = bilinearform.d2W = kwargs["umat"].hessian(kwargs["field"].extract())
... return ddot(ddot(grad(du), d2W[0], mode=(2, 4)), grad(Du))
...
... def a12(du, Dp, **kwargs):
... return ddot(grad(du), bilinearform.d2W[1]) * Dp[0]
...
... def a13(du, DJ, **kwargs):
... return ddot(grad(du), bilinearform.d2W[2]) * DJ[0]
...
... def a22(dp, Dp, **kwargs):
... return dp[0] * bilinearform.d2W[3] * Dp[0]
...
... def a23(dp, DJ, **kwargs):
... return dp[0] * bilinearform.d2W[4] * DJ[0]
...
... def a33(dJ, DJ, **kwargs):
... return dJ[0] * bilinearform.d2W[5] * DJ[0]
...
... return [a11, a12, a13, a22, a23, a33]
>>>
>>> umat = fem.ThreeFieldVariation(fem.NeoHooke(mu=1, bulk=5000))
>>> item = fem.FormItem(bilinearform, linearform, kwargs={"umat": umat, "field": field})
>>> boundaries, loadcase = fem.dof.uniaxial(field, clamped=True, move=-0.3)
>>> step = fem.Step(items=[item], boundaries=boundaries)
>>> job = fem.Job(steps=[step], boundary=boundaries["move"]).evaluate()
See Also
--------
felupe.Form : A function decorator for a linear- or bilinear-form object.
felupe.Step : A Step with multiple substeps.
"""
def __init__(
self, bilinearform=None, linearform=None, sym=False, kwargs=None, ramp_item=0
):
self.bilinearform = bilinearform
self.linearform = linearform
self.sym = sym
self.kwargs = kwargs
self.ramp_item = ramp_item
self.results = Results(stress=False, elasticity=False)
self.assemble = Assemble(vector=self._vector, matrix=self._matrix)
if self.bilinearform is not None:
self.field = self.bilinearform.form.v.field
else:
if self.linearform is not None:
self.field = self.linearform.form.v.field
[docs]
def update(self, value):
key = self.ramp_item
if isinstance(key, int):
key = list(self.kwargs.keys())[key]
self.kwargs[key] = value
def _vector(self, field=None, parallel=False):
if field is not None:
self.field = field
if self.linearform is not None:
self.results.force = self.linearform.assemble(
v=self.field, parallel=parallel, kwargs=self.kwargs
)
else:
from scipy.sparse import csr_matrix
self.results.force = csr_matrix((sum(self.field.fieldsizes), 1))
return self.results.force
def _matrix(self, field=None, parallel=False):
if field is not None:
self.field = field
if self.bilinearform is not None:
self.results.stiffness = self.bilinearform.assemble(
v=self.field,
u=self.field,
parallel=parallel,
sym=self.sym,
kwargs=self.kwargs,
)
else:
from scipy.sparse import csr_matrix
size = sum(self.field.fieldsizes)
self.results.stiffness = csr_matrix((size, size))
return self.results.stiffness