gel.kinematics
Interface to reading displacements from files, continuum mechanics.
The most useful aspect of this submodule is the class Kinematics
which contains definitions of various fields in continuum mechanics,
useful for defining material models in gel.mechanics. It also contains
the proper functionality for allowing differentiating through such
relationships to obtain stress tensors and for projecting $J$ into an
appropriate element-wise basis.
1r"""Interface to reading displacements from files, continuum mechanics. 2 3The most useful aspect of this submodule is the class `Kinematics` 4which contains definitions of various fields in continuum mechanics, 5useful for defining material models in `gel.mechanics`. It also contains 6the proper functionality for allowing differentiating through such 7relationships to obtain stress tensors and for projecting $J$ into an 8appropriate element-wise basis. 9""" 10from .header import * 11from .helper import * 12 13 14class Kinematics: 15 16 def __init__(self, geo, u=None, differentiable_F=False): 17 """ 18 Object that stores variables concerning the kinematics of the problem. 19 20 Needs a geometry geo that all this movement is happening with, and may 21 optionally provide a displacement u to impose. Otherwise, its u will 22 be computed from scratch. 23 24 In order to be able to differentiate computed quantities wrt. F, set 25 differentiable_F=True , ie. for computing stress tensors. 26 """ 27 self.geo = geo 28 """`gel.geometry.Geometry` on which displacements are defined""" 29 30 self.u = None 31 r"""FEniCS function displacement field $\mathbf{u}$""" 32 if u is None: 33 self.u = Function(geo.V) 34 else: 35 self.u = u 36 37 I = Identity(self.u.geometric_dimension()) 38 39 self.F = I + grad(self.u) 40 r"""$\mathbf{F}=\mathbf{I}+\nabla\mathbf{u}$""" 41 42 self.differentiable_F = differentiable_F 43 r"""bool indicating if `F` is equipped for being the argument 44 that another form is differentiated with respect to 45 """ 46 if differentiable_F: 47 # Be able to use it as argument of differentiation 48 self.F = variable(self.F) 49 50 self.C = self.F.T*self.F # Right Cauchy-Green 51 r"""$\mathbf{C}=\mathbf{F}^T\mathbf{F}$""" 52 53 self.Ic = tr(self.C) 54 r"""$I_1=\text{tr}\left(\mathbf{C}\right)$""" 55 56 self.Ju = det(self.F) 57 r"""$J=\det\left(\mathbf{F}\right)$""" 58 59 @property 60 def projected_J(self): 61 r"""Element-wise-DoF $J$ FEniCS function""" 62 return project(self.Ju, self.geo.DG0) 63 64 65def kinematics_from_file(geo, filename, *args, **kwargs): 66 """Returns `Kinematics` object from displacements "u" in xdmf `filename`. 67 68 * `geo`: `gel.geometry.Geometry` object with underlying meshes 69 matching that for the displacements in `filename` 70 * `filename`: str path to full-shape/checkpoing .xdmf file with 71 displacement data "U" 72 * `args`, `kwargs`: other arguments for the `Kinematics` constructor 73 74 Note that this will only read "u" at the first timestep. 75 """ 76 u = load_shape(geo.V, filename, "u") 77 78 return Kinematics(geo, *args, u=u, **kwargs)
class
Kinematics:
15class Kinematics: 16 17 def __init__(self, geo, u=None, differentiable_F=False): 18 """ 19 Object that stores variables concerning the kinematics of the problem. 20 21 Needs a geometry geo that all this movement is happening with, and may 22 optionally provide a displacement u to impose. Otherwise, its u will 23 be computed from scratch. 24 25 In order to be able to differentiate computed quantities wrt. F, set 26 differentiable_F=True , ie. for computing stress tensors. 27 """ 28 self.geo = geo 29 """`gel.geometry.Geometry` on which displacements are defined""" 30 31 self.u = None 32 r"""FEniCS function displacement field $\mathbf{u}$""" 33 if u is None: 34 self.u = Function(geo.V) 35 else: 36 self.u = u 37 38 I = Identity(self.u.geometric_dimension()) 39 40 self.F = I + grad(self.u) 41 r"""$\mathbf{F}=\mathbf{I}+\nabla\mathbf{u}$""" 42 43 self.differentiable_F = differentiable_F 44 r"""bool indicating if `F` is equipped for being the argument 45 that another form is differentiated with respect to 46 """ 47 if differentiable_F: 48 # Be able to use it as argument of differentiation 49 self.F = variable(self.F) 50 51 self.C = self.F.T*self.F # Right Cauchy-Green 52 r"""$\mathbf{C}=\mathbf{F}^T\mathbf{F}$""" 53 54 self.Ic = tr(self.C) 55 r"""$I_1=\text{tr}\left(\mathbf{C}\right)$""" 56 57 self.Ju = det(self.F) 58 r"""$J=\det\left(\mathbf{F}\right)$""" 59 60 @property 61 def projected_J(self): 62 r"""Element-wise-DoF $J$ FEniCS function""" 63 return project(self.Ju, self.geo.DG0)
Kinematics(geo, u=None, differentiable_F=False)
17 def __init__(self, geo, u=None, differentiable_F=False): 18 """ 19 Object that stores variables concerning the kinematics of the problem. 20 21 Needs a geometry geo that all this movement is happening with, and may 22 optionally provide a displacement u to impose. Otherwise, its u will 23 be computed from scratch. 24 25 In order to be able to differentiate computed quantities wrt. F, set 26 differentiable_F=True , ie. for computing stress tensors. 27 """ 28 self.geo = geo 29 """`gel.geometry.Geometry` on which displacements are defined""" 30 31 self.u = None 32 r"""FEniCS function displacement field $\mathbf{u}$""" 33 if u is None: 34 self.u = Function(geo.V) 35 else: 36 self.u = u 37 38 I = Identity(self.u.geometric_dimension()) 39 40 self.F = I + grad(self.u) 41 r"""$\mathbf{F}=\mathbf{I}+\nabla\mathbf{u}$""" 42 43 self.differentiable_F = differentiable_F 44 r"""bool indicating if `F` is equipped for being the argument 45 that another form is differentiated with respect to 46 """ 47 if differentiable_F: 48 # Be able to use it as argument of differentiation 49 self.F = variable(self.F) 50 51 self.C = self.F.T*self.F # Right Cauchy-Green 52 r"""$\mathbf{C}=\mathbf{F}^T\mathbf{F}$""" 53 54 self.Ic = tr(self.C) 55 r"""$I_1=\text{tr}\left(\mathbf{C}\right)$""" 56 57 self.Ju = det(self.F) 58 r"""$J=\det\left(\mathbf{F}\right)$"""
Object that stores variables concerning the kinematics of the problem.
Needs a geometry geo that all this movement is happening with, and may optionally provide a displacement u to impose. Otherwise, its u will be computed from scratch.
In order to be able to differentiate computed quantities wrt. F, set differentiable_F=True , ie. for computing stress tensors.
differentiable_F
bool indicating if F is equipped for being the argument
that another form is differentiated with respect to
def
kinematics_from_file(geo, filename, *args, **kwargs):
66def kinematics_from_file(geo, filename, *args, **kwargs): 67 """Returns `Kinematics` object from displacements "u" in xdmf `filename`. 68 69 * `geo`: `gel.geometry.Geometry` object with underlying meshes 70 matching that for the displacements in `filename` 71 * `filename`: str path to full-shape/checkpoing .xdmf file with 72 displacement data "U" 73 * `args`, `kwargs`: other arguments for the `Kinematics` constructor 74 75 Note that this will only read "u" at the first timestep. 76 """ 77 u = load_shape(geo.V, filename, "u") 78 79 return Kinematics(geo, *args, u=u, **kwargs)
Returns Kinematics object from displacements "u" in xdmf filename.
geo:gel.geometry.Geometryobject with underlying meshes matching that for the displacements infilenamefilename: str path to full-shape/checkpoing .xdmf file with displacement data "U"args,kwargs: other arguments for theKinematicsconstructor
Note that this will only read "u" at the first timestep.