from abc import ABCMeta, abstractmethod
import gem
try:
from firedrake_citations import Citations
Citations().add("Kirby2018zany", """
@Article{Kirby2018zany,
author = {Robert C. Kirby},
title = {A general approach to transforming finite elements},
journal = {SMAI Journal of Computational Mathematics},
year = 2018,
volume = 4,
pages = {197-224},
doi = {10.5802/smai-jcm.33},
archiveprefix ={arXiv},
eprint = {1706.09017},
primaryclass = {math.NA}
}
""")
Citations().add("Kirby2019zany", """
@Article{Kirby:2019,
author = {Robert C. Kirby and Lawrence Mitchell},
title = {Code generation for generally mapped finite
elements},
journal = {ACM Transactions on Mathematical Software},
year = 2019,
volume = 45,
number = 41,
pages = {41:1--41:23},
doi = {10.1145/3361745},
archiveprefix ={arXiv},
eprint = {1808.05513},
primaryclass = {cs.MS}
}""")
Citations().add("Clough1965", """
@inproceedings{Clough1965,
author = {R. W. Clough, J. L. Tocher},
title = {Finite element stiffness matricess for analysis of plate bending},
booktitle = {Proc. of the First Conf. on Matrix Methods in Struct. Mech},
year = 1965,
pages = {515-546},
}
""")
Citations().add("Argyris1968", """
@Article{Argyris1968,
author = {J. H. Argyris and I. Fried and D. W. Scharpf},
title = {{The TUBA family of plate elements for the matrix
displacement method}},
journal = {The Aeronautical Journal},
year = 1968,
volume = 72,
pages = {701-709},
doi = {10.1017/S000192400008489X}
}
""")
Citations().add("Bell1969", """
@Article{Bell1969,
author = {Kolbein Bell},
title = {A refined triangular plate bending finite element},
journal = {International Journal for Numerical Methods in
Engineering},
year = 1969,
volume = 1,
number = 1,
pages = {101-122},
doi = {10.1002/nme.1620010108}
}
""")
Citations().add("Ciarlet1972", r"""
@Article{Ciarlet1972,
author = {P. G. Ciarlet and P. A. Raviart},
title = {{General Lagrange and Hermite interpolation in
$\mathbb{R}^n$ with applications to finite element
methods}},
journal = {Archive for Rational Mechanics and Analysis},
year = 1972,
volume = 46,
number = 3,
pages = {177-199},
doi = {10.1007/BF0025245}
}
""")
Citations().add("Morley1971", """
@Article{Morley1971,
author = {L. S. D. Morley},
title = {The constant-moment plate-bending element},
journal = {The Journal of Strain Analysis for Engineering
Design},
year = 1971,
volume = 6,
number = 1,
pages = {20-24},
doi = {10.1243/03093247V061020}
}
""")
Citations().add("Mardal2002", """
@article{Mardal2002,
doi = {10.1137/s0036142901383910},
year = 2002,
volume = {40},
number = {5},
pages = {1605--1631},
author = {Mardal, K.-A.~ and Tai, X.-C.~ and Winther, R.~},
title = {A robust finite element method for {Darcy--Stokes} flow},
journal = {{SIAM} Journal on Numerical Analysis}
}
""")
Citations().add("Arnold2002", """
@article{Arnold2002,
doi = {10.1007/s002110100348},
year = 2002,
volume = {92},
number = {3},
pages = {401--419},
author = {Arnold, R.~N.~ and Winther, R.~},
title = {Mixed finite elements for elasticity},
journal = {Numerische Mathematik}
}
""")
Citations().add("Arnold2003", """
@article{arnold2003,
doi = {10.1142/s0218202503002507},
year = 2003,
volume = {13},
number = {03},
pages = {295--307},
author = {Arnold, D.~N.~ and Winther, R.~},
title = {Nonconforming mixed elements for elasticity},
journal = {Mathematical Models and Methods in Applied Sciences}
}
""")
Citations().add("Arbogast2017", """
@techreport{Arbogast2017,
title={Direct serendipity finite elements on convex quadrilaterals},
author={Arbogast, T and Tao, Z},
year={2017},
institution={Tech. Rep. ICES REPORT 17-28, Institute for Computational Engineering and Sciences}
}
""")
Citations().add("Gopalakrishnan2024", """
@article{gopalakrishnan2024johnson,
title={{The Johnson-Mercier elasticity element in any dimensions}},
author={Gopalakrishnan, J and Guzman, J and Lee, J J},
journal={arXiv preprint arXiv:2403.13189},
year={2024}
}
""")
Citations().add("Groselj2022", """
@article{groselj2022generalized,
title={{Generalized C1 Clough--Tocher splines for CAGD and FEM}},
author={Gro{\v{s}}elj, Jan and Knez, Marjeta},
journal={Computer Methods in Applied Mechanics and Engineering},
volume={395},
pages={114983},
year={2022},
publisher={Elsevier}
}
""")
Citations().add("PowellSabin1977", """
@article{powell1977piecewise,
title={Piecewise quadratic approximations on triangles},
author={Powell, Michael JD and Sabin, Malcolm A},
journal={ACM Transactions on Mathematical Software},
volume={3},
number={4},
pages={316--325},
year={1977},
publisher={ACM New York, NY, USA}
}
""")
except ImportError:
Citations = None
[docs]class NeedsCoordinateMappingElement(metaclass=ABCMeta):
"""Abstract class for elements that require physical information
either to map or construct their basis functions."""
pass
[docs]class PhysicallyMappedElement(NeedsCoordinateMappingElement):
"""A mixin that applies a "physical" transformation to tabulated
basis functions."""
def __init__(self, *args, **kwargs):
super().__init__(*args, **kwargs)
if Citations is not None:
Citations().register("Kirby2018zany")
Citations().register("Kirby2019zany")
[docs] def basis_evaluation(self, order, ps, entity=None, coordinate_mapping=None):
assert coordinate_mapping is not None
M = self.basis_transformation(coordinate_mapping)
def matvec(table):
i, j = gem.indices(2)
value_indices = self.get_value_indices()
table = gem.Indexed(table, (j, ) + value_indices)
val = gem.ComponentTensor(gem.IndexSum(M[i, j]*table, (j,)), (i,) + value_indices)
# Eliminate zeros
return gem.optimise.aggressive_unroll(val)
result = super().basis_evaluation(order, ps, entity=entity)
return {alpha: matvec(table)
for alpha, table in result.items()}
[docs] def point_evaluation(self, order, refcoords, entity=None):
raise NotImplementedError("TODO: not yet thought about it")
[docs]class DirectlyDefinedElement(NeedsCoordinateMappingElement):
"""Base class for directly defined elements such as direct
serendipity that bypass a coordinate mapping."""
pass
[docs]class PhysicalGeometry(metaclass=ABCMeta):
[docs] @abstractmethod
def cell_size(self):
"""The cell size at each vertex.
:returns: A GEM expression for the cell size, shape (nvertex, ).
"""
[docs] @abstractmethod
def jacobian_at(self, point):
"""The jacobian of the physical coordinates at a point.
:arg point: The point in reference space (on the cell) to
evaluate the Jacobian.
:returns: A GEM expression for the Jacobian, shape (gdim, tdim).
"""
[docs] @abstractmethod
def detJ_at(self, point):
"""The determinant of the jacobian of the physical coordinates at a point.
:arg point: The point in reference space to evaluate the Jacobian determinant.
:returns: A GEM expression for the Jacobian determinant.
"""
[docs] @abstractmethod
def reference_normals(self):
"""The (unit) reference cell normals for each facet.
:returns: A GEM expression for the normal to each
facet (numbered according to FIAT conventions), shape
(nfacet, tdim).
"""
[docs] @abstractmethod
def physical_normals(self):
"""The (unit) physical cell normals for each facet.
:returns: A GEM expression for the normal to each
facet (numbered according to FIAT conventions). These are
all computed by a clockwise rotation of the physical
tangents, shape (nfacet, gdim).
"""
[docs] @abstractmethod
def physical_tangents(self):
"""The (unit) physical cell tangents on each facet.
:returns: A GEM expression for the tangent to each
facet (numbered according to FIAT conventions). These
always point from low to high numbered local vertex, shape
(nfacet, gdim).
"""
[docs] @abstractmethod
def physical_edge_lengths(self):
"""The length of each edge of the physical cell.
:returns: A GEM expression for the length of each
edge (numbered according to FIAT conventions), shape
(nfacet, ).
"""
[docs] @abstractmethod
def physical_points(self, point_set, entity=None):
"""Maps reference element points to GEM for the physical coordinates
:arg point_set: A point_set on the reference cell to push forward to physical space.
:arg entity: Reference cell entity on which the point set is
defined (for example if it is a point set on a facet).
:returns: a GEM expression for the physical locations of the
points, shape (gdim, ) with free indices of the point_set.
"""
[docs] @abstractmethod
def physical_vertices(self):
"""Physical locations of the cell vertices.
:returns: a GEM expression for the physical vertices, shape
(gdim, )."""
