Source code for finat.physically_mapped

import gem
from abc import ABCMeta, abstractmethod

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}
}
""")
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] @abstractmethod def basis_transformation(self, coordinate_mapping): """Transformation matrix for the basis functions. :arg coordinate_mapping: Object providing physical geometry.""" pass
[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, )."""