#
##
## SPDX-FileCopyrightText: © 2007-2023 Benedict Verhegghe <bverheg@gmail.com>
## SPDX-License-Identifier: GPL-3.0-or-later
##
## This file is part of pyFormex 3.4 (Thu Nov 16 18:07:39 CET 2023)
## pyFormex is a tool for generating, manipulating and transforming 3D
## geometrical models by sequences of mathematical operations.
## Home page: https://pyformex.org
## Project page: https://savannah.nongnu.org/projects/pyformex/
## Development: https://gitlab.com/bverheg/pyformex
## Distributed under the GNU General Public License version 3 or later.
##
## This program 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.
##
## This program 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 this program. If not, see http://www.gnu.org/licenses/.
##
"""Definition of elements for the :class:`~mesh.Mesh` model
This modules provides local numbering schemes of element
connectivities for :class:`~mesh.Mesh` models. It allows a consistent
numbering throughout pyFormex.
When interfacing with other programs, one should be aware
that conversions may be necessary. Conversions to/from external programs
are done by the interface modules.
The module defines the :class:`ElementType` class and a whole slew of its
instances, which are the element types used in pyFormex.
Here is also the definition of the :class:`Elems` class, which is a
specialisation of the :class:`~connectivity.Connectivity` using an
ElementType instance as the eltype. The Elems class is one of the basic
data holders in the :class:`Mesh` model.
"""
import re
import numpy as np
import pyformex as pf
from pyformex import arraytools as at
from pyformex.coords import Coords
from pyformex.connectivity import Connectivity
[docs]class InvalidElementType(Exception):
"""Raised for invalid element types."""
pass
[docs]class ElementType():
"""Element types for :class:`Mesh` models.
An ElementType instance stores all the data that define a particular
geometrical entity type. It defines the geometrical meaning of a set
of points in the Mesh geometrical model.
Several subclasses are defined to easily created families of related
element types.
All successfully created element types are stored in the class
:attr:`register`, from where they can be looked up by their name.
Parameters
----------
name: str
The name of the element type. Case is ignored. It is stored in the
ElementType instance in capitalized form. The key in the register
is the name in lower case. Often the name has a numeric last
part equal to the plexitude of the element, but this is not a
requirement.
doc: str
A short description of the element.
ndim: int
The dimensionality (:term:`level`) of the element (0..3):
- 0: point
- 1: line
- 2: surface
- 3: volume
degree: tuple of int
For elements that have a parametric formulation, the degree of
the interpolation functions in the ``ndim`` parametric directions.
nplex: int
The plexitude of the element: this is the number of nodes that define
the geometry of an individual element.
rst: float :term:`array_like`
The position of the nodes in parametric space. This array should have
a shape (ndim, nplex).
h: tuple of lambda
A tuple of lambda functions computing the interpolation of the
element. The tuple should contain exactly nplex functions. Each
function should take ``ndim`` arguments and return an array
of nplex values at the nodes. The functions should fulfil the
finite element requirement:
``h[i](rst[:,j]) = 1 if i==j and 0 if i!=j``. This requirement is
checked when creating the element type.
order: int :term:`array_like`, optional
The order in which the nodes are stored, if different from the order
in ``rst`` and ``h``.
vertices: float :term:`array_like`, optionally
The natural coordinates of the nodes (nplex,3). This should be
be provided if, and only if, the element does not have a parametric
formulation (specified by degree, rst, h).
The nodes of the elements are usually defined in a unit space
[0,1] in each axis direction.
edges: :class:`Elems`
A connectivity table listing the edges of the element with local
node numbers. The eltype of the Elems should be an ElementType
instance of level 1.
*This parameter should be provided if and only if the element is
of level 2 or 3.*
The edges should be the conceptually correct edges of the element.
If the element contains edges of different plexitudes, they
should be specified with the highest plexitude and degenerate
elements should be used for the lower plexitude edges.
faces: :class:`Elems`
A connectivity table listing the faces of the element with local
node numbers. The eltype of the Elems should be an ElementType
instance of level 2.
*This parameter should be provided if and only iff the element is
of level 3.*
The faces should be the conceptually correct faces of the element.
If the element contains faces of different plexitudes, they
should be specified with the highest plexitude and degenerate
elements should be used for the lower plexitude faces.
**kargs: keyword arguments
Other keyword arguments are stored in the ElementType as is.
These can also be set after initialization by direct assignment
to the instance's attribute. Below are some predefined values
used by pyFormex.
reverse: int :term:`array_like`
The order of the nodes for the reversed element. Reversing a line
element reverses its parametric direction. Reversing a surface element
reverses the direction of the positive normal. Reversing a volume
element turns the element inside out (which could possibly be used
to represent holes in space).
gl2faces: :class:`Elems`
A connectivity defining the entities to be used in rendering
the element. This should only be provided if it the rendered geometry
should be different from the element itself or from the element
``faces``. This is used to draw approximate renderings of elements
for which there is no correct functionality available: linear
approximations for higher order elements, triangular subdivisions
for quadrilaterals. Note that although the name suggests `faces`,
this connectivity can be of any level.
gl2edges: :class:`Elems`
A connectivity defining the entities to be used in `wire` rendering
of the element. This is like ``gl2faces`` but defines the edges
to be rendered in the wireframe and smoothwire and flatwire
rendering modes.
conversions: dict
A dict holding the possible strategies for conversion of the element
to other element types. The key is the target element name, possibly
extended with an extra string to discriminate between multiple
strategies leading to the same element. The value is a list of actions
to undertake to get from the initial element type to the target
element type. See more details in the section `Element type conversion`
below.
extruded: dict
A dict with data for how to extrude an element to a higher
level element. Extrusion increases the level of the element with one,
by creating 1 or 2 new planes of nodes in a new direction.
The key in the dict is the degree of the extrusion: 1 or 2. The value
is a tuple (eltype, nodes), where eltype is the target element type
(an existing ElementType instance) and nodes is a list of nodes numbers
as they will appear in the new node planes. If nodes is left out of the
tuple, they will be ordered exactly like in the original element.
degenerate: dict
A dict with data for how to replace a degenerate element with
a reduced element. A degenerate element has one or more coinciding
nodes. The reduced elements replaces coinciding nodes with a single
node yielding an element with lower plexitude.
The keys in the dict are reduced element types. The values are lists
of coinciding node conditions and matching reduction schemes: see
the section **Degenerate element reduction** below for more details.
subdivide_proxy: ElementType, optional
An intermediate element type to help in subdividing an element.
If this is defined, subdivision of the element will be done by
first converting to the subdivide_proxy type, subdivide that
element, and convert back to the original.
Notes
-----
The ordering of the ``rst`` and ``h`` defines a fixed local numbering
scheme of the nodes in the element. It can however be changed by specifying
the 'order' argument.
The ordering of the items in ``edges`` and ``faces`` specify the local
numbering of the edges and faces.
For solid elements, it is guaranteed that the vertices of all faces are
numbered in a consecutive order spinning positively around the outward
normal on the face.
Some of the parameters of an ElementType are instances of :class:`Elems`,
but Elems instances contain themselves an :class:`ElementType` instance.
Therefore care should be taken not to define circular dependencies.
If the ElementType instances are created in order of increasing level,
there is no problem with the ``edges`` and ``faces`` parameters, as these
are of a lower level than the element itself, and will have been defined
before. For other attributes however this might not always be the case.
These attributes can then be defined by direct assignment, after the
needed ElementTypes have been initialized.
**List of elements**
The list of available element types can be found from:
>>> ElementType.list()
['point', 'line2', 'line3', 'line4', 'tri3', 'tri6',
'quad4', 'quad6', 'quad9', 'quad16', 'quad8', 'quad12',
'tet4', 'tet10', 'tet14', 'tet15', 'wedge6',
'hex8', 'hex27', 'hex20', 'hex16', 'octa', 'icosa']
>>> ElementType.printall()
Available Element Types:
0-dimensional elements: ['point']
1-dimensional elements: ['line2', 'line3', 'line4']
2-dimensional elements: ['tri3', 'tri6', 'quad4', 'quad6', 'quad9',
'quad16', 'quad8', 'quad12']
3-dimensional elements: ['tet4', 'tet10', 'tet14', 'tet15', 'wedge6',
'hex8', 'hex27', 'hex20', 'hex16', 'octa', 'icosa']
**Element type conversion**
Element type conversion in pyFormex is a powerful feature to transform
Mesh type objects. While mostly used to change the element type, there
are also conversion types that refine the Mesh.
Available conversion methods are defined in an attribute `conversion`
of the input element type. This attribute should be a dictionary, where
the keys are the name of the conversion method and the values describe
what steps need be taken to achieve this conversion. The method name
should be the name of the target element, optionally followed by a suffix
to discriminate between different methods yielding the same target element
type.
The suffix should always start with a '-'. The part starting at the '-' will
be stripped of to set the final target element name.
E.g., a 'line3' element type is a quadratic line element through three
points.
There are two available methods to convert it to 'line2' (straight line
segments betwee two points), named named 'line2', resp. 'line2-2'.
The first will transform a 'line3' element in a single 'line2' between
the two endpoints (i.e. the chord of the quadratic element);
the second will replace each 'line3' with two straight segments: from
first to central node, and from central node to end node.
The values in the dictionary are a list of execution steps to be performed
in the conversion. Each step is a tuple of a single character defining the
type of the step, and the data needed by this type of step. The steps are
executed one by one to go from the source element type to the target.
Currently, the following step types are defined:
============== =============================================
Type Data
============== =============================================
's' (select) connectivity list of selected nodes
'a' (average) list of tuples of nodes to be averaged
'v' (via) string with name of intermediate element type
'x' (execute) the name of a proper conversion function
'r' (random) list of conversion method names
============== =============================================
The operation of these methods is as follows:
- 's' (select): This is the most common conversion type. It selects a set of
nodes of the input element, and creates one or more new elements with these
nodes. The data field is a list of tuples defining for each created element
which node numbers from the source element should be included. This method
is typically used to reduce the plexitude of the element.
- 'a' (average): Creates new nodes, the position of which is computed as
an average of existing nodes. The data field is a list of tuples with
the numbers of the nodes that should be averaged for each new node. The
resulting new nodes are added in order at the end of the existing nodes.
If this order is not the proper local node numbering, an 's' step should
follow to put the (old and new) nodes in the proper order.
This method will usually increase the plexitude of the elements.
- 'v' (via): The conversion is made via an intermediate element type.
The initial element type is first converted to this intermediate type
and the result is then transformed to the target type.
- 'x' (execute): Calls a function to do the conversion. The function takes
the input Mesh as argument and returns the converted Mesh. Currently this
function should be a global function in the mesh module. Its name is
specified as data in the conversion rule.
- 'r' (random): Chooses a random method between a list of alternatives.
The data field is a list of conversion method names defined for the
same element (and thus inside the same dictionary).
While this could be considered an amusement (e.g. used in the Carpetry
example), there are serious applications for this, e.g. when
transforming a Mesh of squares or rectangles into a Mesh of triangles,
by adding one diagonal in each element.
Results with such a Mesh may however be different dependent on the choice
of diagonal. The converted Mesh has directional preferences, not present
in the original. The Quad4 to Tri3 conversion therefore has the choice to
use either 'up' or 'down' diagonals. But a better choice is often the
'random' method, which will put the diagonals in a random direction, thus
reducing the effect.
**Degenerate element reduction**
Each element can have an attribute :attr:`degenerate`, defining some
rules of how the element can be reduced to a lower plexitude element
in case it becomes degenerate. An element is said to be degenerate if
the same node number is used more than once in the connectivity of a
single element.
Reduction of degenerate elements is usually only done if the element
can be reduced to another element of the same level. For example,
if two node of a quadrilateral element are coinciding, it could be
replaced with a triangle. Both are level 2 elements.
When the triangle has two coinciding nodes however, the element is
normally not reduced to a line (level 1), but rather completely removed
from the (level 2) model. However, nothing prohibits cross-level reductions.
The :attr:`degenerate` attribute of an element is a dict where the key
is a target element and the corresponding value is a list of reduction
rules. Each rule is a tuple of two items: a set of conditions and
a node selector to reduce the element.
The conditions items is itself a tuple of any number of conditions, where
each condition is a tuple of two node indices. If these nodes are equal,
the condition is met. If all conditions in a rule are met, the reduction
rule is applied. The second item in a rule, the node selector, is an index
specifying the indices of the nodes that should be retained in the new
(reduced) elements.
As an example, take the Line3 element, which has 3 nodes defining
a curved line element. Node 1 is the middle node, and nodes 0 and 2 are
the end nodes. The element has four ways of being degenerate: nodes 0 and
1 are the same, nodes 1 and 2 are the same, nodes 0 and 2 are the same,
and all three nodes are the same. For the first two of them, a reduction
scheme is defined, reducing the element to a straight line segment (Line2)
between the two end points::
Line3.degenerate = {
Line2: [(((0, 1), ), (0, 2)),
(((1, 2), ), (0, 2)), ],
}
In this case each of the reduction rules contains only a single
condition, but there exist cases where multiple conditions have to
be net at the same time, which is why the condition (0, 1) is itself
enclosed in a tuple.
But what about the other degenerate cases. If both end points coincide,
it is not clear what to do: reduce to a line segment between the
coincident end points, or between an end point and the middle.
Here pyFormex made the choice to not reduce such case and leave the
degenerate Line3 element. But the user could add a rule to e.g.
reduce the case to a line segment between end point and middle point::
Line3.degenerate[Line2].append((((0, 2), ), (0, 1)))
Also the case of three coinciding points is left unhandled. But the user
could reduce such cases to a Point::
Line3.degenerate[Point] = [(((0, 1), (1, 2)), (0, ))]
Here we need two conditions to check that nodes 0, 1 and 2 are equal.
However, in this case the user probably also wants the third degenerate
case (nodes 0 and 2 are equal) to be reduced to a Point. So he could
just use::
Line3.degenerate[Point] = [(((0, 2), ), (0, ))]
Attributes
----------
register: dict
A dict collecting all the created ElementType instances with their
lower case name as key.
default: dict
A dict with the default ElementType for a given plexitude.
Examples
--------
>>> print(ElementType.default)
{1: Point, 2: Line2, 3: Tri3, 4: Quad4, 6: Wedge6, 8: Hex8}
"""
# Register of the created element types
register = {}
default = {}
########
# Proposed changes in the Element class
# =====================================
# - nodal coordinates are specified as follows:
# - in symmetry directions: between -1 and +1, centered on 0
# - in non-symmetry directions: between 0 and +1, aligned on 0
# - getCoords() : return coords as is
# - getAlignedCoords(): return coords between 0 ad 1 and aligned on 0 in all
# directions
def __new__(clas, *args, **kargs):
"""Create a new ElementType instance"""
# This is here to make sure that all 'ElementType' instances have a
# _name, even when they are created by unpickling (which does
# not call __init__). This helps in solving some problems with
# reading back old pickles, which stored the full ElementType
# with all its data. Newer versions now only store the
# name of the ElementType.
# TODO: We should probably convert the ElementType instances
# to singleton classes. That would have avoided these problems,
# as classes are always pickled by name.
instance = super().__new__(clas)
instance._name = "no_name"
return instance
def _set_elems(self, kargs, attr, ndim):
"""Helper function"""
if attr in kargs:
elems = kargs.pop(attr)
if not isinstance(elems, Elems):
raise ValueError(f"'{attr}' should be of type Elems")
setattr(self, attr, elems)
# def _set_array(self, kargs, attr):
# """Helper function"""
# if attr in kargs:
# arr = kargs.pop(attr)
# arr = np.asanyarray(arr)
# setattr(self, attr, arr)
def __init__(self, name, doc, ndim, nplex, register=True,
reverse=None, **kargs):
"""Create a new ElementType"""
lname = name.lower()
name = name.capitalize()
if lname in self.__class__.register:
raise ValueError(
f"Element type {name} already exists")
self._name = name
self.doc = str(doc)
if ndim not in range(4):
raise ValueError(
f"Invalid ndim {ndim}: should be in range(4)")
self.ndim = ndim
if nplex <= 0:
raise ValueError(
"Plexitude (nplex) should be > 0")
self.nplex = int(nplex)
self._set_elems(kargs, 'edges', 1)
self._set_elems(kargs, 'gl2edges', 1)
self._set_elems(kargs, 'faces', 2)
self._set_elems(kargs, 'gl2faces', 2)
self.reverse = np.s_[::-1] if reverse is None else np.asarray(reverse)
if 'rst' in kargs:
self.degree = tuple(kargs.pop('degree'))
rst = kargs.pop('rst')
rst = np.atleast_2d(rst)
if rst.shape != (self.ndim, self.nplex):
raise ValueError(
f"Element {self.name} has incorrect shape for rst: "
f"{rst.shape} but expected ({self.ndim}, {self.nplex})")
self.rst = rst
if 'h' in kargs:
self.h = kargs.pop('h')
if 'fixh' in kargs:
f = kargs.pop('fixh')
f(self)
# TODO: we could just keep natureal order, and put this
# in an 'abq_order' for output to abaqus
# or make it an option what the internal pyFormex order is
if 'order' in kargs:
self.order = kargs.pop('order')
self.rst = self.rst[:, self.order]
if hasattr(self, 'h'):
self.h = [self.h[i] for i in self.order]
if 'els' in kargs:
self._els = kargs.pop('els')
if 'wts' in kargs:
self._wts = kargs.pop('wts')
elif 'vertices' in kargs:
self._vertices = Coords(kargs['vertices'])
# reject these obsolete options
for a in ['drawedges', 'drawedges2', 'drawfaces', 'drawfaces2']:
if a in kargs:
print(f"Ignoring attribute {a} of element {name}")
del kargs[a]
# other args are added as-is
if kargs:
pf.verbose(3, f"REMAINING KARGS: {list(kargs.keys())}")
self.__dict__.update(kargs)
# register the element
if register:
ElementType.register[name.lower()] = self
pf.verbose(3, f"Registered element {self}")
# check the element
self.check()
@property
def name(self):
return self._name
@property
def lname(self):
return self._name.lower()
@property
def r(self):
return self.rst[0]
@property
def s(self):
return self.rst[1]
@property
def t(self):
return self.rst[2]
@property
def vertices(self):
if hasattr(self, '_vertices'):
return self._vertices
return Coords(self.rst.transpose())
@property
def dr(self):
return self.degree[0]
@property
def ds(self):
return self.degree[1]
@property
def dt(self):
return self.degree[2]
def els(self, *args, **kargs):
return self._els(self, *args, **kargs)
def wts(self, *args, **kargs):
return self._wts(self, *args, **kargs)
def nvertices(self):
return self.nplex
nnodes = nvertices
[docs] def nedges(self):
"""Return the number of edges of the element"""
try:
return self.edges.nelems()
except AttributeError:
return 0
[docs] def nfaces(self):
"""Return the number of faces of the element"""
try:
return self.faces.nelems()
except AttributeError:
return 0
[docs] def getEntities(self, level):
"""Return the type and connectivity table of some element entities.
Parameters
----------
level: int
The :term:`level` of the entities to return. If negative,
it is a value relative to the level of the caller. If non-negative,
it specifies the absolute level. Thus, for an ElementType of
level 3, getEntities(-1) returns the faces, while for a
level 2 ElementType, it returns the edges.
In both cases however, getEntities(1) returns the edges.
Returns
-------
:class:`Elems` | :class:`Connectivity`
The connectivity table and element type of the entities of
the specified level. The type is normally Elems.
If the requested entity level is outside the range 0..ndim,
an empty Connectivity is returned.
Examples
--------
>>> Tri3.getEntities(0)
Elems([[0],
[1],
[2]], eltype=Point)
>>> Tri3.getEntities(1)
Elems([[0, 1],
[1, 2],
[2, 0]], eltype=Line2)
>>> Tri3.getEntities(2)
Elems([[0, 1, 2]], eltype=Tri3)
>>> Tri3.getEntities(3)
Connectivity([], shape=(0, 1))
"""
if level < 0:
level = self.ndim + level
if level < 0 or level > self.ndim:
return Connectivity()
if level == 0:
return Elems(np.arange(self.nplex).reshape((-1, 1)), eltype=Point)
elif level == self.ndim:
return Elems(np.arange(self.nplex).reshape((1, -1)), eltype=self)
elif level == 1:
return self.edges
elif level == 2:
return self.faces
[docs] def getPoints(self):
"""Return the level 0 entities"""
return self.getEntities(0)
[docs] def getEdges(self):
"""Return the level 1 entities"""
return self.getEntities(1)
[docs] def getFaces(self):
"""Return the level 2 entities"""
return self.getEntities(2)
[docs] def getCells(self):
"""Return the level 3 entities"""
return self.getEntities(3)
[docs] def getElement(self):
"""Return the element connectivity: the entity of level self.ndim"""
return self.getEntities(self.ndim)
[docs] def getDrawFaces(self, quadratic=False):
"""Return the local connectivity for drawing the element's faces"""
if not hasattr(self, 'gl2faces'):
self.gl2faces = self.getFaces() # .reduceDegenerate()
return self.gl2faces
[docs] def getDrawEdges(self, quadratic=False):
"""Return the local connectivity for drawing the element's edges"""
if not hasattr(self, 'gl2edges'):
self.gl2edges = self.getEdges() # .reduceDegenerate()
return self.gl2edges
[docs] def toMesh(self, level=None):
"""Convert the element type to a Mesh.
Returns a Mesh with a single element of natural size.
"""
# import here to avoid circular import
from pyformex.mesh import Mesh
if level is None:
level = self.ndim
return Mesh(self.vertices, self.getEntities(level))
[docs] def family(self):
"""Return the element family name.
The element family name is the first part of the lname
that consists only of lower case letters.
"""
m = re.match("[a-z]*", self.lname)
return m[0] if m else ''
def __str__(self):
return self._name
def __repr__(self):
return self._name
def report(self):
return (f"ElementType {self._name}: "
f"ndim={self.ndim}, nplex={self.nplex}, "
f"nedges={self.nedges()}, nfaces={self.nfaces()}")
def isLinear(self):
return getattr(self, 'linear', None) == self.lname
def H(self, rst):
if self.ndim == 0 or not hasattr(self,'h'):
return None
elif self.ndim == 1:
return np.stack([h(rst[0]) for h in self.h])
elif self.ndim == 2:
return np.stack([h(rst[0], rst[1]) for h in self.h])
elif self.ndim == 3:
return np.stack([h(rst[0], rst[1], rst[2]) for h in self.h])
def check(self):
if hasattr(self, 'h'):
H = self.H(self.rst)
E = np.eye(self.nplex)
ok = H.shape == E.shape and np.allclose(H, E)
pf.verbose(3, f"Element {self} is {'not ' if not ok else ''}ok")
if not ok:
print(self.rst)
print(H)
raise ValueError("Some element not ok")
[docs] @classmethod
def list(clas, ndim=None, types=False):
"""Return a list of available ElementTypes.
Parameters
----------
ndim: int, optional
If provided, only return the elements of the specified level.
types: bool, optional
If True, return ElementType instances. The default is to return
element names.
Returns
-------
list
A list of ElementType names (default) or instances.
Examples
--------
>>> ElementType.list()
['point', 'line2', 'line3', 'line4', 'tri3', 'tri6',
'quad4', 'quad6', 'quad9', 'quad16', 'quad8', 'quad12',
'tet4', 'tet10', 'tet14', 'tet15', 'wedge6',
'hex8', 'hex27', 'hex20', 'hex16', 'octa', 'icosa']
>>> ElementType.list(ndim=1)
['line2', 'line3', 'line4']
>>> ElementType.list(ndim=1, types=True)
[Line2, Line3, Line4]
"""
eltypes = list(clas.register.values())
if ndim is not None:
eltypes = [e for e in eltypes if e.ndim == ndim]
if types:
return eltypes
else:
return [e.lname for e in eltypes]
[docs] @classmethod
def printall(clas, lower=False):
"""Print all available element types.
Prints a list of the names of all available element types,
grouped by their dimensionality.
"""
print("Available Element Types:")
for ndim in range(4):
print(f" {ndim}-dimensional elements: {clas.list(ndim)}")
[docs] @classmethod
def has(clas, elname):
"""Check that an ElementType with the given name exists.
Parameters
----------
elname: str
The element name (case insensitive).
Returns
-------
bool
True if the specified ElementType has been registered.
"""
return elname.lower() in clas.register
[docs] @classmethod
def drop(clas, elname):
"""Remove an ElementType from the registry.
Parameters
----------
elname: str
The element name (case insensitive).
"""
elname = elname.lower()
if elname in clas.register:
del clas.register[elname]
[docs] @classmethod
def get(clas, eltype=None, nplex=None, register=True):
"""Find the ElementType matching an element name and/or plexitude.
Parameters
----------
eltype: :class:`ElementType` | str | None
The element type or name. If not provided and ``nplex`` is provided,
the default element type for that plexitude will be used, if it exists.
If a name, it should be the name of one of the existing ElementType
instances (case insensitive).
nplex: int
The :term:`plexitude` of the element. If provided and an eltype was
provided, it should match the eltype plexitude. If no eltype was
provided, the default element type for this plexitude is returned.
Returns
-------
:class:`ElementType`
The ElementType matching the provided eltype and/or nplex
Raises
------
ValueError
If neither ``name`` nor ``nplex`` can resolve into an element type.
Examples
--------
>>> ElementType.get('tri3')
Tri3
>>> ElementType.get(nplex=2).lname
'line2'
>>> ElementType.get('QUAD4')
Quad4
>>> ElementType.get('poly5', register=False)
Poly5
>>> ElementType.get('plex17')
Plex17
"""
if isinstance(eltype, clas):
return eltype
if isinstance(eltype, str):
eltype = eltype.lower()
if eltype in clas.register:
return clas.register[eltype]
elif eltype == 'polygon' and nplex is not None:
return clas.polygon(nplex, register=register)
else:
m = re.match(r"(poly|plex)(\d+)$", eltype)
if m:
# Create an adhoc Poly... or fake Plex... element
name = m.groups()[0]
nplex = int(m.groups()[1])
if name == 'poly':
return clas.polygon(nplex, register=register)
else:
return clas(eltype, 'A catch-all dummy element', ndim=0,
nplex=nplex, register=False)
if eltype is None:
if nplex in clas.default:
return clas.default[nplex]
raise InvalidElementType(f"No such element type: {eltype} {nplex}")
[docs] @classmethod
def polygon(clas, nplex, register=True):
"""Return the Polygon element of the specified plexitude.
Parameters
----------
nplex: int
The plexitude of the element, i.e. the number of vertices
of the polygons. Clearly, nplex should be >= 3.
Returns
-------
ElementType
An ElementType representing a polygon of the specified
plexitude. If nplex < 3, an exception is raised.
For nplex=3 and nplex=4, the returned element type is
:class:`Tri3` and :class:`Quad4`, respectively. For higher
plexitude, an element of the :class:`Polygon` family is
returned, with name Poly## where ## is the plexitude.
Notes
-----
Elements of the :class:`Polygon` family are created on the fly
when needed, and stored in the element database for later reuse
if a polygon element of the same plexitude would be required.
Examples
--------
>>> [ElementType.polygon(i, register=False)
... for i in (3, 4, 5, 6, 8, 137)]
[Tri3, Quad4, Poly5, Poly6, Poly8, Poly137]
>>> ElementType.polygon(2)
Traceback (most recent call last):
...
ValueError: nplex=2 is invalid for Polygon elements
"""
if nplex < 3:
raise ValueError(f"nplex={nplex} is invalid for Polygon elements")
elif nplex==3:
return Tri3
elif nplex==4:
return Quad4
else:
name = f"Poly{nplex}"
if ElementType.has(name):
poly = ElementType.get(name)
else:
poly = Polygon(nplex, register=register)
return poly
#########################################################################
## Elems
############
[docs]class Elems(Connectivity):
"""A Connectivity where the eltype is an ElementType subclass.
This is a :class:`Connectivity` with knowledge about the
geometrical meaning of the element type. It is used to store the
connectivity table of a :class:`Mesh` instance.
It is also used to store some subitems in the :class:`ElementType`.
Parameters
----------
data: int :term:`array_like`
The connectivity data to initialize the Elems. The data should be
2-dim with shape ``(nelems,nplex)``, where ``nelems`` is the number
of elements and ``nplex`` is the plexitude of the elements.
It can be a list, an ndarray or a Connectivity. In the latter case
it usually shares the storage with that object.
eltype: class | str
The element type of the Elems. It is either a subclass of
:class:`elements.ElementType` or the :meth:`name` of such a subclass.
That subclass should have the same plexitude as the connectivity
data. If not provided and the default element type for that plexitude
will be used if such exists in :attr:`Elementtype.default`.
Examples
--------
>>> L = [[0,1,2],[0,1,3]]
>>> Elems(L)
Elems([[0, 1, 2],
[0, 1, 3]], eltype=Tri3)
>>> Elems(L, 'tri3')
Elems([[0, 1, 2],
[0, 1, 3]], eltype=Tri3)
>>> Elems(L, Tri3)
Elems([[0, 1, 2],
[0, 1, 3]], eltype=Tri3)
>>> C = Connectivity(L, eltype='tri3')
>>> Elems(C)
Elems([[0, 1, 2],
[0, 1, 3]], eltype=Tri3)
>>> Elems(C, 'tri3')
Elems([[0, 1, 2],
[0, 1, 3]], eltype=Tri3)
>>> Elems(C, Tri3)
Elems([[0, 1, 2],
[0, 1, 3]], eltype=Tri3)
"""
def __new__(self, data, eltype=None):
"""Create a new Elems"""
if not isinstance(data, Connectivity):
data = Connectivity(data=data, eltype=eltype)
if not isinstance(data.eltype, ElementType):
data.eltype = ElementType.get(data.eltype, data.nplex())
if eltype is not None:
# if eltype == 'polygon':
# eltype = ElementType.polygon(data.nplex())
# else:
eltype = ElementType.get(eltype, data.nplex())
if data.nplex() != eltype.nplex:
if data.nelems() == 0:
# Set correct plexitude for empty Elems
data = data.reshape(0, eltype.nplex)
else:
raise InvalidElementType(
f"Data plexitude ({data.nplex()}) != "
f"eltype plexitude ({eltype.nplex})")
if data.eltype != eltype:
data.eltype = eltype
return data.view(self)
def __repr__(self):
"""String representation of an Elems"""
res = np.ndarray.__repr__(self)
res = re.sub("[)]$", f", eltype={self.eltype._name})", res)
return res
def __setstate__(self, state):
"""Allow an Elems to be properly unpickled"""
# An Elems is pickled like a Connectivity, with the
# ElementType's name as eltype.
# After unpickling, the eltype should be converted
# to an ElementType.
Connectivity.__setstate__(self, state)
if isinstance(self.eltype, str):
self.eltype = ElementType.get(self.eltype)
else:
# This is a FIX for old pickles that stored full ElementType data
name = self.eltype.__dict__.get('_name', None)
if name == 'no_name':
name = None
self.eltype = ElementType.get(name, self.shape[1])
[docs] def levelSelector(self, level):
"""Return a selector for lower level entities.
Parameters
----------
level: int
An specifying one of the hierarchical levels of element
entities. See :meth:`Element.getEntities`.
Returns
-------
:class:`Elems`
A new Elems object with shape
``(self.nelems*selector.nelems,selector.nplex)``.
Examples
--------
>>> E = Elems([[0,1,2],[0,1,3],[0,5,3]],'tri3')
>>> E.levelSelector(1)
Elems([[0, 1],
[1, 2],
[2, 0]], eltype=Line2)
"""
sel = self.eltype.getEntities(level)
return Elems(sel, sel.eltype)
[docs] def selectNodes(self, selector):
"""Return a Connectivity with subsets of the nodes.
Parameters
----------
selector: int | int :term:`array_like`
A single int specifies a relative or absolute hierarchical level
of element entities (See the Element class). A 2-dim int array
selector is then constructed automatically from
``self.eltype.getEntities(selector)``.
Else, it is a 2-dim int array like (often a
:class:`~connectivity.Connectivity` or another :class:`Elems`.
Each row of `selector` holds a list of the local node numbers that
should be retained in the new Connectivity table.
As an example, if the Elems is plex-3 representing triangles,
a selector [[0,1],[1,2],[2,0]] would extract the edges of the
triangle. The same would be obtained with ``selector=-1`` or
``selector=1``.
Returns
-------
:class:`Elems` | :class:`~connectivity.Connectivity`
An Elems or Connectivity object with eltype equal to that of the
selector. This means that if the selector has an eltype that is
one of the elements defined in :mod:`elements`, the return type
will be Elems, else Connectivity.
The shape is ``(self.nelems*selector.nelems,selector.nplex)``.
Duplicate elements created by the selector are retained.
Examples
--------
>>> E = Elems([[0,1,2],[0,1,3],[0,5,3]])
>>> sel = E.levelSelector(1)
>>> E.selectNodes(sel)
Elems([[0, 1],
[1, 2],
[2, 0],
[0, 1],
[1, 3],
[3, 0],
[0, 5],
[5, 3],
[3, 0]], eltype=Line2)
>>> E.selectNodes(Elems([[0,1],[0,2]],eltype=Line2))
Elems([[0, 1],
[0, 2],
[0, 1],
[0, 3],
[0, 5],
[0, 3]], eltype=Line2)
>>> E.selectNodes([[0,1],[0,2]])
Connectivity([[0, 1],
[0, 2],
[0, 1],
[0, 3],
[0, 5],
[0, 3]])
"""
if at.isInt(selector):
selector = self.levelSelector(selector)
else:
selector = Connectivity(selector)
lo = Connectivity.selectNodes(self, selector)
clas = Elems if lo.eltype is not None else Connectivity
return clas(lo, eltype=lo.eltype)
[docs] def insertLevel(self, selector, permutations='all'):
"""Insert an extra hierarchical level in a Connectivity table.
A Connectivity table identifies higher hierarchical entities in
function of lower ones. This method inserts an extra level in the
hierarchy.
For example, if you have volumes defined in function of points,
you can insert an intermediate level of edges, or faces.
Each element may generate multiple instances of the intermediate level.
Parameters
----------
selector: int | int :term:`array_like`
A single int specifies a relative or absolute hierarchical level
of element entities (See the Element class). A 2-dim int array
selector is then constructed automatically from
``self.eltype.getEntities(selector)``.
Else, it is a 2-dim int array like (often a
:class:`~connectivity.Connectivity` or another :class:`Elems`.
Each row of `selector` holds a list of the local node numbers that
should be retained in the new Connectivity table.
permutations: str
Defines which permutations of the row data are allowed while still
considering the rows equal. Equal rows in the intermediate level
are collapsed into single items. Possible values are:
- 'none': no permutations are allowed: rows must match the same date
at the same positions.
- 'roll': rolling is allowed. Rows that can be transformed into
each other by rolling are considered equal;
- 'all': any permutation of the same data will be considered an
equal row. This is the default.
Returns
-------
hi: :class:`~connectivity.Connectivity`
A Connecivity defining the original elements
in function of the intermediate level ones.
lo: :class:`Elems` | :class:`~connectivity.Connectivity`
An Elems or Connectivity object with eltype equal to that of the
selector. This means that if the selector has an eltype that is
one of the elements defined in :mod:`elements`, the return type
will be Elems, else Connectivity.
The resulting node numbering of the created intermediate entities
(the `lo` return value) respects the numbering order of the original
elements and the applied the selector, but in case of collapsing
duplicate rows, it is undefined which of the collapsed sequences is
returned.
Because the precise order of the data in the collapsed rows is lost,
it is in general not possible to restore the exact original table
from the two result tables.
See however :meth:`mesh.Mesh.getBorder` for an application where an
inverse operation is possible, because the border only contains
unique rows.
See also :meth:`mesh.Mesh.combine`, which is an almost inverse operation
for the general case, if the selector is complete.
The resulting rows may however be permutations of the original.
Examples
--------
>>> E = Elems([[0,1,2],[0,1,3],[0,5,3]], 'tri3')
>>> hi, lo = E.insertLevel(1)
>>> hi
Connectivity([[0, 4, 1],
[0, 5, 2],
[3, 6, 2]])
>>> lo
Elems([[0, 1],
[2, 0],
[3, 0],
[0, 5],
[1, 2],
[1, 3],
[5, 3]], eltype=Line2)
>>> hi, lo = E.insertLevel([[0,1],[1,2],[2,0]])
>>> hi
Connectivity([[0, 4, 1],
[0, 5, 2],
[3, 6, 2]])
>>> lo
Connectivity([[0, 1],
[2, 0],
[3, 0],
[0, 5],
[1, 2],
[1, 3],
[5, 3]])
>>> E = Elems([[0,1,2,3]],'quad4')
>>> hi, lo = E.insertLevel([[0,1,2],[1,2,3],[0,1,1],[0,0,1],[1,0,0]])
>>> hi
Connectivity([[1, 2, 0, 0, 0]])
>>> lo
Connectivity([[0, 1, 1],
[0, 1, 2],
[1, 2, 3]])
"""
if at.isInt(selector):
selector = self.levelSelector(selector)
else:
selector = Connectivity(selector)
hi, lo = Connectivity.insertLevel(self, selector,
permutations=permutations)
lo.eltype = selector.eltype
return hi, lo
[docs] def reduceDegenerate(self, target=None, return_indices=False):
"""Reduce degenerate elements to lower plexitude elements.
This will try to reduce the degenerate elements of the Elems
to lower plexitude elements. This is only possible if the ElementType
has an attribute :attr:`degenerate` containing the proper reduction
rules.
Parameters
----------
target: str, optional
Target element name. If provided, only reductions to that element
type are performed. Else, all the target element types for which
a reduction scheme is available, will be tried.
return_indices: bool, optional
If True, also returns the indices of the elements in the
input Elems that are contained in each of the parts returned.
Returns
-------
conn: list of Elems instances
A list of Elems of which the first one contains the originally
non-degenerate elements, the next one(s) contain the reduced
elements (per reduced element type) and the last one contains
elements that could not be reduced (this may be absent).
In the following cases a list with only the original is returned:
- there are no degenerate elements in the Elems;
- the ElementType does not have any reduction scheme defined;
- the ElementType does not have a reduction scheme fot the target.
ind: list of indices, optional
Only returned if ``return_indices`` is True: the indices of the
elements of ``self`` contained in each item in ``conn``.
Note
----
If the Elems is part of a :class:`~mesh.Mesh`, you should
use the :meth:`mesh.Mesh.splitDegenerate` method instead, as that
will preserve the property numbers in the resulting Meshes.
The returned reduced Elems may still be degenerate for
their own element type.
See Also
--------
mesh.Mesh.splitDegenerate: split mesh in non-degenerate and reduced
Examples
--------
>>> Elems([[0,1,2],[0,1,3]],eltype=Line3).reduceDegenerate()
[Elems([[0, 1, 2],
[0, 1, 3]], eltype=Line3)]
>>> C = Elems([[0,1,2],[0,1,1],[0,3,2]],eltype='line3')
>>> C.reduceDegenerate()
[Elems([[0, 1, 2],
[0, 3, 2]], eltype=Line3), \
Elems([[0, 1]], eltype=Line2)]
>>> C = Elems([[0,1,2],[0,1,1],[0,3,2],[1,1,1],[0,0,2]],eltype=Line3)
>>> C.reduceDegenerate()
[Elems([[0, 1, 2],
[0, 3, 2]], eltype=Line3), \
Elems([[1, 1],
[0, 2],
[0, 1]], eltype=Line2)]
>>> conn, ind = C.reduceDegenerate(return_indices=True)
>>> conn
[Elems([[0, 1, 2],
[0, 3, 2]], eltype=Line3),
Elems([[1, 1],
[0, 2],
[0, 1]], eltype=Line2)]
>>> ind
[array([0, 2]), array([3, 4, 1])]
"""
if not hasattr(self.eltype, 'degenerate'):
# Can not reduce
return [self]
degen = self.testDegenerate()
if not degen.any():
# Nothing to reduce
return [self]
# get all reductions for this eltype
strategies = self.eltype.degenerate
# if target, keep only those leading to target
if target is not None:
target = ElementType.get(target)
s = strategies.get(target, [])
if s:
strategies = {target: s}
else:
strategies = {}
if not strategies:
# No matching strategies
return [self]
# non-degenerate: keep
conn = [self[~degen]]
ind = [np.where(~degen)[0]]
# degenerate: reduce
e = self[degen]
i = np.where(degen)[0]
for totype in strategies:
ids = np.array([], dtype=at.Int)
elems = []
for conditions, selector in strategies[totype]:
cond = np.array(conditions)
try:
w = (e[:, cond[:, 0]] == e[:, cond[:, 1]]).all(axis=1)
sel = np.where(w)[0]
except Exception:
raise ValueError(
f"Invalid data in {self.eltype}.degenerate:\n"
f"{totype} = {(conditions, selector)}")
sel = []
if len(sel) > 0:
elems.append(e[sel][:, selector])
ids = np.concatenate([ids, i[sel]])
# remove the reduced elems from e
e = e[~w]
i = i[~w]
if e.nelems() == 0:
break
if elems:
elems = np.concatenate(elems)
conn.append(Elems(elems, eltype=totype))
ind.append(ids)
if e.nelems() == 0:
break
if e.nelems() > 0:
conn.append(e)
ind.append(ids)
if return_indices:
return conn, ind
else:
return conn
[docs] def extrude(self, nnod, degree):
"""Extrude an Elems to a higher level Elems.
Parameters
----------
nnod: int
Node increment for each new node plane. It should be
higher than the highest node number in self.
degree: int
Number of node planes to add. This is also the degree of
the extrusion. Currently it is limited to 1 or 2.
The extrusion adds ``degree`` planes of nodes, each with a node
increment ``nnod``, to the original Elems and then selects
the target nodes from it as defined by the
``self.eltype.extruded[degree]`` value.
Returns
-------
Elems
An Elems for the extruded element.
Examples
--------
>>> a = Elems([[0,1],[1,2]],eltype=Line2).extrude(3,1)
>>> print(a)
[[0 1 4 3]
[1 2 5 4]]
>>> print(a.eltype.lname)
quad4
>>> a = Elems([[0,1,2],[0,2,3]],eltype=Line3).extrude(4,2)
>>> print(a)
[[ 0 2 10 8 1 6 9 4 5]
[ 0 3 11 8 2 7 10 4 6]]
>>> print(a.eltype.lname)
quad9
"""
try:
eltype, reorder = self.eltype.extruded[degree]
except Exception:
try:
eltype = self.eltype.lname
except Exception:
eltype = None
raise ValueError("I don't know how to extrude an Elems"
"of eltype '{eltype}' in degree {degree}")
# create hypermesh Elems
elems = np.concatenate([self+i*nnod for i in range(degree+1)], axis=-1)
# Reorder nodes if necessary
if len(reorder) > 0:
elems = elems[:, reorder]
return Elems(elems, eltype=eltype)
[docs] def getFreeEntities(self, level=-1, return_indices=False):
"""Return the free entities of the specified level.
Parameters
----------
level: int
The :term:`level` of the entities to return. If negative,
it is a value relative to the level of the caller. If non-negative,
it specifies the absolute level.
return_indices: bool
If True, also returns an index array (nentities,2) for inverse
lookup of the higher entity (column 0) and its local
lower entity number (column 1).
Returns
-------
:class:`~elements.Elems`
A connectivity table with the free entities of the
specified level. Free entities are entities
that are only connected to a single element.
See Also
--------
:meth:`Mesh.getFreeEntities`: return the free entities of a Mesh
:meth:`Mesh.getBorder`: return the border entities of a Mesh
Examples
--------
>>> elems = Elems([[0, 1, 3], [3, 2, 0]], eltype='tri3')
>>> elems.getFreeEntities(1)
Elems([[0, 1],
[2, 0],
[1, 3],
[3, 2]], eltype=Line2)
>>> elems.getFreeEntities(1,True)[1]
array([[0, 0],
[1, 1],
[0, 1],
[1, 0]])
"""
hi, lo = self.insertLevel(level)
if hi.size == 0:
if return_indices:
return Connectivity(), []
else:
return Connectivity()
hiinv = hi.inverse(expand=True)
ncon = (hiinv>=0).sum(axis=1)
isbrd = (ncon<=1) # < 1 should not occur
brd = lo[isbrd]
if not return_indices:
return brd
# also return indices where the border elements come from
binv = hiinv[isbrd]
enr = binv[binv >= 0] # element number
a = hi[enr]
b = np.arange(lo.shape[0])[isbrd].reshape(-1, 1)
fnr = np.where(a==b)[1] # local border part number
return brd, np.column_stack([enr, fnr])
#######################################################################
# The collection of default pyFormex elements
#############################################
# A constant we use a lot
e3 = 1./3.
f3 = 2*e3
##########################################
#### Interpolation functions
# 1d, 2d 3d lagrange over a range (0,1)
# 1-dim lagrange formulated directly for degree 1,2,3
lagrange = {
1: [
lambda r: 1.-r,
lambda r: r,
],
2: [
lambda r: 2*(0.5-r)*(1.-r),
lambda r: 4*r*(1.-r),
lambda r: -2*r*(0.5-r),
],
3: [
lambda r: 9/2*(e3-r)*(f3-r)*(1.-r),
lambda r: 27/2*r*(f3-r)*(1.-r),
lambda r: -27/2*r*(e3-r)*(1.-r),
lambda r: 9/2*r*(e3-r)*(f3-r),
]
}
# 2-dim lagrange function factory
def lagrange2(dr, ds, i, j):
return lambda r,s: lagrange[dr][i](r) * lagrange[ds][j](s)
# 3-dim lagrange function factory
def lagrange3(dr, ds, dt, i, j, k):
return lambda r,s,t: lagrange[dr][i](r) * lagrange[ds][j](s) * lagrange[dt][k](t)
def seeds(div, degree):
if at.isInt(div):
r = np.linspace(0., 1., degree*div+1)
else:
r = np.asarray(div)
if degree > 1:
a = np.linspace(0., 1., degree+1)[:-1]
b = [(1.-a)*b + a*c for b,c in zip(r[:-1], r[1:])]
b.append([1.])
r = np.concatenate(b)
return r
##########
# ndim = 0
##########
Point = ElementType(
'point', "A single point",
ndim = 0,
nplex = 1,
vertices = [[0., 0., 0.]]
)
##########
# ndim = 1
##########
class Lagrange1D(ElementType):
def __init__(self, deg, **kargs):
ndim = 1
nplex = deg+1
degree = (deg,)
name = f"Line{nplex}"
doc = f"A {nplex}-node line element (degree {degree})"
rst = np.linspace(0., 1., nplex)
h = lagrange[deg]
super().__init__(name=name, doc=doc, ndim=ndim, degree=degree,
nplex=nplex, rst=rst, h=h, **kargs)
def els(self, nr):
return np.row_stack([(np.arange(self.nplex) + i*self.dr)
for i in range(nr)])
def wts(self, rdiv):
r = seeds(rdiv, self.dr)
H = np.stack([h(r) for h in self.h])
return H.reshape((len(self.h), -1)).transpose()
Line2 = Lagrange1D(1)
Line3 = Lagrange1D(2)
Line4 = Lagrange1D(3)
Line3.gl2faces = Elems([(0, 1), (1, 2)], Line2)
Line4.gl2faces = Elems([(0, 1), (1, 2), (2, 3)], Line2)
##########
# ndim = 2
##########
def tri3_els(self, nr, ns=None):
# ns is not used: always equal to nr
n = nr+1
els1 = [np.row_stack([np.array([0, 1, n-j])
+ i for i in range(nr-j)])
+ j * n - j*(j-1)//2 for j in range(nr)]
els2 = [np.row_stack([np.array([1, 1+n-j, n-j])
+ i for i in range(nr-j-1)])
+ j * n - j*(j-1)//2 for j in range(nr-1)]
elems = np.row_stack(els1+els2)
return elems
# New tri3_wts: accepts unequal divisions.
def tri3_wts(self, rdiv, sdiv=None):
r = seeds(rdiv, self.dr)
nr = len(r)
if sdiv is None:
s, ns = r, nr
else:
s = seeds(sdiv, self.ds)
ns = len(s)
if nr != ns:
raise ValueError("Tri3: number of divisions along r and s"
" directions should be equal")
ij = np.row_stack([(i,j) for j in range(ns) for i in range(nr-j)])
rst = np.column_stack(
[(r[i], s[j]) for j in range(ns) for i in range(nr-j)])
# Fix for unequal divisions
for m in range(ij.shape[0]):
i, j = ij[m]
if i > 0 and j > 0:
l = i+j
a, b = rst[:,m]
d, e = r[l], s[l]
p = 1./(a/d + b/e)
rst[:,m] *= p
return np.column_stack([1.-rst[0]-rst[1], rst[0], rst[1]])
def tri6_els(self, nr, ns=None):
# ns is not used: always equal to nr
n5 = 2*nr+1
n9 = 4*nr+1
els1 = [
np.row_stack([
np.array([0, 2, n9-4*j, 1, n5+1-2*j, n5-2*j])
+ 2*i for i in range(nr-j)])
+ j * n9 - 2*j*(j-1) for j in range(nr)]
els2 = [
np.row_stack([
np.array([2, n9+2-4*j, n9-4*j, n5+2-2*j, n9+1-4*j, n5+1-2*j])
+ 2*i for i in range(nr-j-1)]) +
+ j * n9 - 2*j*(j-1) for j in range(nr-1)]
elems = np.row_stack(els1+els2)
return elems
def tri6_wts(self, rdiv, sdiv=None):
r = seeds(rdiv, self.dr)
nr = len(r)
if sdiv is None:
s, ns = r, nr
else:
s = seeds(sdiv, self.ds)
ns = len(s)
if nr != ns:
raise ValueError("Tri3: number of divisions along r and s"
" directions should be equal")
ij = np.row_stack([(i,j) for j in range(ns) for i in range(nr-j)])
rst = np.column_stack(
[(r[i], s[j]) for j in range(ns) for i in range(nr-j)])
# Fix for unequal divisions
for m in range(ij.shape[0]):
i, j = ij[m]
if i > 0 and j > 0:
l = i+j
a, b = rst[:,m]
d, e = r[l], s[l]
p = 1./(a/d + b/e)
rst[:,m] *= p
r, s = rst
# r, s = np.meshgrid(r, s)
H = np.stack([h(r,s) for h in self.h])
nr = len(r)
H = H.reshape((len(self.h), -1)).transpose()
H = np.concatenate([H[nr*j:nr*j+nr-j] for j in range(nr)])
return H
Tri3 = ElementType(
'tri3', "A 3-node triangle",
ndim = 2,
nplex = 3,
degree = (1,1),
edges = Elems([(0, 1), (1, 2), (2, 0)], Line2),
els = tri3_els,
wts = tri3_wts,
rst = [[0., 1., 0.],
[0., 0., 1.]],
h = [
lambda r, s: 1.-r-s,
lambda r, s: r,
lambda r, s: s,
],
)
Tri6 = ElementType(
'tri6', "A 6-node triangle",
ndim = 2,
nplex = 6,
degree = (2,2),
edges = Elems([(0, 3, 1), (1, 4, 2), (2, 5, 0)], Line3),
reverse = (2, 1, 0, 4, 3, 5),
gl2faces = Elems([(0, 3, 5), (3, 1, 4), (4, 2, 5), (3, 4, 5)], Tri3),
gl2edges = Elems([(0, 3), (3, 1), (1, 4), (4, 2), (2, 5), (5, 0)], Line2),
els = tri6_els,
wts = tri6_wts,
rst = [[0., 1., 0., 0.5, 0.5, 0.0],
[0., 0., 1., 0.0, 0.5, 0.5]],
h = [
lambda r, s: (1.-r-s)*(1.-2*r-2*s),
lambda r, s: r*(2*r-1.),
lambda r, s: s*(2*s-1.),
lambda r, s: 4*(1.-r-s)*r,
lambda r, s: 4*r*s,
lambda r, s: 4*(1.-r-s)*s,
],
)
##############
# Family: quad
##
## Lagrange ##
class Lagrange2D(ElementType):
def __init__(self, dr, ds, name=None, **kargs):
ndim = 2
degree = (dr, ds)
nplex = (dr+1)*(ds+1)
if name is None:
if dr == ds:
name = f"Quad{nplex}"
else:
name = f"Lag2D_{dr}_{ds}"
doc = f"Lagrange quadrilateral, degree {degree}, nplex {nplex}"
r = np.linspace(0, 1, dr+1)
s = np.linspace(0, 1, ds+1)
r, s = np.meshgrid(r, s)
rst = np.row_stack([r.ravel(), s.ravel()])
h =[lagrange2(dr, ds, i, j) for j in range(ds+1) for i in range(dr+1)]
super().__init__(name=name, doc=doc, ndim=ndim, degree=degree,
nplex=nplex, rst=rst, h=h, **kargs)
def els(self, nr, ns=None):
if ns is None:
ns = nr
dr, ds = self.degree
ntr = dr*nr + 1
nts = ds*ns + 1
n = ntr*nts
grid = np.arange(n).reshape(nts,ntr)
stack = []
dr1, ds1 = dr+1, ds+1
for j in range(ns):
jds = j*ds
for i in range(nr):
idr = i*dr
elij = grid[jds:jds+ds1, idr:idr+dr1]
elij = elij.flat
if hasattr(self, 'order'):
elij = elij[self.order]
stack.append(elij)
return np.row_stack(stack)
def wts(self, rdiv, sdiv=None):
r = seeds(rdiv, self.dr)
s = r if sdiv is None else seeds(sdiv, self.ds)
r, s = np.meshgrid(r, s)
H = np.stack([h(r,s) for h in self.h])
return H.reshape((len(self.h), -1)).transpose()
Quad4 = Lagrange2D(
1, 1,
order = [0, 1, 3, 2],
edges = Elems([(0, 1), (1, 2), (2, 3), (3, 0)], Line2),
)
Quad6 = Lagrange2D(
2, 1,
name = 'Quad6',
order = [0, 2, 5, 3, 1, 4],
reverse = [1, 0, 3, 2, 4, 5],
edges = Elems([(0, 4, 1), (1, 1, 2), (2, 5, 3), (3, 3, 0)], Line3),
gl2edges = Elems([(0, 4), (4, 1), (1, 2), (2, 5), (5, 3), (3, 0)], Line2),
gl2faces = Elems([(0, 4, 5, 3), (4, 1, 2, 5)], Quad4),
)
Quad9 = Lagrange2D(
2, 2,
order = [0, 2, 8, 6, 1, 5, 7, 3, 4],
reverse = (3, 2, 1, 0, 6, 5, 4, 7, 8),
edges = Elems([(0, 4, 1), (1, 5, 2), (2, 6, 3), (3, 7, 0)], Line3),
gl2edges = Elems([(0, 4), (4, 1), (1, 5), (5, 2), (2, 6), (6, 3),
(3, 7), (7, 0)], Line2),
gl2faces = Elems([(0, 4, 8, 7), (1, 5, 8, 4), (2, 6, 8, 5),
(3, 7, 8, 6), ], Quad4),
)
Quad16 = Lagrange2D(
3, 3,
reverse = (3, 2, 1, 0, 7, 6, 5, 4, 11, 10, 9, 8, 15, 14, 13, 12),
edges = Elems([(0, 1, 2, 3), (3, 7, 11, 15), (15, 14, 13, 12),
(12, 8, 4, 0)], Line4),
gl2edges = Elems([(0, 1), (1, 2), (2, 3), (3, 7), (7, 11), (11, 15),
(15, 14), (14, 13), (13, 12), (12, 8), (8, 4), (4, 0)],
Line2),
gl2faces = Elems(Quad4.els(3)[:,[(0,1,2),(2,3,0)]].reshape(-1,3), Tri3),
)
## Serendipity ##
Quad8 = ElementType(
'quad8', "An 8-node serendipity quadrilateral of degree {degree}",
ndim = 2,
nplex = 8,
degree = (2,2),
reverse = (3, 2, 1, 0, 6, 5, 4, 7),
edges = Quad9.edges,
gl2edges = Quad9.gl2edges,
gl2faces = Elems([(0, 4, 7), (1, 5, 4), (2, 6, 5), (3, 7, 6),
(4, 5, 6), (4, 6, 7)], Tri3),
rst = Quad9.rst[:,:8],
h = [
lambda r, s: 2*(1.-r)*(1.-s)*(0.5-r-s),
lambda r, s: 2*r*(1.-s)*(r-s-0.5),
lambda r, s: 2*r*s*(r+s-1.5),
lambda r, s: 2*(1.-r)*s*(s-r-0.5),
lambda r, s: 4*r*(1.-r)*(1.-s),
lambda r, s: 4*r*s*(1.-s),
lambda r, s: 4*r*s*(1.-r),
lambda r, s: 4*(1.-r)*s*(1.-s),
],
subdiv_proxy = Quad9,
)
# Quad13 = ElementType(
# 'quad13', "A 13-node serendipity quadrilateral of degree {degree}",
# ndim = 2,
# nplex = 13,
# degree = (3,3),
# reverse = (3, 2, 1, 0, 9, 8, 7, 6, 5, 4, 11, 10, 12),
# edges = Elems([(0, 4, 5, 1), (1, 6, 7, 2), (2, 8, 9, 3), (3, 10, 11, 0)],
# Line4),
# gl2faces = Elems([(0, 4, 11), (1, 6, 5), (2, 8, 7), (3, 10, 9),
# (11, 4, 12), (4, 5, 12), (5, 6, 12), (6, 7, 12),
# (7, 8, 12), (8, 9, 12), (9, 10, 12), (10, 11, 12)], Tri3),
# rst = [[0., 1., 1., 0., e3, f3, 1., 1., f3, e3, 0., 0., 0.5],
# [0., 0., 1., 1., 0., 0., e3, f3, 1., 1., f3, e3, 0.5]],
# h = [
# lambda r, s: 9/2 * (r-1) * (s-1) * (r+s-e3) * (r+s-f3),
# lambda r, s: -9/2 * r * (s-1) * (r-s-e3) * (r-s-f3),
# lambda r, s: 9/2 * r * s * (r+s-4*e3) * (r+s-5*e3),
# lambda r, s: -9/2 * (r-1) * s * (r-s+e3) * (r-s+f3),
# lambda r, s: -27/2 * r * (r-1.) * (r-f3) * (s-1),
# lambda r, s: 27/2 * r * (r-1.) * (r-e3) * (s-1),
# lambda r, s: 27/2 * r * s * (s-1) * (s-f3),
# lambda r, s: -27/2 * r * s * (s-1) * (s-e3),
# lambda r, s: -27/2 * r * (r-1.) * (r-e3) * s,
# lambda r, s: 27/2 * r * (r-1.) * (r-f3) * s,
# lambda r, s: 27/2 * (r-1) * s * (s-1) * (s-e3),
# lambda r, s: -27/2 * (r-1) * s * (s-1) * (s-f3),
# lambda r, s: 16 * r * (r-1) * s * (s-1),
# ],
# )
# Quad13.gl2edges = Quad13.edges.selectNodes(Line4.gl2faces)
# Two not registered Lagrange type elements
# So that we have their automatically generated interpolation functions
Lag2D_3_1 = Lagrange2D(3, 1, register=False)
# reverse = (3, 2, 1, 0, 7, 6, 5, 4),
# edges = Elems([(0, 1, 2, 3), (3, 7, 7, 7), (7, 6, 5, 4),
# (4, 0, 0, 0)], Line4),
# gl2edges = Elems([(0, 1), (1, 2), (2, 3), (3, 7), (7, 6), (6, 5),
# (5, 4), (4, 0)], Line2),
# gl2faces = Elems(Quad4.els(3,1)[:,[(0,1,2),(2,3,0)]].reshape(-1,3), Tri3),
Lag2D_1_3 = Lagrange2D(1, 3, register=False)
# reverse = (3, 2, 1, 0, 7, 6, 5, 4),
# edges = Elems([(0, 1, 1, 1), (1, 3, 5, 7), (7, 6, 6, 6),
# (6, 4, 2, 0)], Line4),
# gl2edges = Elems([(0, 1), (1, 3), (3, 5), (5, 7), (7, 6), (6, 4),
# (4, 2), (2, 0)], Line2),
# gl2faces = Elems(Quad4.els(1,3)[:,[(0,1,2),(2,3,0)]].reshape(-1,3), Tri3),
[docs]def Quad12_fixh(self):
"""Compute the 4 corner interpolation functions"""
for i in range(4):
v = self.h[i](self.r, self.s)
w = np.where(~np.isclose(v, 0.))[0]
w = w[w!=i]
v = v[w]
h = [self.h[j] for j in w]
self.h[i] = lambda r, s, j=i, vv=v, hh=h: Quad4.h[j](r, s) \
- vv[0] * hh[0](r,s) \
- vv[1] * hh[1](r,s) \
- vv[2] * hh[2](r,s) \
- vv[3] * hh[3](r,s)
Quad12 = ElementType(
'quad12', "A 12-node serendipity quadrilateral of degree {degree}",
ndim = 2,
nplex = 12,
degree = (3,3),
reverse = (3, 2, 1, 0, 9, 8, 7, 6, 5, 4, 11, 10),
edges = Elems([(0, 4, 5, 1), (1, 6, 7, 2), (2, 8, 9, 3), (3, 10, 11, 0)],
Line4),
gl2faces = Elems([(0, 4, 11), (1, 6, 5), (2, 8, 7), (3, 10, 9),
(4, 5, 6), (6, 7, 8), (8, 9, 10), (10, 11, 4),
(4, 6, 8), (8, 10, 4)], Tri3),
rst = [[0., 1., 1., 0., e3, f3, 1., 1., f3, e3, 0., 0.],
[0., 0., 1., 1., 0., 0., e3, f3, 1., 1., f3, e3]],
h = [
lambda r, s: Quad4.h[0](r, s),
lambda r, s: Quad4.h[1](r, s),
lambda r, s: Quad4.h[2](r, s),
lambda r, s: Quad4.h[3](r, s),
lambda r, s: -27/2 * r * (r-1.) * (r-f3) * (s-1),
lambda r, s: 27/2 * r * (r-1.) * (r-e3) * (s-1),
lambda r, s: 27/2 * r * s * (s-1) * (s-f3),
lambda r, s: -27/2 * r * s * (s-1) * (s-e3),
lambda r, s: -27/2 * r * (r-1.) * (r-e3) * s,
lambda r, s: 27/2 * r * (r-1.) * (r-f3) * s,
lambda r, s: 27/2 * (r-1) * s * (s-1) * (s-e3),
lambda r, s: -27/2 * (r-1) * s * (s-1) * (s-f3),
],
fixh = Quad12_fixh,
subdiv_proxy = Quad16,
)
Quad12.gl2edges = Quad12.edges.selectNodes(Line4.gl2faces)
######### 3D ###################
#############
# Family: tet
##
def tet4_els(self, nr, ns=None, nt=None):
# ns, nt not used: requires nr = ns = nt
n = nr+1 # number of points along an edge
a = -np.ones((n,n,n), dtype=at.Int)
nodid = 0
for k in range(n):
for j in range(n-k):
for i in range(n-j-k):
a[k,j,i] = nodid
nodid += 1
els1 = np.row_stack([
(a[k,j,i], a[k,j,i+1], a[k,j+1,i], a[k+1,j,i])
for k in range(nr) for j in range(nr-k) for i in range(nr-j-k)])
n1 = nr-1
els2 = [np.row_stack([
(a[k+1,j,i], a[k+1,j+1,i], a[k+1,j,i+1], a[k,j+1,i]),
(a[k,j+1,i], a[k,j+1,i+1], a[k+1,j+1,i], a[k+1,j,i+1]),
(a[k,j+1,i], a[k+1,j,i], a[k,j,i+1], a[k+1,j,i+1]),
(a[k+1,j,i+1], a[k,j+1,i+1], a[k,j,i+1], a[k,j+1,i]),
]) for k in range(n1) for j in range(n1-k) for i in range(n1-j-k)]
return np.concatenate([els1] + els2)
def tet4_wts(self, rdiv, sdiv=None, tdiv=None):
r = seeds(rdiv, self.dr)
nr = len(r)
if sdiv is None:
s, ns = r, nr
else:
s = seeds(sdiv, self.ds)
ns = len(s)
if tdiv is None:
t, nt = s, ns
else:
t = seeds(tdiv, self.dt)
nt = len(t)
if nr != ns:
raise ValueError("Tet4: number of divisions along r, s and t"
" directions should be equal")
ijk = np.row_stack([(i,j,k) for k in range(nt)
for j in range(ns-k) for i in range(nr-k-j)])
rst = np.column_stack([(r[i], s[j], t[k]) for k in range(nt)
for j in range(ns-k) for i in range(nr-k-j)])
# fix for unequal divisions
for m in range(ijk.shape[0]):
i, j, k = ijk[m]
if sum([i > 0, j > 0, k > 0]) > 1:
l = i+j+k
a, b, c = rst[:,m]
d, e, f = r[l], s[l], t[l]
p = 1./(a/d + b/e + c/f)
rst[:,m] *= p
return np.column_stack([1.-rst[0]-rst[1]-rst[2], rst[0], rst[1], rst[2]])
Tet4 = ElementType(
'tet4', "A 4-node tetrahedron",
ndim = 3,
nplex = 4,
degree = (1,1,1),
reverse = (0, 1, 3, 2),
edges = Elems([(0, 1), (1, 2), (2, 0), (0, 3), (1, 3), (2, 3)], Line2),
faces = Elems([(0, 3, 2), (0, 1, 3), (0, 2, 1), (1, 2, 3),], Tri3),
rst = [[0., 1., 0., 0.],
[0., 0., 1., 0.],
[0., 0., 0., 1.]],
h = [
lambda r, s, t: 1.-r-s-t,
lambda r, s, t: r,
lambda r, s, t: s,
lambda r, s, t: t,
],
els = tet4_els,
wts = tet4_wts,
)
Tet10 = ElementType(
'tet10', "A 10-node tetrahedron",
ndim = 3,
nplex = 10,
degree = (2,2,2),
reverse = (0, 1, 3, 2, 4, 6, 5, 9, 8, 7),
edges = Elems([(0, 4, 1), (1, 7, 2), (2, 5, 0), (0, 6, 3), (1, 9, 3),
(2, 8, 3)], Line3),
faces = Elems([(0, 3, 2, 6, 8, 5), (0, 1, 3, 4, 9, 6),
(0, 2, 1, 5, 7, 4), (1, 2, 3, 7, 8, 9)], Tri6),
rst = [[0., 1., 0., 0., 0.5, 0.0, 0.0, 0.5, 0.0, 0.5],
[0., 0., 1., 0., 0.0, 0.5, 0.0, 0.5, 0.5, 0.0],
[0., 0., 0., 1., 0.0, 0.0, 0.5, 0.0, 0.5, 0.5]],
h = [
lambda r, s, t: 2*(r+s+t-1.)*(r+s+t-0.5),
lambda r, s, t: 2*r*(r-0.5),
lambda r, s, t: 2*s*(s-0.5),
lambda r, s, t: 2*t*(t-0.5),
lambda r, s, t: -4*r*(r+s+t-1.),
lambda r, s, t: -4*s*(r+s+t-1.),
lambda r, s, t: -4*t*(r+s+t-1.),
lambda r, s, t: 4*r*s,
lambda r, s, t: 4*s*t,
lambda r, s, t: 4*r*t,
],
subdiv_proxy = Tet4,
)
Tet10.gl2faces = Tet10.faces.selectNodes(Tri6.gl2faces)
Tet10.gl2edges = Tet10.edges.selectNodes(Line3.gl2faces)
Tet14 = ElementType(
'tet14', "A 14-node tetrahedron",
ndim = 3,
nplex = 14,
vertices = Coords.concatenate([
Tet10.vertices,
[(e3, e3, 0.0),
(0.0, e3, e3),
(e3, 0.0, e3),
(e3, e3, e3),
]]),
edges = Tet10.edges,
faces = Tet10.faces, # We do not have a Tri7 yet!
# faces = Elems([(0, 2, 1, 5, 7, 4, 10), (0, 1, 3, 4, 9, 6, 11),
# (0, 3, 2, 6, 8, 5, 12), (1, 2, 3, 2, 7, 8, 13)]), Tri7)
reverse = (0, 1, 3, 2, 4, 6, 5, 9, 8, 7, 12, 11, 10, 13),
gl2faces = Tet10.gl2faces,
gl2edges = Tet10.gl2edges,
subdiv_proxy = Tet4,
)
Tet15 = ElementType(
'tet15', "A 15-node tetrahedron",
ndim = 3,
nplex = 15,
vertices = Coords.concatenate([
Tet14.vertices,
[(0.25, 0.25, 0.25),
]]),
edges = Tet14.edges,
faces = Tet14.faces,
reverse = (0, 1, 3, 2, 4, 6, 5, 9, 8, 7, 12, 11, 10, 13, 14),
gl2faces = Tet14.gl2faces,
gl2edges = Tet14.gl2edges,
subdiv_proxy = Tet4,
)
###############
# Family: wedge
##
Wedge6 = ElementType(
'wedge6', "A 6-node wedge element",
ndim = 3,
nplex = 6,
degree = (1,1,1),
reverse = (3, 4, 5, 0, 1, 2),
rst = [[0., 1., 0., 0., 1., 0.],
[0., 0., 1., 0., 0., 1.],
[0., 0., 0., 1., 1., 1.]],
edges = Elems([(0, 1), (1, 2), (2, 0), (0, 3), (1, 4), (2, 5), (3, 4),
(4, 5), (5, 3)], 'line2'),
faces = Elems([(0, 2, 1, 1), (3, 4, 4, 5), (0, 1, 4, 3), (1, 2, 5, 4),
(0, 3, 5, 2)], Quad4),
)
# Wedge6.gl2faces = Wedge6.faces.selectNodes(Quad4.gl2faces).removeDegenerate()
#############
# Family: hex
##
class Lagrange3D(ElementType):
def __init__(self, dr, ds, dt, **kargs):
ndim = 3
degree = (dr,ds,dt)
nplex = (dr+1)*(ds+1)*(dt+1)
name = f"Hex{nplex}"
doc = f"A {nplex}-node hexahedral of degree {degree}"
r = np.linspace(0, 1, dr+1)
s = np.linspace(0, 1, ds+1)
t = np.linspace(0, 1, dt+1)
t, s, r = np.meshgrid(t, s, r, indexing='ij') # !!! Notice order
rst = np.row_stack([r.ravel(), s.ravel(), t.ravel()])
h = [lagrange3(dr, ds, dt, i, j, k)
for k in range(dt+1) for j in range(ds+1) for i in range(dr+1)]
super().__init__(name=name, doc=doc, ndim=ndim, degree=degree,
nplex=nplex, rst=rst, h=h, **kargs)
def els(self, nr, ns=None, nt=None):
if ns is None:
ns = nr
if nt is None:
nt = ns
dr, ds, dt = self.degree
ntr = dr*nr + 1
nts = ds*ns + 1
ntt = dt*nt + 1
n = ntr*nts*ntt
grid = np.arange(n).reshape(ntt,nts,ntr)
stack = []
dr1, ds1, dt1 = dr+1, ds+1, dt+1
for k in range(nt):
kdt = k*dt
for j in range(ns):
jds = j*ds
for i in range(nr):
idr = i*dr
elij = grid[kdt:kdt+dt1, jds:jds+ds1, idr:idr+dr1]
elij = elij.flat
if hasattr(self, 'order'):
elij = elij[self.order]
stack.append(elij)
return np.row_stack(stack)
def wts(self, rdiv, sdiv=None, tdiv=None):
r = seeds(rdiv, self.dr)
s = r if sdiv is None else seeds(sdiv, self.ds)
t = s if tdiv is None else seeds(tdiv, self.dt)
t, s, r = np.meshgrid(t, s, r, indexing='ij')
H = np.stack([h(r,s,t) for h in self.h])
return H.reshape((len(self.h), -1)).transpose()
Hex8 = Lagrange3D(
1, 1, 1,
order = [0, 1, 3, 2, 4, 5, 7, 6],
reverse = (4, 5, 6, 7, 0, 1, 2, 3),
edges = Elems([(0, 1), (1, 2), (2, 3), (3, 0),
(4, 5), (5, 6), (6, 7), (7, 4),
(0, 4), (1, 5), (2, 6), (3, 7)], Line2),
faces = Elems([(0, 4, 7, 3), (1, 2, 6, 5),
(0, 1, 5, 4), (3, 7, 6, 2),
(0, 3, 2, 1), (4, 5, 6, 7)], Quad4),
)
# Hex8.gl2faces = Hex8.faces.selectNodes(Quad4.gl2faces)
# THIS ELEMENT USES A REGULAR NODE NUMBERING!!
# WE MIGHT SWITCH OTHER ELEMENTS TO THIS REGULAR SCHEME TOO
# AND ADD THE RENUMBERING TO THE FE OUTPUT MODULES
Hex27 = Lagrange3D(
2, 2, 2,
reverse = np.arange(27).reshape((3,9))[::-1].ravel(),
edges = Elems([(0, 1, 2), (6, 7, 8), (18, 19, 20), (24, 25, 26),
(0, 3, 6), (2, 5, 8), (18, 21, 24), (20, 23, 26),
(0, 9, 18), (2, 11, 20), (6, 15, 24), (8, 17, 26)], Line3),
faces = Elems([(0, 18, 24, 6, 9, 21, 15, 3, 12),
(2, 8, 26, 20, 5, 17, 23, 11, 14),
(0, 2, 20, 18, 1, 11, 19, 9, 10),
(6, 24, 26, 8, 15, 25, 17, 7, 16),
(0, 6, 8, 2, 3, 7, 5, 1, 4),
(18, 20, 26, 24, 19, 23, 25, 21, 22)], Quad9),
)
Hex27.gl2edges = Hex27.edges.selectNodes(Line3.gl2faces)
Hex27.gl2faces = Hex27.faces.selectNodes(Quad9.gl2faces)
## Serendipity ##
Hex20 = ElementType(
'hex20', "A 20-node hexahedron",
ndim = 3,
nplex = 20,
degree = (2,2,2),
order = (0, 1, 2, 3, 8, 9, 10, 11, 4, 5, 6, 7, 12, 13, 14, 15, 16,17,18,19),
reverse = (4, 5, 6, 7, 0, 1, 2, 3, 12, 13, 14, 15, 8, 9, 10, 11, 16, 17,
18, 19),
edges = Elems([(0, 8, 1), (1, 9, 2), (2, 10, 3), (3, 11, 0),
(4, 12, 5), (5, 13, 6), (6, 14, 7), (7, 15, 4),
(0, 16, 4), (1, 17, 5), (2, 18, 6), (3, 19, 7)], Line3),
faces = Elems([(0, 4, 7, 3, 16, 15, 19, 11), (1, 2, 6, 5, 9, 18, 13, 17),
(0, 1, 5, 4, 8, 17, 12, 16), (3, 7, 6, 2, 19, 14, 18, 10),
(0, 3, 2, 1, 11, 10, 9, 8), (4, 5, 6, 7, 12, 13, 14, 15)],
Quad8),
rst = np.row_stack([
np.concatenate([Quad8.rst, Quad8.rst, Quad8.rst[:,:4]], axis=-1),
np.concatenate([[0.]*8, [1.]*8, [0.5]*4]),
]),
h = [lambda r,s,t, hj=hi: hj(r,s)*lagrange[2][0](t) for hi in Quad8.h] +
[lambda r,s,t, hj=hi: hj(r,s)*lagrange[2][2](t) for hi in Quad8.h] +
[lambda r,s,t, hj=hi: hj(r,s)*lagrange[2][1](t) for hi in Quad8.h[:4]],
subdiv_proxy = Hex27,
)
Hex20.gl2edges = Hex20.edges.selectNodes(Line3.gl2faces)
Hex20.gl2faces = Hex20.faces.selectNodes(Quad8.gl2faces)
Hex16 = ElementType(
'hex16', "A 16-node serendipity hexahedron of degree (2,2,1)",
ndim = 3,
nplex = 16,
degree = (2,2,1),
order = [0, 1, 2, 3, 8, 9, 10, 11, 4, 5, 6, 7, 12, 13, 14, 15],
reverse = (4, 5, 6, 7, 0, 1, 2, 3, 12, 13, 14, 15, 8, 9, 10, 11),
edges = Elems([(0, 8, 1), (1, 9, 2), (2, 10, 3), (3, 11, 0),
(4, 12, 5), (5, 13, 6), (6, 14, 7), (7, 15, 4),
(0, 0, 4), (1, 1, 5), (2, 2, 6), (3, 3, 7)], Line3),
faces = Elems([(0, 4, 7, 3, 0, 15, 7, 11), (1, 2, 6, 5, 9, 2, 13, 5),
(0, 1, 5, 4, 8, 1, 12, 4), (3, 7, 6, 2, 3, 14, 6, 10),
(0, 3, 2, 1, 11, 10, 9, 8), (4, 5, 6, 7, 12, 13, 14, 15)],
Quad8),
rst = np.row_stack([
np.concatenate([Quad8.rst, Quad8.rst], axis=-1),
np.concatenate([[0.]*8, [1.]*8]),
]),
# !! Notice the use of a local variable hj to replace global hi !!
h = [lambda r,s,t, hj=hi: hj(r,s)*(1.-t) for hi in Quad8.h] +
[lambda r,s,t, hj=hi: hj(r,s)*t for hi in Quad8.h],
subdiv_proxy = Hex20,
)
Hex16.gl2edges = Hex16.edges.selectNodes(Line3.gl2faces).removeDegenerate()
Hex16.gl2faces = Hex16.faces.selectNodes(Quad8.gl2faces).removeDegenerate()
##########################################################
# Some exotic elements not meant for Finite Element applications
# Therefore we only have one of each, with minimal node sets
# TODO: These should probably be moved to another module, as
# they are not really 'elements' but rather meshes.
# But it is easy to use the Mesh methods to get edges and faces :)
##############
# Family: poly
##
# TODO: add conversion to 'tri3'
[docs]class Polygon(ElementType):
"""A Polygon element type.
This element type is useful to define a Mesh of polygons that
all have the same plexitude. The polygon elements are created
on demand.
Examples
--------
>>> P = Polygon(5, register=False)
>>> print(P)
Poly5
>>> print(P.getFaces())
[[0 1 2 3 4]]
>>> print(P.getDrawFaces())
[[0 1 2 3 4]]
>>> print(P.getEdges())
[[0 1]
[1 2]
[2 3]
[3 4]
[4 0]]
>>> print(P.getDrawEdges())
[[0 1]
[1 2]
[2 3]
[3 4]
[4 0]]
"""
def __init__(self, nplex, name=None, **kargs):
from pyformex.formex import Formex
ndim = 2
degree = (1, 1)
if name is None:
name = f"Poly{nplex}"
doc = f"Polygon with {nplex} vertices"
vertices = Formex([[[1.,0.,0.]]]).rosette(nplex).points()
elem = np.arange(nplex, dtype=at.Int)
edges = np.column_stack([elem, np.roll(elem, -1)])
edges = Elems(edges, eltype=Line2)
super().__init__(name=name, doc=doc, ndim=ndim, degree=degree,
nplex=nplex, vertices=vertices, edges=edges,
**kargs)
Octa = ElementType(
'octa',
"""An octahedron: a regular polyhedron with 8 triangular faces.
nfaces = 8, nedges = 12, nvertices = 6
All the faces are equilateral triangles.
All points of the octahedron lie on a sphere with unit radius.
""",
ndim = 3,
nplex = 6,
vertices = [(1.0, 0.0, 0.0),
(0.0, 1.0, 0.0),
(0.0, 0.0, 1.0),
(-1.0, 0.0, 0.0),
(0.0, -1.0, 0.0),
(0.0, 0.0, -1.0),
],
edges = Elems([(0, 1), (1, 3), (3, 4), (4, 0),
(0, 5), (5, 3), (3, 2), (2, 0),
(1, 2), (2, 4), (4, 5), (5, 1)], 'line2'),
faces = Elems([(0, 1, 2), (1, 3, 2), (3, 4, 2), (4, 0, 2),
(1, 0, 5), (3, 1, 5), (4, 3, 5), (0, 4, 5)], Tri3),
reverse = (3, 1, 2, 0, 4, 5),
)
from pyformex.arraytools import golden_ratio as phi # noqa: E402
Icosa = ElementType(
'icosa',
"""An icosahedron: a regular polyhedron with 20 triangular faces.
nfaces = 20, nedges = 30, nvertices = 12
All points of the icosahedron lie on a sphere with unit radius.
""",
ndim = 3,
nplex = 12,
vertices = Coords([(0.0, 1.0, phi),
(0.0, -1.0, phi),
(0.0, 1.0, -phi),
(0.0, -1.0, -phi),
(1.0, phi, 0.0),
(-1.0, phi, 0.0),
(1.0, -phi, 0.0),
(-1.0, -phi, 0.0),
(phi, 0.0, 1.0),
(phi, 0.0, -1.0),
(-phi, 0.0, 1.0),
(-phi, 0.0, -1.0),
]),
edges = Elems([(0, 1), (0, 8), (1, 8), (0, 10), (1, 10),
(2, 3), (2, 9), (3, 9), (2, 11), (3, 11),
(4, 5), (4, 0), (5, 0), (4, 2), (5, 2),
(6, 7), (6, 1), (7, 1), (6, 3), (7, 3),
(8, 9), (8, 4), (9, 4), (8, 6), (9, 6),
(10, 11), (10, 5), (11, 5), (10, 7), (11, 7)], 'line2'),
faces = Elems([(0, 1, 8), (1, 0, 10),
(2, 3, 11), (3, 2, 9),
(4, 5, 0), (5, 4, 2),
(6, 7, 3), (7, 6, 1),
(8, 9, 4), (9, 8, 6),
(10, 11, 7), (11, 10, 5),
(0, 8, 4), (1, 6, 8),
(0, 5, 10), (1, 10, 7),
(2, 11, 5), (3, 7, 11),
(2, 4, 9), (3, 9, 6)], Tri3),
reverse = (2, 3, 0, 1, 4, 5, 6, 7, 9, 8, 11, 10),
)
############################################
# list of default element type per plexitude
ElementType.default = {
1: Point,
2: Line2,
3: Tri3,
4: Quad4,
6: Wedge6,
8: Hex8,
}
######################################################################
# element type conversions
##########################
Line2.linear = 'line2'
Line2.conversions = {
'line3': [('a', [(0, 1)]),
('s', [(0, 2, 1)]),
],
'line2-2': [('v', 'line3'),
('s', [(0, 2), (2, 1)]), ],
}
Line3.linear = 'line2'
Line3.conversions = {
'line2': [('s', [(0, 2)]), ],
'line2-2': [('s', [(0, 1), (1, 2)]), ],
}
Tri3.linear = 'tri3'
Tri3.conversions = {
'tri3-3': [('a', [(0, 1, 2), ]),
('s', [(0, 1, 3), (1, 2, 3), (2, 0, 3)]),
],
'tri3-4': [('v', 'tri6'), ],
'tri6': [('a', [(0, 1), (1, 2), (2, 0)]), ],
'quad4': [('v', 'tri6'), ],
}
Tri6.linear = 'tri3'
Tri6.conversions = {
'tri3': [('s', [(0, 1, 2)]), ],
'tri3-4': [('s', [(0, 3, 5), (3, 1, 4), (4, 2, 5), (3, 4, 5)]), ],
'quad4': [('a', [(0, 1, 2), ]),
('s', [(0, 3, 6, 5), (1, 4, 6, 3), (2, 5, 6, 4)]),
],
}
Quad4.linear = 'quad4'
Quad4.conversions = {
'tri3': 'tri3-u',
'tri3-r': [('r', ['tri3-u', 'tri3-d']), ],
'tri3-u': [('s', [(0, 1, 2), (2, 3, 0)]), ],
'tri3-d': [('s', [(0, 1, 3), (2, 3, 1)]), ],
'tri3-x': [('a', [(0, 1, 2, 3)]),
('s', [(0, 1, 4), (1, 2, 4), (2, 3, 4), (3, 0, 4)]),
],
'tri3-*': [('v', 'quad9'), ],
'quad8': [('a', [(0, 1), (1, 2), (2, 3), (3, 0)])],
'quad4-4': [('v', 'quad9'), ],
'quad9': [('v', 'quad8'), ],
'quad12': [('n', [[e3, f3, 1., 1., f3, e3, 0., 0.],
[0., 0., e3, f3, 1., 1., f3, e3]]), ],
}
Quad6.linear = 'quad4'
Quad6.conversions = {
'tri3': [('v', 'quad4'), ],
'quad4': [('s', [(0, 4, 5, 3), (4, 1, 2, 5)]), ],
'quad8': [('a', [(0, 3), (1, 2)]),
('s', [(0, 1, 2, 3, 4, 7, 5, 6)])],
'quad9': [('a', [(0, 3), (1, 2), (4, 5)]), ],
}
Quad8.linear = 'quad4'
Quad8.conversions = {
'tri3': [('v', 'quad9'), ],
'tri3-v': [('s', [(0, 4, 7), (1, 5, 4), (2, 6, 5),
(3, 7, 6), (5, 6, 4), (7, 4, 6)]), ],
'tri3-h': [('s', [(0, 4, 7), (1, 5, 4), (2, 6, 5),
(3, 7, 6), (4, 5, 7), (6, 7, 5)]), ],
'quad4': [('s', [(0, 1, 2, 3)]), ],
'quad4-4': [('v', 'quad9'), ],
'quad9': [('a', [(4, 5, 6, 7)]), ],
}
Quad9.linear = 'quad4'
Quad9.conversions = {
'quad8': [('s', [(0, 1, 2, 3, 4, 5, 6, 7)]), ('c', None)],
'quad4': [('v', 'quad8'), ],
'quad4-4': [
('s', [(0, 4, 8, 7), (4, 1, 5, 8), (7, 8, 6, 3), (8, 5, 2, 6)]), ],
'tri3': 'tri3-d',
'tri3-d': [('s', [(0, 4, 7), (4, 1, 5), (5, 2, 6), (6, 3, 7),
(7, 4, 8), (4, 5, 8), (5, 6, 8), (6, 7, 8)]), ],
'tri3-*': [('s', [(0, 4, 8), (4, 1, 8), (1, 5, 8), (5, 2, 8),
(2, 6, 8), (6, 3, 8), (3, 7, 8), (7, 0, 8)]), ],
}
Quad16.linear = 'quad4'
Quad16.conversions = {
'quad12': [('s', [(0,3,15,12,1,2,7,11,14,13,8,4)]), ],
'quad4': [('s', [(0,3,15,12)]), ],
'quad4-9': [('s', Quad4.els(3)[:,Quad4.order]), ],
}
Quad12.linear = 'quad4'
Quad12.conversions = {
'quad16': [('n', [[e3, f3, e3, f3], [e3, e3, f3, f3]]),
('s', [(0,4,5,1,11,12,13,6,10,14,15,7,3,9,8,2)]), ],
'quad4': [('s', [(0,1,2,3)]), ],
}
Tet4.linear = 'tet4'
Tet4.conversions = {
'tet4-4': [
('a', [(0, 1, 2, 3)]),
('s', [(0, 1, 2, 4), (0, 3, 1, 4), (1, 3, 2, 4), (2, 3, 0, 4)]), ],
'tet10': [('a', [(0, 1), (0, 2), (0, 3), (1, 2), (2, 3), (1, 3)]), ],
'tet14': [('v', 'tet10'), ],
'tet15': [('v', 'tet14'), ],
'hex8': [('v', 'tet15'), ],
}
Tet10.linear = 'tet4'
Tet10.conversions = {
'tet4': [('s', [(0, 1, 2, 3,)]), ],
'tet14': [('a', [(0, 1, 2), (0, 2, 3), (0, 3, 1), (1, 2, 3), ]), ],
'tet15': [('v', 'tet14'), ],
'hex8': [('v', 'tet15'), ],
}
Tet14.linear = 'tet4'
Tet14.conversions = {
'tet10': [('s', [(0, 1, 2, 3, 4, 5, 6, 7, 8, 9)]), ],
'tet4': [('v', 'tet10'), ],
'tet15': [('a', [(0, 1, 2, 3), ]), ],
'hex8': [('v', 'tet15'), ],
}
Tet15.linear = 'tet4'
Tet15.conversions = {
'tet14': [('s', [(0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13)]), ],
'tet10': [('v', 'tet14'), ],
'tet4': [('v', 'tet10'), ],
'hex8': [
('s', [(0, 4, 10, 5, 6, 12, 14, 11), (4, 1, 7, 10, 12, 9, 13, 14),
(5, 10, 7, 2, 11, 14, 13, 8), (6, 12, 14, 11, 3, 9, 13, 8)]), ],
}
Wedge6.linear = 'wedge6'
Wedge6.conversions = {
'tet4': 'tet4-3',
# tet4-3 is a compatible tet4-3 scheme
'tet4-3': [('x', 'wedge6_tet4'), ],
# tet4-3? has quad diagonals that meet at the same node
# Here are two (out of six) variants.
# The schemes may be incompatible at common quad surfaces
# They are normally only used by tet4-3 conversion
'tet4-3l': [('s', [(1, 2, 0, 5), (1, 0, 4, 5), (0, 4, 5, 3)]), ],
'tet4-3r': [('s', [(1, 2, 0, 4), (2, 0, 4, 5), (0, 4, 5, 3)]), ],
# tet4-8 has no quad diagonals that meet at the same node
# this case requires a Steiner point
'tet4-8': [
('a', [(0, 1, 2, 3, 4, 5), ]),
('s', [(2, 0, 1, 6), (5, 1, 6, 2), (0, 4, 1, 6), (4, 0, 3, 6),
(2, 3, 0, 6), (5, 4, 3, 6), (5, 1, 4, 6), (3, 2, 5, 6)]), ],
# tet4-11 has two diagonals, so is always conforming
'tet4-11': [
('a', [(0, 1, 4, 3), (1, 2, 5, 4), (0, 2, 5, 3)]),
('s', [(0, 1, 2, 6), (0, 6, 2, 8), (1, 2, 6, 7), (2, 6, 7, 8),
(1, 7, 6, 4), (0, 6, 8, 3), (2, 8, 7, 5), (6, 7, 8, 3),
(3, 7, 8, 5), (3, 4, 7, 5), (3, 6, 7, 4)]), ],
}
Hex8.linear = 'hex8'
Hex8.conversions = {
# The default tet4 provides a compatible scheme with 24 tet4's
'tet4': 'tet4-24',
# This scheme only uses 6 tet4's and is compatible
# if the wedge6 subdivision is compatible.
'tet4-w': [('v', 'wedge6'), ],
# The smallest number of tet4 possible, but not compatible.
'tet4-5': [
('s', [(0, 1, 2, 5), (2, 3, 0, 7), (5, 7, 6, 2),
(7, 5, 4, 0), (0, 5, 2, 7)]), ],
# This should not be used: default tet4 is better
# tet4-6 producing compatible triangles on opposite faces
# (but an adjacent hex might not be compatible)
'tet4-6': [
('s', [(0, 2, 3, 7), (0, 2, 7, 6), (0, 4, 6, 7),
(0, 6, 1, 2), (0, 6, 4, 1), (5, 6, 1, 4), ]), ],
# Commented out, because incorrect
# 'tet4-8': [
# ('s', [(0, 1, 3, 4), (1, 2, 0, 5), (2, 3, 1, 6), (3, 0, 2, 7),
# (4, 7, 5, 0), (5, 4, 6, 1), (6, 5, 7, 2), (7, 6, 4, 3), ]), ],
# A scheme that is always compatible
'tet4-24': [
('a', [(0, 3, 2, 1), (0, 1, 5, 4), (0, 4, 7, 3), (1, 2, 6, 5),
(2, 3, 7, 6), (4, 5, 6, 7)]),
('a', [(0, 1, 2, 3, 4, 5, 6, 7)]),
('s', [(0, 1, 8, 14), (1, 2, 8, 14), (2, 3, 8, 14), (3, 0, 8, 14),
(0, 4, 9, 14), (4, 5, 9, 14), (5, 1, 9, 14), (1, 0, 9, 14),
(0, 3, 10, 14), (3, 7, 10, 14), (7, 4, 10, 14), (4, 0, 10, 14),
(1, 5, 11, 14), (5, 6, 11, 14), (6, 2, 11, 14), (2, 1, 11, 14),
(2, 6, 12, 14), (6, 7, 12, 14), (7, 3, 12, 14), (3, 2, 12, 14),
(4, 7, 13, 14), (7, 6, 13, 14), (6, 5, 13, 14), (5, 4, 13, 14),
]), ],
'wedge6': [('s', [(0, 1, 2, 4, 5, 6), (2, 3, 0, 6, 7, 4)]), ],
'wedge6-r': [('s', [(0, 4, 5, 3, 7, 6), (0, 5, 1, 3, 6, 2)]), ],
'hex8-8': [('v', 'hex20'), ],
'hex20': [('a', [(0, 1), (1, 2), (2, 3), (3, 0),
(4, 5), (5, 6), (6, 7), (7, 4),
(0, 4), (1, 5), (2, 6), (3, 7), ]), ],
'hex27': [('v', 'hex20'), ],
}
Hex16.linear = 'hex8'
Hex16.conversions = {
'hex8': [('v', 'hex20'), ],
'hex20': [('a', [(0, 4), (1, 5), (2, 6), (3, 7)]),
('s', [(0, 1, 2, 3, 4, 5, 6, 7, 8, 9,
10, 11, 12, 13, 14, 15, 16, 17, 18, 19)])],
}
Hex20.linear = 'hex8'
Hex20.conversions = {
'hex8': [('s', [(0, 1, 2, 3, 4, 5, 6, 7)]), ],
'hex8-8': [('v', 'hex27'), ],
'hex16': [('s', [(0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15)]), ],
'hex27': [('a', [(8, 9, 10, 11), (8, 17, 12, 16), (11, 19, 15, 16),
(18, 13, 17, 9), (18, 14, 19, 10), (12, 13, 14, 15), ]),
('a', [(20, 21, 22, 23, 24, 25), ]),
('s', [(0, 8, 1, 11, 20, 9, 3, 10, 2, 16, 21, 17, 22, 26, 23,
19, 24, 18, 4, 12, 5, 15, 25, 13, 7, 14, 6), ]),
],
'tet4': [('v', 'hex8'), ],
}
Hex27.linear = 'hex8'
Hex27.conversions = {
'hex8-8': [('s', [(0, 1, 4, 3, 9, 10, 13, 12),
(1, 2, 5, 4, 10, 11, 14, 13),
(3, 4, 7, 6, 12, 13, 16, 15),
(4, 5, 8, 7, 13, 14, 17, 16),
(9, 10, 13, 12, 18, 19, 22, 21),
(10, 11, 14, 13, 19, 20, 23, 22),
(12, 13, 16, 15, 21, 22, 25, 24),
(13, 14, 17, 16, 22, 23, 26, 25), ]), ],
'hex20': [('s', [(0, 2, 8, 6, 18, 20, 26, 24, 1, 5, 7, 3, 19, 23, 25, 21,
9, 11, 17, 15)]), ],
}
##########################################################
# Extrusion database
####################
#
# NEED TO CHECK THIS !!!!:
# For degree 2, the default order is:
# first plane, intermediate plane, last plane.
Point.extruded = {1: (Line2, []),
2: (Line3, [0, 2, 1])}
Line2.extruded = {1: (Quad4, [0, 1, 3, 2])}
Line3.extruded = {1: (Quad6, [0, 2, 5, 3, 1, 4]),
2: (Quad9, [0, 2, 8, 6, 1, 5, 7, 3, 4]), }
Tri3.extruded = {1: (Wedge6, [])}
Quad4.extruded = {1: (Hex8, [])}
Quad8.extruded = {
1: (Hex16, [0, 1, 2, 3, 8, 9, 10, 11, 4, 5, 6, 7, 12, 13, 14, 15]),
2: (Hex20, [0, 1, 2, 3, 16, 17, 18, 19, 4, 5, 6, 7,
20, 21, 22, 23, 8, 9, 10, 11])}
# BV: If Quad9 would be numbered consecutively, extrusion would be as easy as
# Quad9.extruded = { 2: (Hex27, [] }
Quad9.extruded = {2: (Hex27, [0, 4, 1, 7, 8, 5, 3, 6, 2,
9, 13, 10, 16, 17, 14, 12, 15, 11,
18, 22, 19, 25, 26, 23, 21, 24, 20, ])}
############################################################
# Reduction of degenerate elements
##################################
Line3.degenerate = {
Line2: [(((0, 1), ), (0, 2)),
(((1, 2), ), (0, 2)), ],
}
Quad4.degenerate = {
Tri3: [(((0, 1), ), (0, 2, 3)),
(((1, 2), ), (0, 1, 3)),
(((2, 3), ), (0, 1, 2)),
(((3, 0), ), (0, 1, 2)), ],
}
Hex8.degenerate = {
Wedge6: [(((0, 1), (4, 5)), (0, 2, 3, 4, 6, 7)),
(((1, 2), (5, 6)), (0, 1, 3, 4, 5, 7)),
(((2, 3), (6, 7)), (0, 1, 2, 4, 5, 6)),
(((3, 0), (7, 4)), (0, 1, 2, 4, 5, 6)),
(((0, 1), (3, 2)), (0, 4, 5, 3, 7, 6)),
(((1, 5), (2, 6)), (0, 4, 5, 3, 7, 6)),
(((5, 4), (6, 7)), (0, 4, 1, 3, 7, 2)),
(((4, 0), (7, 3)), (0, 5, 1, 3, 6, 2)),
(((0, 3), (1, 2)), (0, 7, 4, 1, 6, 5)),
(((3, 7), (2, 6)), (0, 3, 4, 1, 2, 5)),
(((7, 4), (6, 5)), (0, 3, 4, 1, 2, 5)),
(((4, 0), (5, 1)), (0, 3, 7, 1, 2, 6)), ],
}
Wedge6.degenerate = {
Tet4: [(((0, 1), (0, 2)), (0, 3, 4, 5)),
(((3, 4), (3, 5)), (0, 1, 2, 3)),
(((0, 3), (1, 4)), (0, 1, 2, 5)),
(((0, 3), (2, 5)), (0, 1, 2, 4)),
(((1, 4), (2, 5)), (0, 1, 2, 3)), ],
}
# End
