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py_engine.py
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#!/usr/bin/env python3
__copyright__ = (
"Copyright (©) 2017-2023 EPFL (Ecole Polytechnique Fédérale de Lausanne)"
"Laboratory (LSMS - Laboratoire de Simulation en Mécanique des Solides)"
)
__license__ = "LGPLv3"
__all__ = ['Shapes']
import numpy as np
import numpy.polynomial.polynomial as poly
# pylint: disable=missing-docstring, invalid-name, too-many-instance-attributes
# flake8: noqa
class Shapes:
"""Python version of the shape functions for test purposes."""
# pylint: disable=bad-whitespace, line-too-long
NATURAL_COORDS = {
(1, 'quadrangle'): np.array([[-1.], [1.], [0.]]),
(2, 'quadrangle'): np.array([[-1., -1.], [ 1., -1.], [ 1., 1.], [-1., 1.],
[ 0., -1.], [ 1., 0.], [ 0., 1.], [-1., 0.]]),
(3, 'quadrangle'): np.array([[-1., -1., -1.], [ 1., -1., -1.], [ 1., 1., -1.], [-1., 1., -1.],
[-1., -1., 1.], [ 1., -1., 1.], [ 1., 1., 1.], [-1., 1., 1.],
[ 0., -1., -1.], [ 1., 0., -1.], [ 0., 1., -1.], [-1., 0., -1.],
[-1., -1., 0.], [ 1., -1., 0.], [ 1., 1., 0.], [-1., 1., 0.],
[ 0., -1., 1.], [ 1., 0., 1.], [ 0., 1., 1.], [-1., 0., 1.]]),
(2, 'triangle'): np.array([[0., 0.], [1., 0.], [0., 1], [.5, 0.], [.5, .5], [0., .5]]),
(3, 'triangle'): np.array([[0., 0., 0.], [1., 0., 0.], [0., 1., 0.], [0., 0., 1.],
[.5, 0., 0.], [.5, .5, 0.], [0., .5, 0.],
[0., 0., .5], [.5, 0., .5], [0., .5, .5]]),
(3, 'pentahedron'): np.array([[-1., 1., 0.], [-1., 0., 1.], [-1., 0., 0.],
[ 1., 1., 0.], [ 1., 0., 1.], [ 1., 0., 0.],
[-1., .5, .5], [-1., 0., .5], [-1., .5, 0.],
[ 0., 1., 0.], [ 0., 0., 1.], [ 0., 0., 0.],
[ 1., .5, .5], [ 1., 0., .5], [ 1., .5, 0.],
[ 0., .5, .5], [ 0., 0., .5], [ 0., .5, 0.]]),
}
QUADRATURE_W = {
(1, 'quadrangle', 1): np.array([2.]),
(1, 'quadrangle', 2): np.array([1., 1.]),
(2, 'triangle', 1): np.array([1./2.]),
(2, 'triangle', 2): np.array([1., 1., 1.])/6.,
(3, 'triangle', 1): np.array([1./6.]),
(3, 'triangle', 2): np.array([1., 1., 1., 1.])/24.,
(2, 'quadrangle', 1): np.array([1., 1., 1., 1.]),
(2, 'quadrangle', 2): np.array([1., 1., 1., 1.]),
(3, 'quadrangle', 1): np.array([1., 1., 1., 1.,
1., 1., 1., 1.]),
(3, 'quadrangle', 2): np.array([1., 1., 1., 1.,
1., 1., 1., 1.]),
(3, 'pentahedron', 1): np.array([1., 1., 1., 1., 1., 1.])/6.,
(3, 'pentahedron', 2): np.array([1., 1., 1., 1., 1., 1.])/6.,
}
_tet_a = (5. - np.sqrt(5.))/20.
_tet_b = (5. + 3.*np.sqrt(5.))/20.
QUADRATURE_G = {
(1, 'quadrangle', 1): np.array([[0.]]),
(1, 'quadrangle', 2): np.array([[-1.], [1.]])/np.sqrt(3.),
(2, 'triangle', 1): np.array([[1., 1.]])/3.,
(2, 'triangle', 2): np.array([[1./6., 1./6.], [2./3, 1./6], [1./6., 2./3.]]),
(3, 'triangle', 1): np.array([[1., 1., 1.]])/4.,
(3, 'triangle', 2): np.array([[_tet_a, _tet_a, _tet_a],
[_tet_b, _tet_a, _tet_a],
[_tet_a, _tet_b, _tet_a],
[_tet_a, _tet_a, _tet_b]]),
(2, 'quadrangle', 1): np.array([[-1., -1.], [ 1., -1.],
[-1., 1.], [ 1., 1.]])/np.sqrt(3.),
(2, 'quadrangle', 2): np.array([[-1., -1.], [ 1., -1.],
[-1., 1.], [ 1., 1.]])/np.sqrt(3.),
(3, 'quadrangle', 1): np.array([[-1., -1., -1.],
[ 1., -1., -1.],
[-1., 1., -1.],
[ 1., 1., -1.],
[-1., -1., 1.],
[ 1., -1., 1.],
[-1., 1., 1.],
[ 1., 1., 1.]])/np.sqrt(3.),
(3, 'quadrangle', 2): np.array([[-1., -1., -1.],
[ 1., -1., -1.],
[-1., 1., -1.],
[ 1., 1., -1.],
[-1., -1., 1.],
[ 1., -1., 1.],
[-1., 1., 1.],
[ 1., 1., 1.]])/np.sqrt(3.),
(3, 'pentahedron', 1): np.array([[-1./np.sqrt(3.), 1./6., 1./6.],
[-1./np.sqrt(3.), 2./3., 1./6.],
[-1./np.sqrt(3.), 1./6., 2./3.],
[ 1./np.sqrt(3.), 1./6., 1./6.],
[ 1./np.sqrt(3.), 2./3., 1./6.],
[ 1./np.sqrt(3.), 1./6., 2./3.]]),
(3, 'pentahedron', 2): np.array([[-1./np.sqrt(3.), 1./6., 1./6.],
[-1./np.sqrt(3.), 2./3., 1./6.],
[-1./np.sqrt(3.), 1./6., 2./3.],
[ 1./np.sqrt(3.), 1./6., 1./6.],
[ 1./np.sqrt(3.), 2./3., 1./6.],
[ 1./np.sqrt(3.), 1./6., 2./3.]]),
}
ELEMENT_TYPES = {
'_segment_2': ('quadrangle', 1, 'lagrange', 1, 2),
'_segment_3': ('quadrangle', 2, 'lagrange', 1, 3),
'_triangle_3': ('triangle', 1, 'lagrange', 2, 3),
'_triangle_6': ('triangle', 2, 'lagrange', 2, 6),
'_quadrangle_4': ('quadrangle', 1, 'serendip', 2, 4),
'_quadrangle_8': ('quadrangle', 2, 'serendip', 2, 8),
'_tetrahedron_4': ('triangle', 1, 'lagrange', 3, 4),
'_tetrahedron_10': ('triangle', 2, 'lagrange', 3, 10),
'_pentahedron_6': ('pentahedron', 1, 'lagrange', 3, 6),
'_pentahedron_15': ('pentahedron', 2, 'lagrange', 3, 15),
'_hexahedron_8': ('quadrangle', 1, 'serendip', 3, 8),
'_hexahedron_20': ('quadrangle', 2, 'serendip', 3, 20),
}
MONOMES = {(1, 'quadrangle'): np.array([[0], [1], [2], [3], [4], [5]]),
(2, 'triangle'): np.array([[0, 0], # 1
[1, 0], [0, 1], # x y
[2, 0], [1, 1], [0, 2]]), # x^2 x.y y^2
(2, 'quadrangle'): np.array([[0, 0],
[1, 0], [1, 1], [0, 1],
[2, 0], [2, 1], [1, 2], [0, 2]]),
(3, 'triangle'): np.array([[0, 0, 0],
[1, 0, 0], [0, 1, 0], [0, 0, 1],
[2, 0, 0], [1, 1, 0], [0, 2, 0], [0, 1, 1], [0, 0, 2],
[1, 0, 1]]),
(3, 'quadrangle'): np.array([[0, 0, 0],
[1, 0, 0], [0, 1, 0], [0, 0, 1],
[1, 1, 0], [1, 0, 1],
[0, 1, 1], [1, 1, 1],
[2, 0, 0], [0, 2, 0], [0, 0, 2],
[2, 1, 0], [2, 0, 1], [2, 1, 1],
[1, 2, 0], [0, 2, 1], [1, 2, 1],
[1, 0, 2], [0, 1, 2], [1, 1, 2]])}
SHAPES = {
(3, 'pentahedron', 1): np.array([
[[[ 0., 0.], [ 1., 0.]], [[ 0., 0.], [-1., 0.]]],
[[[ 0., 1.], [ 0., 0.]], [[ 0., -1.], [ 0., 0.]]],
[[[ 1., -1.], [-1., 0.]], [[-1., 1.], [ 1., 0.]]],
[[[ 0., 0.], [ 1., 0.]], [[ 0., 0.], [ 1., 0.]]],
[[[ 0., 1.], [ 0., 0.]], [[ 0., 1.], [ 0., 0.]]],
[[[ 1., -1.], [-1., 0.]], [[ 1., -1.], [-1., 0.]]]
])/2.,
(3, 'pentahedron', 2): np.array([
# 0
[[[ 0. , 0. , 0. ], [-1. , 0. , 0. ], [ 1. , 0. , 0. ]],
[[ 0. , 0. , 0. ], [ 0.5, 0. , 0. ], [-1. , 0. , 0. ]],
[[ 0. , 0. , 0. ], [ 0.5, 0. , 0. ], [ 0. , 0. , 0. ]]],
# 1
[[[ 0. , -1. , 1. ], [ 0. , 0. , 0. ], [ 0. , 0. , 0. ]],
[[ 0. , 0.5, -1. ], [ 0. , 0. , 0. ], [ 0. , 0. , 0. ]],
[[ 0. , 0.5, 0. ], [ 0. , 0. , 0. ], [ 0. , 0. , 0. ]]],
# 2
[[[ 0. , -1. , 1. ], [-1. , 2. , 0. ], [ 1. , 0. , 0. ]],
[[-0.5, 1.5, -1. ], [ 1.5, -2. , 0. ], [-1. , 0. , 0. ]],
[[ 0.5, -0.5, 0. ], [-0.5, 0. , 0. ], [ 0. , 0. , 0. ]]],
# 3
[[[ 0. , 0. , 0. ], [-1. , 0. , 0. ], [ 1. , 0. , 0. ]],
[[ 0. , 0. , 0. ], [-0.5, 0. , 0. ], [ 1. , 0. , 0. ]],
[[ 0. , 0. , 0. ], [ 0.5, 0. , 0. ], [ 0. , 0. , 0. ]]],
# 4
[[[ 0. , -1. , 1. ], [ 0. , 0. , 0. ], [ 0. , 0. , 0. ]],
[[ 0. , -0.5, 1. ], [ 0. , 0. , 0. ], [ 0. , 0. , 0. ]],
[[ 0. , 0.5, 0. ], [ 0. , 0. , 0. ], [ 0. , 0. , 0. ]]],
# 5
[[[ 0. , -1. , 1. ], [-1. , 2. , 0. ], [ 1. , 0. , 0. ]],
[[ 0.5, -1.5, 1. ], [-1.5, 2. , 0. ], [ 1. , 0. , 0. ]],
[[ 0.5, -0.5, 0. ], [-0.5, 0. , 0. ], [ 0. , 0. , 0. ]]],
# 6
[[[ 0. , 0. , 0. ], [ 0. , 2. , 0. ], [ 0. , 0. , 0. ]],
[[ 0. , 0. , 0. ], [ 0. , -2. , 0. ], [ 0. , 0. , 0. ]],
[[ 0. , 0. , 0. ], [ 0. , 0. , 0. ], [ 0. , 0. , 0. ]]],
# 7
[[[ 0. , 2. , -2. ], [ 0. , -2. , 0. ], [ 0. , 0. , 0. ]],
[[ 0. , -2. , 2. ], [ 0. , 2. , 0. ], [ 0. , 0. , 0. ]],
[[ 0. , 0. , 0. ], [ 0. , 0. , 0. ], [ 0. , 0. , 0. ]]],
# 8
[[[ 0. , 0. , 0. ], [ 2. , -2. , 0. ], [-2. , 0. , 0. ]],
[[ 0. , 0. , 0. ], [-2. , 2. , 0. ], [ 2. , 0. , 0. ]],
[[ 0. , 0. , 0. ], [ 0. , 0. , 0. ], [ 0. , 0. , 0. ]]],
# 9
[[[ 0. , 0. , 0. ], [ 1. , 0. , 0. ], [ 0. , 0. , 0. ]],
[[ 0. , 0. , 0. ], [ 0. , 0. , 0. ], [ 0. , 0. , 0. ]],
[[ 0. , 0. , 0. ], [-1. , 0. , 0. ], [ 0. , 0. , 0. ]]],
# 10
[[[ 0. , 1. , 0. ], [ 0. , 0. , 0. ], [ 0. , 0. , 0. ]],
[[ 0. , 0. , 0. ], [ 0. , 0. , 0. ], [ 0. , 0. , 0. ]],
[[ 0. , -1. , 0. ], [ 0. , 0. , 0. ], [ 0. , 0. , 0. ]]],
# 11
[[[ 1. , -1. , 0. ], [-1. , 0. , 0. ], [ 0. , 0. , 0. ]],
[[ 0. , 0. , 0. ], [ 0. , 0. , 0. ], [ 0. , 0. , 0. ]],
[[-1. , 1. , 0. ], [ 1. , 0. , 0. ], [ 0. , 0. , 0. ]]],
# 12
[[[ 0. , 0. , 0. ], [ 0. , 2. , 0. ], [ 0. , 0. , 0. ]],
[[ 0. , 0. , 0. ], [ 0. , 2. , 0. ], [ 0. , 0. , 0. ]],
[[ 0. , 0. , 0. ], [ 0. , 0. , 0. ], [ 0. , 0. , 0. ]]],
# 13
[[[ 0. , 2. , -2. ], [ 0. , -2. , 0. ], [ 0. , 0. , 0. ]],
[[ 0. , 2. , -2. ], [ 0. , -2. , 0. ], [ 0. , 0. , 0. ]],
[[ 0. , 0. , 0. ], [ 0. , 0. , 0. ], [ 0. , 0. , 0. ]]],
# 14
[[[ 0. , 0. , 0. ], [ 2. , -2. , 0. ], [-2. , 0. , 0. ]],
[[ 0. , 0. , 0. ], [ 2. , -2. , 0. ], [-2. , 0. , 0. ]],
[[ 0. , 0. , 0. ], [ 0. , 0. , 0. ], [ 0. , 0. , 0. ]]],
])}
# pylint: enable=bad-whitespace, line-too-long
def __init__(self, element):
self._shape, self._order, self._inter_poly, self._dim, \
self._nnodes = self.ELEMENT_TYPES[element]
self._ksi = self.NATURAL_COORDS[(self._dim, self._shape)][:self._nnodes]
self._g = self.QUADRATURE_G[(self._dim, self._shape, self._order)]
self._w = self.QUADRATURE_W[(self._dim, self._shape, self._order)]
self._poly_shape = ()
self._monome = []
def polyval(self, x, p):
if self._dim == 1:
return poly.polyval(x[0], p)
if self._dim == 2:
return poly.polyval2d(x[0], x[1], p)
if self._dim == 3:
return poly.polyval3d(x[0], x[1], x[2], p)
return None
def shape_from_monomes(self):
momo = self.MONOMES[(self._dim, self._shape)][:self._nnodes]
_shapes = list(momo[0])
for s, _ in enumerate(_shapes):
_shapes[s] = max(momo[:, s])+1
self._poly_shape = tuple(_shapes)
self._monome = []
for m in momo:
p = np.zeros(self._poly_shape)
p[tuple(m)] = 1
self._monome.append(p)
# evaluate polynomial constant for shapes
_x = self._ksi
_xe = np.zeros((self._nnodes, self._nnodes))
for n in range(self._nnodes):
_xe[:, n] = [self.polyval(_x[n], m) for m in self._monome]
_a = np.linalg.inv(_xe)
_n = np.zeros((self._nnodes,) + self._monome[0].shape)
# set shapes polynomials
for n in range(self._nnodes):
for m in range(len(self._monome)):
_n[n] += _a[n, m] * self._monome[m]
return _n
def compute_shapes(self):
if (self._dim, self._shape) in self.MONOMES:
return self.shape_from_monomes()
_n = self.SHAPES[(self._dim, self._shape, self._order)]
self._poly_shape = _n[0].shape
return _n
# pylint: disable=too-many-locals,too-many-branches
def precompute(self, **kwargs):
X = np.array(kwargs["X"], copy=False)
nb_element = X.shape[0]
X = X.reshape(nb_element, self._nnodes, self._dim)
_x = self._ksi
_n = self.compute_shapes()
# sanity check on shapes
for n in range(self._nnodes):
for m in range(self._nnodes):
v = self.polyval(_x[n], _n[m])
ve = 1. if n == m else 0.
test = np.isclose(v, ve)
if not test:
raise Exception("Most probably an error in the shapes evaluation")
# compute shapes derivatives
_b = np.zeros((self._dim, self._nnodes,) + self._poly_shape)
for d in range(self._dim):
for n in range(self._nnodes):
_der = poly.polyder(_n[n], axis=d)
_mshape = np.array(self._poly_shape)
_mshape[d] = _mshape[d] - _der.shape[d]
_mshape = tuple(_mshape)
_comp = np.zeros(_mshape)
if self._dim == 1:
_bt = np.hstack((_der, _comp))
else:
if d == 0:
_bt = np.vstack((_der, _comp))
if d == 1:
_bt = np.hstack((_der, _comp))
if d == 2:
_bt = np.dstack((_der, _comp))
_b[d, n] = _bt
_nb_quads = len(self._g)
_nq = np.zeros((_nb_quads, self._nnodes))
_bq = np.zeros((_nb_quads, self._dim, self._nnodes))
# evaluate shapes and shapes derivatives on gauss points
for q in range(_nb_quads):
_g = self._g[q]
for n in range(self._nnodes):
_nq[q, n] = self.polyval(_g, _n[n])
for d in range(self._dim):
_bq[q, d, n] = self.polyval(_g, _b[d, n])
_j = np.array(kwargs['j'], copy=False).reshape((nb_element, _nb_quads))
_B = np.array(kwargs['B'], copy=False).reshape((nb_element, _nb_quads,
self._nnodes, self._dim))
_N = np.array(kwargs['N'], copy=False).reshape((nb_element, _nb_quads, self._nnodes))
_Q = kwargs['Q']
if np.linalg.norm(_Q - self._g.T) > 1e-15:
raise Exception('Not using the same quadrature'
' points norm({0} - {1}) = {2}'.format(_Q, self._g.T,
np.linalg.norm(_Q - self._g.T)))
for e in range(nb_element):
for q in range(_nb_quads):
_J = np.matmul(_bq[q], X[e])
if np.linalg.norm(_N[e, q] - _nq[q]) > 1e-10:
print("{0},{1}".format(e, q))
print(_N[e, q])
print(_nq[q])
_N[e, q] = _nq[q]
_tmp = np.matmul(np.linalg.inv(_J), _bq[q])
_B[e, q] = _tmp.T
_j[e, q] = np.linalg.det(_J) * self._w[q]

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