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rAKA/test/test_fe_engine/py_engine/py_engine.py
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py_engine.py
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#!/usr/bin/env python3
__all__ = ['Shapes']
import numpy as np
import numpy.polynomial.polynomial as poly
import aka_test
class Shapes(object):
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]]),
}
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([]),
(3, 'triangle', 2): np.array([]),
(2, 'quadrangle', 1): np.array([1., 1., 1., 1.]),
(2, 'quadrangle', 2): np.array([]),
(3, 'quadrangle', 1): np.array([1., 1., 1., 1.,
1., 1., 1., 1.]),
(3, 'quadrangle', 2): np.array([]),
}
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.]]),
(2, 'quadrangle', 1): np.array([[-1., -1.], [-1., 1.],
[ 1., 1.], [ 1., -1.]])/np.sqrt(3.),
(2, 'quadrangle', 2): np.array([]),
(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([]),
}
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], [0, 1, 1], [1, 0, 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]]),
}
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)]
momo = self.MONOMES[(self._dim, self._shape)][:self._nnodes]
_shape = list(momo[0])
for s in range(len(_shape)):
_shape[s] = max(momo[:,s])+1
self._poly_shape = tuple(_shape)
self._monome = []
for m in momo:
p = np.zeros(self._poly_shape)
p[tuple(m)] = 1
self._monome.append(p)
def polyval(self, x, p):
if 1 == self._dim:
return poly.polyval(x[0], p)
if 2 == self._dim:
return poly.polyval2d(x[0], x[1], p)
if 3 == self._dim:
return poly.polyval3d(x[0], x[1], x[2], p)
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)
# 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]
# 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("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 1 == self._dim:
_bt = np.hstack((_der, _comp))
else:
if 0 == d:
_bt = np.vstack((_der, _comp))
if 1 == d:
_bt = np.hstack((_der, _comp))
if 2 == d:
_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._dim, self._nnodes))
_N = np.array(kwargs['N'], copy=False).reshape((nb_element, _nb_quads, self._nnodes))
for e in range(nb_element):
for q in range(_nb_quads):
_J = np.matmul(_bq[q], X[e])
_N[e, q] = _nq[q]
_B[e, q] = np.matmul(np.linalg.inv(_J), _bq[q])
_j[e, q] = np.linalg.det(_J) * self._w[q]
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