import sympy as sm
import numpy as nm
import scipy.sparse as sps
from sfepy.base.base import dict_to_struct
import sfepy.base.testing as tst
conf = {
'name' : 'semismooth_newton',
'kind' : 'nls.semismooth_newton',
'semismooth' : True,
'i_max' : 10,
'eps_a' : 1e-8,
'eps_r' : 1e-2,
'macheps' : 1e-16,
'lin_red' : 1e-2, # Linear system error < (eps_a * lin_red).
'ls_red' : {'regular' : 0.1, 'steepest_descent' : 0.01},
'ls_red_warp' : 0.001,
'ls_on' : 2.0,
'ls_min' : 1e-10,
## 'log' : {'plot' : 'aux.png'},
}
ls_conf = {
'name' : 'ls',
'kind' : 'ls.scipy_direct',
}
alpha = 30.0 * nm.pi / 180.0
ca = nm.cos(alpha)
sa = nm.sin(alpha)
# Coefficient of friction.
fc = 0.01
# Spring (bar) stiffnesses.
ks = nm.ones((7,), dtype=nm.float64)
# Load vector.
fs = nm.array([0, 0, 1, 1, -1, -1], dtype=nm.float64)
[docs]def eval_matrix(mtx, **kwargs):
num_mtx = nm.zeros(mtx.shape, dtype=nm.float64)
mtx = mtx.subs(kwargs)
for ir in range(mtx.shape[0]):
for ic in range(mtx.shape[1]):
num_mtx[ir,ic] = float(mtx[ir,ic])
return num_mtx
[docs]def convert_to_csr(m_in):
m_out = {}
for key, mtx in m_in.items():
m_out[key] = sps.csr_matrix(mtx)
return m_out
[docs]def define_matrices():
e = sm.zeros(7, 1)
e[0] = sm.sympify('u1')
e[1] = sm.sympify('c * u3 + s * v3')
e[2] = sm.sympify('u2 - u1')
e[3] = sm.sympify('v4 + u2n')
e[4] = sm.sympify('u4 - u3')
e[5] = sm.sympify('v3 + u1n')
e[6] = sm.sympify('c * u4 + s * v4 - c * u1 + s * u1n')
k = sm.Matrix(7, 1, lambda i, j: sm.Symbol('k%d' % i))
psi = (k.T * e.applyfunc(lambda x: x**2))[0]
work = sm.sympify('u1 * g1 + u2 * g2')
w = ['u1', 'u2', 'u3', 'u4', 'v3', 'v4']
wbar = w[:2]
g = ['g1', 'g2']
mtx_a = sm.zeros(6, 6)
for ir in range(mtx_a.shape[0]):
for ic in range(mtx_a.shape[1]):
mtx_a[ir,ic] = sm.diff(sm.diff(psi, w[ir]), w[ic])
mtx_b_bar = sm.zeros(2, 6)
for ir in range(mtx_b_bar.shape[0]):
for ic in range(mtx_b_bar.shape[1]):
mtx_b_bar[ir,ic] = sm.diff(sm.diff(work, g[ir]), w[ic])
mtx_b = sm.zeros(2, 2)
for ir in range(mtx_b.shape[0]):
for ic in range(mtx_b.shape[1]):
mtx_b[ir,ic] = sm.diff(sm.diff(work, wbar[ir]), g[ic])
mtx_c = sm.eye(2)
sn1 = sm.diff(psi, 'u1n').subs('u1n', 0)
sn2 = sm.diff(psi, 'u2n').subs('u2n', 0)
sn = sm.Matrix([sn1, sn2])
mtx_d = sm.zeros(2, 6)
for ir in range(mtx_d.shape[0]):
for ic in range(mtx_d.shape[1]):
mtx_d[ir,ic] = sm.diff(sn[ir], w[ic])
subsd = {'c' : ca, 's' : sa}
for ii, _k in enumerate(k):
subsd['k%d' % ii] = ks[ii]
m = {
'A' : eval_matrix(mtx_a, **subsd),
'Bb' : eval_matrix(mtx_b_bar),
'B' : eval_matrix(mtx_b),
'C' : eval_matrix(mtx_c),
}
ms = convert_to_csr(m)
m['sn'] = sn.subs(subsd)
m['D'] = mtx_d.subs(subsd)
m['fs'] = fs
return m, ms, w
[docs]def test_semismooth_newton():
import numpy as nm
from sfepy.solvers import Solver
m, ms, w_names = define_matrices()
ns = [0, 6, 2, 2]
offsets = nm.cumsum(ns)
nx = offsets[-1]
iw = slice(offsets[0], offsets[1])
ig = slice(offsets[1], offsets[2])
il = slice(offsets[2], offsets[3])
def fun_smooth(vec_x):
xw = vec_x[iw]
xg = vec_x[ig]
xl = vec_x[il]
rw = ms['A'] * xw - ms['Bb'].T * xg - m['fs']
rg = ms['Bb'] * xw + xl * xg
rwg = nm.r_[rw, rg]
return rwg
def fun_smooth_grad(vec_x):
xl = vec_x[il]
xg = vec_x[ig]
mzl = nm.zeros((6, 2), dtype=nm.float64)
mw = nm.c_[m['A'], -m['Bb'].T, mzl]
mg = nm.c_[m['Bb'], nm.diag(xl), nm.diag(xg)]
mx = nm.r_[mw, mg]
mx = sps.csr_matrix(mx)
return mx
def fun_a(vec_x):
xw = vec_x[iw]
xg = vec_x[ig]
subsd = {}
for ii, key in enumerate(w_names):
subsd[key] = xw[ii]
sn = eval_matrix(m['sn'], **subsd).squeeze()
ra = nm.abs(xg) - fc * nm.abs(sn)
return -ra
def fun_a_grad(vec_x):
xw = vec_x[iw]
xg = vec_x[ig]
xl = vec_x[il]
subsd = {}
for ii, key in enumerate(w_names):
subsd[key] = xw[ii]
md = eval_matrix(m['D'], **subsd)
sn = eval_matrix(m['sn'], **subsd).squeeze()
ma = nm.zeros((xl.shape[0], nx), dtype=nm.float64)
ma[:,iw] = - fc * nm.sign(sn)[:,None] * md
ma[:,ig] = nm.sign(xg)[:,None] * m['C']
return -sps.csr_matrix(ma)
def fun_b(vec_x):
xl = vec_x[il]
return xl
def fun_b_grad(vec_x):
xl = vec_x[il]
mb = nm.zeros((xl.shape[0], nx), dtype=nm.float64)
mb[:,il] = m['C']
return sps.csr_matrix(mb)
vec_x0 = 0.1 * nm.ones((nx,), dtype=nm.float64)
lin_solver = Solver.any_from_conf(dict_to_struct(ls_conf))
status = {}
solver = Solver.any_from_conf(dict_to_struct(conf),
fun_smooth=fun_smooth,
fun_smooth_grad=fun_smooth_grad,
fun_a=fun_a,
fun_a_grad=fun_a_grad,
fun_b=fun_b,
fun_b_grad=fun_b_grad,
lin_solver=lin_solver,
status=status)
vec_x = solver(vec_x0)
xw = vec_x[iw]
xg = vec_x[ig]
xl = vec_x[il]
tst.report('x:', xw)
tst.report('g:', xg)
tst.report('l:', xl)
subsd = {}
for ii, key in enumerate(w_names):
subsd[key] = xw[ii]
sn = eval_matrix(m['sn'], **subsd).squeeze()
tst.report('sn:', sn)
assert status['condition'] == 0