DAMASK_EICMD/processing/misc/yieldSurface.py

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#!/usr/bin/python
# -*- coding: UTF-8 no BOM -*-
import threading,time,os,subprocess,shlex,string
import numpy as np
from scipy.linalg import svd
from optparse import OptionParser
import damask
from damask.util import leastsqBound
scriptID = string.replace('$Id$','\n','\\n')
scriptName = scriptID.split()[1][:-3]
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def execute(cmd,streamIn=None,wd='./'):
'''
executes a command in given directory and returns stdout and stderr for optional stdin
'''
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initialPath=os.getcwd()
os.chdir(wd)
process = subprocess.Popen(shlex.split(cmd),stdout=subprocess.PIPE,stderr = subprocess.PIPE,stdin=subprocess.PIPE)
if streamIn != None:
out,error = process.communicate(streamIn.read())
else:
out,error = process.communicate()
2014-07-07 19:47:44 +05:30
os.chdir(initialPath)
return out,error
def principalStresses(sigmas):
'''
computes principal stresses (i.e. eigenvalues) for a set of Cauchy stresses.
sorted in descending order.
'''
lambdas=np.zeros(0,'d')
for i in xrange(np.shape(sigmas)[1]):
eigenvalues = np.linalg.eigvalsh(sym6to33(sigmas[:,i]))
lambdas = np.append(lambdas,np.sort(eigenvalues)[::-1]) #append eigenvalues in descending order
lambdas = np.transpose(lambdas.reshape(np.shape(sigmas)[1],3))
return lambdas
def stressInvariants(lambdas):
'''
computes stress invariants (i.e. eigenvalues) for a set of principal Cauchy stresses.
'''
Is=np.zeros(0,'d')
for i in xrange(np.shape(lambdas)[1]):
I = np.array([lambdas[0,i]+lambdas[1,i]+lambdas[2,i],\
lambdas[0,i]*lambdas[1,i]+lambdas[1,i]*lambdas[2,i]+lambdas[2,i]*lambdas[0,i],\
lambdas[0,i]*lambdas[1,i]*lambdas[2,i]])
Is = np.append(Is,I)
Is = Is.reshape(3,np.shape(lambdas)[1])
return Is
def formatOutput(n, type='%-14.6f'):
return ''.join([type for i in xrange(n)])
def sym6to33(sigma6):
''' Shape the symmetric stress tensor(6,1) into (3,3) '''
sigma33 = np.empty((3,3))
sigma33[0,0] = sigma6[0]; sigma33[1,1] = sigma6[1]; sigma33[2,2] = sigma6[2];
sigma33[0,1] = sigma6[3]; sigma33[1,0] = sigma6[3]
sigma33[1,2] = sigma6[4]; sigma33[2,1] = sigma6[4]
sigma33[2,0] = sigma6[5]; sigma33[0,2] = sigma6[5]
return sigma33
def array2tuple(array):
'''transform numpy.array into tuple'''
try:
return tuple(array2tuple(i) for i in array)
except TypeError:
return array
def get_weight(ndim):
#more to do
return np.ones(ndim)
# ---------------------------------------------------------------------------------------------
# isotropic yield surfaces
# ---------------------------------------------------------------------------------------------
class Tresca(object):
'''
residuum of Tresca yield criterion (eq. 2.26)
'''
def __init__(self, uniaxialStress):
self.stress0 = uniaxialStress
def fun(self,sigma0, ydata, sigmas):
lambdas = principalStresses(sigmas)
r = np.amax(np.array([abs(lambdas[2,:]-lambdas[1,:]),\
abs(lambdas[1,:]-lambdas[0,:]),\
abs(lambdas[0,:]-lambdas[2,:])]),0) - sigma0
return r.ravel()
def jac(self,sigma0, ydata, sigmas):
return np.ones(len(ydata)) * (-1.0)
class vonMises(object):
'''
residuum of Huber-Mises-Hencky yield criterion (eq. 2.37)
'''
def __init__(self, uniaxialStress):
self.stress0 = uniaxialStress
def fun(self, sigma0, ydata, sigmas):
return HosfordBasis(sigma0, 1.0,1.0,1.0, 2.0, sigmas)
def jac(self, sigma0, ydata, sigmas):
return HosfordBasis(sigma0, 1.0,1.0,1.0, 2.0, sigmas, Jac=True, nParas=1)
class Drucker(object):
'''
residuum of Drucker yield criterion (eq. 2.41, F = sigma0)
'''
def __init__(self, uniaxialStress):
self.stress0 = uniaxialStress
def fun(self, (sigma0, C_D), ydata, sigmas):
return DruckerBasis(sigma0, C_D, 1.0, sigmas)
def jac(self, (sigma0, C_D), ydata, sigmas):
return DruckerBasis(sigma0, C_D, 1.0, sigmas, Jac=True, nParas=2)
class generalDrucker(object):
'''
residuum of general Drucker yield criterion (eq. 2.42, F = sigma0)
'''
def __init__(self, uniaxialStress):
self.stress0 = uniaxialStress
def fun(self, (sigma0, C_D, p), ydata, sigmas):
return DruckerBasis(sigma0, C_D, p, sigmas)
def jac(self, (sigma0, C_D, p), ydata, sigmas):
return DruckerBasis(sigma0, C_D, p, sigmas, Jac=True, nParas=3)
class Hosford(object):
'''
residuum of Hershey yield criterion (eq. 2.43, Y = sigma0)
'''
def __init__(self, uniaxialStress):
self.stress0 = uniaxialStress
def fun(self, (sigma0, a), ydata, sigmas):
return HosfordBasis(sigma0, 1.0,1.0,1.0, a, sigmas)
def jac(self, (sigma0, a), ydata, sigmas):
return HosfordBasis(sigma0, 1.0,1.0,1.0, a, sigmas, Jac=True, nParas=2)
class Hill1948(object):
'''
residuum of Hill 1948 quadratic yield criterion (eq. 2.48)
'''
def __init__(self, uniaxialStress):
self.stress0 = uniaxialStress
def fun(self, (F,G,H,L,M,N), ydata, sigmas):
r = F*(sigmas[1]-sigmas[2])**2.0 + G*(sigmas[2]-sigmas[0])**2.0 + H*(sigmas[0]-sigmas[1])**2.0\
+ 2.0*L*sigmas[4]**2.0 + 2.0*M*sigmas[5]**2.0 + 2.0*N*sigmas[3]**2.0 - 1.0
return r.ravel()/2.0
def jac(self, (F,G,H,L,M,N), ydata, sigmas):
jF=(sigmas[1]-sigmas[2])**2.0; jG=(sigmas[2]-sigmas[0])**2.0; jH=(sigmas[0]-sigmas[1])**2.0
jL=2.0*sigmas[4]**2.0; jM=2.0*sigmas[5]**2.0; jN=2.0*sigmas[3]**2.0
jaco = []
for jacv in zip(jF, jG, jH, jL, jM, jN): jaco.append(jacv)
return np.array(jaco)
class Hill1979(object):
'''
residuum of Hill 1979 non-quadratic yield criterion (eq. 2.48)
'''
def __init__(self, uniaxialStress):
self.stress0 = uniaxialStress
def fun(self, (f,g,h,a,b,c,m), ydata, sigmas):
return Hill1979Basis(self.stress0, f,g,h,a,b,c,m, sigmas)
def jac(self, (f,g,h,a,b,c,m), ydata, sigmas):
return Hill1979Basis(self.stress0, f,g,h,a,b,c,m, sigmas, Jac=True)
class generalHosford(object):
'''
residuum of Hershey yield criterion (eq. 2.104, sigmas = sigma0)
'''
def __init__(self, uniaxialStress):
self.stress0 = uniaxialStress
def fun(self, (sigma0, F, G, H, a), ydata, sigmas, nParas=5):
return HosfordBasis(sigma0, F, G, H, a, sigmas)
def jac(self, (sigma0, F, G, H, a), ydata, sigmas):
return HosfordBasis(sigma0, F,G,H, a, sigmas, Jac=True, nParas=5)
class Barlat1991iso(object):
'''
residuum of isotropic Barlat 1991 yield criterion (eq. 2.37)
'''
def __init__(self, uniaxialStress):
self.stress0 = uniaxialStress
def fun(self, (sigma0, m), ydata, sigmas):
return Barlat1991Basis(sigma0, 1.0,1.0,1.0,1.0,1.0,1.0, m, sigmas)
def jac(self, (sigma0, m), ydata, sigmas):
return Barlat1991Basis(sigma0, 1.0,1.0,1.0,1.0,1.0,1.0, m, sigmas, Jac=True, nParas=2)
class Barlat1991aniso(object):
'''
residuum of anisotropic Barlat 1991 yield criterion (eq. 2.37)
'''
def __init__(self, uniaxialStress):
self.stress0 = uniaxialStress
def fun(self, (sigma0, a,b,c,f,g,h, m), ydata, sigmas):
return Barlat1991Basis(sigma0, a,b,c,f,g,h, m, sigmas)
def jac(self, (sigma0, a,b,c,f,g,h, m), ydata, sigmas):
return Barlat1991Basis(sigma0, a,b,c,f,g,h, m, sigmas, Jac=True, nParas=8)
class Yld200418p(object):
'''
residuum of anisotropic Barlat 1991 yield criterion (eq. 2.37)
'''
def __init__(self, uniaxialStress):
self.stress0 = uniaxialStress
def fun(self, (sigma0, c12,c21,c23,c32,c31,c13,c44,c55,c66,
d12,d21,d23,d32,d31,d13,d44,d55,d66, m), ydata, sigmas):
return Yld200418pBasis(sigma0, c12,c21,c23,c32,c31,c13,c44,c55,c66,
d12,d21,d23,d32,d31,d13,d44,d55,d66, m, sigmas)
def jac(self, (sigma0, c12,c21,c23,c32,c31,c13,c44,c55,c66,
d12,d21,d23,d32,d31,d13,d44,d55,d66, m), ydata, sigmas):
return Yld200418pBasis(sigma0, c12,c21,c23,c32,c31,c13,c44,c55,c66,
d12,d21,d23,d32,d31,d13,d44,d55,d66, m, sigmas, Jac=True)
class KarafillisBoyce(object):
'''
residuum of Karafillis-Boyce yield criterion
'''
def __init__(self, uniaxialStress):
self.stress0 = uniaxialStress
def fun(self, (c11,c12,c13,c14,c15,c16,c21,c22,c23,c24,c25,c26,
b1, b2, a, alpha), ydata, sigmas):
return KarafillisBoyceBasis(self.stress0, c11,c12,c13,c14,c15,c16,c21,c22,c23,c24,c25,c26,
b1, b2, a, alpha, sigmas)
def jac(self, (c11,c12,c13,c14,c15,c16,c21,c22,c23,c24,c25,c26,
b1, b2, a, alpha), ydata, sigmas):
return KarafillisBoyceBasis(self.stress0, c11,c12,c13,c14,c15,c16,c21,c22,c23,c24,c25,c26,
b1, b2, a, alpha, sigmas, Jac=True)
class BBC2003(object):
'''
residuum of anisotropic Barlat 1991 yield criterion (eq. 2.37)
'''
def __init__(self, uniaxialStress):
self.stress0 = uniaxialStress
def fun(self, (sigma0, a,b,c, d,e,f,g, k), ydata, sigmas):
return BBC2003Basis(sigma0, a,b,c, d,e,f,g, k, sigmas)
def jac(self, (sigma0, a,b,c, d,e,f,g, k), ydata, sigmas):
return BBC2003Basis(sigma0, a,b,c, d,e,f,g, k, sigmas, Jac=True)
class BBC2005(object):
'''
residuum of anisotropic Barlat 1991 yield criterion (eq. 2.37)
'''
def __init__(self, uniaxialStress):
self.stress0 = uniaxialStress
def fun(self, (a,b,L, M, N, P, Q, R, k), ydata, sigmas):
return BBC2005Basis(self.stress0, a,b,L, M, N, P, Q, R, k, sigmas)
def jac(self, (a,b,L, M, N, P, Q, R, k), ydata, sigmas):
return BBC2005Basis(self.stress0, a,b,L, M, N, P, Q, R, k, sigmas, Jac=True)
class Cazacu_Barlat2D(object):
'''
'''
def __init__(self, uniaxialStress):
self.stress0 = uniaxialStress
def fun(self, (a1,a2,a3,a4,b1,b2,b3,b4,b5,b10,c), ydata, sigmas):
return Cazacu_Barlat2DBasis(a1,a2,a3,a4,b1,b2,b3,b4,b5,b10,c,
self.stress0, sigmas)
def jac(self, (a1,a2,a3,a4,b1,b2,b3,b4,b5,b10,c), ydata, sigmas):
return Cazacu_Barlat2DBasis(a1,a2,a3,a4,b1,b2,b3,b4,b5,b10,c,
self.stress0, sigmas,Jac=True)
class Cazacu_Barlat3D(object):
'''
'''
def __init__(self, uniaxialStress):
self.stress0 = uniaxialStress
def fun(self, (a1,a2,a3,a4,a5,a6,b1,b2,b3,b4,b5,b6,b7,b8,b9,b10,b11,c),ydata, sigmas):
return Cazacu_Barlat3DBasis(a1,a2,a3,a4,a5,a6,b1,b2,b3,b4,b5,b6,b7,b8,b9,b10,b11,c,
self.stress0, sigmas)
def jac(self, (a1,a2,a3,a4,a5,a6,b1,b2,b3,b4,b5,b6,b7,b8,b9,b10,b11,c),ydata, sigmas):
return Cazacu_Barlat3DBasis(a1,a2,a3,a4,a5,a6,b1,b2,b3,b4,b5,b6,b7,b8,b9,b10,b11,c,
self.stress0, sigmas,Jac=True)
class Vegter(object):
'''
Vegter yield criterion
'''
def __init__(self, refPts, refNormals,nspace=11):
self.refPts, self.refNormals = self._getRefPointsNormals(refPts, refNormals)
self.hingePts = self._getHingePoints()
self.nspace = nspace
def _getRefPointsNormals(self,refPtsQtr,refNormalsQtr):
if len(refPtsQtr) == 12:
refPts = refPtsQtr
refNormals = refNormalsQtr
else:
refPts = np.empty([13,2])
refNormals = np.empty([13,2])
refPts[12] = refPtsQtr[0]
refNormals[12] = refNormalsQtr[0]
for i in xrange(3):
refPts[i] = refPtsQtr[i]
refPts[i+3] = refPtsQtr[3-i][::-1]
refPts[i+6] =-refPtsQtr[i]
refPts[i+9] =-refPtsQtr[3-i][::-1]
refNormals[i] = refNormalsQtr[i]
refNormals[i+3] = refNormalsQtr[3-i][::-1]
refNormals[i+6] =-refNormalsQtr[i]
refNormals[i+9] =-refNormalsQtr[3-i][::-1]
return refPts,refNormals
def _getHingePoints(self):
'''
calculate the hinge point B according to the reference points A,C and the normals n,m
refPoints = np.array([[p1_x, p1_y], [p2_x, p2_y]]);
refNormals = np.array([[n1_x, n1_y], [n2_x, n2_y]])
'''
def hingPoint(points, normals):
A1 = points[0][0]; A2 = points[0][1]
C1 = points[1][0]; C2 = points[1][1]
n1 = normals[0][0]; n2 = normals[0][1]
m1 = normals[1][0]; m2 = normals[1][1]
B1 = (m2*(n1*A1 + n2*A2) - n2*(m1*C1 + m2*C2))/(n1*m2-m1*n2)
B2 = (n1*(m1*C1 + m2*C2) - m1*(n1*A1 + n2*A2))/(n1*m2-m1*n2)
return np.array([B1,B2])
return np.array([hingPoint(self.refPts[i:i+2],self.refNormals[i:i+2]) for i in xrange(len(self.refPts)-1)])
def getBezier(self):
def bezier(R,H):
b = []
for mu in np.linspace(0.0,1.0,self.nspace):
b.append(np.array(R[0]*np.ones_like(mu) + 2.0*mu*(H - R[0]) + mu**2*(R[0]+R[1] - 2.0*H)))
return b
return np.array([bezier(self.refPts[i:i+2],self.hingePts[i]) for i in xrange(len(self.refPts)-1)])
def VetgerCriterion(stress,lankford, rhoBi0, theta=0.0):
'''
0-pure shear; 1-uniaxial; 2-plane strain; 3-equi-biaxial
'''
def getFourierParas(r):
# get the value after Fourier transformation
nset = len(r)
lmatrix = np.empty([nset,nset])
theta = np.linspace(0.0,np.pi/2,nset)
for i,th in enumerate(theta):
lmatrix[i] = np.array([np.cos(2*j*th) for j in xrange(nset)])
return np.linalg.solve(lmatrix, r)
nps = len(stress)
if nps%4 != 0:
print ('Warning: the number of stress points is uncorrect, stress points of %s are missing in set %i'%(
['eq-biaxial, plane strain & uniaxial', 'eq-biaxial & plane strain','eq-biaxial'][nps%4-1],nps/4+1))
else:
nset = nps/4
strsSet = stress.reshape(nset,4,2)
refPts = np.empty([4,2])
fouriercoeffs = np.array([np.cos(2.0*i*theta) for i in xrange(nset)])
for i in xrange(2):
refPts[3,i] = sum(strsSet[:,3,i])/nset
for j in xrange(3):
refPts[j,i] = np.dot(getFourierParas(strsSet[:,j,i]), fouriercoeffs)
rhoUn = np.dot(getFourierParas(-lankford/(lankford+1)), fouriercoeffs)
rhoBi = (rhoBi0+1 + (rhoBi0-1)*np.cos(2.0*theta))/(rhoBi0+1 - (rhoBi0-1)*np.cos(2.0*theta))
nVec = lambda rho : np.array([1.0,rho]/np.sqrt(1.0+rho**2))
refNormals = np.array([nVec(-1.0),nVec(rhoUn),nVec(0.0),nVec(rhoBi)])
vegter = Vegter(refPts, refNormals)
def Cazacu_Barlat3DBasis(a1,a2,a3,a4,a5,a6,b1,b2,b3,b4,b5,b6,b7,b8,b9,b10,b11,c,
sigma0,sigmas, Jac = False):
'''
residuum of the 3D Cazacu<EFBFBD>Barlat (CZ) yield criterion
'''
s11 = sigmas[0]; s22 = sigmas[1]; s33 = sigmas[2]
s12 = sigmas[3]; s23 = sigmas[4]; s31 = sigmas[5]
s123, s321 = s11*s22*s33, s12*s23*s31
s1_2, s2_2, s3_2 = s11**2, s22**2, s33**2
s1_3, s2_3, s3_3 = s11*s1_2, s22*s2_2, s33*s3_2
s12_2, s23_2, s31_2 = s12**2, s23**2, s31**2
d12_2, d23_2, d31_2 = (s11-s22)**2, (s22-s33)**2, (s33-s11)**2
J20 = ( a1*d12_2 + a2*d23_2 + a3*d31_2 )/6.0 + a4*s12_2 + a5*s23_2 + a6*s31_2
J30 = ( (b1 +b2 )*s1_3 + (b3 +b4 )*s2_3 + ( b1+b4-b2 + b1+b4-b3 )*s3_3 )/27.0- \
( (b1*s22+b2*s33)*s1_2 + (b3*s33+b4*s11)*s2_2 + ((b1+b4-b2)*s11 + (b1+b4-b3)*s22)*s3_2 )/9.0 + \
( (b1+b4)*s123/9.0 + b11*s321 )*2.0 - \
( ( 2.0*b9 *s22 - b8*s33 - (2.0*b9 -b8)*s11 )*s31_2 +
( 2.0*b10*s33 - b5*s22 - (2.0*b10-b5)*s11 )*s12_2 +
( (b6+b7)*s11 - b6*s22 - b7*s33 )*s23_2
)/3.0
f0 = (J20**3 - c*J30**2)/18.0
r = f0**(1.0/6.0)*(3.0/sigma0)
if not Jac:
return (r - 1.0).ravel()
else:
drdf = r/f0/108.0
dj2 = drdf*3.0*J20**2.0
dj3 = -drdf*2.0*J30*c
jc = -drdf*J30**2
ja1,ja2,ja3 = dj2*d12_2/6.0, dj2*d23_2/6.0, dj2*d31_2/6.0
ja4,ja5,ja6 = dj2*s12_2, dj2*s23_2, dj2*s31_2
jb1 = dj3*( (s1_3 + 2.0*s3_3)/27.0 - s22*s1_2/9.0 - (s11+s22)*s3_2/9.0 + s123/4.5 )
jb2 = dj3*( (s1_3 - s3_3)/27.0 - s33*s1_2/9.0 + s11 *s3_2/9.0 )
jb3 = dj3*( (s2_3 - s3_3)/27.0 - s33*s2_2/9.0 + s22 *s3_2/9.0 )
jb4 = dj3*( (s2_3 + 2.0*s3_3)/27.0 - s11*s2_2/9.0 - (s11+s22)*s3_2/9.0 + s123/4.5 )
jb5, jb10 = dj3*(s22 - s11)*s12_2/3.0, dj3*(s11 - s33)*s12_2/3.0*2.0
jb6, jb7 = dj3*(s22 - s11)*s23_2/3.0, dj3*(s33 - s11)*s23_2/3.0
jb8, jb9 = dj3*(s33 - s11)*s31_2/3.0, dj3*(s11 - s22)*s31_2/3.0*2.0
jb11 = dj3*s321*2.0
jaco = []
for jacv in zip(ja1,ja2,ja3,ja4,ja5,ja6,jb1,jb2,jb3,jb4,jb5,jb6,jb7,jb8,jb9,jb10,jb11,jc):
jaco.append(jacv)
return np.array(jaco)
def Cazacu_Barlat2DBasis(a1,a2,a3,a4,b1,b2,b3,b4,b5,b10,c,
sigma0,sigmas, Jac = False):
'''
residuum of the 2D Cazacu<EFBFBD>Barlat (CZ) yield criterion for plain stress
'''
s11 = sigmas[0]; s22 = sigmas[1]; s12 = sigmas[3]
s1_2, s2_2 = s11**2, s22**2
s1_3, s2_3 = s11*s1_2, s22*s2_2
s12_2 = s12**2
J20 = ( a1*(s11-s22)**2 + a2*s2_2 + a3*s1_2 )/6.0 + a4*s12_2
J30 = ( (b1+b2)*s1_3 + (b3+b4)*s2_3 )/27.0 - ( (b1*s11 + b4*s22)*s11*s22 )/9.0 + \
( b5*s22 + (2*b10-b5)*s11 )*s12_2/3.0
f0 = (J20**3 - c*J30**2)/18.0
r = f0**(1.0/6.0)*(3.0/sigma0)
if not Jac:
return (r - 1.0).ravel()
else:
drdf = r/f0/108.0
dj2 = drdf*3.0*J20**2.0
dj3 = -drdf*2.0*J30*c
jc = -drdf*J30**2
ja1,ja2,ja3,ja4 = dj2*(s11-s22)**2/6.0, dj2*s2_2/6.0, dj2*s1_2/6.0, dj2*s12_2
jb1, jb2 = s1_3/27.0 - s1_2*s22/9.0, s1_3/27.0
jb4, jb3 = s2_3/27.0 - s2_2*s11/9.0, s2_3/27.0
jb5, jb10= -s12_2*(s11 - s22)/3.0, s12_2*s11*2.0/3.0
jaco = []
for jacv in zip(ja1,ja2,ja3,ja4,jb1,jb2,jb3,jb4,jb5,jb10,jc):
jaco.append(jacv)
return np.array(jaco)
def DruckerBasis(sigma0, C_D, p, sigmas, Jac=False, nParas=2):
s11 = sigmas[0]; s22 = sigmas[1]; s33 = sigmas[2]
s12 = sigmas[3]; s23 = sigmas[4]; s31 = sigmas[5]
I1 = s11 + s22 + s33
I2 = s11*s22 + s22*s33 + s33*s11 - s12**2 - s23**2 - s31**2
I3 = s11*s22*s33 + 2.0*s12*s23*s31 - s12**2*s33 - s23**2*s11 - s31**2*s22
J2 = I1**2/3.0 - I2
J3 = I1**3/13.5 - I1*I2/3.0 + I3
left= J2**(3.0*p) - C_D*J3**(2.0*p); right = 3.0**(0.5)/sigma0
expo= 1.0/(6.0*p)
if not Jac:
return (left**expo*right - 1.0).ravel()
else:
jaco = []
dfdl = expo*left**(expo-1.0)
js = -left**expo*right/sigma0
jC = -dfdl*J3**(2*p)*right
if nParas == 2:
for jacv in zip(js, jC): jaco.append(jacv)
return np.array(jaco)
else:
ln = lambda x : np.log(x + 1.0e-32)
dldp = 3.0*J2**(3.0*p)*ln(J2) - 2.0*C_D*J3**(2.0*p)*ln(J3)
jp = dfdl*dldp*right + (left**expo)*ln(left)*expo/(-p)*right
for jacv in zip(js, jC, jp): jaco.append(jacv)
return np.array(jaco)
def Hill1979Basis(sigma0, f,g,h,a,b,c,m, sigmas, Jac=False):
s1,s2,s3 = principalStresses(sigmas)
d23 = s2-s3; d123 = 2.0*s1 - s2 - s3
d31 = s3-s1; d231 = 2.0*s2 - s3 - s1
d12 = s1-s2; d312 = 2.0*s3 - s1 - s2
d23s = d23**2; d123s = d123**2
d31s = d31**2; d231s = d231**2
d12s = d12**2; d312s = d312**2
m2 = m/2.0; mi = 1.0/m
base = f* d23s**m2 + g* d31s**m2 + h* d12s**m2 + \
a*d123s**m2 + b*d231s**m2 + c*d312s**m2
left = base**mi
r = left/sigma0
if not Jac:
return (r-1.0).ravel()
else:
ln = lambda x : np.log(x + 1.0e-32)
drdb = r/base*mi
dbdm = ( f* d23s**m2*ln( d23s) + g* d31s**m2*ln( d31s) + h*d12s**m2*ln( d12s) +
a*d123s**m2*ln(d123s) + b*d231s**m2*ln(d231s) + c*d312s**m2*ln(d312s) )*0.5
jf = drdb*d23s**m2; ja = drdb*d123s**m2
jg = drdb*d31s**m2; jb = drdb*d231s**m2
jh = drdb*d12s**m2; jc = drdb*d312s**m2
jm = drdb*dbdm + r*ln(base)*(-mi*mi)
jaco = []
for jacv in zip(jf,jg,jh,ja,jb,jc,jm):
jaco.append(jacv)
return np.array(jaco)
def HosfordBasis(sigma0, F,G,H, a, sigmas, Jac=False, nParas=1):
'''
residuum of Hershey yield criterion (eq. 2.43, Y = sigma0)
'''
lambdas = principalStresses(sigmas)
diff23 = abs(lambdas[1,:] - lambdas[2,:])
diff31 = abs(lambdas[2,:] - lambdas[0,:])
diff12 = abs(lambdas[0,:] - lambdas[1,:])
base = F*diff23**a + G*diff31**a + H*diff12**a; expo = 1.0/a
left = base**expo
right = 2.0**expo*sigma0
if not Jac:
if nParas == 1: return (left - right).ravel()
else: return (left/right - 1.0).ravel()
else:
ones = np.ones(np.shape(sigmas)[1])
if nParas > 1:
ln = lambda x : np.log(x + 1.0e-32)
dbda = F*ln(diff23)*diff23**a + G*ln(diff31)*diff31**a + H*ln(diff12)*diff12**a
deda = -expo*expo
drda = sigma0*(2.0**expo)*ln(2.0)*deda
dldb = expo*left/base
jaco = []
if nParas == 1: # von Mises
return ones*(-2.0**0.5)
elif nParas == 2: # isotropic Hosford
js = ones*(-2.0**expo) # d[]/dsigma0
ja = dldb*dbda + left*ln(base)*deda - drda # d[]/da
for jacv in zip(js, ja):
jaco.append(jacv)
return np.array(jaco)
elif nParas == 5: # anisotropic Hosford
js = -left/right/sigma0 #ones*(-2.0**expo) # d[]/dsigma0
jF = dldb*diff23**a/right
jG = dldb*diff31**a/right
jH = dldb*diff12**a/right
ja =(dldb*dbda + left*ln(base)*deda)/right + left*(-right**(-2))*drda # d[]/da
for jacv in zip(js, jF,jG,jH,ja):
jaco.append(jacv)
return np.array(jaco)
def Barlat1991Basis(sigma0, a, b, c, f, g, h, m, sigmas, Jac=False, nParas=2):
'''
residuum of Barlat 1997 yield criterion
'''
cos = np.cos; sin = np.sin; pi = np.pi; abs = np.abs
dAda = sigmas[1] - sigmas[2]; A = a*dAda
dBdb = sigmas[2] - sigmas[0]; B = b*dBdb
dCdc = sigmas[0] - sigmas[1]; C = c*dCdc
dFdf = sigmas[4]; F = f*dFdf
dGdg = sigmas[5]; G = g*dGdg
dHdh = sigmas[3]; H = h*dHdh
I2 = (F*F + G*G + H*H)/3.0 + ((A-C)**2+(C-B)**2+(B-A)**2)/54.0
I3 = (C-B)*(A-C)*(B-A)/54.0 + F*G*H - \
( (C-B)*F*F + (A-C)*G*G + (B-A)*H*H )/6.0
theta = np.arccos(I3/I2**1.5)
phi1 = (2.0*theta + pi)/6.0
phi2 = (2.0*theta + pi*3.0)/6.0
phi3 = (2.0*theta + pi*5.0)/6.0
cos1 = 2.0*cos(phi1); absc1 = abs(cos1)
cos2 = 2.0*cos(phi2); absc2 = abs(cos2)
cos3 = 2.0*cos(phi3); absc3 = abs(cos3)
ratio= np.sqrt(3.0*I2)/sigma0; expo = 1.0/m
left = ( absc1**m + absc2**m + absc3**m )/2.0
leftNorm = left**expo
r = ratio*leftNorm - 1.0
if not Jac:
return r.ravel()
else:
ln = lambda x : np.log(x + 1.0e-32)
jaco = []
dfdl = expo*leftNorm/left
js = -(r + 1.0)/sigma0
jm = (r+1.0)*ln(left)*(-expo*expo) + ratio*dfdl*0.5*(
absc1**m*ln(absc1) + absc2**m*ln(absc2) + absc3**m*ln(absc3) )
if nParas == 2:
for jacv in zip(js, jm): jaco.append(jacv)
return np.array(jaco)
else:
dI2da = (2.0*A-B-C)*dAda/27.0
dI2db = (2.0*B-C-A)*dBdb/27.0
dI2dc = (2.0*C-A-B)*dCdc/27.0
dI2df = 2.0*F*dFdf/3.0
dI2dg = 2.0*G*dGdg/3.0
dI2dh = 2.0*H*dHdh/3.0
dI3da = dI2da*(B-C)/2.0 + (H**2 - G**2)*dAda/6.0
dI3db = dI2db*(C-A)/2.0 + (F**2 - H**2)*dBdb/6.0
dI3dc = dI2dc*(A-B)/2.0 + (G**2 - F**2)*dCdc/6.0
dI3df = ( (H*G + (B-C)) * F/3.0 )*dFdf
dI3dg = ( (F*H + (C-A)) * G/3.0 )*dGdg
dI3dh = ( (G*F + (A-B)) * H/3.0 )*dHdh
darccos = -(1.0 - I3**2/I2**3)**(-0.5)
dthedI2 = darccos*I3*(-1.5)*I2**(-2.5)
dthedI3 = darccos*I2**(-1.5)
dc1dthe = -sin(phi1)/3.0
dc2dthe = -sin(phi2)/3.0
dc3dthe = -sin(phi3)/3.0
dfdc = ratio * dfdl * 0.5 * m
dfdc1 = dfdc* absc1**(expo-1.0)*np.sign(cos1)
dfdc2 = dfdc* absc2**(expo-1.0)*np.sign(cos2)
dfdc3 = dfdc* absc3**(expo-1.0)*np.sign(cos3)
dfdthe= (dfdc1*dc1dthe + dfdc2*dc2dthe + dfdc2*dc2dthe)*2.0
dfdI2 = dfdthe*dthedI2; dfdI3 = dfdthe*dthedI3
ja = dfdI2*dI2da + dfdI3*dI3da
jb = dfdI2*dI2db + dfdI3*dI3db
jc = dfdI2*dI2dc + dfdI3*dI3dc
jf = dfdI2*dI2df + dfdI3*dI3df
jg = dfdI2*dI2dg + dfdI3*dI3dg
jh = dfdI2*dI2dh + dfdI3*dI3dh
for jacv in zip(js,ja,jb,jc,jf,jg,jh,jm):
jaco.append(jacv)
return np.array(jaco)
def BBC2003Basis(sigma0, a,b,c, d,e,f,g, k, sigmas, Jac=False):
'''
residuum of the BBC2003 yield criterion for plain stress
'''
s11 = sigmas[0]; s22 = sigmas[1]; s12 = sigmas[3]
k2 = 2.0*k
M = d+e; N = e+f; P = (d-e)/2.0; Q = (e-f)/2.0; R = g**2
Gamma = M*s11 + N*s22
Psi = ( (P*s11 + Q*s22)**2 + s12**2*R )**0.5
l1 = b*Gamma + c*Psi; l2 = b*Gamma - c*Psi; l3 = 2.0*c*Psi
l1s = l1**2; l2s = l2**2; l3s = l3**2
left = a*l1s**k + a*l2s**k + (1-a)*l3s**k
sBar = left**(1.0/k2); r = sBar/sigma0 - 1.0
if not Jac:
return r.ravel()
else:
temp = (P*s11 + Q*s22)/Psi
dPsidP = temp*s11; dPsidQ = temp*s22; dPsidR = 0.5*s12**2/Psi
ln = lambda x : np.log(x + 1.0e-32)
jaco = []
expo = 0.5/k; k1 = k-1.0
dsBardl = expo*sBar/left/sigma0
dsBarde = sBar*ln(left); dedk = expo/(-k)
dldl1 = a *k*(l1s**k1)*(2.0*l1)
dldl2 = a *k*(l2s**k1)*(2.0*l2)
dldl3 = (1-a)*k*(l3s**k1)*(2.0*l3)
dldGama = (dldl1 + dldl2)*b
dldPsi = (dldl1 - dldl2 + 2.0*dldl3)*c
dlda = l1s**k + l2s**k - l3s**k
dldb = dldl1*Gamma + dldl2*Gamma
dldc = dldl1*Psi - dldl2*Psi + dldl3*2.0*Psi
dldk = a*ln(l1s)*l1s**k + a*ln(l2s)*l2s**k + (1-a)*ln(l3s)*l3s**k
js = -(r + 1.0)/sigma0
ja = dsBardl * dlda
jb = dsBardl * dldb
jc = dsBardl * dldc
jd = dsBardl *(dldGama*s11 + dldPsi*dPsidP*0.5)
je = dsBardl *(dldGama*(s11+s22) + dldPsi*(dPsidP*(-0.5) + dPsidQ*0.5) )
jf = dsBardl *(dldGama*s22 + dldPsi*dPsidQ*(-0.5))
jg = dsBardl * dldPsi * dPsidR * 2.0*g
jk = dsBardl * dldk + dsBarde * dedk
for jacv in zip(js,ja,jb,jc,jd, je, jf,jg,jk):
jaco.append(jacv)
return np.array(jaco)
def BBC2005Basis(sigma0, a,b,L, M, N, P, Q, R, k, sigmas, Jac=False):
'''
residuum of the BBC2005 yield criterion for plain stress
'''
s11 = sigmas[0]; s22 = sigmas[1]; s12 = sigmas[3]
k2 = 2.0*k
Gamma = L*s11 + M*s22
Lambda = ( (N*s11 - P*s22)**2 + s12**2 )**0.5
Psi = ( (Q*s11 - R*s22)**2 + s12**2 )**0.5
l1 = Lambda + Gamma; l2 = Lambda - Gamma; l3 = Lambda + Psi; l4 = Lambda - Psi
l1s = l1**2; l2s = l2**2; l3s = l3**2; l4s = l4**2
left = a*l1s**k + a*l2s**k + b*l3s**k + b*l4s**k
sBar = left**(1.0/k2); r = sBar/sigma0 - 1.0
if not Jac:
return r.ravel()
else:
ln = lambda x : np.log(x + 1.0e-32)
jaco = []
expo = 0.5/k; k1 = k-1.0
dsBardl = expo*sBar/left/sigma0
dsBarde = sBar*ln(left); dedk = expo/(-k)
dldl1 = a*k*(l1s**k1)*(2.0*l1)
dldl2 = a*k*(l2s**k1)*(2.0*l2)
dldl3 = b*k*(l3s**k1)*(2.0*l3)
dldl4 = b*k*(l4s**k1)*(2.0*l4)
dldLambda = dldl1 + dldl2 + dldl3 + dldl4
dldGama = dldl1 - dldl2
dldPsi = dldl3 - dldl4
temp = (N*s11 - P*s22)/Lambda
dLambdadN = s11*temp; dLambdadP = -s22*temp
temp = (Q*s11 - R*s22)/Psi
dPsidQ = s11*temp; dPsidR = -s22*temp
dldk = a*ln(l1s)*l1s**k + a*ln(l2s)*l2s**k + b*ln(l3s)*l3s**k + b*ln(l4s)*l4s**k
ja = dsBardl * (l1s**k + l2s**k)
jb = dsBardl * (l3s**k + l4s**k)
jL = dsBardl * dldGama*s11
jM = dsBardl * dldGama*s22
jN = dsBardl * dldLambda*dLambdadN
jP = dsBardl * dldLambda*dLambdadP
jQ = dsBardl * dldPsi*dPsidQ
jR = dsBardl * dldPsi*dPsidR
jk = dsBardl * dldk + dsBarde * dedk
for jacv in zip(ja,jb,jL,jM, jN, jP,jQ,jR,jk):
jaco.append(jacv)
return np.array(jaco)
def principalStress(p):
sin = np.sin; cos = np.cos
s11 = p[0]; s22 = p[1]; s33 = p[2]
s12 = p[3]; s23 = p[4]; s31 = p[5]
I1 = s11 + s22 + s33
I2 = s11*s22 + s22*s33 + s33*s11 - s12**2 - s23**2 - s31**2
I3 = s11*s22*s33 + 2.0*s12*s23*s31 - s12**2*s33 - s23**2*s11 - s31**2*s22
third = 1.0/3.0
I1s3I2= (I1**2 - 3.0*I2)**0.5
numer = 2.0*I1**3 - 9.0*I1*I2 + 27.0*I3
denom = I1s3I2**(-3.0)
cs = 0.5*numer*denom
phi = np.arccos(cs)/3.0
t1 = I1/3.0; t2 = 2.0/3.0*I1s3I2
S = np.array( [t1 + t2*cos(phi), t1+t2*cos(phi+np.pi*2.0/3.0), t1+t2*cos(phi+np.pi*4.0/3.0)])
return S, np.array([I1,I2,I3])
def principalStrs_Der(p, Invariant, s1, s2, s3, s4, s5, s6, Karafillis=False):
sin = np.sin; cos = np.cos
I1 = Invariant[0,:]; I2 = Invariant[1,:]; I3 = Invariant[2,:]
third = 1.0/3.0
I1s3I2= (I1**2 - 3.0*I2)**0.5
numer = 2.0*I1**3 - 9.0*I1*I2 + 27.0*I3
denom = I1s3I2**(-3.0)
cs = 0.5*numer*denom
phi = np.arccos(cs)*third
dphidcs = -third/np.sqrt(1.0 - cs**2)
dcsddenom = 0.5*numer*(-1.5)*I1s3I2**(-5.0)
dcsdI1 = 0.5*(6.0*I1**2 - 9.0*I2)*denom + dcsddenom*(2.0*I1)
dcsdI2 = 0.5*( - 9.0*I1)*denom + dcsddenom*(-3.0)
dcsdI3 = 13.5*denom
dphidI1, dphidI2, dphidI3 = dphidcs*dcsdI1, dphidcs*dcsdI2, dphidcs*dcsdI3
dI1s3I2dI1= I1/I1s3I2; dI1s3I2dI2 = -1.5/I1s3I2
third2 = 2.0*third; tcoeff = third2*I1s3I2
dSidIj = lambda theta : ( tcoeff*(-sin(theta))*dphidI1 + third2*dI1s3I2dI1*cos(theta) + third,
tcoeff*(-sin(theta))*dphidI2 + third2*dI1s3I2dI2*cos(theta),
tcoeff*(-sin(theta))*dphidI3)
dSdI = np.array([dSidIj(phi),dSidIj(phi+np.pi*2.0/3.0),dSidIj(phi+np.pi*4.0/3.0)]) # i=1,2,3; j=1,2,3
# calculate the derivation of principal stress with regards to the anisotropic coefficients
one = np.ones_like(p[0]); zero = np.zeros_like(p[0])
dIdp = np.array([ [one, one, one, zero, zero, zero],
[p[1]+p[2], p[2]+p[0], p[0]+p[1], -2.0*p[3], -2.0*p[4], -2.0*p[5]],
[p[1]*p[2]-p[4]**2, p[2]*p[0]-p[5]**2, p[0]*p[1]-p[3]**2,
-2.0*p[3]*p[2]+2.0*p[4]*p[5], -2.0*p[4]*p[0]+2.0*p[5]*p[3], -2.0*p[5]*p[1]+2.0*p[3]*p[4]]
])
if Karafillis:
dSdp = np.empty_like(dIdp)
dSdc = np.empty_like(dIdp)
zero = np.zeros_like(s1)
dpdc = np.array([[zero,s2-s3,s3-s2], [s1-s3,zero,s3-s1], [s1-s2,s2-s1,zero]])
for i in xrange(3):
for j in xrange(6):
dSdp[i,j] = dSdI[i,0]*dIdp[0,j]+dSdI[i,1]*dIdp[1,j]+dSdI[i,2]*dIdp[2,j]
for j in xrange(3):
dSdc[i,j] = (dSdp[i,0]*dpdc[j,0]+dSdp[i,1]*dpdc[j,1]+dSdp[i,2]*dpdc[j,2])/3.0
dSdc[i,3:6] = dSdp[i,3]*s4,dSdp[i,4]*s5,dSdp[i,5]*s6
return dSdc
else:
dIdc = np.empty([3,9,len(s1)])
dSdc = np.empty_like(dIdc)
for i in xrange(3):
dIdc[i]=np.array([dIdp[i,0]*(-s2), dIdp[i,1]*(-s1), dIdp[i,1]*(-s3),
dIdp[i,2]*(-s2), dIdp[i,2]*(-s1), dIdp[i,0]*(-s3),
dIdp[i,3]* s4, dIdp[i,4]* s5, dIdp[i,5]* s6 ])
for i in xrange(3):
for j in xrange(9):
dSdc[i,j] = dSdI[i,0]*dIdc[0,j]+dSdI[i,1]*dIdc[1,j]+dSdI[i,2]*dIdc[2,j]
return dSdc
def Yld200418pBasis(sigma0, c12,c21,c23,c32,c31,c13,c44,c55,c66,
d12,d21,d23,d32,d31,d13,d44,d55,d66, m, sigmas, Jac=False):
sv = (sigmas[0] + sigmas[1] + sigmas[2])/3.0
s1 = sigmas[0]-sv; s2 = sigmas[1]-sv; s3 = sigmas[2]-sv
s4 = sigmas[3]; s5 = sigmas[4]; s6 = sigmas[5]
ys = lambda s1,s2,s3,s4,s5,s6,c12,c21,c23,c32,c13,c31,c44,c55,c66: np.array( [
-c12*s2-c13*s3, -c21*s1-c23*s3, -c31*s1-c32*s2, c44*s4, c55*s5, c66*s6 ])
p = ys(s1,s2,s3,s4,s5,s6,c12,c21,c23,c32,c13,c31,c44,c55,c66)
q = ys(s1,s2,s3,s4,s5,s6,d12,d21,d23,d32,d13,d31,d44,d55,d66)
plambdas, pInvariant = principalStress(p) # no sort
qlambdas, qInvariant = principalStress(q) # no sort
P1 = plambdas[0,:]; P2 = plambdas[1,:]; P3 = plambdas[2,:]
Q1 = qlambdas[0,:]; Q2 = qlambdas[1,:]; Q3 = qlambdas[2,:]
m2 = m/2.0; m1 = 1.0/m; m21 = m2-1.0
P1Q1s = (P1-Q1)**2; P1Q2s = (P1-Q2)**2; P1Q3s = (P1-Q3)**2
P2Q1s = (P2-Q1)**2; P2Q2s = (P2-Q2)**2; P2Q3s = (P2-Q3)**2
P3Q1s = (P3-Q1)**2; P3Q2s = (P3-Q2)**2; P3Q3s = (P3-Q3)**2
phi= P1Q1s**m2 + P1Q2s**m2 + P1Q3s**m2 + \
P2Q1s**m2 + P2Q2s**m2 + P2Q3s**m2 + \
P3Q1s**m2 + P3Q2s**m2 + P3Q3s**m2
r = (0.25*phi)**m1/sigma0 - 1.0
if not Jac:
return r.ravel()
else:
ln = lambda x : np.log(x + 1.0e-32)
drdphi = (r+1.0)*m1/phi
dphidm =( (P1Q1s**m2)*ln(P1Q1s) + (P1Q2s**m2)*ln(P1Q2s) + (P1Q3s**m2)*ln(P1Q3s) +
(P2Q1s**m2)*ln(P2Q1s) + (P2Q2s**m2)*ln(P2Q2s) + (P2Q3s**m2)*ln(P2Q3s) +
(P3Q1s**m2)*ln(P3Q1s) + (P3Q2s**m2)*ln(P3Q2s) + (P3Q3s**m2)*ln(P3Q3s) )*0.5
jm = drdphi*dphidm + (r+1.0)*ln(0.25*phi)*(-m1*m1)
dPdc = principalStrs_Der(p, pInvariant, s1,s2,s3,s4,s5,s6)
dQdd = principalStrs_Der(q, qInvariant, s1,s2,s3,s4,s5,s6)
dphidP = m*np.array([ P1Q1s**m21*(P1-Q1) + P1Q2s**m21*(P1-Q2) + P1Q3s**m21*(P1-Q3),
P2Q1s**m21*(P2-Q1) + P2Q2s**m21*(P2-Q2) + P2Q3s**m21*(P2-Q3),
P3Q1s**m21*(P3-Q1) + P3Q2s**m21*(P3-Q2) + P3Q3s**m21*(P3-Q3) ])
dphidQ = m*np.array([ P1Q1s**m21*(Q1-P1) + P2Q1s**m21*(Q1-P2) + P3Q1s**m21*(Q1-P3),
P1Q2s**m21*(Q2-P1) + P2Q2s**m21*(Q2-P2) + P3Q2s**m21*(Q2-P3),
P1Q3s**m21*(Q3-P1) + P2Q3s**m21*(Q3-P2) + P3Q3s**m21*(Q3-P3)])
jc = drdphi*(dphidP[0]*dPdc[0]+dphidP[1]*dPdc[1]+dphidP[2]*dPdc[2])
jd = drdphi*(dphidQ[0]*dQdd[0]+dphidQ[1]*dQdd[1]+dphidQ[2]*dQdd[2])
return np.vstack((jc,jd, jm)).T
def KarafillisBoyceBasis(sigma0, c11,c12,c13,c14,c15,c16,c21,c22,c23,c24,c25,c26,
b1, b2, a, alpha , sigmas, Jac=False):
s1 = sigmas[0]; s2 = sigmas[1]; s3 = sigmas[2]
s4 = sigmas[3]; s5 = sigmas[4]; s6 = sigmas[5]
ks = lambda s1,s2,s3,s4,s5,s6,c11,c12,c13,c14,c15,c16: np.array( [
((c12+c13)*s1-c13*s2-c12*s3)/3.0, ((c13+c11)*s2-c13*s1-c11*s3)/3.0,
((c11+c12)*s3-c12*s1-c11*s2)/3.0, c14*s4, c15*s5, c16*s6 ])
p = ks(s1,s2,s3,s4,s5,s6,c11,c12,c13,c14,c15,c16)
q = ks(s1,s2,s3,s4,s5,s6,c21,c22,c23,c24,c25,c26)
plambdas, pInvariant = principalStress(p)
qlambdas, qInvariant = principalStress(q)
P1 = plambdas[0,:]; P2 = plambdas[1,:]; P3 = plambdas[2,:]
Q1 = qlambdas[0,:]; Q2 = qlambdas[1,:]; Q3 = qlambdas[2,:]
b1h = b1/2.0; b1h1 = b1h-1.0; b2h = b2/2.0; b2h1 = b2h-1.0
b1i = 1.0/b1; b2i = 1.0/b2
ai = 1.0/a
P2P3s = (P2-P3)**2; Q1s = Q1**2
P3P1s = (P3-P1)**2; Q2s = Q2**2
P1P2s = (P1-P2)**2; Q3s = Q3**2
phi10 = P2P3s**b1h + P3P1s**b1h + P1P2s**b1h
phi20 = Q1s**b2h+Q2s**b2h+Q3s**b2h; rb2 = 3.0**b2/(2.0**b2+2.0)
phi1 = (0.5*phi10)**b1i
phi2 = (rb2*phi20)**b2i
Stress = alpha*phi1**a + (1.0-alpha)*phi2**a; EqStress = Stress**ai
r = EqStress/sigma0 - 1.0
if not Jac:
return r.ravel()
else:
ln = lambda x : np.log(x + 1.0e-32)
drds = (r+1.0)*ai/Stress
drdphi1 = drds* alpha *a*phi1**(a-1.0)
drdphi2 = drds*(1.0-alpha)*a*phi2**(a-1.0)
dsda = alpha*phi1**a*ln(phi1) + (1.0-alpha)*phi2**a*ln(phi2)
dphi1dphi10 = phi1/phi10/b1; dphi2dphi20 = phi2/phi20/b2
dphi1dP = np.array([ dphi1dphi10*b1*( P3P1s**b1h1*(P1-P3) + P1P2s**b1h1*(P1-P2)),
dphi1dphi10*b1*( P2P3s**b1h1*(P2-P3) + P1P2s**b1h1*(P2-P1)),
dphi1dphi10*b1*( P3P1s**b1h1*(P3-P1) + P2P3s**b1h1*(P3-P2)) ])
dphi2dQ = np.array([ dphi2dphi20*b2*Q1s*b2h1*Q1,
dphi2dphi20*b2*Q2s*b2h1*Q2,
dphi2dphi20*b2*Q3s*b2h1*Q3 ])
dPdc= principalStrs_Der(p, pInvariant, s1,s2,s3,s4,s5,s6, Karafillis=True)
dQdc= principalStrs_Der(q, qInvariant, s1,s2,s3,s4,s5,s6, Karafillis=True)
dphi10db1 = ( (P2P3s**b1h)*ln(P2P3s)+(P3P1s**b1h)*ln(P3P1s)+(P1P2s**b1h)*ln(P1P2s) )*0.5
dphi20db2 = ( (P2P3s**b1h)*ln(P2P3s)+(P3P1s**b1h)*ln(P3P1s)+(P1P2s**b1h)*ln(P1P2s) )*0.5
drb2db2 = rb2*ln(3.0) - rb2*ln(2.0)/(1.0+2.0**(1.0-b2))
dphi1db1 = phi1*ln(phi10)*(-b1i*b1i) + b1i*phi1/(0.5*phi10)* 0.5*dphi10db1
dphi2db2 = phi2*ln(phi20)*(-b2i*b2i) + b2i*phi2/(rb2*phi20)*(rb2*dphi20db2 + drb2db2*phi20)
ja = drds*dsda - (r+1.0)*ln(Stress)/a/a #drda
jb1 = drds * ( alpha *a*phi1**(a-1)) * dphi1db1
jb2 = drds * ((1.0-alpha)*a*phi2**(a-1)) * dphi2db2
jalpha = drds * (phi1**a - phi2**a)
jc1 = drdphi1*(dphi1dP[0]*dPdc[0]+dphi1dP[1]*dPdc[1]+dphi1dP[2]*dPdc[2])
jc2 = drdphi2*(dphi2dQ[0]*dQdc[0]+dphi2dQ[1]*dQdc[1]+dphi2dQ[2]*dQdc[2])
return np.vstack((jc1,jc2,jb1,jb2,ja,jalpha)).T
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fittingCriteria = {
'tresca' :{'func' : Tresca,
'num' : 1,
'name' : 'Tresca',
'paras': 'Initial yield stress:',
'text' : '\nCoefficient of Tresca criterion:\nsigma0: ',
'error': 'The standard deviation error is: '
},
'vonmises' :{'func' : vonMises,
'num' : 1,
'name' : 'Huber-Mises-Hencky(von Mises)',
'paras': 'Initial yield stress:',
'text' : '\nCoefficient of Huber-Mises-Hencky criterion:\nsigma0: ',
'error': 'The standard deviation error is: '
},
'hosfordiso' :{'func' : Hosford,
'num' : 2,
'name' : 'Gerenal isotropic Hosford',
'paras': 'Initial yield stress, a:',
'text' : '\nCoefficients of Hosford criterion:\nsigma0, a: ',
'error': 'The standard deviation errors are: '
},
'hosfordaniso' :{'func' : generalHosford,
'num' : 5,
'name' : 'Gerenal isotropic Hosford',
'paras': 'Initial yield stress, F, G, H, a:',
'text' : '\nCoefficients of Hosford criterion:\nsigma0, F, G, H, a: ',
'error': 'The standard deviation errors are: '
},
'hill1948' :{'func' : Hill1948,
'num' : 6,
'name' : 'Hill1948',
'paras': 'Normalized [F, G, H, L, M, N]:',
'text' : '\nCoefficients of Hill1948 criterion:\n[F, G, H, L, M, N]:'+' '*16,
'error': 'The standard deviation errors are: '
},
'hill1979' :{'func' : Hill1979,
'num' : 7,
'name' : 'Hill1979',
'paras': 'f,g,h,a,b,c,m:',
'text' : '\nCoefficients of Hill1979 criterion:\n f,g,h,a,b,c,m:\n',
'error': 'The standard deviation errors are: '
},
'drucker' :{'func' : Drucker,
'num' : 2,
'name' : 'Drucker',
'paras': 'Initial yield stress, C_D:',
'text' : '\nCoefficients of Drucker criterion:\nsigma0, C_D: ',
'error': 'The standard deviation errors are: '
},
'gdrucker' :{'func' : generalDrucker,
'num' : 3,
'name' : 'General Drucker',
'paras': 'Initial yield stress, C_D, p:',
'text' : '\nCoefficients of Drucker criterion:\nsigma0, C_D, p: ',
'error': 'The standard deviation errors are: '
},
'barlat1991iso' :{'func' : Barlat1991iso,
'num' : 2,
'name' : 'Barlat1991iso',
'paras': 'Initial yield stress, m:',
'text' : '\nCoefficients of isotropic Barlat 1991 criterion:\nsigma0, m:\n',
'error': 'The standard deviation errors are: '
},
'barlat1991aniso':{'func' : Barlat1991aniso,
'num' : 8,
'name' : 'Barlat1991aniso',
'paras': 'Initial yield stress, a, b, c, f, g, h, m:',
'text' : '\nCoefficients of anisotropic Barlat 1991 criterion:\nsigma0, a, b, c, f, g, h, m:\n',
'error': 'The standard deviation errors are: '
},
'bbc2003' :{'func' : BBC2003,
'num' : 9,
'name' : 'Banabic-Balan-Comsa 2003',
'paras': 'Initial yield stress, a, b, c, d, e, f, g, k:',
'text' : '\nCoefficients of anisotropic Barlat 1991 criterion:\nsigma0, a, b, c, d, e, f, g, k:\n',
'error': 'The standard deviation errors are: '
},
'bbc2005' :{'func' : BBC2005,
'num' : 9,'err':np.inf,
'name' : 'Banabic-Balan-Comsa 2003',
'paras': 'a, b, L ,M, N, P, Q, R, k:',
'text' : '\nCoefficients of Banabic-Balan-Comsa 2005 criterion: a, b, L ,M, N, P, Q, R, k:\n',
'error': 'The standard deviation errors are: '
},
'Cazacu_Barlat2D':{'func' : Cazacu_Barlat2D,
'num' : 11,
'name' : 'Cazacu Barlat for plain stress',
'paras': 'a1,a2,a3,a6; b1,b2,b3,b4,b5,b10; c:',
'text' : '\nCoefficients of Cazacu Barlat yield criterion for plane stress: \
\n a1,a2,a3,a6; b1,b2,b3,b4,b5,b10; c:\n',
'error': 'The standard deviation errors are: '
},
'Cazacu_Barlat3D':{'func' : Cazacu_Barlat3D,
'num' : 18,
'name' : 'Cazacu Barlat',
'paras': 'a1,a2,a3,a4,a5,a6; b1,b2,b3,b4,b5,b6,b7,b8,b9,b10,b11; c:',
'text' : '\nCoefficients of Cazacu Barlat yield criterion for plane stress: \
\n a1,a2,a3,a4,a5,a6; b1,b2,b3,b4,b5,b6,b7,b8,b9,b10,b11; c\n',
'error': 'The standard deviation errors are: '
},
'yld200418p' :{'func' : Yld200418p,
'num' : 20,
'name' : 'Yld200418p',
'paras': 'Equivalent stress,c12,c21,c23,c32,c31,c13,c44,c55,c66,d12,d21,d23,d32,d31,d13,d44,d55,d66,m:',
'text' : '\nCoefficients of Yld2004-18p yield criterion: \
\n Y, c12,c21,c23,c32,c31,c13,c44,c55,c66,d12,d21,d23,d32,d31,d13,d44,d55,d66,m\n',
'error': 'The standard deviation errors are: '
},
'karafillis' :{'func' : KarafillisBoyce,
'num' : 16,
'name' : 'Yld200418p',
'paras': 'c11,c12,c13,c14,c15,c16,c21,c22,c23,c24,c25,c26,b1,b2,a,alpha',
'text' : '\nCoefficients of Karafillis-Boyce yield criterion: \
\n c11,c12,c13,c14,c15,c16,c21,c22,c23,c24,c25,c26,b1,b2,a,alpha\n',
'error': 'The standard deviation errors are: '
}
}
for key in fittingCriteria.keys():
if 'num' in fittingCriteria[key].keys():
fittingCriteria[key]['bound']=[(None,None)]*fittingCriteria[key]['num']
fittingCriteria[key]['guess']=np.ones(fittingCriteria[key]['num'],'d')
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thresholdParameter = ['totalshear','equivalentStrain']
#---------------------------------------------------------------------------------------------------
class Loadcase():
#---------------------------------------------------------------------------------------------------
'''
Class for generating load cases for the spectral solver
'''
# ------------------------------------------------------------------
def __init__(self,finalStrain,incs,time,ND=3,RD=1,nSet=1,dimension=3,vegter=False):
print('using the random load case generator')
self.finalStrain = finalStrain
self.incs = incs
self.time = time
self.ND = ND
self.RD = RD
self.nSet = nSet
self.dimension = dimension
self.vegter = vegter
self.NgeneratedLoadCases = 0
if self.vegter:
self.vegterLoadcase = self._vegterLoadcase()
def getLoadcase(self,number):
if self.dimension == 3:
print 'generate random 3D load case'
return self._getLoadcase3D()
else:
if self.vegter is True:
print 'generate load case for Vegter'
return self._getLoadcase2dVegter(number)
else:
print 'generate random 2D load case'
return self._getLoadcase2dRandom()
def getLoadcase3D(self):
self.NgeneratedLoadCases+=1
defgrad=['*']*9
stress =[0]*9
values=(np.random.random_sample(9)-.5)*self.finalStrain*2
main=np.array([0,4,8])
np.random.shuffle(main)
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for i in main[:2]: # fill 2 out of 3 main entries
defgrad[i]=1.+values[i]
stress[i]='*'
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for off in [[1,3,0],[2,6,0],[5,7,0]]: # fill 3 off-diagonal pairs of defgrad (1 or 2 entries)
off=np.array(off)
np.random.shuffle(off)
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for i in off[0:2]:
if i != 0:
defgrad[i]=values[i]
stress[i]='*'
return 'f '+' '.join(str(c) for c in defgrad)+\
' p '+' '.join(str(c) for c in stress)+\
' incs %s'%self.incs+\
' time %s'%self.time
def _getLoadcase2dVegter(self,number): #for a 2D simulation, I would use this generator before switching to a random 2D generator
NDzero=[[1,2,3,6],[1,3,5,7],[2,5,6,7]] # no deformation / * for stress
# biaxial f1 = f2
# shear f1 = -f2
# unixaial f1 , f2 =0
# plane strain f1 , s2 =0
# modulo to get one out of 4
stress =['*', '*', '0']*3
defgrad = self.vegterLoadcase[number-1]
return 'f '+' '.join(str(c) for c in defgrad)+\
' p '+' '.join(str(c) for c in stress)+\
' incs %s'%self.incs+\
' time %s'%self.time
def _vegterLoadcase(self):
'''
generate the stress points for Vegter criteria
'''
theta = np.linspace(0.0,np.pi/2.0,self.nSet)
f = [0.0, 0.0, '*']*3; loadcase = []
for i in xrange(self.nSet*4): loadcase.append(f)
# more to do for F
F = np.array([ [[1.1, 0.1], [0.1, 1.1]], # uniaxial tension
[[1.1, 0.1], [0.1, 1.1]], # shear
[[1.1, 0.1], [0.1, 1.1]], # eq-biaxial
[[1.1, 0.1], [0.1, 1.1]], # eq-biaxial
])
for i,t in enumerate(theta):
R = np.array([np.cos(t), np.sin(t), -np.sin(t), np.cos(t)]).reshape(2,2)
for j in xrange(4):
loadcase[i*4+j][0],loadcase[i*4+j][1],loadcase[i*4+j][3],loadcase[i*4+j][4] = np.dot(R.T,np.dot(F[j],R)).reshape(4)
return loadcase
def _getLoadcase2dRandom(self):
'''
generate random stress points for 2D tests
'''
self.NgeneratedLoadCases+=1
defgrad=['0', '0', '*']*3
stress =['*', '*', '0']*3
defgrad[0],defgrad[1],defgrad[3],defgrad[4] = (np.random.random_sample(4)-.5)*self.finalStrain*2.0 + np.eye(2).reshape(4)
return 'f '+' '.join(str(c) for c in defgrad)+\
' p '+' '.join(str(c) for c in stress)+\
' incs %s'%self.incs+\
' time %s'%self.time
def _defgradScale(self, defgrad, finalStrain):
'''
'''
defgrad0 = (np.array([ 0.0 if i is '*' else i for i in defgrad ]))
det0 = 1.0 - numpy.linalg.det(defgrad0.reshape(3,3))
if defgrad0[0] == 0.0: defgrad0[0] = det0/(defgrad0[4]*defgrad0[8]-defgrad0[5]*defgrad0[7])
if defgrad0[4] == 0.0: defgrad0[4] = det0/(defgrad0[0]*defgrad0[8]-defgrad0[2]*defgrad0[6])
if defgrad0[8] == 0.0: defgrad0[8] = det0/(defgrad0[0]*defgrad0[4]-defgrad0[1]*defgrad0[3])
strain = np.dot(defgrad0.reshape(3,3).T,defgrad0.reshape(3,3)) - np.eye(3)
eqstrain = 2.0/3.0*np.sqrt( 1.5*(strain[0][0]**2+strain[1][1]**2+strain[2][2]**2) +
3.0*(strain[0][1]**2+strain[1][2]**2+strain[2][0]**2) )
r = finalStrain*1.25/eqstrain
# if r>1.0: defgrad =( np.array([i*r if i is not '*' else i for i in defgrad]))
#---------------------------------------------------------------------------------------------------
class Criterion(object):
#---------------------------------------------------------------------------------------------------
'''
Fitting to certain criterion
'''
def __init__(self,name='worst'):
self.name = name
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self.results = fittingCriteria
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if self.name.lower() not in map(str.lower, self.results.keys()):
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raise Exception('no suitable fitting criterion selected')
else:
print('fitting to the %s criterion'%name)
def fit(self,stress):
global fitResults
nameCriterion = self.name.lower()
criteriaClass = fittingCriteria[nameCriterion]['func']
numParas = fittingCriteria[nameCriterion]['num']
textParas = fittingCriteria[nameCriterion]['text'] + formatOutput(numParas)
textError = fittingCriteria[nameCriterion]['error']+ formatOutput(numParas,'%-14.8f')+'\n'
bounds = fittingCriteria[nameCriterion]['bound'] # Default bounds, no bound
guess0 = fittingCriteria[nameCriterion]['guess'] # Default initial guess, depends on bounds
criteria = criteriaClass(0.0)
if fitResults == [] : initialguess = guess0
else : initialguess = np.array(fitResults[-1])
weight = get_weight(np.shape(stress)[1])
ydata = np.zeros(np.shape(stress)[1])
try:
popt, pcov, infodict, errmsg, ierr = \
leastsqBound (criteria.fun, initialguess, args=(ydata,stress),
bounds=bounds, Dfun=criteria.jac, full_output=True)
if ierr not in [1, 2, 3, 4]:
raise RuntimeError("Optimal parameters not found: " + errmsg)
if (len(ydata) > len(initialguess)) and pcov is not None:
s_sq = (criteria.fun(popt, *(ydata,stress))**2).sum()/(len(ydata)-len(initialguess))
pcov = pcov * s_sq
perr = np.sqrt(np.diag(pcov))
fitResults.append(popt.tolist())
print (textParas%array2tuple(popt))
print (textError%array2tuple(perr))
print('Number of function calls =', infodict['nfev'])
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except Exception as detail:
print detail
pass
#---------------------------------------------------------------------------------------------------
class myThread (threading.Thread):
#---------------------------------------------------------------------------------------------------
'''
Runner class
'''
def __init__(self, threadID):
threading.Thread.__init__(self)
self.threadID = threadID
def run(self):
s.acquire()
conv=converged()
s.release()
while not conv:
doSim(4.,self.name)
s.acquire()
conv=converged()
s.release()
def doSim(delay,thread):
s.acquire()
me=loadcaseNo()
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if not os.path.isfile('%s.load'%me):
print('generating loadcase for sim %s from %s'%(me,thread))
f=open('%s.load'%me,'w')
f.write(myLoad.getLoadcase(me))
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f.close()
s.release()
else: s.release()
s.acquire()
if not os.path.isfile('%s_%i.spectralOut'%(options.geometry,me)):
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print('starting simulation %s from %s'%(me,thread))
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s.release()
execute('DAMASK_spectral -g %s -l %i'%(options.geometry,me))
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else: s.release()
s.acquire()
if not os.path.isfile('./postProc/%s_%i.txt'%(options.geometry,me)):
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print('starting post processing for sim %i from %s'%(me,thread))
s.release()
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try:
execute('postResults --cr f,p --co totalshear %s_%i.spectralOut'%(options.geometry,me))
except:
execute('postResults --cr f,p %s_%i.spectralOut'%(options.geometry,me))
execute('addCauchy ./postProc/%s_%i.txt'%(options.geometry,me))
execute('addStrainTensors -l -v ./postProc/%s_%i.txt'%(options.geometry,me))
execute('addMises -s Cauchy -e ln(V) ./postProc/%s_%i.txt'%(options.geometry,me))
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else: s.release()
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s.acquire()
print('-'*10)
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print('reading values for sim %i from %s'%(me,thread))
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s.release()
refFile = open('./postProc/%s_%i.txt'%(options.geometry,me))
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table = damask.ASCIItable(refFile)
table.head_read()
if options.fitting =='equivalentStrain':
thresholdKey = 'Mises(ln(V))'
elif options.fitting =='totalshear':
thresholdKey = 'totalshear'
s.acquire()
for l in [thresholdKey,'1_Cauchy']:
if l not in table.labels: print '%s not found'%l
s.release()
table.data_readArray(['%i_Cauchy'%(i+1) for i in xrange(9)]+[thresholdKey]+['%i_ln(V)'%(i+1) for i in xrange(9)])
line = 0
lines = np.shape(table.data)[0]
yieldStress = np.empty((int(options.yieldValue[2]),6),'d')
deformationRate = np.empty((int(options.yieldValue[2]),6),'d')
for i,threshold in enumerate(np.linspace(options.yieldValue[0],options.yieldValue[1],options.yieldValue[2])):
while line < lines:
if table.data[line,9]>= threshold:
upper,lower = table.data[line,9],table.data[line-1,9] # values for linear interpolation
stress = np.array(table.data[line-1,0:9] * (upper-threshold)/(upper-lower) + \
table.data[line ,0:9] * (threshold-lower)/(upper-lower)).reshape(3,3) # linear interpolation of stress values
dstrain= np.array(table.data[line,10:] - table.data[line-1,10:]).reshape(3,3)
yieldStress[i,0]= stress[0,0]; yieldStress[i,1]=stress[1,1]; yieldStress[i,2]=stress[2,2]
yieldStress[i,3]=(stress[0,1] + stress[1,0])/2.0 # 0 3 5
yieldStress[i,4]=(stress[1,2] + stress[2,1])/2.0 # * 1 4 yieldStress
yieldStress[i,5]=(stress[2,0] + stress[0,2])/2.0 # * * 2
# D*dt = 0.5(L+L^T)*dt = 0.5*d(lnF + lnF^T) = dlnV
deformationRate[i,0]= dstrain[0,0]; deformationRate[i,1]=dstrain[1,1]; deformationRate[i,2]=dstrain[2,2]
deformationRate[i,3]=(dstrain[0,1] + dstrain[1,0])/2.0 # 0 3 5
deformationRate[i,4]=(dstrain[1,2] + dstrain[2,1])/2.0 # * 1 4
deformationRate[i,5]=(dstrain[2,0] + dstrain[0,2])/2.0 # * * 2
break
else:
line+=1
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s.acquire()
global stressAll, strainAll
print('number of yield points of sim %i: %i'%(me,len(yieldStress)))
print('starting fitting for sim %i from %s'%(me,thread))
try:
for i in xrange(int(options.yieldValue[2])):
stressAll[i]=np.append(stressAll[i], yieldStress[i]/unitGPa)
strainAll[i]=np.append(strainAll[i], deformationRate[i])
myFit.fit(stressAll[i].reshape(len(stressAll[i])//6,6).transpose())
except Exception as detail:
print('could not fit for sim %i from %s'%(me,thread))
print detail
s.release()
return
s.release()
def loadcaseNo():
global N_simulations
N_simulations+=1
return N_simulations
def converged():
global N_simulations
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if N_simulations < options.max:
return False
else:
return True
# --------------------------------------------------------------------
# MAIN
# --------------------------------------------------------------------
parser = OptionParser(option_class=damask.extendableOption, usage='%prog options [file[s]]', description = """
Performs calculations with various loads on given geometry file and fits yield surface.
""", version=string.replace(scriptID,'\n','\\n')
)
# maybe make an option to specifiy if 2D/3D fitting should be done?
parser.add_option('-l','--load' , dest='load', type='float', nargs=3,
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help='load: final strain; increments; time %default', metavar='float int float')
parser.add_option('-g','--geometry', dest='geometry', type='string',
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help='name of the geometry file [%default]', metavar='string')
parser.add_option('-c','--criterion', dest='criterion', choices=fittingCriteria.keys(),
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help='criterion for stopping simulations [%default]', metavar='string')
parser.add_option('-f','--fitting', dest='fitting', choices=thresholdParameter,
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help='yield criterion [%default]', metavar='string')
parser.add_option('-y','--yieldvalue', dest='yieldValue', type='float', nargs=3,
help='yield points: start; end; count %default', metavar='float float int')
parser.add_option('--min', dest='min', type='int',
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help='minimum number of simulations [%default]', metavar='int')
parser.add_option('--max', dest='max', type='int',
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help='maximum number of iterations [%default]', metavar='int')
parser.add_option('-t','--threads', dest='threads', type='int',
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help='number of parallel executions [%default]', metavar='int')
parser.add_option('-d','--dimension', dest='dimension', type='int',
help='dimension of the virtual test [%default]', metavar='int')
parser.add_option('-v', '--vegter', dest='vegter', action='store_true',
help='Vegter criteria [%default]')
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parser.set_defaults(min = 12)
parser.set_defaults(max = 30)
parser.set_defaults(threads = 4)
parser.set_defaults(yieldValue = (0.002,0.004,2))
parser.set_defaults(load = (0.010,100,100.0))
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parser.set_defaults(criterion = 'worst')
parser.set_defaults(fitting = 'totalshear')
parser.set_defaults(geometry = '20grains16x16x16')
parser.set_defaults(dimension = 3)
parser.set_defaults(vegter = 'False')
options = parser.parse_args()[0]
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if not os.path.isfile(options.geometry+'.geom'):
parser.error('geometry file %s.geom not found'%options.geometry)
if not os.path.isfile('material.config'):
parser.error('material.config file not found')
if options.threads<1:
parser.error('invalid number of threads %i'%options.threads)
if options.min<0:
parser.error('invalid minimum number of simulations %i'%options.min)
if options.max<options.min:
parser.error('invalid maximum number of simulations (below minimum)')
if options.yieldValue[0]>options.yieldValue[1]:
parser.error('invalid yield start (below yield end)')
if options.yieldValue[2] != int(options.yieldValue[2]):
parser.error('count must be an integer')
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if not os.path.isfile('numerics.config'):
print('numerics.config file not found')
if not os.path.isfile('material.config'):
print('material.config file not found')
if options.vegter is True:
options.dimension = 2
unitGPa = 10.e8
N_simulations=0
fitResults = []
s=threading.Semaphore(1)
stressAll=[np.zeros(0,'d').reshape(0,0) for i in xrange(int(options.yieldValue[2]))]
strainAll=[np.zeros(0,'d').reshape(0,0) for i in xrange(int(options.yieldValue[2]))]
myLoad = Loadcase(options.load[0],options.load[1],options.load[2],
nSet = 10, dimension = options.dimension, vegter = options.vegter)
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myFit = Criterion(options.criterion)
threads=[]
for i in range(options.threads):
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threads.append(myThread(i))
threads[i].start()
for i in range(options.threads):
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threads[i].join()
print 'finished fitting to yield criteria'