Newer
Older
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
import numpy as np
import matplotlib.pyplot as plt
import sympy as sym
import math
import os
import subprocess
import fileinput
import re
import matlab.engine
import sys
# print(sys.executable)
# from subprocess import Popen, PIPE
# --------------------------------------------------
# 'classifyMin' classifies Minimizers by utilizing the result of
# Lemma1.6
#
#
#
#
# 'classifyMin_ana': (Special-Case : Lemma1.4)
# ..additionally assumes Poisson-ratio=0 => q12==0
#
#
#
# Output : MinimizingMatrix, Angle, Type, Curvature
def get_gstar(q1,q2,q12,q3,b1,b2):
# a = np.array([a1,a2])
b = np.array([b1,b2])
H = np.array([[2*q1, q12+2*q3], [q12+2*q3,2*q2] ])
A = np.array([[q1,q12/2], [q12/2,q2] ])
# print('det(H)=', np.linalg.det(H))
# check if g* is in G^*_R^2
tmp = A.dot(b)
## compute inverse of H :
inv_H = np.linalg.inv(H)
g_star = 2*inv_H.dot(tmp)
# print('g_star=', g_star)
return g_star
def determine_b(q1,q2,q12,q3,g_star):
##
# Input: g_star
# Output : b such that g_star is minimizer, i.e A*b = (1/2)*H*g_star
# q1=1;
# q2=2;
# q12=1/2;
# q3=((4*q1*q2)**0.5-q12)/2;
# H=[2*q1,q12+2*q3;q12+2*q3,2*q2];
H = np.array([[2*q1, q12+2*q3], [q12+2*q3,2*q2] ])
A = np.array([[q1,1/2*q12], [1/2*q12,q2] ])
abar = np.array([q12+2*q3, 2*q2])
rhs = (1/2)*H.dot(g_star)
print('rhs:', rhs)
b = np.linalg.lstsq(A, rhs)[0]
print('b',b)
b1=b[0]
b2=b[1]
return b
## ---------------
def get_minimizer(q1,q2,q3,b1,b2):
# In the case if q12 == 0:
quotient = (q1*q2-q3**2)
g_star = np.array([(q1*q2*b1-q3*q2*b2)/quotient, (q1*q2*b2-q3*q1*b1)/quotient])
print('g_star=', g_star)
return g_star
def energy(a1,a2,q1,q2,q12,q3,b1,b2):
a = np.array([a1,a2])
b = np.array([b1,b2])
H = np.array([[2*q1, q12+2*q3], [q12+2*q3,2*q2] ])
A = np.array([[q1,1/2*q12], [1/2*q12,q2] ])
tmp = H.dot(a)
# print('H',H)
# print('A',A)
# print('b',b)
# print('a',a)
# print('tmp',tmp)
tmp = (1/2)*a.dot(tmp)
# print('tmp',tmp)
tmp2 = A.dot(b)
# print('tmp2',tmp2)
tmp2 = 2*a.dot(tmp2)
# print('tmp2',tmp2)
energy = tmp - tmp2
# print('energy',energy)
# energy_axial1.append(energy_1)
return energy
def determinant(q1,q2,q3,q12): # TODO General:Matrix
return q1*q2 - (q3**2 + 2*q3*q12 + q12**2)
def harmonicMean(mu_1, beta, theta):
return mu_1*(beta/(theta+((1-theta)*beta)))
def arithmeticMean(mu_1, beta, theta):
return mu_1*((1-theta)+theta*beta)
def prestrain_b1(rho_1, beta, alpha, theta):
return (3.0*rho_1/2.0)*(1-(theta*(1+alpha)))
# return (3.0*rho_1/2.0)*beta*(1-(theta*(1+alpha)))
def prestrain_b2(rho_1, beta, alpha, theta):
return (3.0*rho_1/(2.0*((1.0-theta) + theta*beta)))*(1-theta*(1+beta*alpha))
# return (3.0*rho_1/(4.0*((1.0-theta) + theta*beta)))*(1-theta*(1+beta*alpha))
# Define function to be minimized
def f(a1, a2, q1, q2, q3, q12, b1, b2):
A = np.array([[q1, q3 + q12/2.0], [q3 + q12/2.0, q2]])
B = np.array([-2.0*q1*b1-q12*b2, -2.0*q2*b2-q12*b1])
a = np.array([a1, a2])
tmp = np.dot(A, a)
tmp2 = np.dot(a, tmp)
tmpB = np.dot(B, a)
return tmp2 + tmpB + q1*(b1**2) + q2*(b2**2) + q12*b1*b2
# ---- Alternative Version using alpha,beta,theta ,mu_1,rho_1
def classifyMin_ana(alpha,beta,theta,q3,mu_1,rho_1,print_Cases=False, print_Output=False):
# Assumption of Classification-Lemma1.6:
# 1. [b3 == 0]
# 2. Q is orthotropic i.e. q13 = q31 = q23 = q32 == 0
# 3. This additionally assumes that Poisson-Ratio = 0 => q12 == 0
q12 = 0.0
q1 = (1.0/6.0)*harmonicMean(mu_1, beta, theta)
q2 = (1.0/6.0)*arithmeticMean(mu_1, beta, theta)
# print('q1: ', q1)
# print('q2: ', q2)
b1 = prestrain_b1(rho_1, beta, alpha,theta)
b2 = prestrain_b2(rho_1, beta, alpha,theta)
# print('alpha:',alpha)
# print('beta:',beta)
# print('theta:',theta)
return classifyMin(q1, q2, q3, q12, b1, b2, print_Cases, print_Output)
# Matrix Version that just gets matrices Q & B
def classifyMin_mat(Q,B,print_Cases=False, print_Output=False):
q1 = Q[0][0]
q2 = Q[1][1]
q3 = Q[2][2]
q12 = Q[0][1]
b1 = B[0]
b2 = B[1]
b3 = B[2]
return classifyMin(q1, q2, q3, q12, b1, b2, print_Cases, print_Output)
# --------------------------------------------------------------------
# Classify Type of minimizer 1 = R1 , 2 = R2 , 3 = R3 # before : destinction between which axis.. (4Types )
# where
# R1 : unique local (global) minimizer which is not axial
# R2 : continuum of local (global) minimizers which are not axial
# R3 : one or two local (global) minimizers which are axial
# Partition given by
# R1 = E1
# R2 = P1.2
# R3 = E2 U E3 U P1.1 U P2 U H
# -------------------------------------------------------------------
def classifyMin(q1, q2, q3, q12, b1, b2, print_Cases=False, print_Output=False): #ClassifyMin_hom?
# Assumption of Classification-Lemma1.6:
# 1. [b3 == 0]
# 2. Q is orthotropic i.e. q13 = q31 = q23 = q32 == 0
# TODO: check if Q is orthotropic here - assert()
if print_Output: print("Run ClassifyMin_NEW...")
CaseCount = 0
epsilon = sys.float_info.epsilon #Machine epsilon
# print('epsilon:',epsilon)
b = np.array([b1,b2])
H = np.array([[2*q1, q12+2*q3], [q12+2*q3,2*q2] ])
A = np.array([[q1,q12/2], [q12/2,q2] ])
# B = np.array([-2.0*q1*b1-q12*b2, -2.0*q2*b2-q12*b1])
# A = np.array([[q1, q3 + q12/2.0], [q3 + q12/2.0, q2]])
determinant = np.linalg.det(H)
# print('determinant:',determinant)
# determinant = q1*q2 - (q3**2 + 2*q3*q12 + q12**2)
if print_Cases: print("determinant:", determinant)
# Define values for axial-Solutions (b1*,0) & (0,b2*)
# b1_star = (2.0*q1*b1 + b2*q12)/(2*q1)
# b2_star = (2.0*q2*b2 + b1*q12)/(2*q2)
# ------------------------------------ Parabolic Case -----------------------------------
# if abs(determinant) < epsilon:
if abs(determinant) < epsilon:
if print_Cases: print('P : parabolic case (determinant equal zero)')
print('P : PARABOLIC CASE (determinant equal zero)')
# ------------------------------------ Elliptic Case -----------------------------------
if determinant >= epsilon:
if print_Cases: print('E : elliptic case (determinant greater zero)')
g_star = get_gstar(q1,q2,q12,q3,b1,b2)
# a1_star = (b1*(q12**2) + 2*b1*q3*q12 - 4*b1*q1*q2 + 4*b2*q2*q3) / \
# (4*(q3**2) + 4*q3*q12 + (q12**2) - 4*q1*q2)
# a2_star = (b2*(q12**2) + 2*b2*q3*q12 + 4*b1*q1*q3 - 4*b2*q1*q2) / \
# (4*(q3**2) + 4*q3*q12 + (q12**2) - 4*q1*q2)
# prod = a1_star*a2_star
prod = g_star[0]*g_star[1]
if prod >= epsilon:
if print_Cases: print('(E1) - inside Lambda ')
# a1 = a1_star
# a2 = a2_star
# type = 1 # non-axial Minimizer
# CaseCount += 1
a1 = g_star[0]
a2 = g_star[1]
CaseCount += 1
if prod < epsilon: # same as prod = 0 ? or better use <=epsilon ?
if abs(((b2**2)/q1 - (b1**2)/q2)) < epsilon :
print('((b2**2)/q1 - (b1**2)/q2)', ((b2**2)/q1 - (b1**2)/q2))
print('two minimizers')
a1 = b1
a2 = 0.0
# type = 3 # 1
CaseCount += 1
else:
#compare energy values
if energy(b1, 0, q1, q2, q3, q12, b1, b2) > energy(0, b2, q1, q2, q3, q12, b1, b2):
a1 = 0
a2 = b2
CaseCount += 1
else :
a1 = b1
a2 = 0
CaseCount += 1
else :
# g_star = get_gstar(q1,q2,q12,q3,b1,b2)
# prod = g_star[0]*g_star[1]
if abs((b2**2)/q1 - (b1**2)/q2) < epsilon :
print('two minimizers')
a1 = b1
a2 = 0.0
# type = 3 # 1
CaseCount += 1
else:
#compare energy values
if energy(b1, 0, q1, q2, q3, q12, b1, b2) > energy(0, b2, q1, q2, q3, q12, b1, b2):
a1 = 0
a2 = b2
CaseCount += 1
else :
a1 = b1
a2 = 0
CaseCount += 1
#
# if print_Cases: print('(E2) - on the boundary of Lambda ')
# a1 = a1_star
# a2 = a2_star
# type = 3 # could check which axis: if a1_star or a2_star close to zero.. ?
# CaseCount += 1
#
# if prod <= -1.0*epsilon:
# if print_Cases: print('(E3) - Outside Lambda ')
# if f(b1_star, 0, q1, q2, q3, q12, b1, b2) < f(0, b2_star, q1, q2, q3, q12, b1, b2):
# a1 = b1_star
# a2 = 0.0
# type = 3 # 1
# CaseCount += 1
# if f(b1_star, 0, q1, q2, q3, q12, b1, b2) > f(0, b2_star, q1, q2, q3, q12, b1, b2):
# a1 = 0
# a2 = b2_star
# type = 3 # 2
# CaseCount += 1
#
# # TODO Problem: angle depends on how you choose... THE angle is not defined for this case
# if f(b1_star, 0, q1, q2, q3, q12, b1, b2) == f(0, b2_star, q1, q2, q3, q12, b1, b2):
# # Two Minimizers pick one
# a1 = b1_star
# a2 = 0.0
# type = 3 # 4
# CaseCount += 1
# ------------------------------------ Hyperbolic Case -----------------------------------
# if determinant <= -1.0*epsilon:
# if print_Cases: print('H : hyperbolic case (determinant smaller zero)')
# # One or two minimizers wich are axial
# type = 3 # (always type 3)
# if f(b1_star, 0, q1, q2, q3, q12, b1, b2) < f(0, b2_star, q1, q2, q3, q12, b1, b2):
# a1 = b1_star
# a2 = 0.0
# # type = 3 # 1
# CaseCount += 1
# if f(b1_star, 0, q1, q2, q3, q12, b1, b2) > f(0, b2_star, q1, q2, q3, q12, b1, b2):
# a1 = 0
# a2 = b2_star
# # type = 3 # 2
# CaseCount += 1
# # TODO can add this case to first or second ..
# if f(b1_star, 0, q1, q2, q3, q12, b1, b2) == f(0, b2_star, q1, q2, q3, q12, b1, b2):
# # Two Minimizers pick one
# a1 = b1_star
# a2 = 0.0
# # type = 3 # 4
# CaseCount += 1
# ---------------------------------------------------------------------------------------
if (CaseCount > 1):
print('Error: More than one Case happened!')
# compute a3
# a3 = math.sqrt(2.0*a1*a2) # never needed?
# print('a1:', a1)
# print('a2:', a2)
# compute the angle <(e,e_1) where Minimizer = kappa* (e (x) e)
# e = [math.sqrt((a1/(a1+a2))), math.sqrt((a2/(a1+a2)))]
e = [((a1/(a1+a2)))**0.5, ((a2/(a1+a2)))**0.5]
angle = math.atan2(e[1], e[0])
type = 1 # ToDO..
# compute kappa
kappa = (a1 + a2)
# Minimizer G
# Minimizer = np.array([[a1, math.sqrt(a1*a2)], [math.sqrt(a1*a2), a2]],dtype=object)
Minimizer = np.array([[a1, (a1*a2)**0.5], [(a1*a2)**0.5, a2]],dtype=object)
# Minimizer = np.array([[a1, math.sqrt(a1*a2)], [math.sqrt(a1*a2), a2]])
# MinimizerVec = np.array([a1, a2],dtype=object)
MinimizerVec = np.array([a1, a2])
if print_Output:
print('--- Output ClassifyMin ---')
print("Minimizing Matrix G:")
print(Minimizer)
print("angle = ", angle)
print("type: ", type)
print("kappa = ", kappa)
return Minimizer, angle, type, kappa
# return MinimizerVec, angle, type, kappa #return Minimizer Vector instead
# ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
# ---------------------------------------------- Main ---------------------
# --- Input Parameters ----
# mu_1 = 1.0
# rho_1 = 1.0
# alpha = 9.0
# beta = 2.0
# theta = 1.0/8.0
# # define q1, q2 , mu_gamma, q12
# # 1. read from Cell-Output
# # 2. define values from analytic formulas (expect for mu_gamma)
# q1 = (1.0/6.0)*harmonicMean(mu_1, beta, theta)
# q2 = (1.0/6.0)*arithmeticMean(mu_1, beta, theta)
# # TEST
# q12 = 0.0 # (analytical example)
# # q12 = 12.0 # (analytical example)
# set mu_gamma to value or read from Cell-Output
# mu_gamma = q1 # TODO read from Cell-Output
# b1 = prestrain_b1(rho_1, beta, alpha, theta)
# b2 = prestrain_b2(rho_1, beta, alpha, theta)
# print('---- Input parameters: -----')
# print('mu_1: ', mu_1)
# print('rho_1: ', rho_1)
# print('alpha: ', alpha)
# print('beta: ', beta)
# print('theta: ', theta)
# print("q1: ", q1)
# print("q2: ", q2)
# print("mu_gamma: ", mu_gamma)
# print("q12: ", q12)
# print("b1: ", b1)
# print("b2: ", b2)
# print('----------------------------')
# # print("machine epsilon", sys.float_info.epsilon)
#
#
# # ------- Options --------
# print_Cases = True
# print_Output = True
# G, angle, type, kappa = classifyMin(q1, q2, mu_gamma, q12, b1, b2, print_Cases, print_Output)
#
# G, angle, type, kappa = classifyMin_ana(alpha, beta, theta, mu_gamma, q12, print_Cases, print_Output)
#
# Out = classifyMin_ana(alpha, beta, theta, mu_gamma, q12, print_Cases, print_Output)
# print('TEST:')
# Out = classifyMin_ana(alpha, beta, theta)
# print('Out[0]', Out[0])
# print('Out[1]', Out[1])
# print('Out[2]', Out[2])
# print('Out[3]', Out[3])
# #supress certain Outout..
# _,_,T,_ = classifyMin_ana(alpha, beta, theta, mu_gamma, q12, print_Cases, print_Output)
# print('Output only type..:', T)
# Test = f(1,2 ,q1,q2,mu_gamma,q12,b1,b2)
# print("Test", Test)
# -----------------------------------------------------------------------------------------------------------------------------------------------------------------