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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 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):
Klaus Böhnlein
committed
# 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
Klaus Böhnlein
committed
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)
Klaus Böhnlein
committed
# 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)
Klaus Böhnlein
committed
# --------------------------------------------------------------------
# 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
Klaus Böhnlein
committed
# -------------------------------------------------------------------
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...")
CaseCount = 0
epsilon = sys.float_info.epsilon #Machine epsilon
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]])
# print('Matrix A:', A)
# print('Matrix B:', B)
# print('Matrix rank of A:', np.linalg.matrix_rank(A))
# print('shape of A:', A.shape)
# print('shape of B:', B.shape)
# print('Matrix [A,B]:', np.c_[A, B])
# print('Matrix rank of [A,B]:', np.linalg.matrix_rank(np.c_[A, B]))
# print('shape of [A,B]:', C.shape)
#
# x = np.linalg.solve(A, B) # works only if A is not singular!!
# print("Solution LGS:", x)
# # print("sym Matrix", sym.Matrix(([A],[B])))
#
# # Test
# C = np.array([[1, 0], [0, 0]])
# d = np.array([5, 0])
# y = np.linalg.lstsq(C, d)[0]
# print("Solution LGS:", y)
# T = np.c_[C, d]
# print('T:', T)
# Trref = sym.Matrix(T).rref()[0]
# Out = np.array(Trref, dtype=float)
# print('rref:', Out)
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 print_Cases: print('P : parabolic case (determinant equal zero)')
print('P : parabolic case (determinant equal zero)')
# if print_Cases: print('P : parabolic case (determinant equal zero)')
# check if rank(A) == rank([A,B])
# OK this way? (or use Sympy?)
if np.linalg.matrix_rank(A) == np.linalg.matrix_rank(np.c_[A, B]):
if print_Cases: print('P1 (B is in the range of A)')
if (q12+q3)/2.0 <= -1.0*epsilon:
print('should not happen(not compatible with det = 0)')
if (abs(B[0]) < epsilon and abs(B[1]) < epsilon) and (q12+q3)/2.0 >= epsilon:
if print_Cases: print('P1.1')
a1 = 0.0
a2 = 0.0
type = 3
CaseCount += 1
if (abs(B[0]) >= epsilon or abs(B[1]) >= epsilon) and (q12+q3)/2.0 >= epsilon:
# Continuum of minimizers located along a line of negative slope in Lambda
if print_Cases: print('P1.2 (Continuum of minimizers located along a line of negative slope in Lambda) ')
# Just solve Aa* = b (alternatively using SymPy ?)
# we know that A is singular (det A = 0 ) here..
# & we know that there are either infinitely many solutions or a unique solution ...
# ---- determine one via Least Squares
# "If A is square and of full rank, then x (but for round-off error)
# is the “exact” solution of the equation. Else, x minimizes the
# Euclidean 2-norm || b-Ax ||. If there are multiple minimizing solutions,
# the one with the smallest 2-norm is returned. ""
# a = np.linalg.lstsq(A, B)[0] # TODO check is this Ok ?
a = np.linalg.lstsq(A, -B/2)[0] # TODO check is this Ok ? (UPDATE 13-10-21)
print("Solution LGS: a =", a)
a1 = a[0]
a2 = a[1]
type = 2
CaseCount += 1
else:
if print_Cases: print('P2 (B is not in the range of A)')
# local Minimizers occur on the boundary of Lambda...
# There are at most two local minima and they are either
# (b1_star, 0) or (0, b2_star)
# could also outsource this to another function..
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
# ------------------------------------ Elliptic Case -----------------------------------
if determinant >= epsilon:
if print_Cases: print('E : elliptic case (determinant greater zero)')
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
if prod >= epsilon:
if print_Cases: print('(E1) - inside Lambda ')
a1 = a1_star
a2 = a2_star
type = 1 # non-axial Minimizer
CaseCount += 1
if abs(prod) < epsilon: # same as prod = 0 ? or better use <=epsilon ?
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
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
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
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
CaseCount += 1
# ---------------------------------------------------------------------------------------
if (CaseCount > 1):
print('Error: More than one Case happened!')
# compute a3
# 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])
# compute kappa
kappa = (a1 + a2)
# 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
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# 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:')
# 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)
# -----------------------------------------------------------------------------------------------------------------------------------------------------------------