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3dfdy.py
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3dfdy.py
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# -*- coding: utf-8 -*-
"""
Created on Sat Mar 10 23:24:23 2018
@author: bshri_etybhvn
fixme: The formula to calculate the array of weights is incorrect (--line 169, works only because of 1s)
: Exhaustive check of derivatives at Gauss-points (--line 167, did a random check though!)
: Determinant (and consequent inverse) calculation, remove generator (--line 171)
: Determinant (and consequent inverse) calculation for the deformation gradient, DONOT use la.inv and la.det (--line 183)
: Applying the boundary conditions in a more general sense (node->dof on the node!!)
"""
import numpy as np
import numpy.linalg as la
import matplotlib.pyplot as plt
from matplotlib import rc
import matplotlib
from mpl_toolkits.mplot3d import Axes3D
from matplotlib.ticker import MaxNLocator
from scipy.optimize import fsolve,minimize
import pandas as pd
import matplotlib.tri as mptri
from matplotlib.ticker import AutoMinorLocator
import scipy.sparse as sp
from scipy.interpolate import interp1d,splrep,splder,splev
from scipy.integrate import solve_bvp #verify the FEM solution
##################################################################
rc('font',**{'family':'lmodern','sans-serif':['Helvetica']})
rc('text', usetex=True)
matplotlib.rcParams['xtick.direction']='in'
matplotlib.rcParams['ytick.direction']='in'
matplotlib.rcParams['xtick.top']=True
matplotlib.rcParams['ytick.right']=True
rc('xtick',labelsize=18)
rc('ytick',labelsize=18)
mx=AutoMinorLocator(10)
my=AutoMinorLocator(10)
##################################################################
class geometry() : #1D geometry
def __init__(self,Eltype):
self.tolNR=1.e-6
self.maxiter=500
if Eltype[0]=='L':
self.A=8.e-1
self.B=1.
ne=10
self.nLnodes=ne+1
self.nQnodes=2*ne+1
self.mu=1.e5
kap=1.e1*self.mu
self.lam=kap+2.*self.mu/3
self.Po=2.5e5*np.linspace(0,1,100)
self.epS=2.*np.linspace(1,20,100)
elif Eltype[0]=='Q':
self.xlength=2.
self.ylength=2.
self.nx=1
self.ny=1
self.nDim=2 #No. of dof per node, it is essentially the dimension of the problem
self.thck=1.
self.nSteps=10
E=10.;nu=0.25
self.mu=E/(2*(1+nu))
self.lam=2*nu*self.mu/(1-2*nu)
self.kap=self.lam+2./3*self.mu
elif Eltype[0]=='H':
self.xlength=1.
self.ylength=1.
self.zlength=1.
self.nx=1
self.ny=1
self.nz=1
self.nDim=3 #No. of dof per node, it is essentially the dimension of the problem
self.nSteps=100
E=11000.;nu=0.25
self.mu=E/(2*(1+nu))
self.lam=2*nu*self.mu/(1-2*nu)
self.kap=self.lam+2./3*self.mu
self.Sf=10 #Scale factor for deformed plot
self.bta=2.
def meshgn():
xs=0.;ys=0.;zs=0.;
xe=float(format(xs+geom.xlength+geom.xlength/geom.nx,'.15f'))
ye=float(format(ys+geom.ylength+geom.ylength/geom.ny,'.15f'))
ze=float(format(zs+geom.zlength+geom.zlength/geom.nz,'.15f'))
stepx=float(format(geom.xlength/geom.nx,'.15f'))
stepy=float(format(geom.ylength/geom.ny,'.15f'))
stepz=float(format(geom.zlength/geom.nz,'.15f'))
mesh=np.einsum('ijkl->ikjl',np.mgrid[xs:xe:stepx,ys:ye:stepy,zs:ze:stepz]).reshape(3,-1).T
# connectivity
col1=np.hstack(((np.arange(geom.nz*i+(i+1),(geom.nz+1)*(i+1),1) for i in range(geom.nx)))) #Layer 1 on the first face
col1=np.hstack((( (geom.nz+1)*(geom.nx+1) )*i+col1 for i in range(geom.ny)))
conn=np.vstack((col1,
col1+1,
col1+geom.nz+1,
col1+geom.nz+2,
col1+(geom.nz+1)*(geom.nx+1),
col1+1+(geom.nz+1)*(geom.nx+1),
col1+2+(geom.nz+1)*(geom.nx+1),
col1+3+(geom.nz+1)*(geom.nx+1))).T-1 #stack scalar sum to generate connectivity
return {'msh':mesh,
'con':conn
}
class GPXi():
def __init__(self,ordr):
from numpy.polynomial.legendre import leggauss #Gauss-Legendre Quadrature for 1D (proxy 2D quads -- check, 3D hex -- not checked)
self.xi=leggauss(ordr)[0] #nodes
self.wght=leggauss(ordr)[1] #weights
class basis(): # defined on the canonical element (1D : [-1,1], 2D (Q): [-1,1] x [-1,1], 3D (H): [-1,1]^3 )
def __init__(self,eltype,deg):
from sympy import Symbol,diff,Array,lambdify,tensorproduct,Matrix,flatten
if eltype=='L': #L: 1D FE
z=Symbol('z')
if deg==2.: # denotes the number of nodes
N=1/2*Array([1-z,1+z])
dfN=diff(N,z)
self.Ns=lambdify(z,N,'numpy')
self.dN=lambdify(z,dfN,'numpy')
elif deg==3.:
N=1/2*Array([z*(z-1),2*(1+z)*(1-z),z*(1+z)])
dfN=diff(N,z)
self.Ns=lambdify(z,N,'numpy')
self.dN=lambdify(z,dfN,'numpy')
else:
raise Exception('Element type not implemented yet')
elif eltype=='Q': #Q: 2D FE : Node-numbering <-- "tensor-product" starting from bottom left corner
if deg==4.:
xi=Symbol('xi');eta=Symbol('eta')
arr1=1/2*Array([1-eta,1+eta]);arr2=1/2*Array([1-xi,1+xi])
N=tensorproduct(arr1,arr2);
dfN=Matrix(flatten(diff(N,xi))).col_join(Matrix(flatten(diff(N,eta))))
self.Ns=lambdify((xi,eta),flatten(N),'numpy')
self.dN=lambdify((xi,eta),dfN,'numpy')
elif deg==9.:
xi=Symbol('xi');eta=Symbol('eta')
arr1=Array([eta*(eta-1)/2,(1-eta**2),eta*(eta+1)/2]);arr2=Array([xi*(xi-1)/2,(1-xi**2),xi*(xi+1)/2])
N=tensorproduct(arr1,arr2);
dfN=Matrix(flatten(diff(N,xi))).col_join(Matrix(flatten(diff(N,eta))))
self.Ns=lambdify((xi,eta),flatten(N),'numpy')
self.dN=lambdify((xi,eta),dfN,'numpy')
elif eltype=='H':
if deg==8.:
xi=Symbol('xi');eta=Symbol('eta');rho=Symbol('rho')
arr1=1/2*Array([1-eta,1+eta]);arr2=1/2*Array([1-xi,1+xi]);arr3=1/2*Array([1-rho,1+rho])
N=tensorproduct(arr1,arr2,arr3);
dfN=Matrix(flatten(diff(N,xi))).col_join(Matrix(flatten(diff(N,eta)))).col_join(Matrix(flatten(diff(N,rho))))
self.Ns=lambdify((xi,eta,rho),flatten(N),'numpy')
self.dN=lambdify((xi,eta,rho),dfN,'numpy')
else:
raise Exception('Only 1D, 2D and 3D continuum elements implemented')
class DWDIi(): # the substitution changes for a 3D element
def __init__(self,ndim):
from sympy import Symbol,diff,lambdify,log,transpose,Matrix
I1=Symbol('I1');J=Symbol('J');
W = 1/2*geom.mu*(I1-3)-geom.mu*log(J)+geom.lam/2*(J-1)**2 #change W here to include the modified Neo-Hookean
dWdI1=diff(W,I1);
# dWdI2=diff(W,I2);
dWdJ=diff(W,J);
d2WdI12=diff(dWdI1,I1);
d2WdJ2=diff(dWdJ,J);
if ndim==2 or ndim==3:
f11=Symbol('f11');f12=Symbol('f12');f13=Symbol('f13');
f21=Symbol('f21');f22=Symbol('f22');f23=Symbol('f23');
f31=Symbol('f31');f32=Symbol('f32');f33=Symbol('f33')
f=Matrix([f11,f12,f13,f21,f22,f23,f31,f32,f33]);
dWdI1=dWdI1.subs(I1,transpose(f).dot(f)) #substituting I1, in terms of
d2WdI12=d2WdI12.subs(I1,transpose(f).dot(f))
# dWdI2.subs(0.5*((transpose(f).dot(f)+f[0]**2)**2
# - (f[1]**2 + f[0]**2)**2
# + 2*f[0]**2*(f[1]
# + f[2])**2
# + (f[0]**2
# + f[2]**2)**2 )) # cannot get expression of I2 directly in terms of vector representation of F
Jf=f[0]*(f[4]*f[8]-f[7]*f[5])-f[1]*(f[3]*f[8]-f[6]*f[5])+f[2]*(f[3]*f[7]-f[6]*f[4])
dWdJ=dWdJ.subs(J,Jf)
d2WdJ2=d2WdJ2.subs(J,Jf)
self.DWDI1=lambdify(f,dWdI1,'numpy')
self.DWDJ=lambdify(f,dWdJ,'numpy') #output the derivative of invariants at the given F (input) as lambda function
self.D2WDI12=lambdify(f,d2WdI12,'numpy')
self.D2WDJ2=lambdify(f,d2WdJ2,'numpy')
def locmat(nodes,de): #local stiffness (jacobian) and force (residual) over the reference element
"""
Storing the Gauss-points, local basis-functions, local gradients, and global gradients.
Forming the B-matrix using kron (trick -- check notes!)
nodes --- all xs, followed by all ys followed by all zs
de --- all dx, followed by all dy, followed by all dz
"""
Xi=np.tile(np.repeat(GP.xi,OrdGauss),OrdGauss)
Eta=np.repeat(GP.xi.T,2*OrdGauss).flatten()
Rho=np.tile(GP.xi,2*OrdGauss) #Generating Gauss-Points through numpy (--check with jupyter notebook)
dof=de.reshape(de.size,-1,1).repeat(len(Xi),axis=-1) #arranging dof for (dot) product with B (len(Xi) and not len(GP.xi)) !!!
Wg=np.repeat(GP.wght,2*OrdGauss)
Nshp=np.kron(np.eye(geom.nDim).reshape(geom.nDim,geom.nDim,-1),B.Ns(Xi,Eta,Rho)) #kron has to be taken on the nDim (and not OrdGauss)
aaa.append(np.array(B.Ns(Xi,Eta,Rho)))
gDshpL=np.array(B.dN(Xi,Eta,Rho)).reshape(geom.nDim,len(Xi),-1) #local derivatives
Je=np.einsum('ilk,lj->ijk',gDshpL,nodes.reshape(geom.nDim,-1).T) #computing the jacobian (remains the same, even for 3D ? -- check ?)
# print(Wg)
detJ=np.dstack((la.det(Je[:,:,i]) for i in range(Xi.size))) # 1x1xNgP # try making it faster by removing the generator
Jeinv=np.dstack((la.inv(Je[:,:,i]) for i in range(Xi.size))) # 3x3xNgP #avoid computing inverse on a loop (--check ?)
gDshpG=np.einsum('ilk,ljk->ijk',Jeinv,gDshpL) #global derivatives (remains the same, even for 3D ? )
Bmat=np.kron(np.eye(geom.nDim).reshape(geom.nDim,geom.nDim,-1),gDshpG)
gradU=np.einsum('ilk,ljk->ijk',Bmat,dof) # 9x1xNgP #remember that gradU is never symmetric !!!
# print(gradU.reshape(geom.nDim,geom.nDim,-1))
"""
Computing the deformation gradient (F11,F12,F13...,F33).T = B*de, and first piola (S) --> (S11,S12,S13,.....,S33),
Multiplying by the Gauss-weights, and calculating the element residual (res)
"""
F=gradU+np.eye(geom.nDim).reshape(-1,1,1).repeat(len(Xi),axis=-1) #convert to 3x3xNgP and the take det(F) and inv(F)
detF=np.dstack((la.det(F.reshape(geom.nDim,geom.nDim,-1)[:,:,i]) for i in range(Xi.size))) # 1x1xNgP
WpI1=dWdIi.DWDI1(*F)
WpJ=dWdIi.DWDJ(*F)
WppI1=dWdIi.D2WDI12(*F)
WppJ=dWdIi.D2WDJ2(*F)
Finv=(np.dstack((la.inv(F.reshape(geom.nDim,geom.nDim,-1)[:,:,i]) for i in range(Xi.size)))).reshape(-1,1,Xi.size) #avoid computing inverse on the loop for the deformation gradient
S=WpI1*2*F+(WpJ*detF).reshape(1,1,-1)*Finv[np.array([0,3,6,1,4,7,2,5,8],int)] #notice the swap of axes for transpose (of the inverse)
fac=Wg*detJ
res=np.einsum('lik,ljk->ij',Bmat,fac*S) #double contraction along axis 1 and 2 (of B)
"""
Computing the Consistent Tangent: B^T *C *B <-- Cijkl, check notes
Cijkl = 4*W''_(I1) Fij Fkl + 2 W'_(I1) delik deljl + J**2*W''(J) F-1ji F-1lk +J*W'(J) F-1ji F-1lk - J W'_(J) F-1jk F-1li
F = (F12,F11,F22)
"""
# Helpful variables:
F11=F[0];F12=F[1];F13=F[2];F21=F[3];F22=F[4];F23=F[5];F31=F[6];F32=F[7];F33=F[8]
Fi11=Finv[0];Fi12=Finv[1];Fi13=Finv[2];Fi21=Finv[3];Fi22=Finv[4];Fi23=Finv[5];
Fi31=Finv[6];Fi32=Finv[7];Fi33=Finv[8]
#This C does not have minor symmetry (relates S to F) , only major symmetry
C1111=4*WppI1*F11*F11+2*WpI1+detF**2*WppJ*Fi11**2 #scalar addition to multi-dimensional array (--check??)
C1112=4*WppI1*F11*F12+detF**2*WppJ*Fi11*Fi21
C1121=4*WppI1*F11*F21+detF**2*WppJ*Fi11*Fi12
C1122=4*WppI1*F11*F22+detF**2*WppJ*Fi11*Fi22+detF*WpJ*(Fi11*Fi22-Fi12*Fi21)
C1212=4*WppI1*F12*F12+2*WpI1+detF**2*WppJ*Fi21**2
C1221=4*WppI1*F12*F21+detF**2*WppJ*Fi21*Fi12+detF*WpJ*(Fi21*Fi12 -Fi22*Fi11)
C1222=4*WppI1*F12*F22+detF**2*WppJ*Fi21*Fi22
C2121=4*WppI1*F21*F21+2*WpI1+detF**2*WppJ*Fi12**2
C2122=4*WppI1*F21*F22+detF**2*WppJ*Fi12*Fi22
C2222=4*WppI1*F22*F22+2*WpI1+detF**2*WppJ*Fi22**2
C1113=4*WppI1*F11*F13+detF**2*WppJ*Fi11*Fi31
C1123=4*WppI1*F11*F23+detF**2*WppJ*Fi11*Fi32+detF*WpJ*(Fi11*Fi32-Fi12*Fi31)
C1131=4*WppI1*F11*F31+detF**2*WppJ*Fi11*Fi13
C1132=4*WppI1*F11*F32+detF**2*WppJ*Fi11*Fi23+detF*WpJ*(Fi11*Fi23-Fi13*Fi21)
C1133=4*WppI1*F11*F33+detF**2*WppJ*Fi11*Fi33+detF*WpJ*(Fi11*Fi33-Fi13*Fi31)
C1213=4*WppI1*F12*F13+detF**2*WppJ*Fi21*Fi31
C1223=4*WppI1*F12*F23+detF**2*WppJ*Fi21*Fi32+detF*WpJ*(Fi21*Fi32-Fi22*Fi31)
C1231=4*WppI1*F12*F31+detF**2*WppJ*Fi21*Fi13+detF*WpJ*(Fi21*Fi13-Fi23*Fi11)
C1232=4*WppI1*F12*F32+detF**2*WppJ*Fi21*Fi23
C1233=4*WppI1*F12*F33+detF**2*WppJ*Fi21*Fi33+detF*WpJ*(Fi21*Fi33-Fi23*Fi31)
C1313=4*WppI1*F13*F13+2*WpI1+detF**2*WppJ*Fi31**2
C1322=4*WppI1*F13*F22+detF**2*WppJ*Fi31*Fi22+detF*WpJ*(Fi31*Fi22-Fi32*Fi21)
C1323=4*WppI1*F13*F23+detF**2*WppJ*Fi31*Fi32
C1331=4*WppI1*F13*F31+detF**2*WppJ*Fi31*Fi13+detF*WpJ*(Fi31*Fi13-Fi33*Fi11)
C1332=4*WppI1*F13*F32+detF**2*WppJ*Fi31*Fi23+detF*WpJ*(Fi31*Fi23-Fi33*Fi21)
C1333=4*WppI1*F13*F33+detF**2*WppJ*Fi31*Fi33
C2113=4*WppI1*F21*F13+detF**2*WppJ*Fi12*Fi31+detF*WpJ*(Fi12*Fi31-Fi11*Fi32)
C2123=4*WppI1*F21*F23+detF**2*WppJ*Fi12*Fi32
C2131=4*WppI1*F21*F31+detF**2*WppJ*Fi12*Fi13
C2132=4*WppI1*F21*F32+detF**2*WppJ*Fi12*Fi23+detF*WpJ*(Fi23*Fi12-Fi22*Fi13)
C2133=4*WppI1*F21*F33+detF**2*WppJ*Fi12*Fi33+detF*WpJ*(Fi12*Fi33-Fi13*Fi32)
C2223=4*WppI1*F22*F23+detF**2*WppJ*Fi22*Fi32
C2231=4*WppI1*F22*F31+detF**2*WppJ*Fi22*Fi13+detF*WpJ*(Fi22*Fi13-Fi23*Fi12)
C2232=4*WppI1*F22*F32+detF**2*WppJ*Fi22*Fi23
C2233=4*WppI1*F22*F33+detF**2*WppJ*Fi22*Fi33+detF*WpJ*(Fi22*Fi33-Fi23*Fi32)
C2323=4*WppI1*F23*F23+2*WpI1+detF**2*WppJ*Fi32**2
C2331=4*WppI1*F23*F31+detF**2*WppJ*Fi32*Fi13+detF*WpJ*(Fi32*Fi13-Fi33*Fi12)
C2332=4*WppI1*F23*F32+detF**2*WppJ*Fi32*Fi23+detF*WpJ*(Fi32*Fi23-Fi33*Fi22)
C2333=4*WppI1*F23*F33+detF**2*WppJ*Fi32*Fi33
C3131=4*WppI1*F31*F31+2*WpI1+detF**2*WppJ*Fi13**2
C3132=4*WppI1*F31*F32+detF**2*WppJ*Fi13*Fi23
C3133=4*WppI1*F31*F33+detF**2*WppJ*Fi13*Fi33
C3232=4*WppI1*F32*F32+2*WpI1+detF**2*WppJ*Fi23**2
C3233=4*WppI1*F32*F33+detF**2*WppJ*Fi23*Fi33
C3333=4*WppI1*F33*F33+2*WpI1+detF**2*WppJ*Fi33**2
# C1111=C1111.flatten()
# C1112=C1112.flatten()
# C1121=C1121.flatten()
# C1122=C1122.flatten()
# C1212=C1212.flatten()
# C1221=C1221.flatten()
# C1222=C1222.flatten()
# C2121=C2121.flatten()
# C2122=C2122.flatten()
# C2222=C2222.flatten()
# This part changes when C loses major symmetry (non-conservative systems ?)
C1211=C1112.copy(); C1311=C1113.copy(); C1312=C1213.copy()
C1321=C2113.copy(); C2111=C1121.copy(); C2112=C1221.copy()
C2211=C1122.copy(); C2212=C1222.copy(); C2213=C1322.copy()
C2221=C2122.copy(); C2311=C1123.copy(); C2312=C1223.copy()
C2313=C1323.copy(); C2321=C2123.copy(); C2322=C2223.copy()
C3111=C1131.copy(); C3112=C1231.copy(); C3113=C1331.copy()
C3121=C2131.copy(); C3122=C2231.copy(); C3123=C2331.copy()
C3211=C1132.copy(); C3212=C1232.copy(); C3213=C1332.copy()
C3221=C2132.copy(); C3222=C2232.copy(); C3223=C2332.copy()
C3231=C3132.copy(); C3311=C1133.copy(); C3312=C1233.copy()
C3313=C1333.copy(); C3321=C2133.copy(); C3322=C2233.copy()
C3323=C2333.copy(); C3331=C3133.copy(); C3332=C3233.copy()
C=np.array([[C1111,C1112,C1113,C1121,C1122,C1123,C1131,C1132,C1133],
[C1211,C1212,C1213,C1221,C1222,C1223,C1231,C1232,C1233],
[C1311,C1312,C1313,C1321,C1322,C1323,C1331,C1332,C1333],
[C2111,C2112,C2113,C2121,C2122,C2123,C2131,C2132,C2133],
[C2211,C2212,C2213,C2221,C2222,C2223,C2231,C2232,C2233],
[C2311,C2312,C2313,C2321,C2322,C2323,C2331,C2332,C2333],
[C3111,C3112,C3113,C3121,C3122,C3123,C3131,C3132,C3133],
[C3211,C3212,C3213,C3221,C3222,C3223,C3231,C3232,C3233],
[C3311,C3312,C3313,C3321,C3322,C3323,C3331,C3332,C3333]]).reshape(9,9,-1)
# print(C[-1,-1,-1])
C *= fac
D=np.einsum('lik,lpk,pjk->ij',Bmat,C,Bmat) #Check the multiplication once for a simple case!
return {'K':D,
'F':res.flatten(),
'Stress':S,
'DefGrad':F,
'InptGlobal':np.einsum('ilj,l->ij',Nshp,nodes)}
Eltype='H8'
OrdGauss=2 #No. of Gauss-points (in 2D: # of points in each direction counted the same way as local nodes)
geom=geometry(Eltype)
B=basis(Eltype[0],float(Eltype[1]))
GP=GPXi(OrdGauss)
dWdIi=DWDIi(geom.nDim)
meshxyz=meshgn()['msh']
conVxyz=meshgn()['con']
aaa=([])
dof=1.e9*np.ones(meshxyz.size) #initializing dofs (displacement of nodes)
def assembly(disp):
globK=0.*np.eye(disp.size)
globF=np.zeros(disp.size)
for i in range(len(conVxyz)):
elnodes=conVxyz[i]
# print(elnodes)
globdof=np.array([3*elnodes,3*elnodes+1,3*elnodes+2]).flatten()#.T.flatten() : gets the elemental dofs in the order u1.....,w3
# print(globdof)
nodexy=meshxyz[elnodes]
locdisp=disp[globdof]
# print(locdisp)
globK[np.ix_(globdof,globdof)] += locmat(nodexy.T.flatten(),locdisp)['K']
# print(globK)
globF[globdof] += locmat(nodexy.T.flatten(),locdisp)['F']
strs=locmat(nodexy.T.flatten(),locdisp)['Stress']
DG=locmat(nodexy.T.flatten(),locdisp)['DefGrad']
# calculate strains and integration point coordinates
Strn=(np.einsum('lik,ljk->ijk',DG.reshape(geom.nDim,geom.nDim,-1),DG.reshape(geom.nDim,geom.nDim,-1))-np.eye(geom.nDim).reshape(geom.nDim,geom.nDim,-1).repeat(8,axis=-1))/2
epsStrn=(np.einsum('ijk->jik',DG.reshape(geom.nDim,geom.nDim,-1))+DG.reshape(geom.nDim,geom.nDim,-1)-2*np.eye(geom.nDim).reshape(geom.nDim,geom.nDim,-1).repeat(8,axis=-1))/2
intpt.append(locmat(nodexy.T.flatten(),locdisp)['InptGlobal'])
# print(epsStrn[:,:,-1])
return {'Jac':globK,
'rhs':globF,
'S':strs.reshape(geom.nDim,geom.nDim,-1),
'F':DG.reshape(geom.nDim,geom.nDim,-1),
'E':Strn,
'SmlStrain':epsStrn,
'IntP':intpt}
#def bcassign(nodes):
# dofpres=np.array([[1.,1,1,0.02*geom.ylength],
# [0.,1,1,0.02*geom.ylength]]) #xcoor,ycoor,dir (x:0, y:1),val
#
prescribed_dofs=np.array([[0,0.],
[1,0],
[2,0],
[8,0],
[12,0],
[14,0],
[20,0.],
[5,geom.bta*geom.zlength],
[11,geom.bta*geom.zlength],
[17,geom.bta*geom.zlength],
[23,geom.bta*geom.zlength]])
dof[(prescribed_dofs[:,0]).astype(int)]=0.
fdof=dof==1.e9 #free dofs flags: further initialization to zeros needed only for the first step
nfdof=np.invert(fdof) #fixed dofs flags
dof[fdof]=0.
lineardof=dof.copy()
DfGrn=([]) # Deformation Gradient
Strs=([]) # First-PK Stress
LagStrain=([]) # Lagrangian Strain
epss=([])
intpt=([])
# Linear FEM (as a special case of Nonlinear)
intpt1=assembly(dof)['IntP'][0].T #global gauss-point
Ks=assembly(lineardof)['Jac'] #linear stiffness
lineardof[(prescribed_dofs[:,0]).astype(int)]=prescribed_dofs[:,1]
Fs=assembly(lineardof)['rhs']
lineardof[fdof]=la.solve(Ks[np.ix_(fdof,fdof)],-Ks[np.ix_(fdof,nfdof)]@lineardof[nfdof])
linearStress=assembly(lineardof)['S']
linearStrain=assembly(lineardof)['SmlStrain']
##
dofstore=np.zeros(dof.shape);
#Strn=np.zeros((3,3,8))
#Strs=np.zeros((3,3,8))
#Fs=assembly(dof.reshape(-1,3).T.flatten())['rhs']
#
for i in range(geom.nSteps):
dof[(prescribed_dofs[:,0]).astype(int)]=(i+1)/(geom.nSteps)*prescribed_dofs[:,1]
Ks1=assembly(dof)['Jac']
Fs1=assembly(dof)['rhs']
normres=la.norm(Fs1[fdof],2)
iterNR=0;
while normres >= geom.tolNR and iterNR<=geom.maxiter:
dof[fdof] += la.solve(Ks1[np.ix_(fdof,fdof)],-Fs1[fdof]) #external force add (-- not required here, only for this case though)
Ks1=assembly(dof)['Jac']
Fs1=assembly(dof)['rhs']
normres=la.norm(Fs1[fdof],2)
print('Iter: {}'.format(iterNR))
iterNR += 1
dofstore=np.vstack((dofstore,dof))
DfGrn.append(assembly(dof)['F'])
Strs.append(assembly(dof)['S'])
LagStrain.append(assembly(dof)['E'])
epss.append(assembly(dof)['SmlStrain'])
#
##intpt=
DfGrn=np.array(DfGrn)
Strs=np.array(Strs)
LagStrain=np.array(LagStrain)
epss=np.array(epss)
##intpt=np.array(intpt)
##
#################################################################################
###Output the Strain in respective excel sheets
###Linear Stresses and Strains at all integration points
###pd.DataFrame(linearStress.reshape(-1,8).T).to_excel('LinearStress.xlsx',index=False,header=False)
###pd.DataFrame(linearStrain.reshape(-1,8).T).to_excel('LinearStrain.xlsx',index=False,header=False)
###pd.DataFrame(Strs[:,-1,-1,-1]).to_excel('StrsHist.xlsx',index=False,header=False)
###pd.DataFrame(dofstore[:,[-3,-2,-1]]).to_excel('Disp.xlsx',index=False,header=False)
plt.figure()
plt.plot(np.linspace(0,1,Strs[:,-1,-1,-1].size),Strs[:,-1,-1,-1],label=r'$\sigma_{33}$')
plt.xlabel(r'$t$ (Time)',fontsize=20)
plt.ylabel(r'$\sigma_{33}$ (Axial Stress)',fontsize=20)
plt.legend(loc=0,fontsize=20)
#plt.grid(True)
plt.legend(loc=0,fontsize=14)
plt.ticklabel_format(style='sci',axis='y',scilimits=(0,0))
ax=plt.gca()
ax.xaxis.set_minor_locator(mx)
ax.set_title(r'Stress vs time',fontsize=20)
ax.yaxis.set_minor_locator(my)
##plt.savefig('stress.eps')
#plt.close()
#plt.figure(3)
# plt.plot(xp,fexact(xp)['stress'],label=r'Exact Solution')
# if El[1]=='2':
# plt.plot(xp,fsample(xp,dof)['stress'],label=r'Linear Basis')
# elif El[1]=='3':
# plt.plot(xp,fsample(xp,dof)['stress'],label=r'Quadratic Basis')
#
###############################################################################
#Visualization using Scipy.interpolate's n-D griddata
#from scipy.interpolate import griddata
#mgD=np.mgrid[-1:1.25:0.25,-1:1.25:0.25,-1:1.25:0.25]
#Strs33=griddata(intpt1,Strs[-1,-1,-1,:],
# np.einsum('ijkl->ikjl',mgD).reshape(3,-1).T,method='nearest') #last step stress33 component
#xx=np.arange(-1.,1.25,0.25);yy=xx.copy()
###############################################################################
##Create vtk data file for visualization in Paraview
#filename='CIHW3.vtk'
#filenamedef='CIHW3def.vtk'
#filestrsstrn='CIHW31.vtk'
#name='Hex8'
#if name=='Hex8':
# ParaviewID=12
#output_file=open(filename,'w')
#output_filedef=open(filenamedef,'w')
#output_filestrsstrn=open(filestrsstrn,'w')
#
#output_file.write('# vtk DataFile Version 2.0\n')
#output_file.write('%s\n' %name);
#output_file.write('ASCII\n');
#output_file.write('\n');
##nodes
#output_file.write('DATASET UNSTRUCTURED_GRID\n')
#output_file.write('POINTS %d float\n' %(len(meshxyz))); #list of nodes
#for i in range(len(meshxyz)):
# output_file.write('%15.10f %15.10f %15.10f \n'%( meshxyz[i,0],
# meshxyz[i,1],
# meshxyz[i,2]))
#
#output_file.write('\n');
#output_file.write('CELLS %d %d \n' %(len(conVxyz), (len(conVxyz[0])+1)*len(conVxyz))) #connectivity
#output_file.write('8 %d %d %d %d %d %d %d %d'%(conVxyz[0,0],
# conVxyz[0,2],
# conVxyz[0,6],
# conVxyz[0,4],
# conVxyz[0,1],
# conVxyz[0,3],
# conVxyz[0,7],
# conVxyz[0,5]
# ))
#output_file.write('\n')
#output_file.write('CELL_TYPES %d\n' %(len(conVxyz)))
#a=ParaviewID*np.ones(len(conVxyz),int)
#output_file.write('\n'.join(map(str,a)))
#output_file.write('\nPOINT_DATA %d\n' %len(meshxyz)) #data fields
#output_file.write('SCALARS U float 3\n')
#output_file.write('Lookup_table default\n')
#
#for i in range (len(meshxyz)):
# output_file.write('%12.10f %12.10f %12.10f\n' %(dofstore[-1].reshape(-1,3)[i,0],
# dofstore[-1].reshape(-1,3)[i,1],
# dofstore[-1].reshape(-1,3)[i,2]))
#output_file.write('\n')
#output_file.close()
#
## Deformed data
#output_filedef.write('# vtk DataFile Version 2.0\n')
#output_filedef.write('%s\n' %name);
#output_filedef.write('ASCII\n');
#output_filedef.write('\n');
##nodes
#output_filedef.write('DATASET UNSTRUCTURED_GRID\n')
#output_filedef.write('POINTS %d float\n' %(len(meshxyz))); #list of nodes
#
#for i in range(len(meshxyz)):
# output_filedef.write('%15.10f %15.10f %15.10f \n'%( meshxyz[i,0]+geom.Sf*dofstore[-1].reshape(-1,3)[i,0],
# meshxyz[i,1]+geom.Sf*dofstore[-1].reshape(-1,3)[i,1],
# meshxyz[i,2]+geom.Sf*dofstore[-1].reshape(-1,3)[i,2]))
#
#output_filedef.write('\n');
#output_filedef.write('CELLS %d %d \n' %(len(conVxyz), (len(conVxyz[0])+1)*len(conVxyz))) #connectivity
#output_filedef.write('8 %d %d %d %d %d %d %d %d'%(conVxyz[0,0],
# conVxyz[0,2],
# conVxyz[0,6],
# conVxyz[0,4],
# conVxyz[0,1],
# conVxyz[0,3],
# conVxyz[0,7],
# conVxyz[0,5]
# ))
#output_filedef.write('\n')
#output_filedef.write('CELL_TYPES %d\n' %(len(conVxyz)))
#a=ParaviewID*np.ones(len(conVxyz),int)
#output_filedef.write('\n'.join(map(str,a)))
#output_filedef.write('\nPOINT_DATA %d\n' %len(meshxyz)) #data fields
#output_filedef.write('SCALARS Udef float 3\n')
#output_filedef.write('Lookup_table default\n')
#
#for i in range (len(meshxyz)):
# output_filedef.write('%12.10f %12.10f %12.10f\n' %(dofstore[-1].reshape(-1,3)[i,0],
# dofstore[-1].reshape(-1,3)[i,1],
# dofstore[-1].reshape(-1,3)[i,2]))
#output_filedef.write('\n')
#
#output_filedef.close()
#
##Stress-Strain File
#
#output_filestrsstrn.write('# vtk DataFile Version 2.0\n')
#output_filestrsstrn.write('%s\n' %name);
#output_filestrsstrn.write('ASCII\n');
#output_filestrsstrn.write('\n');
##nodes
#output_filestrsstrn.write('DATASET UNSTRUCTURED_GRID\n')
#output_filestrsstrn.write('POINTS %d float\n' %(len(meshxyz))); #list of integration point
#for i in range(len(intpt1)):
# output_filestrsstrn.write('%15.10f %15.10f %15.10f \n'%( intpt1[i,0],
# intpt1[i,1],
# intpt1[i,2])) #write global integration point
#
#output_filestrsstrn.write('\n');
#output_filestrsstrn.write('CELLS %d %d \n' %(len(conVxyz), (len(conVxyz[0])+1)*len(conVxyz))) #connectivity
#output_filestrsstrn.write('8 %d %d %d %d %d %d %d %d'%(conVxyz[0,0],
# conVxyz[0,2],
# conVxyz[0,6],
# conVxyz[0,4],
# conVxyz[0,1],
# conVxyz[0,3],
# conVxyz[0,7],
# conVxyz[0,5]
# ))
#output_filestrsstrn.write('\n')
#output_filestrsstrn.write('CELL_TYPES %d\n' %(len(conVxyz)))
#a=ParaviewID*np.ones(len(conVxyz),int)
#output_filestrsstrn.write('\n'.join(map(str,a)))
#output_filestrsstrn.write('\nPOINT_DATA %d\n' %len(meshxyz)) #data fields
#output_filestrsstrn.write('SCALARS E33 float 1\n')
#output_filestrsstrn.write('Lookup_table default\n')
#
#for i in range (len(intpt1)):
# output_filestrsstrn.write('%12.10f \n' %(LagStrain[-1,-1,-1,i]))
#output_filestrsstrn.write('\n')
#
##output_filestrsstrn.write('\nPOINT_DATA %d\n' %len(meshxyz)) #data fields
#output_filestrsstrn.write('SCALARS S33 float 1\n')
#output_filestrsstrn.write('Lookup_table default\n')
#
#for i in range (len(intpt1)):
# output_filestrsstrn.write('%12.10f \n' %(Strs[-1,-1,-1,i]))
#output_filestrsstrn.write('\n')
#output_filestrsstrn.close()