midtermprob3m.py

# -*- 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
matplotlib.rcParams['lines.linewidth']=3
rc('xtick',labelsize=18)
rc('ytick',labelsize=18)
mx=AutoMinorLocator(10)
my=AutoMinorLocator(10)
##################################################################
class geometry() : #1D geometry
    def __init__(self,Eltype,Tf,E,nu):
        self.tolNR=1.e-12
        self.maxiter=50
        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=1.
            self.ylength=1.
            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
            self.mu=40.
            kap=1.e1*self.mu 
            self.lam=40.
        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
            self.mu=E/(2*(1+nu))
            self.lam=2*nu*self.mu/(1-2*nu)
            self.kap=self.lam+2./3*self.mu
            self.Ko=120.
            self.Kp=900.
            self.Hp=2.5
            self.NGp=8    # number of gauss point, remember to change this if you change integration order 
            self.Sf=10                #Scale factor for deformed plot
            Ee=np.eye(3)
            self.Eye=0.5*(np.einsum('ik,jl->ijkl',Ee,Ee)+np.einsum('il,jk->ijkl',Ee,Ee))
            self.dt=Tf/self.nSteps
            self.T=np.linspace(0.,Tf,self.nSteps+1)

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,flatten
        I1=Symbol('I1');I2=Symbol('I2');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 f(eta,K):
#    return la.norm(eta,2)-(2/3.)**0.5*K    #yield surface boundary 
#          
init=True
def locmat(nodes,de,qe,alphe,eplast):                     #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)
#    print(nodes.shape)
    gDshpL=np.array(B.dN(Xi,Eta,Rho)).reshape(geom.nDim,int(Eltype[1]),-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 ?)
    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)
    """
    Checking if the stress-state is admissible, f(....) <= 0
    Check for the yield surface, the backstress and the plastic strain
    Compute Cep_ijkl separately
    """
    F3b3=F.reshape(geom.nDim,geom.nDim,-1)
    epSS=0.5*(F3b3+np.einsum('ijk->jik',F3b3)-2*np.eye(geom.nDim).reshape(3,3,-1).repeat(geom.NGp,axis=-1))
    S = (2*geom.mu*epSS+geom.lam*np.einsum('iik->k',epSS)*np.eye(geom.nDim).reshape(3,3,-1).repeat(geom.NGp,axis=-1)).reshape(-1,1,geom.NGp)
    edev=epSS-1./3*np.einsum('iik->k',epSS)*np.eye(geom.nDim).reshape(3,3,-1).repeat(geom.NGp,axis=-1)  #    print(epSS[:,:,0])
    strial=2*geom.mu*(edev-eplast)
    sEff = strial - qe
    ftrial = la.norm(sEff,2,axis=(0,1)) - (2./3)**0.5*(geom.Ko+geom.Kp*alphe)  
    plsTidx=ftrial>0
    thta=np.ones(geom.NGp)
    thtab=np.zeros(geom.NGp)
#    if la.norm(sEff[:,:,0],2)==0.:          #trial check for the first step 
#        nNp1=np.zeros((3,3,geom.NGp))
#    else:
#        nNp1=sEff/la.norm(sEff,2,axis=(0,1))    #n_{n+1}
    for kgp in range(geom.NGp):
        if ftrial[kgp] > 0:
            nNp1=sEff[:,:,kgp]/la.norm(sEff,2,axis=(0,1))[kgp]
            dgam=3*ftrial[kgp]/(2*(geom.Hp+geom.Kp+3*geom.mu))
            stemp=strial[:,:,kgp]+geom.kap*np.trace(epSS[:,:,kgp])-2*geom.mu*dgam*nNp1
            S[:,:,kgp] = stemp.reshape(-1,1)
            thta[kgp]=1.-2*geom.mu*dgam/la.norm(sEff,2,axis=(0,1))[kgp]
            thtab[kgp]=(1.+1/(3*geom.mu)*(geom.Kp+geom.Hp))**(-1) + thta[kgp] - 1. 
            qe[:,:,kgp] += 2./3*dgam*geom.Hp
            eplast[:,:,kgp] += dgam*nNp1
        else:
            nNp1=np.zeros((3,3))
            dgam=0.
            stemp=strial[:,:,kgp]+geom.kap*np.trace(epSS[:,:,kgp])-2*geom.mu*dgam*nNp1
            S[:,:,kgp] = stemp.reshape(-1,1)
#            thta[kgp]=1.-2*geom.mu*dgam/la.norm(sEff,2,axis=(0,1))[kgp]
#            thtab[kgp]=(1.+1/(3*geom.mu)*(geom.Kp+geom.Hp))**(-1) + thta[kgp] - 1. 
            qe[:,:,kgp] += 2./3*dgam*geom.Hp
            eplast[:,:,kgp] += dgam*nNp1
               
#    if dgam[0]==0:
#        thta=np.ones(geom.NGp)
#        thtab=np.zeros(geom.NGp)
#    else:
#        thta=1.-2*geom.mu*dgam/la.norm(sEff,2,axis=(0,1)); 
#        thtab=(1.+1/(3*geom.mu)*(geom.Kp+geom.Hp))**(-1) + thta - 1.
    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 
    """
#    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

# 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) 
    
#    if la.norm(sEff[:,:,0],2)==0.:
#        thta=np.ones(geom.NGp)
#        thtab=np.zeros(geom.NGp)
#        print('thta:\n',thta)
#        thtab=((1.+1/(3*geom.mu)*(geom.Kp+geom.Hp))**(-1) + thta - 1.)*0.
    
#        print('thtab:\n',thtab)
    IDGp = (np.einsum('ij,kl->ijkl',np.eye(geom.nDim),np.eye(geom.nDim))[:,:,:,:,np.newaxis]).repeat(geom.NGp,axis=-1)
    EyeGp = geom.Eye[:,:,:,:,np.newaxis].repeat(geom.NGp,axis=-1)
    Cep=(geom.kap*IDGp+2*geom.mu*thta*(EyeGp-1./3*IDGp) - 2.*geom.mu*thtab*np.einsum('ijg,klg->ijklg',nNp1,nNp1)).reshape(9,9,-1,order='A')
    C[:,:,plsTidx]=Cep[:,:,plsTidx].copy()           #only take values for those gauss points which are plastified
    # Start computation here tomorrow 
    """
    Some useful variables and shapes of previous variables: 
        nNp1 = 3x3xNGp
        dgam = NGp     (hoping that scalar multiplication along axis3 works!)
        thta = NGp
        thtab= NGp
        la.norm(sEff,2,axis=(0,1)) = NGp 
        Now start doing einsum to generate Cep, by first introducing a 
        new axis at the end that keeps track of Gauss-points for the above
        n-d arrays
        IdGp  : del ij del kl x NGp
        EyeGP : Eye extended along 5th dimension
    """
    C *= fac
    D=np.einsum('lik,lpk,pjk->ij',Bmat,C,Bmat)                                #Check the multiplication once for a simple case!
#    print(eplast.shape)
#    print('mtwhat:',S[-1,-1,-1])
    return D,res.flatten(),S,F,np.einsum('ilj,l->ij',Nshp,nodes),qe,alphe,eplast

Eltype='H8'
OrdGauss=2           #No. of Gauss-points (in 2D: # of points in each direction counted the same way as local nodes)
Tf = 1.2
globE=11000.
globnu=0.25
geom=geometry(Eltype,Tf,globE,globnu)
B=basis(Eltype[0],float(Eltype[1]))
GP=GPXi(OrdGauss) 
dWdIi=DWDIi(geom.nDim)

meshxyz=meshgn()['msh']
conVxyz=meshgn()['con']

dof=1.e9*np.ones(meshxyz.size) #initializing dofs (displacement of nodes)

def assembly(disp,q,alph,Ep):
    globK=0.*np.eye(disp.size)
    globF=np.zeros(disp.size) 
    strs=np.zeros((len(conVxyz),9,1,geom.NGp))
    Strn=np.zeros((len(conVxyz),geom.nDim,geom.nDim,geom.NGp))
    DG=np.zeros((len(conVxyz),9,1,geom.NGp))
    epsStrn=np.zeros((len(conVxyz),geom.nDim,geom.nDim,geom.NGp)) #geom.nDim=3 in this case
    for i in range(len(conVxyz)):
        elnodes=conVxyz[i]
        ep=Ep[i]
        qe=q[i]
        alphe=alph[i]
        globdof=np.array([3*elnodes,3*elnodes+1,3*elnodes+2]).flatten()#.T.flatten()  : gets the elemental dofs in the order u1.....,w3
        nodexy=meshxyz[elnodes]
        locdisp=disp[globdof]
        elemK,elemF,strs[i],DG[i],gipt,q[i],alph[i],Ep[i]=locmat(nodexy.T.flatten(),locdisp,qe,alphe,ep)
        globK[np.ix_(globdof,globdof)] += elemK
        globF[globdof] += elemF  
        intpt.append(gipt)

#        globK[np.ix_(globdof,globdof)] += locmat(nodexy.T.flatten(),locdisp,qe,alphe,ep)['K']
#        globF[globdof] += locmat(nodexy.T.flatten(),locdisp,qe,alphe,ep)['F']
#        strs[i]=locmat(nodexy.T.flatten(),locdisp,qe,alphe,ep)['Stress']
#        DG[i]=locmat(nodexy.T.flatten(),locdisp,qe,alphe,ep)['DefGrad']
#        calculate strains and integration point coordinates
        Strn[i]=(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[i]=(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
        strs=strs.reshape(len(conVxyz),geom.nDim,geom.nDim,-1)    
        DG=DG.reshape(len(conVxyz),geom.nDim,geom.nDim,-1)
#        print('globK=',globK)
    return globK,globF,strs,DG,Strn,epsStrn,intpt,q,Ep,alph
#{'Jac':globK,
#            'rhs':globF,
#            'S':strs.reshape(len(conVxyz),geom.nDim,geom.nDim,-1),
#            'F':DG.reshape(len(conVxyz),geom.nDim,geom.nDim,-1),
#            'E':Strn,
#            'SmlStrain':epsStrn,
#            'IntP':intpt,
#            'qGlob':q,
#            'EP':Ep,
#            'Alpha':alph}
        
prescribed_dofs=np.array([[0,0.],
                          [1,0],
                          [2,0],
                          [8,0],
                          [12,0],
                          [14,0],
                          [20,0.],
                          [5,0.02*geom.zlength],
                          [11,0.02*geom.zlength],
                          [17,0.02*geom.zlength],
                          [23,0.02*geom.zlength]])           #apply 2% strain

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=np.zeros((len(conVxyz),geom.nDim,geom.nDim,geom.NGp,geom.nSteps+1))                  # Deformation Gradient
Strs=np.zeros((len(conVxyz),geom.nDim,geom.nDim,geom.NGp,geom.nSteps+1))                   # First-PK Stress
LagStrain=np.zeros((len(conVxyz),geom.nDim,geom.nDim,geom.NGp,geom.nSteps+1))              # Lagrangian Strain
epss=np.zeros((len(conVxyz),geom.nDim,geom.nDim,geom.NGp,geom.nSteps+1))
intpt=([])
qGlobal=np.zeros((len(conVxyz),geom.nDim,geom.nDim,geom.NGp,geom.nSteps+1))    #check dimensions once again though
alphGlobal=np.zeros((len(conVxyz),geom.NGp,geom.nSteps+1))                     #just a scalar
Eplastic=np.zeros((len(conVxyz),geom.nDim,geom.nDim,geom.NGp,geom.nSteps+1))                     #initialize the plastic strain
#_,_,_,_,_,_,intpt1,_,_,_=assembly(dof,qGlobal[:,:,:,:,0],alphGlobal[:,:,0],Eplastic[:,:,:,:,0])    #global gauss-point locations (?)
#intpt1=intpt1[0].T
dofstore=np.zeros(dof.shape);
Ks3,_,_,_,_,_,_,_,_,_=assembly(dof,qGlobal[:,:,:,:,0],alphGlobal[:,:,0],Eplastic[:,:,:,:,0])  #linear stiffness

#Newton-Steps
for istep in range(geom.nSteps):
    dof[(prescribed_dofs[:,0]).astype(int)]=(istep+1)/(geom.nSteps)*prescribed_dofs[:,1]
    Ks1,Fs1,_,_,_,_,_,_,_,_=assembly(dof,qGlobal[:,:,:,:,istep],alphGlobal[:,:,istep],Eplastic[:,:,:,:,istep])
    normres=la.norm(Fs1[fdof],2)
    iterNR=0;
#    print(normres)
#    break
    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)
#        print(dof[6])
        Ks1,Fs1,_,_,_,_,_,_,_,_=assembly(dof,qGlobal[:,:,:,:,istep],alphGlobal[:,:,istep],Eplastic[:,:,:,:,istep]) 
        normres=la.norm(Fs1[fdof],2)
        print('Iter: {}'.format(iterNR+1))
        iterNR += 1
    print('Step: {}'.format(istep+1))
    dofstore=np.vstack((dofstore,dof))
    _,_,Strs[:,:,:,:,istep+1],DfGrn[:,:,:,:,istep+1],LagStrain[:,:,:,:,istep+1],epss[:,:,:,:,istep+1],_,qGlobal[:,:,:,:,istep+1],Eplastic[:,:,:,:,istep+1],alphGlobal[:,:,istep+1]=assembly(dof,qGlobal[:,:,:,:,istep],alphGlobal[:,:,istep],Eplastic[:,:,:,:,istep])
###############################################################################
#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)
plot_it=False
if plot_it:
    plt.figure(figsize=(10,10))
    plt.plot(geom.T,Strs[0,0,0,-1,:],label=r'$\sigma_{11}$')
    plt.plot(geom.T,Strs[0,0,1,-1,:],label=r'$\sigma_{12}$')
    plt.plot(geom.T,Strs[0,1,1,-1,:],label=r'$\sigma_{22}$')
    plt.plot(geom.T,Strs[0,-1,-1,-1,:],label=r'$\sigma_{33}$')
    plt.xlabel(r'$t$ (Time)',fontsize=20)
    plt.ylabel(r'$\sigma_{ij}$ ',fontsize=20)
    plt.legend(loc=0,fontsize=20)
    #plt.grid(True)
    plt.legend(loc=0,fontsize=18)
    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.figure(figsize=(10,10))
    plt.plot(geom.T,qGlobal[0,0,0,-1,:],label=r'$q_{11}$')
    plt.plot(geom.T,qGlobal[0,0,1,-1,:],label=r'$q_{12}$')
    plt.plot(geom.T,qGlobal[0,1,1,-1,:],label=r'$q_{22}$')
    plt.plot(geom.T,qGlobal[0,-1,-1,-1,:],label=r'$q_{33}$')
    plt.xlabel(r'$t$ (Time)',fontsize=20)
    plt.ylabel(r'$q_{ij}$',fontsize=20)
    plt.legend(loc=0,fontsize=20)
    #plt.grid(True)
    plt.legend(loc=0,fontsize=18)
    plt.ticklabel_format(style='sci',axis='y',scilimits=(0,0))
    ax=plt.gca()
    ax.xaxis.set_minor_locator(mx)
    ax.set_title(r'Back-Stress vs time',fontsize=20)
    ax.yaxis.set_minor_locator(my) 
    
    
    plt.figure(figsize=(10,10))
    plt.plot(geom.T,alphGlobal[0,-1,:],label=r'$\alpha$')
    #plt.plot(geom.T,Strs[:,0,1,-1],label=r'$\sigma_{12}$')
    #plt.plot(geom.T,Strs[:,1,1,-1],label=r'$\sigma_{22}$')
    #plt.plot(geom.T,Strs[:,-1,-1,-1],label=r'$\sigma_{33}$')
    plt.xlabel(r'$t$ (Time)',fontsize=20)
    plt.ylabel(r'$\alpha$',fontsize=20)
    plt.legend(loc=0,fontsize=20)
    #plt.grid(True)
    plt.legend(loc=0,fontsize=18)
    plt.ticklabel_format(style='sci',axis='y',scilimits=(0,0))
    ax=plt.gca()
    ax.xaxis.set_minor_locator(mx)
    ax.set_title(r'$\alpha$ 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()
#