Matrix generation

Init symbols for sympy



In [1]:

    
from sympy import *
from geom_util import *
from sympy.vector import CoordSys3D
N = CoordSys3D('N')
alpha1, alpha2, alpha3 = symbols("alpha_1 alpha_2 alpha_3", real = True, positive=True)
init_printing()

%matplotlib inline

%reload_ext autoreload
%autoreload 2
%aimport geom_util

Lame params



In [2]:

    
H1=symbols('H1')
H2=S(1)
H3=S(1)

H=[H1, H2, H3]
DIM=3
dH = zeros(DIM,DIM)
for i in range(DIM):
    for j in range(DIM):
        if (i == 0 and j != 1):
            dH[i,j]=Symbol('H_{{{},{}}}'.format(i+1,j+1))

dH









    Out[2]:





$$\left[\begin{matrix}H_{1,1} & 0 & H_{1,3}\\0 & 0 & 0\\0 & 0 & 0\end{matrix}\right]$$

Metric tensor

${\displaystyle \hat{G}=\sum_{i,j} g^{ij}\vec{R}_i\vec{R}_j}$



In [3]:

    
G_up = getMetricTensorUpLame(H1, H2, H3)

${\displaystyle \hat{G}=\sum_{i,j} g_{ij}\vec{R}^i\vec{R}^j}$



In [4]:

    
G_down = getMetricTensorDownLame(H1, H2, H3)

Christoffel symbols



In [5]:

    
DIM=3

G_down_diff = MutableDenseNDimArray.zeros(DIM, DIM, DIM)
for i in range(DIM):
    for j in range(DIM):
        for k in range(DIM):
                
            G_down_diff[i,i,k]=2*H[i]*dH[i,k]
                

GK = getChristoffelSymbols2(G_up, G_down_diff, (alpha1, alpha2, alpha3))
GK









    Out[5]:





$$\left[\begin{matrix}\left[\begin{matrix}\frac{H_{1,1}}{H_{1}} & 0 & - H_{1} H_{1,3}\\0 & 0 & 0\\\frac{H_{1,3}}{H_{1}} & 0 & 0\end{matrix}\right] & \left[\begin{matrix}0 & 0 & 0\\0 & 0 & 0\\0 & 0 & 0\end{matrix}\right] & \left[\begin{matrix}\frac{H_{1,3}}{H_{1}} & 0 & 0\\0 & 0 & 0\\0 & 0 & 0\end{matrix}\right]\end{matrix}\right]$$

Gradient of vector

$ \left( \begin{array}{c} \nabla_1 u_1 \\ \nabla_2 u_1 \\ \nabla_3 u_1 \\ \nabla_1 u_2 \\ \nabla_2 u_2 \\ \nabla_3 u_2 \\ \nabla_1 u_3 \\ \nabla_2 u_3 \\ \nabla_3 u_3 \\ \end{array}

\right)

B \cdot \left( \begin{array}{c} u_1 \\ \frac { \partial u_1 } { \partial \alpha_1} \\ \frac { \partial u_1 } { \partial \alpha_2} \\ \frac { \partial u_1 } { \partial \alpha_3} \\ u_2 \\ \frac { \partial u_2 } { \partial \alpha_1} \\ \frac { \partial u_2 } { \partial \alpha_2} \\ \frac { \partial u_2 } { \partial \alpha_3} \\ u_3 \\ \frac { \partial u_3 } { \partial \alpha_1} \\ \frac { \partial u_3 } { \partial \alpha_2} \\ \frac { \partial u_3 } { \partial \alpha_3} \\ \end{array} \right) = B \cdot D \cdot \left( \begin{array}{c} u^1 \\ \frac { \partial u^1 } { \partial \alpha_1} \\ \frac { \partial u^1 } { \partial \alpha_2} \\ \frac { \partial u^1 } { \partial \alpha_3} \\ u^2 \\ \frac { \partial u^2 } { \partial \alpha_1} \\ \frac { \partial u^2 } { \partial \alpha_2} \\ \frac { \partial u^2 } { \partial \alpha_3} \\ u^3 \\ \frac { \partial u^3 } { \partial \alpha_1} \\ \frac { \partial u^3 } { \partial \alpha_2} \\ \frac { \partial u^3 } { \partial \alpha_3} \\ \end{array} \right) $



In [6]:

    
def row_index_to_i_j_grad(i_row):
    return i_row // 3, i_row % 3
        

B = zeros(9, 12)
B[0,1] = S(1)
B[1,2] = S(1)

B[2,3] = S(1)

B[3,5] = S(1)
B[4,6] = S(1)
B[5,7] = S(1)

B[6,9] = S(1)
B[7,10] = S(1)
B[8,11] = S(1)

for row_index in range(9):
    i,j=row_index_to_i_j_grad(row_index)
    B[row_index, 0] = -GK[i,j,0]
    B[row_index, 4] = -GK[i,j,1]
    B[row_index, 8] = -GK[i,j,2]

B









    Out[6]:





$$\left[\begin{array}{cccccccccccc}- \frac{H_{1,1}}{H_{1}} & 1 & 0 & 0 & 0 & 0 & 0 & 0 & H_{1} H_{1,3} & 0 & 0 & 0\\0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\- \frac{H_{1,3}}{H_{1}} & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\- \frac{H_{1,3}}{H_{1}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{array}\right]$$

Physical coordinates

$u_i=u_{[i]} H_i$



In [7]:

    
P=zeros(12,12)
P[0,0]=H[0]
P[1,0]=dH[0,0]
P[1,1]=H[0]
P[2,0]=dH[0,1]
P[2,2]=H[0]
P[3,0]=dH[0,2]
P[3,3]=H[0]

P[4,4]=H[1]
P[5,4]=dH[1,0]
P[5,5]=H[1]
P[6,4]=dH[1,1]
P[6,6]=H[1]
P[7,4]=dH[1,2]
P[7,7]=H[1]

P[8,8]=H[2]
P[9,8]=dH[2,0]
P[9,9]=H[2]
P[10,8]=dH[2,1]
P[10,10]=H[2]
P[11,8]=dH[2,2]
P[11,11]=H[2]
P=simplify(P)
P









    Out[7]:





$$\left[\begin{array}{cccccccccccc}H_{1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\H_{1,1} & H_{1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & H_{1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\H_{1,3} & 0 & 0 & H_{1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{array}\right]$$



In [8]:

    
B_P = zeros(9,9)

for i in range(3):
    for j in range(3):
        
        row_index = i*3+j
        
        B_P[row_index, row_index] = 1/(H[i]*H[j])
        


Grad_U_P = simplify(B_P*B*P)
Grad_U_P









    Out[8]:





$$\left[\begin{array}{cccccccccccc}0 & \frac{1}{H_{1}} & 0 & 0 & 0 & 0 & 0 & 0 & \frac{H_{1,3}}{H_{1}} & 0 & 0 & 0\\0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & \frac{1}{H_{1}} & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\- \frac{H_{1,3}}{H_{1}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{1}{H_{1}} & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{array}\right]$$

Strain tensor

$ \left( \begin{array}{c} \varepsilon_{11} \\ \varepsilon_{22} \\ \varepsilon_{33} \\ 2\varepsilon_{12} \\ 2\varepsilon_{13} \\ 2\varepsilon_{23} \\ \end{array}

\right)

\left(E + E_{NL} \left( \nabla \vec{u} \right) \right) \cdot \left( \begin{array}{c} \nabla_1 u_1 \\ \nabla_2 u_1 \\ \nabla_3 u_1 \\ \nabla_1 u_2 \\ \nabla_2 u_2 \\ \nabla_3 u_2 \\ \nabla_1 u_3 \\ \nabla_2 u_3 \\ \nabla_3 u_3 \\ \end{array} \right)$



In [9]:

    
E=zeros(6,9)
E[0,0]=1
E[1,4]=1
E[2,8]=1
E[3,1]=1
E[3,3]=1
E[4,2]=1
E[4,6]=1
E[5,5]=1
E[5,7]=1
E









    Out[9]:





$$\left[\begin{matrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0\end{matrix}\right]$$



In [10]:

    
StrainL=simplify(E*Grad_U_P)
StrainL









    Out[10]:





$$\left[\begin{array}{cccccccccccc}0 & \frac{1}{H_{1}} & 0 & 0 & 0 & 0 & 0 & 0 & \frac{H_{1,3}}{H_{1}} & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\0 & 0 & 1 & 0 & 0 & \frac{1}{H_{1}} & 0 & 0 & 0 & 0 & 0 & 0\\- \frac{H_{1,3}}{H_{1}} & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & \frac{1}{H_{1}} & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0\end{array}\right]$$



In [11]:

    
def E_NonLinear(grad_u):
    N = 3

    du = zeros(N, N)

    #    print("===Deformations===")

    for i in range(N):
        for j in range(N):
            index = i*N+j
            du[j,i] = grad_u[index]

    #    print("========")
    
    I = eye(3)

    a_values = S(1)/S(2) * du * G_up


    E_NL = zeros(6,9)
    E_NL[0,0] = a_values[0,0]
    E_NL[0,3] = a_values[0,1]
    E_NL[0,6] = a_values[0,2]

    E_NL[1,1] = a_values[1,0]
    E_NL[1,4] = a_values[1,1]
    E_NL[1,7] = a_values[1,2]

    E_NL[2,2] = a_values[2,0]
    E_NL[2,5] = a_values[2,1]
    E_NL[2,8] = a_values[2,2]

    E_NL[3,1] = 2*a_values[0,0]
    E_NL[3,4] = 2*a_values[0,1]
    E_NL[3,7] = 2*a_values[0,2]

    E_NL[4,0] = 2*a_values[2,0]
    E_NL[4,3] = 2*a_values[2,1]
    E_NL[4,6] = 2*a_values[2,2]

    E_NL[5,2] = 2*a_values[1,0]
    E_NL[5,5] = 2*a_values[1,1]
    E_NL[5,8] = 2*a_values[1,2]


    return E_NL


%aimport geom_util
u=getUHat3DPlane(alpha1, alpha2, alpha3)


# u=getUHatU3Main(alpha1, alpha2, alpha3)

gradu=B*u


E_NL = E_NonLinear(gradu)*B

E_NL









    Out[11]:





$$\left[\begin{array}{cccccccccccc}- \frac{H_{1,3}}{H_{1}} \left(\frac{u_{3,1}}{2} - \frac{H_{1,3} u_{1}}{2 H_{1}}\right) - \frac{H_{1,1}}{H_{1}^{3}} \left(\frac{H_{1} H_{1,3}}{2} u_{3} + \frac{u_{1,1}}{2} - \frac{H_{1,1} u_{1}}{2 H_{1}}\right) & \frac{1}{H_{1}^{2}} \left(\frac{H_{1} H_{1,3}}{2} u_{3} + \frac{u_{1,1}}{2} - \frac{H_{1,1} u_{1}}{2 H_{1}}\right) & 0 & 0 & 0 & 0 & 0 & 0 & \frac{H_{1,3}}{H_{1}} \left(\frac{H_{1} H_{1,3}}{2} u_{3} + \frac{u_{1,1}}{2} - \frac{H_{1,1} u_{1}}{2 H_{1}}\right) & \frac{u_{3,1}}{2} - \frac{H_{1,3} u_{1}}{2 H_{1}} & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\- \frac{H_{1,3}}{H_{1}^{3}} \left(\frac{u_{1,3}}{2} - \frac{H_{1,3} u_{1}}{2 H_{1}}\right) & 0 & 0 & \frac{1}{H_{1}^{2}} \left(\frac{u_{1,3}}{2} - \frac{H_{1,3} u_{1}}{2 H_{1}}\right) & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{u_{3,3}}{2}\\0 & 0 & \frac{2}{H_{1}^{2}} \left(\frac{H_{1} H_{1,3}}{2} u_{3} + \frac{u_{1,1}}{2} - \frac{H_{1,1} u_{1}}{2 H_{1}}\right) & 0 & 0 & 0 & 0 & 0 & 0 & 0 & u_{3,1} - \frac{H_{1,3} u_{1}}{H_{1}} & 0\\- \frac{H_{1,3} u_{3,3}}{H_{1}} - \frac{2 H_{1,1}}{H_{1}^{3}} \left(\frac{u_{1,3}}{2} - \frac{H_{1,3} u_{1}}{2 H_{1}}\right) & \frac{2}{H_{1}^{2}} \left(\frac{u_{1,3}}{2} - \frac{H_{1,3} u_{1}}{2 H_{1}}\right) & 0 & 0 & 0 & 0 & 0 & 0 & \frac{2 H_{1,3}}{H_{1}} \left(\frac{u_{1,3}}{2} - \frac{H_{1,3} u_{1}}{2 H_{1}}\right) & u_{3,3} & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\end{array}\right]$$



In [26]:

    
%aimport geom_util


u=getUHatU3MainPlane(alpha1, alpha2, alpha3)

gradup=Grad_U_P*u

# e=E*gradup
# e


E_NLp = E_NonLinear(gradup)*gradup


simplify(E_NLp)









    Out[26]:





$$\left[\begin{matrix}\frac{1}{2 H_{1}^{4}} \left(H_{1}^{2} \left(H_{1,3} \operatorname{u_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{\partial}{\partial \alpha_{1}} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right)^{2} + H_{1,3}^{2} \operatorname{u_{3}}^{2}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right)\\0\\0\\0\\0\\0\end{matrix}\right]$$



In [ ]:

    
w

Virtual work



In [24]:

    
%aimport geom_util
C_tensor = getIsotropicStiffnessTensor()
C = convertStiffnessTensorToMatrix(C_tensor)
C









    Out[24]:





$$\left[\begin{matrix}\lambda + 2 \mu & \lambda & \lambda & 0 & 0 & 0\\\lambda & \lambda + 2 \mu & \lambda & 0 & 0 & 0\\\lambda & \lambda & \lambda + 2 \mu & 0 & 0 & 0\\0 & 0 & 0 & \mu & 0 & 0\\0 & 0 & 0 & 0 & \mu & 0\\0 & 0 & 0 & 0 & 0 & \mu\end{matrix}\right]$$



In [25]:

    
StrainL.T*C*StrainL*H1









    Out[25]:





$$\left[\begin{array}{cccccccccccc}\frac{H_{1,3}^{2} \mu}{H_{1}} & 0 & 0 & - H_{1,3} \mu & 0 & 0 & 0 & 0 & 0 & - \frac{H_{1,3} \mu}{H_{1}} & 0 & 0\\0 & \frac{1}{H_{1}} \left(\lambda + 2 \mu\right) & 0 & 0 & 0 & 0 & \lambda & 0 & \frac{H_{1,3}}{H_{1}} \left(\lambda + 2 \mu\right) & 0 & 0 & \lambda\\0 & 0 & H_{1} \mu & 0 & 0 & \mu & 0 & 0 & 0 & 0 & 0 & 0\\- H_{1,3} \mu & 0 & 0 & H_{1} \mu & 0 & 0 & 0 & 0 & 0 & \mu & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & \mu & 0 & 0 & \frac{\mu}{H_{1}} & 0 & 0 & 0 & 0 & 0 & 0\\0 & \lambda & 0 & 0 & 0 & 0 & H_{1} \left(\lambda + 2 \mu\right) & 0 & H_{1,3} \lambda & 0 & 0 & H_{1} \lambda\\0 & 0 & 0 & 0 & 0 & 0 & 0 & H_{1} \mu & 0 & 0 & H_{1} \mu & 0\\0 & \frac{H_{1,3}}{H_{1}} \left(\lambda + 2 \mu\right) & 0 & 0 & 0 & 0 & H_{1,3} \lambda & 0 & \frac{H_{1,3}^{2}}{H_{1}} \left(\lambda + 2 \mu\right) & 0 & 0 & H_{1,3} \lambda\\- \frac{H_{1,3} \mu}{H_{1}} & 0 & 0 & \mu & 0 & 0 & 0 & 0 & 0 & \frac{\mu}{H_{1}} & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & H_{1} \mu & 0 & 0 & H_{1} \mu & 0\\0 & \lambda & 0 & 0 & 0 & 0 & H_{1} \lambda & 0 & H_{1,3} \lambda & 0 & 0 & H_{1} \left(\lambda + 2 \mu\right)\end{array}\right]$$

Tymoshenko theory

$u_1 \left( \alpha_1, \alpha_2, \alpha_3 \right)=u\left( \alpha_1 \right)+\alpha_3\gamma \left( \alpha_1 \right) $

$u_2 \left( \alpha_1, \alpha_2, \alpha_3 \right)=0 $

$u_3 \left( \alpha_1, \alpha_2, \alpha_3 \right)=w\left( \alpha_1 \right) $

$ \left( \begin{array}{c} u_1 \\ \frac { \partial u_1 } { \partial \alpha_1} \\ \frac { \partial u_1 } { \partial \alpha_2} \\ \frac { \partial u_1 } { \partial \alpha_3} \\ u_2 \\ \frac { \partial u_2 } { \partial \alpha_1} \\ \frac { \partial u_2 } { \partial \alpha_2} \\ \frac { \partial u_2 } { \partial \alpha_3} \\ u_3 \\ \frac { \partial u_3 } { \partial \alpha_1} \\ \frac { \partial u_3 } { \partial \alpha_2} \\ \frac { \partial u_3 } { \partial \alpha_3} \\ \end{array} \right) = T \cdot \left( \begin{array}{c} u \\ \frac { \partial u } { \partial \alpha_1} \\ \gamma \\ \frac { \partial \gamma } { \partial \alpha_1} \\ w \\ \frac { \partial w } { \partial \alpha_1} \\ \end{array} \right) $



In [15]:

    
T=zeros(12,6)
T[0,0]=1
T[0,2]=alpha3
T[1,1]=1
T[1,3]=alpha3
T[3,2]=1

T[8,4]=1
T[9,5]=1
T









    Out[15]:





$$\left[\begin{matrix}1 & 0 & \alpha_{3} & 0 & 0 & 0\\0 & 1 & 0 & \alpha_{3} & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 1 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 1 & 0\\0 & 0 & 0 & 0 & 0 & 1\\0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0\end{matrix}\right]$$



In [16]:

    
D_p_T = StrainL*T
simplify(D_p_T)









    Out[16]:





$$\left[\begin{matrix}0 & \frac{1}{H_{1}} & 0 & \frac{\alpha_{3}}{H_{1}} & \frac{H_{1,3}}{H_{1}} & 0\\0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0\\- \frac{H_{1,3}}{H_{1}} & 0 & \frac{1}{H_{1}} \left(H_{1} - H_{1,3} \alpha_{3}\right) & 0 & 0 & \frac{1}{H_{1}}\\0 & 0 & 0 & 0 & 0 & 0\end{matrix}\right]$$



In [17]:

    
u = Function("u")
t = Function("theta")
w = Function("w")

u1=u(alpha1)+alpha3*t(alpha1)
u3=w(alpha1)

gu = zeros(12,1) 
gu[0] = u1
gu[1] = u1.diff(alpha1)
gu[3] = u1.diff(alpha3)


gu[8] = u3
gu[9] = u3.diff(alpha1)



gradup=Grad_U_P*gu



# E_NLp = E_NonLinear(gradup)*gradup


# simplify(E_NLp)
# gradup=Grad_U_P*gu

# o20=(K*u(alpha1)-w(alpha1).diff(alpha1)+t(alpha1))/2
# o21=K*t(alpha1)
# O=1/2*o20*o20+alpha3*o20*o21-alpha3*K/2*o20*o20
# O=expand(O)
# O=collect(O,alpha3)
# simplify(O)

StrainNL = E_NonLinear(gradup)*gradup
StrainL*gu+simplify(StrainNL)









    Out[17]:





$$\left[\begin{matrix}\frac{H_{1,3}}{H_{1}} w{\left (\alpha_{1} \right )} + \frac{1}{H_{1}} \left(\alpha_{3} \frac{d}{d \alpha_{1}} \theta{\left (\alpha_{1} \right )} + \frac{d}{d \alpha_{1}} u{\left (\alpha_{1} \right )}\right) + \frac{1}{2 H_{1}^{4}} \left(H_{1}^{2} \left(H_{1,3} \left(\alpha_{3} \theta{\left (\alpha_{1} \right )} + u{\left (\alpha_{1} \right )}\right) - \frac{d}{d \alpha_{1}} w{\left (\alpha_{1} \right )}\right)^{2} + \left(H_{1,3} w{\left (\alpha_{1} \right )} + \alpha_{3} \frac{d}{d \alpha_{1}} \theta{\left (\alpha_{1} \right )} + \frac{d}{d \alpha_{1}} u{\left (\alpha_{1} \right )}\right)^{2}\right)\\0\\\frac{\theta^{2}{\left (\alpha_{1} \right )}}{2 H_{1}^{2}}\\0\\\theta{\left (\alpha_{1} \right )} - \frac{H_{1,3}}{H_{1}} \left(\alpha_{3} \theta{\left (\alpha_{1} \right )} + u{\left (\alpha_{1} \right )}\right) + \frac{1}{H_{1}} \frac{d}{d \alpha_{1}} w{\left (\alpha_{1} \right )} + \frac{1}{H_{1}^{3}} \left(H_{1,3} w{\left (\alpha_{1} \right )} + \alpha_{3} \frac{d}{d \alpha_{1}} \theta{\left (\alpha_{1} \right )} + \frac{d}{d \alpha_{1}} u{\left (\alpha_{1} \right )}\right) \theta{\left (\alpha_{1} \right )}\\0\end{matrix}\right]$$

Square theory

$u^1 \left( \alpha_1, \alpha_2, \alpha_3 \right)=u_{10}\left( \alpha_1 \right)p_0\left( \alpha_3 \right)+u_{11}\left( \alpha_1 \right)p_1\left( \alpha_3 \right)+u_{12}\left( \alpha_1 \right)p_2\left( \alpha_3 \right) $

$u^2 \left( \alpha_1, \alpha_2, \alpha_3 \right)=0 $

$u^3 \left( \alpha_1, \alpha_2, \alpha_3 \right)=u_{30}\left( \alpha_1 \right)p_0\left( \alpha_3 \right)+u_{31}\left( \alpha_1 \right)p_1\left( \alpha_3 \right)+u_{32}\left( \alpha_1 \right)p_2\left( \alpha_3 \right) $

$ \left( \begin{array}{c} u^1 \\ \frac { \partial u^1 } { \partial \alpha_1} \\ \frac { \partial u^1 } { \partial \alpha_2} \\ \frac { \partial u^1 } { \partial \alpha_3} \\ u^2 \\ \frac { \partial u^2 } { \partial \alpha_1} \\ \frac { \partial u^2 } { \partial \alpha_2} \\ \frac { \partial u^2 } { \partial \alpha_3} \\ u^3 \\ \frac { \partial u^3 } { \partial \alpha_1} \\ \frac { \partial u^3 } { \partial \alpha_2} \\ \frac { \partial u^3 } { \partial \alpha_3} \\ \end{array} \right) = L \cdot \left( \begin{array}{c} u_{10} \\ \frac { \partial u_{10} } { \partial \alpha_1} \\ u_{11} \\ \frac { \partial u_{11} } { \partial \alpha_1} \\ u_{12} \\ \frac { \partial u_{12} } { \partial \alpha_1} \\ u_{30} \\ \frac { \partial u_{30} } { \partial \alpha_1} \\ u_{31} \\ \frac { \partial u_{31} } { \partial \alpha_1} \\ u_{32} \\ \frac { \partial u_{32} } { \partial \alpha_1} \\ \end{array} \right) $



In [18]:

    
L=zeros(12,12)
h=Symbol('h')
p0=1/2-alpha3/h
p1=1/2+alpha3/h
p2=1-(2*alpha3/h)**2

L[0,0]=p0
L[0,2]=p1
L[0,4]=p2

L[1,1]=p0
L[1,3]=p1
L[1,5]=p2

L[3,0]=p0.diff(alpha3)
L[3,2]=p1.diff(alpha3)
L[3,4]=p2.diff(alpha3)

L[8,6]=p0
L[8,8]=p1
L[8,10]=p2

L[9,7]=p0
L[9,9]=p1
L[9,11]=p2

L[11,6]=p0.diff(alpha3)
L[11,8]=p1.diff(alpha3)
L[11,10]=p2.diff(alpha3)

L









    Out[18]:





$$\left[\begin{array}{cccccccccccc}- \frac{\alpha_{3}}{h} + 0.5 & 0 & \frac{\alpha_{3}}{h} + 0.5 & 0 & - \frac{4 \alpha_{3}^{2}}{h^{2}} + 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & - \frac{\alpha_{3}}{h} + 0.5 & 0 & \frac{\alpha_{3}}{h} + 0.5 & 0 & - \frac{4 \alpha_{3}^{2}}{h^{2}} + 1 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\- \frac{1}{h} & 0 & \frac{1}{h} & 0 & - \frac{8 \alpha_{3}}{h^{2}} & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & - \frac{\alpha_{3}}{h} + 0.5 & 0 & \frac{\alpha_{3}}{h} + 0.5 & 0 & - \frac{4 \alpha_{3}^{2}}{h^{2}} + 1 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & - \frac{\alpha_{3}}{h} + 0.5 & 0 & \frac{\alpha_{3}}{h} + 0.5 & 0 & - \frac{4 \alpha_{3}^{2}}{h^{2}} + 1\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & - \frac{1}{h} & 0 & \frac{1}{h} & 0 & - \frac{8 \alpha_{3}}{h^{2}} & 0\end{array}\right]$$



In [19]:

    
D_p_L = StrainL*L
simplify(D_p_L)









    Out[19]:





$$\left[\begin{array}{cccccccccccc}0 & \frac{1}{H_{1} h} \left(- \alpha_{3} + 0.5 h\right) & 0 & \frac{\alpha_{3} + 0.5 h}{H_{1} h} & 0 & \frac{1}{H_{1} h^{2}} \left(- 4 \alpha_{3}^{2} + h^{2}\right) & - \frac{H_{1,3}}{H_{1} h} \left(\alpha_{3} - 0.5 h\right) & 0 & \frac{H_{1,3}}{H_{1} h} \left(\alpha_{3} + 0.5 h\right) & 0 & \frac{H_{1,3}}{H_{1} h^{2}} \left(- 4 \alpha_{3}^{2} + h^{2}\right) & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & - \frac{1}{h} & 0 & \frac{1}{h} & 0 & - \frac{8 \alpha_{3}}{h^{2}} & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\frac{1}{H_{1} h} \left(- H_{1} + H_{1,3} \left(\alpha_{3} - 0.5 h\right)\right) & 0 & \frac{1}{H_{1} h} \left(H_{1} - H_{1,3} \left(\alpha_{3} + 0.5 h\right)\right) & 0 & \frac{1}{H_{1} h^{2}} \left(- 8 H_{1} \alpha_{3} + H_{1,3} \left(4 \alpha_{3}^{2} - h^{2}\right)\right) & 0 & 0 & \frac{1}{H_{1} h} \left(- \alpha_{3} + 0.5 h\right) & 0 & \frac{\alpha_{3} + 0.5 h}{H_{1} h} & 0 & \frac{1}{H_{1} h^{2}} \left(- 4 \alpha_{3}^{2} + h^{2}\right)\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\end{array}\right]$$



In [20]:

    
h = 0.5
exp=(0.5-alpha3/h)*(1-(2*alpha3/h)**2)#/(1+alpha3*0.8)
p02=integrate(exp, (alpha3, -h/2, h/2))
integral = expand(simplify(p02))
integral









    Out[20]:





$$0.166666666666667$$

Mass matrix



In [21]:

    
rho=Symbol('rho')
B_h=zeros(3,12)
B_h[0,0]=1
B_h[1,4]=1
B_h[2,8]=1
M=simplify(rho*P.T*B_h.T*G_up*B_h*P)
M









    Out[21]:





$$\left[\begin{array}{cccccccccccc}\rho & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & \rho & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \rho & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\end{array}\right]$$



In [22]:

    
M_p = L.T*M*L
integrate(M_p, (alpha3, -h/2, h/2))









    Out[22]:





$$\left[\begin{array}{cccccccccccc}0.125 \rho + \frac{0.0104166666666667 \rho}{h^{2}} & 0 & 0.125 \rho - \frac{0.0104166666666667 \rho}{h^{2}} & 0 & 0.25 \rho - \frac{0.0208333333333333 \rho}{h^{2}} & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0.125 \rho - \frac{0.0104166666666667 \rho}{h^{2}} & 0 & 0.125 \rho + \frac{0.0104166666666667 \rho}{h^{2}} & 0 & 0.25 \rho - \frac{0.0208333333333333 \rho}{h^{2}} & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0.25 \rho - \frac{0.0208333333333333 \rho}{h^{2}} & 0 & 0.25 \rho - \frac{0.0208333333333333 \rho}{h^{2}} & 0 & 0.5 \rho - \frac{0.0833333333333333 \rho}{h^{2}} + \frac{0.00625 \rho}{h^{4}} & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0.125 \rho + \frac{0.0104166666666667 \rho}{h^{2}} & 0 & 0.125 \rho - \frac{0.0104166666666667 \rho}{h^{2}} & 0 & 0.25 \rho - \frac{0.0208333333333333 \rho}{h^{2}} & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0.125 \rho - \frac{0.0104166666666667 \rho}{h^{2}} & 0 & 0.125 \rho + \frac{0.0104166666666667 \rho}{h^{2}} & 0 & 0.25 \rho - \frac{0.0208333333333333 \rho}{h^{2}} & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0.25 \rho - \frac{0.0208333333333333 \rho}{h^{2}} & 0 & 0.25 \rho - \frac{0.0208333333333333 \rho}{h^{2}} & 0 & 0.5 \rho - \frac{0.0833333333333333 \rho}{h^{2}} + \frac{0.00625 \rho}{h^{4}} & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\end{array}\right]$$