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
In [47]:
# h1 = Function("H1")
# h2 = Function("H2")
# h3 = Function("H3")
# H1 = h1(alpha1, alpha2, alpha3)
# H2 = h2(alpha1, alpha2, alpha3)
# H3 = h3(alpha1, alpha2, alpha3)
H1,H2,H3=symbols('H1,H2,H3')
H=[H1, H2, H3]
DIM=3
dH = zeros(DIM,DIM)
for i in range(DIM):
for j in range(DIM):
dH[i,j]=Symbol('H_{{{},{}}}'.format(i+1,j+1))
dH
Out[47]:
$$\left[\begin{matrix}H_{1,1} & H_{1,2} & H_{1,3}\\H_{2,1} & H_{2,2} & H_{2,3}\\H_{3,1} & H_{3,2} & H_{3,3}\end{matrix}\right]$$
${\displaystyle \hat{G}=\sum_{i,j} g^{ij}\vec{R}_i\vec{R}_j}$
In [48]:
G_up = getMetricTensorUpLame(H1, H2, H3)
${\displaystyle \hat{G}=\sum_{i,j} g_{ij}\vec{R}^i\vec{R}^j}$
In [49]:
G_down = getMetricTensorDownLame(H1, H2, H3)
In [50]:
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[50]:
$$\left[\begin{matrix}\left[\begin{matrix}\frac{H_{1,1}}{H_{1}} & - \frac{H_{1} H_{1,2}}{H_{2}^{2}} & - \frac{H_{1} H_{1,3}}{H_{3}^{2}}\\\frac{H_{1,2}}{H_{1}} & \frac{H_{2,1}}{H_{2}} & 0\\\frac{H_{1,3}}{H_{1}} & 0 & \frac{H_{3,1}}{H_{3}}\end{matrix}\right] & \left[\begin{matrix}\frac{H_{1,2}}{H_{1}} & \frac{H_{2,1}}{H_{2}} & 0\\- \frac{H_{2} H_{2,1}}{H_{1}^{2}} & \frac{H_{2,2}}{H_{2}} & - \frac{H_{2} H_{2,3}}{H_{3}^{2}}\\0 & \frac{H_{2,3}}{H_{2}} & \frac{H_{3,2}}{H_{3}}\end{matrix}\right] & \left[\begin{matrix}\frac{H_{1,3}}{H_{1}} & 0 & \frac{H_{3,1}}{H_{3}}\\0 & \frac{H_{2,3}}{H_{2}} & \frac{H_{3,2}}{H_{3}}\\- \frac{H_{3} H_{3,1}}{H_{1}^{2}} & - \frac{H_{3} H_{3,2}}{H_{2}^{2}} & \frac{H_{3,3}}{H_{3}}\end{matrix}\right]\end{matrix}\right]$$
$ \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}
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 [51]:
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[51]:
$$\left[\begin{array}{cccccccccccc}- \frac{H_{1,1}}{H_{1}} & 1 & 0 & 0 & \frac{H_{1} H_{1,2}}{H_{2}^{2}} & 0 & 0 & 0 & \frac{H_{1} H_{1,3}}{H_{3}^{2}} & 0 & 0 & 0\\- \frac{H_{1,2}}{H_{1}} & 0 & 1 & 0 & - \frac{H_{2,1}}{H_{2}} & 0 & 0 & 0 & 0 & 0 & 0 & 0\\- \frac{H_{1,3}}{H_{1}} & 0 & 0 & 1 & 0 & 0 & 0 & 0 & - \frac{H_{3,1}}{H_{3}} & 0 & 0 & 0\\- \frac{H_{1,2}}{H_{1}} & 0 & 0 & 0 & - \frac{H_{2,1}}{H_{2}} & 1 & 0 & 0 & 0 & 0 & 0 & 0\\\frac{H_{2} H_{2,1}}{H_{1}^{2}} & 0 & 0 & 0 & - \frac{H_{2,2}}{H_{2}} & 0 & 1 & 0 & \frac{H_{2} H_{2,3}}{H_{3}^{2}} & 0 & 0 & 0\\0 & 0 & 0 & 0 & - \frac{H_{2,3}}{H_{2}} & 0 & 0 & 1 & - \frac{H_{3,2}}{H_{3}} & 0 & 0 & 0\\- \frac{H_{1,3}}{H_{1}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - \frac{H_{3,1}}{H_{3}} & 1 & 0 & 0\\0 & 0 & 0 & 0 & - \frac{H_{2,3}}{H_{2}} & 0 & 0 & 0 & - \frac{H_{3,2}}{H_{3}} & 0 & 1 & 0\\\frac{H_{3} H_{3,1}}{H_{1}^{2}} & 0 & 0 & 0 & \frac{H_{3} H_{3,2}}{H_{2}^{2}} & 0 & 0 & 0 & - \frac{H_{3,3}}{H_{3}} & 0 & 0 & 1\end{array}\right]$$
$ \left( \begin{array}{c} \varepsilon_{11} \\ \varepsilon_{22} \\ \varepsilon_{33} \\ 2\varepsilon_{12} \\ 2\varepsilon_{13} \\ 2\varepsilon_{23} \\ \end{array}
\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 [52]:
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[52]:
$$\left[\begin{matrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0\end{matrix}\right]$$
In [53]:
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=getUHat3D(alpha1, alpha2, alpha3)
# u=getUHatU3Main(alpha1, alpha2, alpha3)
gradu=B*u
E_NL = E_NonLinear(gradu)*B
E_NL
Out[53]:
$$\left[\begin{array}{cccccccccccc}- \frac{H_{1,3}}{H_{1} H_{3}^{2}} \left(\frac{1}{2} \frac{\partial}{\partial \alpha_{1}} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{3,1}}{2 H_{3}} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{1,3}}{2 H_{1}} \operatorname{u_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right) - \frac{H_{1,2}}{H_{1} H_{2}^{2}} \left(\frac{1}{2} \frac{\partial}{\partial \alpha_{1}} \operatorname{u_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{2,1}}{2 H_{2}} \operatorname{u_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{1,2}}{2 H_{1}} \operatorname{u_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right) - \frac{H_{1,1}}{H_{1}^{3}} \left(\frac{H_{1} H_{1,3}}{2 H_{3}^{2}} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \frac{H_{1} H_{1,2}}{2 H_{2}^{2}} \operatorname{u_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \frac{1}{2} \frac{\partial}{\partial \alpha_{1}} \operatorname{u_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{1,1}}{2 H_{1}} \operatorname{u_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right) & \frac{1}{H_{1}^{2}} \left(\frac{H_{1} H_{1,3}}{2 H_{3}^{2}} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \frac{H_{1} H_{1,2}}{2 H_{2}^{2}} \operatorname{u_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \frac{1}{2} \frac{\partial}{\partial \alpha_{1}} \operatorname{u_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{1,1}}{2 H_{1}} \operatorname{u_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right) & 0 & 0 & - \frac{H_{2,1}}{H_{2}^{3}} \left(\frac{1}{2} \frac{\partial}{\partial \alpha_{1}} \operatorname{u_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{2,1}}{2 H_{2}} \operatorname{u_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{1,2}}{2 H_{1}} \operatorname{u_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right) + \frac{H_{1,2}}{H_{1} H_{2}^{2}} \left(\frac{H_{1} H_{1,3}}{2 H_{3}^{2}} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \frac{H_{1} H_{1,2}}{2 H_{2}^{2}} \operatorname{u_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \frac{1}{2} \frac{\partial}{\partial \alpha_{1}} \operatorname{u_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{1,1}}{2 H_{1}} \operatorname{u_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right) & \frac{1}{H_{2}^{2}} \left(\frac{1}{2} \frac{\partial}{\partial \alpha_{1}} \operatorname{u_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{2,1}}{2 H_{2}} \operatorname{u_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{1,2}}{2 H_{1}} \operatorname{u_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right) & 0 & 0 & - \frac{H_{3,1}}{H_{3}^{3}} \left(\frac{1}{2} \frac{\partial}{\partial \alpha_{1}} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{3,1}}{2 H_{3}} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{1,3}}{2 H_{1}} \operatorname{u_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right) + \frac{H_{1,3}}{H_{1} H_{3}^{2}} \left(\frac{H_{1} H_{1,3}}{2 H_{3}^{2}} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \frac{H_{1} H_{1,2}}{2 H_{2}^{2}} \operatorname{u_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \frac{1}{2} \frac{\partial}{\partial \alpha_{1}} \operatorname{u_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{1,1}}{2 H_{1}} \operatorname{u_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right) & \frac{1}{H_{3}^{2}} \left(\frac{1}{2} \frac{\partial}{\partial \alpha_{1}} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{3,1}}{2 H_{3}} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{1,3}}{2 H_{1}} \operatorname{u_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right) & 0 & 0\\\frac{H_{2,1}}{H_{1}^{2} H_{2}} \left(\frac{H_{2} H_{2,3}}{2 H_{3}^{2}} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \frac{1}{2} \frac{\partial}{\partial \alpha_{2}} \operatorname{u_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{2,2}}{2 H_{2}} \operatorname{u_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \frac{H_{2} H_{2,1}}{2 H_{1}^{2}} \operatorname{u_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right) - \frac{H_{1,2}}{H_{1}^{3}} \left(\frac{1}{2} \frac{\partial}{\partial \alpha_{2}} \operatorname{u_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{2,1}}{2 H_{2}} \operatorname{u_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{1,2}}{2 H_{1}} \operatorname{u_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right) & 0 & \frac{1}{H_{1}^{2}} \left(\frac{1}{2} \frac{\partial}{\partial \alpha_{2}} \operatorname{u_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{2,1}}{2 H_{2}} \operatorname{u_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{1,2}}{2 H_{1}} \operatorname{u_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right) & 0 & - \frac{H_{2,3}}{H_{2} H_{3}^{2}} \left(\frac{1}{2} \frac{\partial}{\partial \alpha_{2}} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{3,2}}{2 H_{3}} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{2,3}}{2 H_{2}} \operatorname{u_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right) - \frac{H_{2,2}}{H_{2}^{3}} \left(\frac{H_{2} H_{2,3}}{2 H_{3}^{2}} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \frac{1}{2} \frac{\partial}{\partial \alpha_{2}} \operatorname{u_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{2,2}}{2 H_{2}} \operatorname{u_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \frac{H_{2} H_{2,1}}{2 H_{1}^{2}} \operatorname{u_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right) - \frac{H_{2,1}}{H_{1}^{2} H_{2}} \left(\frac{1}{2} \frac{\partial}{\partial \alpha_{2}} \operatorname{u_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{2,1}}{2 H_{2}} \operatorname{u_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{1,2}}{2 H_{1}} \operatorname{u_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right) & 0 & \frac{1}{H_{2}^{2}} \left(\frac{H_{2} H_{2,3}}{2 H_{3}^{2}} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \frac{1}{2} \frac{\partial}{\partial \alpha_{2}} \operatorname{u_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{2,2}}{2 H_{2}} \operatorname{u_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \frac{H_{2} H_{2,1}}{2 H_{1}^{2}} \operatorname{u_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right) & 0 & - \frac{H_{3,2}}{H_{3}^{3}} \left(\frac{1}{2} \frac{\partial}{\partial \alpha_{2}} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{3,2}}{2 H_{3}} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{2,3}}{2 H_{2}} \operatorname{u_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right) + \frac{H_{2,3}}{H_{2} H_{3}^{2}} \left(\frac{H_{2} H_{2,3}}{2 H_{3}^{2}} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \frac{1}{2} \frac{\partial}{\partial \alpha_{2}} \operatorname{u_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{2,2}}{2 H_{2}} \operatorname{u_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \frac{H_{2} H_{2,1}}{2 H_{1}^{2}} \operatorname{u_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right) & 0 & \frac{1}{H_{3}^{2}} \left(\frac{1}{2} \frac{\partial}{\partial \alpha_{2}} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{3,2}}{2 H_{3}} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{2,3}}{2 H_{2}} \operatorname{u_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right) & 0\\\frac{H_{3,1}}{H_{1}^{2} H_{3}} \left(\frac{1}{2} \frac{\partial}{\partial \alpha_{3}} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{3,3}}{2 H_{3}} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \frac{H_{3} H_{3,2}}{2 H_{2}^{2}} \operatorname{u_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \frac{H_{3} H_{3,1}}{2 H_{1}^{2}} \operatorname{u_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right) - \frac{H_{1,3}}{H_{1}^{3}} \left(\frac{1}{2} \frac{\partial}{\partial \alpha_{3}} \operatorname{u_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{3,1}}{2 H_{3}} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{1,3}}{2 H_{1}} \operatorname{u_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right) & 0 & 0 & \frac{1}{H_{1}^{2}} \left(\frac{1}{2} \frac{\partial}{\partial \alpha_{3}} \operatorname{u_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{3,1}}{2 H_{3}} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{1,3}}{2 H_{1}} \operatorname{u_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right) & \frac{H_{3,2}}{H_{2}^{2} H_{3}} \left(\frac{1}{2} \frac{\partial}{\partial \alpha_{3}} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{3,3}}{2 H_{3}} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \frac{H_{3} H_{3,2}}{2 H_{2}^{2}} \operatorname{u_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \frac{H_{3} H_{3,1}}{2 H_{1}^{2}} \operatorname{u_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right) - \frac{H_{2,3}}{H_{2}^{3}} \left(\frac{1}{2} \frac{\partial}{\partial \alpha_{3}} \operatorname{u_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{3,2}}{2 H_{3}} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{2,3}}{2 H_{2}} \operatorname{u_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right) & 0 & 0 & \frac{1}{H_{2}^{2}} \left(\frac{1}{2} \frac{\partial}{\partial \alpha_{3}} \operatorname{u_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{3,2}}{2 H_{3}} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{2,3}}{2 H_{2}} \operatorname{u_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right) & - \frac{H_{3,3}}{H_{3}^{3}} \left(\frac{1}{2} \frac{\partial}{\partial \alpha_{3}} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{3,3}}{2 H_{3}} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \frac{H_{3} H_{3,2}}{2 H_{2}^{2}} \operatorname{u_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \frac{H_{3} H_{3,1}}{2 H_{1}^{2}} \operatorname{u_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right) - \frac{H_{3,2}}{H_{2}^{2} H_{3}} \left(\frac{1}{2} \frac{\partial}{\partial \alpha_{3}} \operatorname{u_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{3,2}}{2 H_{3}} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{2,3}}{2 H_{2}} \operatorname{u_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right) - \frac{H_{3,1}}{H_{1}^{2} H_{3}} \left(\frac{1}{2} \frac{\partial}{\partial \alpha_{3}} \operatorname{u_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{3,1}}{2 H_{3}} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{1,3}}{2 H_{1}} \operatorname{u_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right) & 0 & 0 & \frac{1}{H_{3}^{2}} \left(\frac{1}{2} \frac{\partial}{\partial \alpha_{3}} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{3,3}}{2 H_{3}} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \frac{H_{3} H_{3,2}}{2 H_{2}^{2}} \operatorname{u_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \frac{H_{3} H_{3,1}}{2 H_{1}^{2}} \operatorname{u_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right)\\\frac{2 H_{2,1}}{H_{1}^{2} H_{2}} \left(\frac{1}{2} \frac{\partial}{\partial \alpha_{1}} \operatorname{u_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{2,1}}{2 H_{2}} \operatorname{u_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{1,2}}{2 H_{1}} \operatorname{u_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right) - \frac{2 H_{1,2}}{H_{1}^{3}} \left(\frac{H_{1} H_{1,3}}{2 H_{3}^{2}} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \frac{H_{1} H_{1,2}}{2 H_{2}^{2}} \operatorname{u_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \frac{1}{2} \frac{\partial}{\partial \alpha_{1}} \operatorname{u_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{1,1}}{2 H_{1}} \operatorname{u_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right) & 0 & \frac{2}{H_{1}^{2}} \left(\frac{H_{1} H_{1,3}}{2 H_{3}^{2}} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \frac{H_{1} H_{1,2}}{2 H_{2}^{2}} \operatorname{u_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \frac{1}{2} \frac{\partial}{\partial \alpha_{1}} \operatorname{u_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{1,1}}{2 H_{1}} \operatorname{u_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right) & 0 & - \frac{2 H_{2,3}}{H_{2} H_{3}^{2}} \left(\frac{1}{2} \frac{\partial}{\partial \alpha_{1}} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{3,1}}{2 H_{3}} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{1,3}}{2 H_{1}} \operatorname{u_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right) - \frac{2 H_{2,2}}{H_{2}^{3}} \left(\frac{1}{2} \frac{\partial}{\partial \alpha_{1}} \operatorname{u_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{2,1}}{2 H_{2}} \operatorname{u_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{1,2}}{2 H_{1}} \operatorname{u_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right) - \frac{2 H_{2,1}}{H_{1}^{2} H_{2}} \left(\frac{H_{1} H_{1,3}}{2 H_{3}^{2}} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \frac{H_{1} H_{1,2}}{2 H_{2}^{2}} \operatorname{u_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \frac{1}{2} \frac{\partial}{\partial \alpha_{1}} \operatorname{u_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{1,1}}{2 H_{1}} \operatorname{u_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right) & 0 & \frac{2}{H_{2}^{2}} \left(\frac{1}{2} \frac{\partial}{\partial \alpha_{1}} \operatorname{u_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{2,1}}{2 H_{2}} \operatorname{u_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{1,2}}{2 H_{1}} \operatorname{u_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right) & 0 & - \frac{2 H_{3,2}}{H_{3}^{3}} \left(\frac{1}{2} \frac{\partial}{\partial \alpha_{1}} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{3,1}}{2 H_{3}} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{1,3}}{2 H_{1}} \operatorname{u_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right) + \frac{2 H_{2,3}}{H_{2} H_{3}^{2}} \left(\frac{1}{2} \frac{\partial}{\partial \alpha_{1}} \operatorname{u_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{2,1}}{2 H_{2}} \operatorname{u_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{1,2}}{2 H_{1}} \operatorname{u_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right) & 0 & \frac{2}{H_{3}^{2}} \left(\frac{1}{2} \frac{\partial}{\partial \alpha_{1}} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{3,1}}{2 H_{3}} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{1,3}}{2 H_{1}} \operatorname{u_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right) & 0\\- \frac{2 H_{1,3}}{H_{1} H_{3}^{2}} \left(\frac{1}{2} \frac{\partial}{\partial \alpha_{3}} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{3,3}}{2 H_{3}} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \frac{H_{3} H_{3,2}}{2 H_{2}^{2}} \operatorname{u_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \frac{H_{3} H_{3,1}}{2 H_{1}^{2}} \operatorname{u_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right) - \frac{2 H_{1,2}}{H_{1} H_{2}^{2}} \left(\frac{1}{2} \frac{\partial}{\partial \alpha_{3}} \operatorname{u_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{3,2}}{2 H_{3}} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{2,3}}{2 H_{2}} \operatorname{u_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right) - \frac{2 H_{1,1}}{H_{1}^{3}} \left(\frac{1}{2} \frac{\partial}{\partial \alpha_{3}} \operatorname{u_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{3,1}}{2 H_{3}} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{1,3}}{2 H_{1}} \operatorname{u_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right) & \frac{2}{H_{1}^{2}} \left(\frac{1}{2} \frac{\partial}{\partial \alpha_{3}} \operatorname{u_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{3,1}}{2 H_{3}} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{1,3}}{2 H_{1}} \operatorname{u_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right) & 0 & 0 & - \frac{2 H_{2,1}}{H_{2}^{3}} \left(\frac{1}{2} \frac{\partial}{\partial \alpha_{3}} \operatorname{u_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{3,2}}{2 H_{3}} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{2,3}}{2 H_{2}} \operatorname{u_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right) + \frac{2 H_{1,2}}{H_{1} H_{2}^{2}} \left(\frac{1}{2} \frac{\partial}{\partial \alpha_{3}} \operatorname{u_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{3,1}}{2 H_{3}} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{1,3}}{2 H_{1}} \operatorname{u_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right) & \frac{2}{H_{2}^{2}} \left(\frac{1}{2} \frac{\partial}{\partial \alpha_{3}} \operatorname{u_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{3,2}}{2 H_{3}} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{2,3}}{2 H_{2}} \operatorname{u_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right) & 0 & 0 & - \frac{2 H_{3,1}}{H_{3}^{3}} \left(\frac{1}{2} \frac{\partial}{\partial \alpha_{3}} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{3,3}}{2 H_{3}} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \frac{H_{3} H_{3,2}}{2 H_{2}^{2}} \operatorname{u_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \frac{H_{3} H_{3,1}}{2 H_{1}^{2}} \operatorname{u_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right) + \frac{2 H_{1,3}}{H_{1} H_{3}^{2}} \left(\frac{1}{2} \frac{\partial}{\partial \alpha_{3}} \operatorname{u_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{3,1}}{2 H_{3}} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{1,3}}{2 H_{1}} \operatorname{u_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right) & \frac{2}{H_{3}^{2}} \left(\frac{1}{2} \frac{\partial}{\partial \alpha_{3}} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{3,3}}{2 H_{3}} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \frac{H_{3} H_{3,2}}{2 H_{2}^{2}} \operatorname{u_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \frac{H_{3} H_{3,1}}{2 H_{1}^{2}} \operatorname{u_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right) & 0 & 0\\\frac{2 H_{3,1}}{H_{1}^{2} H_{3}} \left(\frac{1}{2} \frac{\partial}{\partial \alpha_{2}} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{3,2}}{2 H_{3}} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{2,3}}{2 H_{2}} \operatorname{u_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right) - \frac{2 H_{1,3}}{H_{1}^{3}} \left(\frac{1}{2} \frac{\partial}{\partial \alpha_{2}} \operatorname{u_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{2,1}}{2 H_{2}} \operatorname{u_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{1,2}}{2 H_{1}} \operatorname{u_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right) & 0 & 0 & \frac{2}{H_{1}^{2}} \left(\frac{1}{2} \frac{\partial}{\partial \alpha_{2}} \operatorname{u_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{2,1}}{2 H_{2}} \operatorname{u_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{1,2}}{2 H_{1}} \operatorname{u_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right) & \frac{2 H_{3,2}}{H_{2}^{2} H_{3}} \left(\frac{1}{2} \frac{\partial}{\partial \alpha_{2}} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{3,2}}{2 H_{3}} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{2,3}}{2 H_{2}} \operatorname{u_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right) - \frac{2 H_{2,3}}{H_{2}^{3}} \left(\frac{H_{2} H_{2,3}}{2 H_{3}^{2}} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \frac{1}{2} \frac{\partial}{\partial \alpha_{2}} \operatorname{u_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{2,2}}{2 H_{2}} \operatorname{u_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \frac{H_{2} H_{2,1}}{2 H_{1}^{2}} \operatorname{u_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right) & 0 & 0 & \frac{2}{H_{2}^{2}} \left(\frac{H_{2} H_{2,3}}{2 H_{3}^{2}} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \frac{1}{2} \frac{\partial}{\partial \alpha_{2}} \operatorname{u_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{2,2}}{2 H_{2}} \operatorname{u_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \frac{H_{2} H_{2,1}}{2 H_{1}^{2}} \operatorname{u_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right) & - \frac{2 H_{3,3}}{H_{3}^{3}} \left(\frac{1}{2} \frac{\partial}{\partial \alpha_{2}} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{3,2}}{2 H_{3}} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{2,3}}{2 H_{2}} \operatorname{u_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right) - \frac{2 H_{3,2}}{H_{2}^{2} H_{3}} \left(\frac{H_{2} H_{2,3}}{2 H_{3}^{2}} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \frac{1}{2} \frac{\partial}{\partial \alpha_{2}} \operatorname{u_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{2,2}}{2 H_{2}} \operatorname{u_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \frac{H_{2} H_{2,1}}{2 H_{1}^{2}} \operatorname{u_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right) - \frac{2 H_{3,1}}{H_{1}^{2} H_{3}} \left(\frac{1}{2} \frac{\partial}{\partial \alpha_{2}} \operatorname{u_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{2,1}}{2 H_{2}} \operatorname{u_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{1,2}}{2 H_{1}} \operatorname{u_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right) & 0 & 0 & \frac{2}{H_{3}^{2}} \left(\frac{1}{2} \frac{\partial}{\partial \alpha_{2}} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{3,2}}{2 H_{3}} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \frac{H_{2,3}}{2 H_{2}} \operatorname{u_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right)\end{array}\right]$$
$u_i=u_{[i]} H_i$
In [9]:
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[9]:
$$\left[\begin{array}{cccccccccccc}\operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\frac{\partial}{\partial \alpha_{1}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\frac{\partial}{\partial \alpha_{2}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & 0 & \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\frac{\partial}{\partial \alpha_{3}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & 0 & 0 & \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & \operatorname{H_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & \frac{\partial}{\partial \alpha_{1}} \operatorname{H_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & \operatorname{H_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & \frac{\partial}{\partial \alpha_{2}} \operatorname{H_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & 0 & \operatorname{H_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & \frac{\partial}{\partial \alpha_{3}} \operatorname{H_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & 0 & 0 & \operatorname{H_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{\partial}{\partial \alpha_{1}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{\partial}{\partial \alpha_{2}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & 0 & \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{\partial}{\partial \alpha_{3}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & 0 & 0 & \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\end{array}\right]$$
In [36]:
B_P = zeros(9,9)
for i in range(3):
for j in range(3):
ratio=1
if (i==0):
ratio = ratio*H1
elif (i==1):
ratio = ratio*H2
elif (i==2):
ratio = ratio*H3
if (j==0):
ratio = ratio*H1
elif (j==1):
ratio = ratio*H2
elif (j==2):
ratio = ratio*H3
row_index = i*3+j
B_P[row_index, row_index] = ratio
Grad_U_P = simplify(B_P*B*P)
B_P
Out[36]:
$$\left[\begin{matrix}\operatorname{H_{1}}^{2}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & \operatorname{H_{2}}^{2}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & \operatorname{H_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & \operatorname{H_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \operatorname{H_{3}}^{2}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\end{matrix}\right]$$
In [37]:
StrainL=simplify(E*Grad_U_P)
StrainL
Out[37]:
$$\left[\begin{array}{cccccccccccc}0 & \operatorname{H_{1}}^{3}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & 0 & 0 & \frac{\operatorname{H_{1}}^{3}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{\operatorname{H_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} \frac{\partial}{\partial \alpha_{2}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & 0 & 0 & 0 & \frac{\operatorname{H_{1}}^{3}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{\operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} \frac{\partial}{\partial \alpha_{3}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & 0 & 0 & 0\\\frac{\operatorname{H_{2}}^{3}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{\operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} \frac{\partial}{\partial \alpha_{1}} \operatorname{H_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & 0 & 0 & 0 & 0 & 0 & \operatorname{H_{2}}^{3}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & 0 & \frac{\operatorname{H_{2}}^{3}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{\operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} \frac{\partial}{\partial \alpha_{3}} \operatorname{H_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & 0 & 0 & 0\\\frac{\operatorname{H_{3}}^{3}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{\operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} \frac{\partial}{\partial \alpha_{1}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & 0 & 0 & 0 & \frac{\operatorname{H_{3}}^{3}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{\operatorname{H_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} \frac{\partial}{\partial \alpha_{2}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & 0 & 0 & 0 & 0 & 0 & 0 & \operatorname{H_{3}}^{3}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\\- \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{2}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & 0 & \operatorname{H_{1}}^{2}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & 0 & - \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{1}} \operatorname{H_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{2}}^{2}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & 0 & 0 & 0 & 0 & 0 & 0\\- \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{3}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & 0 & 0 & \operatorname{H_{1}}^{2}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & 0 & 0 & 0 & 0 & - \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{1}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{3}}^{2}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & 0 & 0\\0 & 0 & 0 & 0 & - \operatorname{H_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{3}} \operatorname{H_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & 0 & 0 & \operatorname{H_{2}}^{2}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & - \operatorname{H_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{2}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & 0 & \operatorname{H_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{3}}^{2}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & 0\end{array}\right]$$
In [40]:
%aimport geom_util
u=getUHatU3Main(alpha1, alpha2, alpha3)
gradup=B_P*B*P*u
E_NLp = E_NonLinear(gradup)*B*P*u
simplify(E_NLp)
Out[40]:
$$\left[\begin{matrix}\frac{\operatorname{H_{1}}^{2}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \left(\frac{\partial}{\partial \alpha_{3}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right)^{2}}{2 \operatorname{H_{3}}^{2}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} \operatorname{u_{3}}^{2}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \frac{\operatorname{H_{1}}^{2}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{2}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{3}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{\operatorname{H_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} \operatorname{u_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \frac{\operatorname{H_{1}}^{2}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \left(\frac{\partial}{\partial \alpha_{2}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right)^{2}}{2 \operatorname{H_{2}}^{2}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} \operatorname{u_{2}}^{2}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \frac{1}{2} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \left(\frac{\partial}{\partial \alpha_{1}} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right)^{2} - \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{u_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{3}} \operatorname{H_{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 )} + \frac{\operatorname{u_{1}}^{2}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \left(\frac{\partial}{\partial \alpha_{3}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right)^{2}}{2 \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \frac{\operatorname{u_{1}}^{2}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \left(\frac{\partial}{\partial \alpha_{2}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right)^{2}}{2 \operatorname{H_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\\\frac{\operatorname{H_{2}}^{2}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \left(\frac{\partial}{\partial \alpha_{3}} \operatorname{H_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right)^{2}}{2 \operatorname{H_{3}}^{2}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} \operatorname{u_{3}}^{2}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \frac{1}{2} \operatorname{H_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \left(\frac{\partial}{\partial \alpha_{2}} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right)^{2} - \operatorname{H_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{u_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{3}} \operatorname{H_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{2}} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \frac{\operatorname{u_{2}}^{2}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \left(\frac{\partial}{\partial \alpha_{3}} \operatorname{H_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right)^{2}}{2 \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} \operatorname{H_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \frac{\operatorname{H_{2}}^{2}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{1}} \operatorname{H_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{3}} \operatorname{H_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{\operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} \operatorname{u_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \frac{\operatorname{u_{2}}^{2}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \left(\frac{\partial}{\partial \alpha_{1}} \operatorname{H_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right)^{2}}{2 \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} \operatorname{H_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \frac{\operatorname{H_{2}}^{2}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \left(\frac{\partial}{\partial \alpha_{1}} \operatorname{H_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right)^{2}}{2 \operatorname{H_{1}}^{2}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} \operatorname{u_{1}}^{2}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\\\frac{\operatorname{u_{3}}^{2}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \left(\frac{\partial}{\partial \alpha_{2}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right)^{2}}{2 \operatorname{H_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \frac{\operatorname{H_{3}}^{2}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \left(\frac{\partial}{\partial \alpha_{2}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right)^{2}}{2 \operatorname{H_{2}}^{2}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} \operatorname{u_{2}}^{2}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \frac{\operatorname{u_{3}}^{2}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \left(\frac{\partial}{\partial \alpha_{1}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right)^{2}}{2 \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \frac{\operatorname{H_{3}}^{2}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{1}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{2}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{\operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} \operatorname{u_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{u_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \frac{\operatorname{H_{3}}^{2}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \left(\frac{\partial}{\partial \alpha_{1}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right)^{2}}{2 \operatorname{H_{1}}^{2}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} \operatorname{u_{1}}^{2}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\\\frac{1}{\operatorname{H_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} \left(\left(\operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{1}} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \operatorname{u_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{3}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right) \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{2}} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \left(\left(\operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{1}} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \operatorname{u_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{3}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right) \operatorname{H_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{3}} \operatorname{H_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \left(\operatorname{H_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{3}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{u_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{2}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right) \frac{\partial}{\partial \alpha_{1}} \operatorname{H_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right) \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{u_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{u_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{2}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{3}} \operatorname{H_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \operatorname{H_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{u_{1}}^{2}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{2}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{1}} \operatorname{H_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right)\\\frac{1}{\operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} \left(\left(- \left(\operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{u_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{2}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \operatorname{H_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{u_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{1}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right) \frac{\partial}{\partial \alpha_{3}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{2}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{2}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right) \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{u_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \left(\operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{u_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{2}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \operatorname{H_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{u_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{1}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right) \operatorname{H_{3}}^{2}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{1}} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{u_{3}}^{2}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{3}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{1}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{u_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{2}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{1}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right)\\\frac{1}{\operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} \left(\left(- \left(\operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{3}} \operatorname{H_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{u_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{1}} \operatorname{H_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right) \frac{\partial}{\partial \alpha_{2}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{u_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{1}} \operatorname{H_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{1}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right) \operatorname{H_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \left(\operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{2}} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \operatorname{u_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{3}} \operatorname{H_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right) \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{u_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{2}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \left(\operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{2}} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \operatorname{u_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{3}} \operatorname{H_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right) \operatorname{H_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{u_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{1}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right)\end{matrix}\right]$$
$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 [13]:
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[13]:
$$\left[\begin{matrix}1 & 0 & \alpha_{3} & 0 & 0 & 0\\0 & 1 & 0 & \alpha_{3} & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 1 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 1 & 0\\0 & 0 & 0 & 0 & 0 & 1\\0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0\end{matrix}\right]$$
In [41]:
D_p_T = StrainL*T
simplify(D_p_T)
Out[41]:
$$\left[\begin{matrix}0 & \operatorname{H_{1}}^{3}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & 0 & \alpha_{3} \operatorname{H_{1}}^{3}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & \frac{\operatorname{H_{1}}^{3}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{\operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} \frac{\partial}{\partial \alpha_{3}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & 0\\\frac{\operatorname{H_{2}}^{3}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{\operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} \frac{\partial}{\partial \alpha_{1}} \operatorname{H_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & 0 & \frac{\alpha_{3} \operatorname{H_{2}}^{3}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{\operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} \frac{\partial}{\partial \alpha_{1}} \operatorname{H_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & 0 & \frac{\operatorname{H_{2}}^{3}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{\operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} \frac{\partial}{\partial \alpha_{3}} \operatorname{H_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & 0\\\frac{\operatorname{H_{3}}^{3}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{\operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} \frac{\partial}{\partial \alpha_{1}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & 0 & \frac{\alpha_{3} \operatorname{H_{3}}^{3}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{\operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} \frac{\partial}{\partial \alpha_{1}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & 0 & 0 & 0\\- \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{2}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & 0 & - \alpha_{3} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{2}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & 0 & 0 & 0\\- \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{3}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & 0 & \left(- \alpha_{3} \frac{\partial}{\partial \alpha_{3}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right) \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & 0 & - \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{1}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{3}}^{2}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\\0 & 0 & 0 & 0 & - \operatorname{H_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{2}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & 0\end{matrix}\right]$$
In [42]:
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
# 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
simplify(StrainNL)
Out[42]:
$$\left[\begin{matrix}\frac{\operatorname{H_{1}}^{2}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{2 \operatorname{H_{3}}^{2}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} \left(\left(\alpha_{3} \theta{\left (\alpha_{1} \right )} + u{\left (\alpha_{1} \right )}\right)^{2} \operatorname{H_{3}}^{2}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \left(\frac{\partial}{\partial \alpha_{2}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right)^{2} + \left(\left(\alpha_{3} \theta{\left (\alpha_{1} \right )} + u{\left (\alpha_{1} \right )}\right) \frac{\partial}{\partial \alpha_{3}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{d}{d \alpha_{1}} w{\left (\alpha_{1} \right )}\right)^{2} \operatorname{H_{3}}^{2}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \left(\left(\alpha_{3} \frac{d}{d \alpha_{1}} \theta{\left (\alpha_{1} \right )} + \frac{d}{d \alpha_{1}} u{\left (\alpha_{1} \right )}\right) \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + w{\left (\alpha_{1} \right )} \frac{\partial}{\partial \alpha_{3}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right)^{2} \operatorname{H_{1}}^{2}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right)\\\frac{\operatorname{H_{2}}^{4}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{2 \operatorname{H_{1}}^{2}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{3}}^{2}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} \left(\left(\alpha_{3} \theta{\left (\alpha_{1} \right )} + u{\left (\alpha_{1} \right )}\right) \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{1}} \operatorname{H_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} w{\left (\alpha_{1} \right )} \frac{\partial}{\partial \alpha_{3}} \operatorname{H_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right)^{2}\\\frac{\left(\alpha_{3} \theta{\left (\alpha_{1} \right )} + u{\left (\alpha_{1} \right )}\right)^{2}}{2 \operatorname{H_{1}}^{2}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} \operatorname{H_{3}}^{4}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \left(\frac{\partial}{\partial \alpha_{1}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right)^{2} + \frac{1}{2} \left(\operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \theta{\left (\alpha_{1} \right )} - w{\left (\alpha_{1} \right )} \frac{\partial}{\partial \alpha_{1}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right)^{2} \operatorname{H_{3}}^{2}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \frac{1}{2} \operatorname{H_{3}}^{2}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} w^{2}{\left (\alpha_{1} \right )} \left(\frac{\partial}{\partial \alpha_{2}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right)^{2}\\- \frac{\operatorname{H_{2}}^{2}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{\operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} \left(\alpha_{3} \theta{\left (\alpha_{1} \right )} + u{\left (\alpha_{1} \right )}\right) \left(\left(\alpha_{3} \theta{\left (\alpha_{1} \right )} + u{\left (\alpha_{1} \right )}\right) \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{1}} \operatorname{H_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} w{\left (\alpha_{1} \right )} \frac{\partial}{\partial \alpha_{3}} \operatorname{H_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right) \frac{\partial}{\partial \alpha_{2}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\\- \left(\alpha_{3} \theta{\left (\alpha_{1} \right )} + u{\left (\alpha_{1} \right )}\right) \left(\left(\alpha_{3} \theta{\left (\alpha_{1} \right )} + u{\left (\alpha_{1} \right )}\right) \frac{\partial}{\partial \alpha_{3}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{d}{d \alpha_{1}} w{\left (\alpha_{1} \right )}\right) \operatorname{H_{3}}^{2}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{1}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \left(\alpha_{3} \theta{\left (\alpha_{1} \right )} + u{\left (\alpha_{1} \right )}\right) \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} w{\left (\alpha_{1} \right )} \frac{\partial}{\partial \alpha_{2}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{2}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \left(\left(\alpha_{3} \frac{d}{d \alpha_{1}} \theta{\left (\alpha_{1} \right )} + \frac{d}{d \alpha_{1}} u{\left (\alpha_{1} \right )}\right) \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + w{\left (\alpha_{1} \right )} \frac{\partial}{\partial \alpha_{3}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right) \left(\operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \theta{\left (\alpha_{1} \right )} - w{\left (\alpha_{1} \right )} \frac{\partial}{\partial \alpha_{1}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right) \operatorname{H_{1}}^{2}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\\- \frac{\operatorname{H_{2}}^{2}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} w{\left (\alpha_{1} \right )}}{\operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} \left(\left(\alpha_{3} \theta{\left (\alpha_{1} \right )} + u{\left (\alpha_{1} \right )}\right) \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{1}} \operatorname{H_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} w{\left (\alpha_{1} \right )} \frac{\partial}{\partial \alpha_{3}} \operatorname{H_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right) \frac{\partial}{\partial \alpha_{2}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\end{matrix}\right]$$
$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 [43]:
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[43]:
$$\left[\begin{array}{cccccccccccc}- \frac{\alpha_{3}}{h} + 0.5 & 0 & \frac{\alpha_{3}}{h} + 0.5 & 0 & - \frac{4 \alpha_{3}^{2}}{h^{2}} + 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & - \frac{\alpha_{3}}{h} + 0.5 & 0 & \frac{\alpha_{3}}{h} + 0.5 & 0 & - \frac{4 \alpha_{3}^{2}}{h^{2}} + 1 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\- \frac{1}{h} & 0 & \frac{1}{h} & 0 & - \frac{8 \alpha_{3}}{h^{2}} & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & - \frac{\alpha_{3}}{h} + 0.5 & 0 & \frac{\alpha_{3}}{h} + 0.5 & 0 & - \frac{4 \alpha_{3}^{2}}{h^{2}} + 1 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & - \frac{\alpha_{3}}{h} + 0.5 & 0 & \frac{\alpha_{3}}{h} + 0.5 & 0 & - \frac{4 \alpha_{3}^{2}}{h^{2}} + 1\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & - \frac{1}{h} & 0 & \frac{1}{h} & 0 & - \frac{8 \alpha_{3}}{h^{2}} & 0\end{array}\right]$$
In [17]:
B_General = zeros(9, 12)
B_General[0,1] = S(1)
B_General[1,2] = S(1)
B_General[2,3] = S(1)
B_General[3,5] = S(1)
B_General[4,6] = S(1)
B_General[5,7] = S(1)
B_General[6,9] = S(1)
B_General[7,10] = S(1)
B_General[8,11] = S(1)
for row_index in range(9):
i,j=row_index_to_i_j_grad(row_index)
B_General[row_index, 0] = -Symbol("G_{{{}{}}}^1".format(i+1,j+1))
B_General[row_index, 4] = -Symbol("G_{{{}{}}}^2".format(i+1,j+1))
B_General[row_index, 8] = -Symbol("G_{{{}{}}}^3".format(i+1,j+1))
B_General
Out[17]:
$$\left[\begin{array}{cccccccccccc}- G_{11}^1 & 1 & 0 & 0 & - G_{11}^2 & 0 & 0 & 0 & - G_{11}^3 & 0 & 0 & 0\\- G_{12}^1 & 0 & 1 & 0 & - G_{12}^2 & 0 & 0 & 0 & - G_{12}^3 & 0 & 0 & 0\\- G_{13}^1 & 0 & 0 & 1 & - G_{13}^2 & 0 & 0 & 0 & - G_{13}^3 & 0 & 0 & 0\\- G_{21}^1 & 0 & 0 & 0 & - G_{21}^2 & 1 & 0 & 0 & - G_{21}^3 & 0 & 0 & 0\\- G_{22}^1 & 0 & 0 & 0 & - G_{22}^2 & 0 & 1 & 0 & - G_{22}^3 & 0 & 0 & 0\\- G_{23}^1 & 0 & 0 & 0 & - G_{23}^2 & 0 & 0 & 1 & - G_{23}^3 & 0 & 0 & 0\\- G_{31}^1 & 0 & 0 & 0 & - G_{31}^2 & 0 & 0 & 0 & - G_{31}^3 & 1 & 0 & 0\\- G_{32}^1 & 0 & 0 & 0 & - G_{32}^2 & 0 & 0 & 0 & - G_{32}^3 & 0 & 1 & 0\\- G_{33}^1 & 0 & 0 & 0 & - G_{33}^2 & 0 & 0 & 0 & - G_{33}^3 & 0 & 0 & 1\end{array}\right]$$
In [18]:
simplify(B_General*L)
Out[18]:
$$\left[\begin{array}{cccccccccccc}\frac{G_{11}^1}{h} \left(\alpha_{3} - 0.5 h\right) & - \frac{\alpha_{3}}{h} + 0.5 & - \frac{G_{11}^1}{h} \left(\alpha_{3} + 0.5 h\right) & \frac{\alpha_{3}}{h} + 0.5 & \frac{4 G_{11}^1}{h^{2}} \alpha_{3}^{2} - G_{11}^1 & - \frac{4 \alpha_{3}^{2}}{h^{2}} + 1 & \frac{G_{11}^3}{h} \left(\alpha_{3} - 0.5 h\right) & 0 & - \frac{G_{11}^3}{h} \left(\alpha_{3} + 0.5 h\right) & 0 & \frac{4 G_{11}^3}{h^{2}} \alpha_{3}^{2} - G_{11}^3 & 0\\\frac{G_{12}^1}{h} \left(\alpha_{3} - 0.5 h\right) & 0 & - \frac{G_{12}^1}{h} \left(\alpha_{3} + 0.5 h\right) & 0 & \frac{4 G_{12}^1}{h^{2}} \alpha_{3}^{2} - G_{12}^1 & 0 & \frac{G_{12}^3}{h} \left(\alpha_{3} - 0.5 h\right) & 0 & - \frac{G_{12}^3}{h} \left(\alpha_{3} + 0.5 h\right) & 0 & \frac{4 G_{12}^3}{h^{2}} \alpha_{3}^{2} - G_{12}^3 & 0\\\frac{1}{h} \left(G_{13}^1 \left(\alpha_{3} - 0.5 h\right) - 1\right) & 0 & \frac{1}{h} \left(- G_{13}^1 \left(\alpha_{3} + 0.5 h\right) + 1\right) & 0 & \frac{1}{h^{2}} \left(G_{13}^1 \left(4 \alpha_{3}^{2} - h^{2}\right) - 8 \alpha_{3}\right) & 0 & \frac{G_{13}^3}{h} \left(\alpha_{3} - 0.5 h\right) & 0 & - \frac{G_{13}^3}{h} \left(\alpha_{3} + 0.5 h\right) & 0 & \frac{4 G_{13}^3}{h^{2}} \alpha_{3}^{2} - G_{13}^3 & 0\\\frac{G_{21}^1}{h} \left(\alpha_{3} - 0.5 h\right) & 0 & - \frac{G_{21}^1}{h} \left(\alpha_{3} + 0.5 h\right) & 0 & \frac{4 G_{21}^1}{h^{2}} \alpha_{3}^{2} - G_{21}^1 & 0 & \frac{G_{21}^3}{h} \left(\alpha_{3} - 0.5 h\right) & 0 & - \frac{G_{21}^3}{h} \left(\alpha_{3} + 0.5 h\right) & 0 & \frac{4 G_{21}^3}{h^{2}} \alpha_{3}^{2} - G_{21}^3 & 0\\\frac{G_{22}^1}{h} \left(\alpha_{3} - 0.5 h\right) & 0 & - \frac{G_{22}^1}{h} \left(\alpha_{3} + 0.5 h\right) & 0 & \frac{4 G_{22}^1}{h^{2}} \alpha_{3}^{2} - G_{22}^1 & 0 & \frac{G_{22}^3}{h} \left(\alpha_{3} - 0.5 h\right) & 0 & - \frac{G_{22}^3}{h} \left(\alpha_{3} + 0.5 h\right) & 0 & \frac{4 G_{22}^3}{h^{2}} \alpha_{3}^{2} - G_{22}^3 & 0\\\frac{G_{23}^1}{h} \left(\alpha_{3} - 0.5 h\right) & 0 & - \frac{G_{23}^1}{h} \left(\alpha_{3} + 0.5 h\right) & 0 & \frac{4 G_{23}^1}{h^{2}} \alpha_{3}^{2} - G_{23}^1 & 0 & \frac{G_{23}^3}{h} \left(\alpha_{3} - 0.5 h\right) & 0 & - \frac{G_{23}^3}{h} \left(\alpha_{3} + 0.5 h\right) & 0 & \frac{4 G_{23}^3}{h^{2}} \alpha_{3}^{2} - G_{23}^3 & 0\\\frac{G_{31}^1}{h} \left(\alpha_{3} - 0.5 h\right) & 0 & - \frac{G_{31}^1}{h} \left(\alpha_{3} + 0.5 h\right) & 0 & \frac{4 G_{31}^1}{h^{2}} \alpha_{3}^{2} - G_{31}^1 & 0 & \frac{G_{31}^3}{h} \left(\alpha_{3} - 0.5 h\right) & - \frac{\alpha_{3}}{h} + 0.5 & - \frac{G_{31}^3}{h} \left(\alpha_{3} + 0.5 h\right) & \frac{\alpha_{3}}{h} + 0.5 & \frac{4 G_{31}^3}{h^{2}} \alpha_{3}^{2} - G_{31}^3 & - \frac{4 \alpha_{3}^{2}}{h^{2}} + 1\\\frac{G_{32}^1}{h} \left(\alpha_{3} - 0.5 h\right) & 0 & - \frac{G_{32}^1}{h} \left(\alpha_{3} + 0.5 h\right) & 0 & \frac{4 G_{32}^1}{h^{2}} \alpha_{3}^{2} - G_{32}^1 & 0 & \frac{G_{32}^3}{h} \left(\alpha_{3} - 0.5 h\right) & 0 & - \frac{G_{32}^3}{h} \left(\alpha_{3} + 0.5 h\right) & 0 & \frac{4 G_{32}^3}{h^{2}} \alpha_{3}^{2} - G_{32}^3 & 0\\\frac{G_{33}^1}{h} \left(\alpha_{3} - 0.5 h\right) & 0 & - \frac{G_{33}^1}{h} \left(\alpha_{3} + 0.5 h\right) & 0 & \frac{4 G_{33}^1}{h^{2}} \alpha_{3}^{2} - G_{33}^1 & 0 & \frac{1}{h} \left(G_{33}^3 \left(\alpha_{3} - 0.5 h\right) - 1\right) & 0 & \frac{1}{h} \left(- G_{33}^3 \left(\alpha_{3} + 0.5 h\right) + 1\right) & 0 & \frac{1}{h^{2}} \left(G_{33}^3 \left(4 \alpha_{3}^{2} - h^{2}\right) - 8 \alpha_{3}\right) & 0\end{array}\right]$$
In [44]:
D_p_L = StrainL*L
simplify(D_p_L)
Out[44]:
$$\left[\begin{array}{cccccccccccc}0 & \frac{1}{h} \left(- \alpha_{3} + 0.5 h\right) \operatorname{H_{1}}^{3}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & 0 & \frac{1}{h} \left(\alpha_{3} + 0.5 h\right) \operatorname{H_{1}}^{3}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & 0 & \frac{1}{h^{2}} \left(- 4 \alpha_{3}^{2} + h^{2}\right) \operatorname{H_{1}}^{3}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & - \frac{\operatorname{H_{1}}^{3}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{3}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{h \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} \left(\alpha_{3} - 0.5 h\right) & 0 & \frac{\operatorname{H_{1}}^{3}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{3}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{h \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} \left(\alpha_{3} + 0.5 h\right) & 0 & \frac{\operatorname{H_{1}}^{3}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{3}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{h^{2} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} \left(- 4 \alpha_{3}^{2} + h^{2}\right) & 0\\- \frac{\operatorname{H_{2}}^{3}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{1}} \operatorname{H_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{h \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} \left(\alpha_{3} - 0.5 h\right) & 0 & \frac{\operatorname{H_{2}}^{3}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{1}} \operatorname{H_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{h \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} \left(\alpha_{3} + 0.5 h\right) & 0 & \frac{\operatorname{H_{2}}^{3}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{1}} \operatorname{H_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{h^{2} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} \left(- 4 \alpha_{3}^{2} + h^{2}\right) & 0 & - \frac{\operatorname{H_{2}}^{3}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{3}} \operatorname{H_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{h \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} \left(\alpha_{3} - 0.5 h\right) & 0 & \frac{\operatorname{H_{2}}^{3}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{3}} \operatorname{H_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{h \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} \left(\alpha_{3} + 0.5 h\right) & 0 & \frac{\operatorname{H_{2}}^{3}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{3}} \operatorname{H_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{h^{2} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} \left(- 4 \alpha_{3}^{2} + h^{2}\right) & 0\\- \frac{\operatorname{H_{3}}^{3}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{1}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{h \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} \left(\alpha_{3} - 0.5 h\right) & 0 & \frac{\operatorname{H_{3}}^{3}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{1}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{h \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} \left(\alpha_{3} + 0.5 h\right) & 0 & \frac{\operatorname{H_{3}}^{3}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{1}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{h^{2} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} \left(- 4 \alpha_{3}^{2} + h^{2}\right) & 0 & - \frac{1}{h} \operatorname{H_{3}}^{3}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & 0 & \frac{1}{h} \operatorname{H_{3}}^{3}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & 0 & - \frac{8 \alpha_{3}}{h^{2}} \operatorname{H_{3}}^{3}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & 0\\\frac{1}{h} \left(\alpha_{3} - 0.5 h\right) \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{2}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & 0 & - \frac{1}{h} \left(\alpha_{3} + 0.5 h\right) \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{2}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & 0 & \frac{1}{h^{2}} \left(4 \alpha_{3}^{2} - h^{2}\right) \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{2}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\frac{1}{h} \left(\left(\alpha_{3} - 0.5 h\right) \frac{\partial}{\partial \alpha_{3}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right) \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & 0 & \frac{1}{h} \left(- \left(\alpha_{3} + 0.5 h\right) \frac{\partial}{\partial \alpha_{3}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right) \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & 0 & \frac{1}{h^{2}} \left(- 8 \alpha_{3} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \left(4 \alpha_{3}^{2} - h^{2}\right) \frac{\partial}{\partial \alpha_{3}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right) \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & 0 & \frac{1}{h} \left(\alpha_{3} - 0.5 h\right) \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{1}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & - \frac{1}{h} \left(\alpha_{3} - 0.5 h\right) \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{3}}^{2}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & - \frac{1}{h} \left(\alpha_{3} + 0.5 h\right) \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{1}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & \frac{1}{h} \left(\alpha_{3} + 0.5 h\right) \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{3}}^{2}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & \frac{1}{h^{2}} \left(4 \alpha_{3}^{2} - h^{2}\right) \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{1}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & \frac{1}{h^{2}} \left(- 4 \alpha_{3}^{2} + h^{2}\right) \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{3}}^{2}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\\0 & 0 & 0 & 0 & 0 & 0 & \frac{1}{h} \left(\alpha_{3} - 0.5 h\right) \operatorname{H_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{2}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & 0 & - \frac{1}{h} \left(\alpha_{3} + 0.5 h\right) \operatorname{H_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{2}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & 0 & \frac{1}{h^{2}} \left(4 \alpha_{3}^{2} - h^{2}\right) \operatorname{H_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{2}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & 0\end{array}\right]$$
In [ ]:
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
In [35]:
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[35]:
$$\left[\begin{array}{cccccccccccc}\rho & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & \rho & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \rho & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\end{array}\right]$$
In [32]:
e1 = L.T*M*L/(1+alpha3*0.8)
e2 = L.T*M*L
e1r=integrate(e1, (alpha3, -thick/2, thick/2))
e2r=integrate(e2, (alpha3, -thick/2, thick/2))
In [34]:
thick=0.1
e1r.subs(h, thick)-e2r.subs(h, thick)
Out[34]:
$$\left[\begin{array}{cccccccccccc}- 16.6441298646453 \rho & 0 & 15.3644532083905 \rho & 0 & 811.435180731817 \rho & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\15.3644532083905 \rho & 0 & - 14.1832132180015 \rho & 0 & - 749.017089906293 \rho & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\811.435180731817 \rho & 0 & - 749.017089906293 \rho & 0 & - 38958.2380522155 \rho & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & - 16.6441298646453 \rho & 0 & 15.3644532083905 \rho & 0 & 811.435180731817 \rho & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 15.3644532083905 \rho & 0 & - 14.1832132180015 \rho & 0 & - 749.017089906293 \rho & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 811.435180731817 \rho & 0 & - 749.017089906293 \rho & 0 & - 38958.2380522155 \rho & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\end{array}\right]$$
Content source: tarashor/vibrations
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