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Linear Shell 2 order solution

Init symbols for sympy





In [2]:



    


from sympy import *
from geom_util import *
from sympy.vector import CoordSys3D
import matplotlib.pyplot as plt
import sys
sys.path.append("../")

%matplotlib inline

%reload_ext autoreload
%autoreload 2
%aimport geom_util



    











In [3]:



    


# Any tweaks that normally go in .matplotlibrc, etc., should explicitly go here
%config InlineBackend.figure_format='retina'
plt.rcParams['figure.figsize'] = (12, 12)

plt.rc('text', usetex=True)
    
plt.rc('font', family='serif')

init_printing()



    











In [4]:



    


N = CoordSys3D('N')
alpha1, alpha2, alpha3 = symbols("alpha_1 alpha_2 alpha_3", real = True, positive=True)
A,K,rho, h = symbols("A K rho h")

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 [5]:



    


L=zeros(12,12)
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[5]:







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
















In [6]:



    


B=Matrix([[0, 1/(A*(K*alpha3 + 1)), 0, 0, 0, 0, 0, 0, K/(K*alpha3 + 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, 1/(A*(K*alpha3 + 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], [-K/(K*alpha3 + 1), 0, 0, 0, 0, 0, 0, 0, 0, 1/(A*(K*alpha3 + 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]])
B



    


















    
Out[6]:







$$\left[\begin{array}{cccccccccccc}0 & \frac{1}{A \left(K \alpha_{3} + 1\right)} & 0 & 0 & 0 & 0 & 0 & 0 & \frac{K}{K \alpha_{3} + 1} & 0 & 0 & 0\\0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & \frac{1}{A \left(K \alpha_{3} + 1\right)} & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\- \frac{K}{K \alpha_{3} + 1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{1}{A \left(K \alpha_{3} + 1\right)} & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{array}\right]$$
















In [7]:



    


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[7]:







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
















In [8]:



    


simplify(E*B*L)



    


















    
Out[8]:







$$\left[\begin{array}{cccccccccccc}0 & \frac{- \alpha_{3} + 0.5 h}{A h \left(K \alpha_{3} + 1\right)} & 0 & \frac{\alpha_{3} + 0.5 h}{A h \left(K \alpha_{3} + 1\right)} & 0 & \frac{- 4 \alpha_{3}^{2} + h^{2}}{A h^{2} \left(K \alpha_{3} + 1\right)} & - \frac{K \left(\alpha_{3} - 0.5 h\right)}{h \left(K \alpha_{3} + 1\right)} & 0 & \frac{K \left(\alpha_{3} + 0.5 h\right)}{h \left(K \alpha_{3} + 1\right)} & 0 & \frac{K \left(- 4 \alpha_{3}^{2} + h^{2}\right)}{h^{2} \left(K \alpha_{3} + 1\right)} & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & - \frac{1}{h} & 0 & \frac{1}{h} & 0 & - \frac{8 \alpha_{3}}{h^{2}} & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\- \frac{0.5 K h + 1.0}{h \left(K \alpha_{3} + 1\right)} & 0 & \frac{- 0.5 K h + 1.0}{h \left(K \alpha_{3} + 1\right)} & 0 & - \frac{4 K \alpha_{3}^{2} + K h^{2} + 8 \alpha_{3}}{h^{2} \left(K \alpha_{3} + 1\right)} & 0 & 0 & \frac{- \alpha_{3} + 0.5 h}{A h \left(K \alpha_{3} + 1\right)} & 0 & \frac{\alpha_{3} + 0.5 h}{A h \left(K \alpha_{3} + 1\right)} & 0 & \frac{- 4 \alpha_{3}^{2} + h^{2}}{A h^{2} \left(K \alpha_{3} + 1\right)}\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\end{array}\right]$$
















In [9]:



    


mu = Symbol('mu')
la = Symbol('lambda')
C_tensor = getIsotropicStiffnessTensor(mu, la)
C = convertStiffnessTensorToMatrix(C_tensor)
C



    


















    
Out[9]:







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
















In [14]:



    


S=L.T*B.T*E.T*C*E*B*L*A*(1+alpha3*K)
S=simplify(S)
S



    


















    
Out[14]:







$$\left[\begin{array}{cccccccccccc}\frac{1.0 A \mu}{h^{2} \left(K \alpha_{3} + 1\right)} \left(0.25 K^{2} h^{2} + 1.0 K h + 1.0\right) & 0 & \frac{1.0 A \mu \left(0.25 K^{2} h^{2} - 1.0\right)}{h^{2} \left(K \alpha_{3} + 1\right)} & 0 & - \frac{A \mu}{h^{3} \left(K \alpha_{3} + 1\right)} \left(K \left(4 \alpha_{3}^{2} - h^{2}\right) - 8 \alpha_{3} \left(K \alpha_{3} + 1\right)\right) \left(K \alpha_{3} - K \left(\alpha_{3} - 0.5 h\right) + 1\right) & 0 & 0 & \frac{\mu}{h^{2} \left(K \alpha_{3} + 1\right)} \left(\alpha_{3} - 0.5 h\right) \left(K \alpha_{3} - K \left(\alpha_{3} - 0.5 h\right) + 1\right) & 0 & - \frac{\mu}{h^{2} \left(K \alpha_{3} + 1\right)} \left(\alpha_{3} + 0.5 h\right) \left(K \alpha_{3} - K \left(\alpha_{3} - 0.5 h\right) + 1\right) & 0 & \frac{\mu}{h^{3} \left(K \alpha_{3} + 1\right)} \left(4 \alpha_{3}^{2} - h^{2}\right) \left(K \alpha_{3} - K \left(\alpha_{3} - 0.5 h\right) + 1\right)\\0 & \frac{\left(\alpha_{3} - 0.5 h\right)^{2} \left(\lambda + 2 \mu\right)}{A h^{2} \left(K \alpha_{3} + 1\right)} & 0 & - \frac{\left(\alpha_{3} - 0.5 h\right) \left(\alpha_{3} + 0.5 h\right) \left(\lambda + 2 \mu\right)}{A h^{2} \left(K \alpha_{3} + 1\right)} & 0 & \frac{\left(\alpha_{3} - 0.5 h\right) \left(4 \alpha_{3}^{2} - h^{2}\right) \left(\lambda + 2 \mu\right)}{A h^{3} \left(K \alpha_{3} + 1\right)} & \frac{1}{h^{2} \left(K \alpha_{3} + 1\right)} \left(\alpha_{3} - 0.5 h\right) \left(K \left(\alpha_{3} - 0.5 h\right) \left(\lambda + 2 \mu\right) + \lambda \left(K \alpha_{3} + 1\right)\right) & 0 & - \frac{1}{h^{2} \left(K \alpha_{3} + 1\right)} \left(\alpha_{3} - 0.5 h\right) \left(K \left(\alpha_{3} + 0.5 h\right) \left(\lambda + 2 \mu\right) + \lambda \left(K \alpha_{3} + 1\right)\right) & 0 & \frac{1}{h^{3} \left(K \alpha_{3} + 1\right)} \left(\alpha_{3} - 0.5 h\right) \left(K \left(4 \alpha_{3}^{2} - h^{2}\right) \left(\lambda + 2 \mu\right) + 8 \alpha_{3} \lambda \left(K \alpha_{3} + 1\right)\right) & 0\\\frac{1.0 A \mu \left(0.25 K^{2} h^{2} - 1.0\right)}{h^{2} \left(K \alpha_{3} + 1\right)} & 0 & \frac{1.0 A \mu}{h^{2} \left(K \alpha_{3} + 1\right)} \left(0.25 K^{2} h^{2} - 1.0 K h + 1.0\right) & 0 & \frac{A \mu}{h^{3} \left(K \alpha_{3} + 1\right)} \left(K \left(4 \alpha_{3}^{2} - h^{2}\right) - 8 \alpha_{3} \left(K \alpha_{3} + 1\right)\right) \left(K \alpha_{3} - K \left(\alpha_{3} + 0.5 h\right) + 1\right) & 0 & 0 & \frac{\mu}{h^{2} \left(K \alpha_{3} + 1\right)} \left(\alpha_{3} - 0.5 h\right) \left(- K \alpha_{3} + K \left(\alpha_{3} + 0.5 h\right) - 1\right) & 0 & \frac{\mu}{h^{2} \left(K \alpha_{3} + 1\right)} \left(\alpha_{3} + 0.5 h\right) \left(K \alpha_{3} - K \left(\alpha_{3} + 0.5 h\right) + 1\right) & 0 & \frac{\mu}{h^{3} \left(K \alpha_{3} + 1\right)} \left(4 \alpha_{3}^{2} - h^{2}\right) \left(- K \alpha_{3} + K \left(\alpha_{3} + 0.5 h\right) - 1\right)\\0 & - \frac{\left(\alpha_{3} - 0.5 h\right) \left(\alpha_{3} + 0.5 h\right) \left(\lambda + 2 \mu\right)}{A h^{2} \left(K \alpha_{3} + 1\right)} & 0 & \frac{\left(\alpha_{3} + 0.5 h\right)^{2} \left(\lambda + 2 \mu\right)}{A h^{2} \left(K \alpha_{3} + 1\right)} & 0 & - \frac{\left(\alpha_{3} + 0.5 h\right) \left(4 \alpha_{3}^{2} - h^{2}\right) \left(\lambda + 2 \mu\right)}{A h^{3} \left(K \alpha_{3} + 1\right)} & - \frac{1}{h^{2} \left(K \alpha_{3} + 1\right)} \left(\alpha_{3} + 0.5 h\right) \left(K \left(\alpha_{3} - 0.5 h\right) \left(\lambda + 2 \mu\right) + \lambda \left(K \alpha_{3} + 1\right)\right) & 0 & \frac{1}{h^{2} \left(K \alpha_{3} + 1\right)} \left(\alpha_{3} + 0.5 h\right) \left(K \left(\alpha_{3} + 0.5 h\right) \left(\lambda + 2 \mu\right) + \lambda \left(K \alpha_{3} + 1\right)\right) & 0 & - \frac{1}{h^{3} \left(K \alpha_{3} + 1\right)} \left(\alpha_{3} + 0.5 h\right) \left(K \left(4 \alpha_{3}^{2} - h^{2}\right) \left(\lambda + 2 \mu\right) + 8 \alpha_{3} \lambda \left(K \alpha_{3} + 1\right)\right) & 0\\- \frac{A \mu}{h^{3} \left(K \alpha_{3} + 1\right)} \left(K \left(4 \alpha_{3}^{2} - h^{2}\right) - 8 \alpha_{3} \left(K \alpha_{3} + 1\right)\right) \left(K \alpha_{3} - K \left(\alpha_{3} - 0.5 h\right) + 1\right) & 0 & \frac{A \mu}{h^{3} \left(K \alpha_{3} + 1\right)} \left(K \left(4 \alpha_{3}^{2} - h^{2}\right) - 8 \alpha_{3} \left(K \alpha_{3} + 1\right)\right) \left(K \alpha_{3} - K \left(\alpha_{3} + 0.5 h\right) + 1\right) & 0 & \frac{A \mu}{h^{4} \left(K \alpha_{3} + 1\right)} \left(K \left(4 \alpha_{3}^{2} - h^{2}\right) - 8 \alpha_{3} \left(K \alpha_{3} + 1\right)\right)^{2} & 0 & 0 & \frac{\mu}{h^{3} \left(K \alpha_{3} + 1\right)} \left(\alpha_{3} - 0.5 h\right) \left(- K \left(4 \alpha_{3}^{2} - h^{2}\right) + 8 \alpha_{3} \left(K \alpha_{3} + 1\right)\right) & 0 & \frac{\mu}{h^{3} \left(K \alpha_{3} + 1\right)} \left(\alpha_{3} + 0.5 h\right) \left(K \left(4 \alpha_{3}^{2} - h^{2}\right) - 8 \alpha_{3} \left(K \alpha_{3} + 1\right)\right) & 0 & \frac{\mu}{h^{4} \left(K \alpha_{3} + 1\right)} \left(16 K \alpha_{3}^{4} - K h^{4} + 32 \alpha_{3}^{3} - 8 \alpha_{3} h^{2}\right)\\0 & \frac{\left(\alpha_{3} - 0.5 h\right) \left(4 \alpha_{3}^{2} - h^{2}\right) \left(\lambda + 2 \mu\right)}{A h^{3} \left(K \alpha_{3} + 1\right)} & 0 & - \frac{\left(\alpha_{3} + 0.5 h\right) \left(4 \alpha_{3}^{2} - h^{2}\right) \left(\lambda + 2 \mu\right)}{A h^{3} \left(K \alpha_{3} + 1\right)} & 0 & \frac{\left(4 \alpha_{3}^{2} - h^{2}\right)^{2} \left(\lambda + 2 \mu\right)}{A h^{4} \left(K \alpha_{3} + 1\right)} & \frac{1}{h^{3} \left(K \alpha_{3} + 1\right)} \left(4 \alpha_{3}^{2} - h^{2}\right) \left(K \left(\alpha_{3} - 0.5 h\right) \left(\lambda + 2 \mu\right) + \lambda \left(K \alpha_{3} + 1\right)\right) & 0 & - \frac{1}{h^{3} \left(K \alpha_{3} + 1\right)} \left(4 \alpha_{3}^{2} - h^{2}\right) \left(K \left(\alpha_{3} + 0.5 h\right) \left(\lambda + 2 \mu\right) + \lambda \left(K \alpha_{3} + 1\right)\right) & 0 & \frac{1}{h^{4} \left(K \alpha_{3} + 1\right)} \left(4 \alpha_{3}^{2} - h^{2}\right) \left(K \left(4 \alpha_{3}^{2} - h^{2}\right) \left(\lambda + 2 \mu\right) + 8 \alpha_{3} \lambda \left(K \alpha_{3} + 1\right)\right) & 0\\0 & \frac{1}{h^{2} \left(K \alpha_{3} + 1\right)} \left(\alpha_{3} - 0.5 h\right) \left(K \left(\alpha_{3} - 0.5 h\right) \left(\lambda + 2 \mu\right) + \lambda \left(K \alpha_{3} + 1\right)\right) & 0 & - \frac{1}{h^{2} \left(K \alpha_{3} + 1\right)} \left(\alpha_{3} + 0.5 h\right) \left(K \left(\alpha_{3} - 0.5 h\right) \left(\lambda + 2 \mu\right) + \lambda \left(K \alpha_{3} + 1\right)\right) & 0 & \frac{1}{h^{3} \left(K \alpha_{3} + 1\right)} \left(4 \alpha_{3}^{2} - h^{2}\right) \left(K \left(\alpha_{3} - 0.5 h\right) \left(\lambda + 2 \mu\right) + \lambda \left(K \alpha_{3} + 1\right)\right) & \frac{A}{h^{2} \left(K \alpha_{3} + 1\right)} \left(K \left(\alpha_{3} - 0.5 h\right) \left(K \left(\alpha_{3} - 0.5 h\right) \left(\lambda + 2 \mu\right) + \lambda \left(K \alpha_{3} + 1\right)\right) + \left(K \alpha_{3} + 1\right) \left(K \lambda \left(\alpha_{3} - 0.5 h\right) + \left(\lambda + 2 \mu\right) \left(K \alpha_{3} + 1\right)\right)\right) & 0 & - \frac{A}{h^{2} \left(K \alpha_{3} + 1\right)} \left(K \left(\alpha_{3} + 0.5 h\right) \left(K \left(\alpha_{3} - 0.5 h\right) \left(\lambda + 2 \mu\right) + \lambda \left(K \alpha_{3} + 1\right)\right) + \left(K \alpha_{3} + 1\right) \left(K \lambda \left(\alpha_{3} - 0.5 h\right) + \left(\lambda + 2 \mu\right) \left(K \alpha_{3} + 1\right)\right)\right) & 0 & \frac{A}{h^{3} \left(K \alpha_{3} + 1\right)} \left(K \left(4 \alpha_{3}^{2} - h^{2}\right) \left(K \left(\alpha_{3} - 0.5 h\right) \left(\lambda + 2 \mu\right) + \lambda \left(K \alpha_{3} + 1\right)\right) + 8 \alpha_{3} \left(K \alpha_{3} + 1\right) \left(K \lambda \left(\alpha_{3} - 0.5 h\right) + \left(\lambda + 2 \mu\right) \left(K \alpha_{3} + 1\right)\right)\right) & 0\\\frac{\mu}{h^{2} \left(K \alpha_{3} + 1\right)} \left(\alpha_{3} - 0.5 h\right) \left(K \alpha_{3} - K \left(\alpha_{3} - 0.5 h\right) + 1\right) & 0 & \frac{\mu}{h^{2} \left(K \alpha_{3} + 1\right)} \left(\alpha_{3} - 0.5 h\right) \left(- K \alpha_{3} + K \left(\alpha_{3} + 0.5 h\right) - 1\right) & 0 & \frac{\mu}{h^{3} \left(K \alpha_{3} + 1\right)} \left(\alpha_{3} - 0.5 h\right) \left(- K \left(4 \alpha_{3}^{2} - h^{2}\right) + 8 \alpha_{3} \left(K \alpha_{3} + 1\right)\right) & 0 & 0 & \frac{\mu \left(\alpha_{3} - 0.5 h\right)^{2}}{A h^{2} \left(K \alpha_{3} + 1\right)} & 0 & - \frac{\mu \left(\alpha_{3} - 0.5 h\right) \left(\alpha_{3} + 0.5 h\right)}{A h^{2} \left(K \alpha_{3} + 1\right)} & 0 & \frac{\mu \left(\alpha_{3} - 0.5 h\right) \left(4 \alpha_{3}^{2} - h^{2}\right)}{A h^{3} \left(K \alpha_{3} + 1\right)}\\0 & - \frac{1}{h^{2} \left(K \alpha_{3} + 1\right)} \left(\alpha_{3} - 0.5 h\right) \left(K \left(\alpha_{3} + 0.5 h\right) \left(\lambda + 2 \mu\right) + \lambda \left(K \alpha_{3} + 1\right)\right) & 0 & \frac{1}{h^{2} \left(K \alpha_{3} + 1\right)} \left(\alpha_{3} + 0.5 h\right) \left(K \left(\alpha_{3} + 0.5 h\right) \left(\lambda + 2 \mu\right) + \lambda \left(K \alpha_{3} + 1\right)\right) & 0 & - \frac{1}{h^{3} \left(K \alpha_{3} + 1\right)} \left(4 \alpha_{3}^{2} - h^{2}\right) \left(K \left(\alpha_{3} + 0.5 h\right) \left(\lambda + 2 \mu\right) + \lambda \left(K \alpha_{3} + 1\right)\right) & - \frac{A}{h^{2} \left(K \alpha_{3} + 1\right)} \left(K \left(\alpha_{3} - 0.5 h\right) \left(K \left(\alpha_{3} + 0.5 h\right) \left(\lambda + 2 \mu\right) + \lambda \left(K \alpha_{3} + 1\right)\right) + \left(K \alpha_{3} + 1\right) \left(K \lambda \left(\alpha_{3} + 0.5 h\right) + \left(\lambda + 2 \mu\right) \left(K \alpha_{3} + 1\right)\right)\right) & 0 & \frac{A}{h^{2} \left(K \alpha_{3} + 1\right)} \left(K \left(\alpha_{3} + 0.5 h\right) \left(K \left(\alpha_{3} + 0.5 h\right) \left(\lambda + 2 \mu\right) + \lambda \left(K \alpha_{3} + 1\right)\right) + \left(K \alpha_{3} + 1\right) \left(K \lambda \left(\alpha_{3} + 0.5 h\right) + \left(\lambda + 2 \mu\right) \left(K \alpha_{3} + 1\right)\right)\right) & 0 & - \frac{A}{h^{3} \left(K \alpha_{3} + 1\right)} \left(K \left(4 \alpha_{3}^{2} - h^{2}\right) \left(K \left(\alpha_{3} + 0.5 h\right) \left(\lambda + 2 \mu\right) + \lambda \left(K \alpha_{3} + 1\right)\right) + 8 \alpha_{3} \left(K \alpha_{3} + 1\right) \left(K \lambda \left(\alpha_{3} + 0.5 h\right) + \left(\lambda + 2 \mu\right) \left(K \alpha_{3} + 1\right)\right)\right) & 0\\\frac{\mu}{h^{2} \left(K \alpha_{3} + 1\right)} \left(\alpha_{3} + 0.5 h\right) \left(- K \alpha_{3} + K \left(\alpha_{3} - 0.5 h\right) - 1\right) & 0 & \frac{\mu}{h^{2} \left(K \alpha_{3} + 1\right)} \left(\alpha_{3} + 0.5 h\right) \left(K \alpha_{3} - K \left(\alpha_{3} + 0.5 h\right) + 1\right) & 0 & \frac{\mu}{h^{3} \left(K \alpha_{3} + 1\right)} \left(\alpha_{3} + 0.5 h\right) \left(K \left(4 \alpha_{3}^{2} - h^{2}\right) - 8 \alpha_{3} \left(K \alpha_{3} + 1\right)\right) & 0 & 0 & - \frac{\mu \left(\alpha_{3} - 0.5 h\right) \left(\alpha_{3} + 0.5 h\right)}{A h^{2} \left(K \alpha_{3} + 1\right)} & 0 & \frac{\mu \left(\alpha_{3} + 0.5 h\right)^{2}}{A h^{2} \left(K \alpha_{3} + 1\right)} & 0 & - \frac{\mu \left(\alpha_{3} + 0.5 h\right) \left(4 \alpha_{3}^{2} - h^{2}\right)}{A h^{3} \left(K \alpha_{3} + 1\right)}\\0 & \frac{1}{h^{3} \left(K \alpha_{3} + 1\right)} \left(\alpha_{3} - 0.5 h\right) \left(K \left(4 \alpha_{3}^{2} - h^{2}\right) \left(\lambda + 2 \mu\right) + 8 \alpha_{3} \lambda \left(K \alpha_{3} + 1\right)\right) & 0 & - \frac{1}{h^{3} \left(K \alpha_{3} + 1\right)} \left(\alpha_{3} + 0.5 h\right) \left(K \left(4 \alpha_{3}^{2} - h^{2}\right) \left(\lambda + 2 \mu\right) + 8 \alpha_{3} \lambda \left(K \alpha_{3} + 1\right)\right) & 0 & \frac{1}{h^{4} \left(K \alpha_{3} + 1\right)} \left(4 \alpha_{3}^{2} - h^{2}\right) \left(K \left(4 \alpha_{3}^{2} - h^{2}\right) \left(\lambda + 2 \mu\right) + 8 \alpha_{3} \lambda \left(K \alpha_{3} + 1\right)\right) & \frac{A}{h^{3} \left(K \alpha_{3} + 1\right)} \left(K \left(\alpha_{3} - 0.5 h\right) \left(K \left(4 \alpha_{3}^{2} - h^{2}\right) \left(\lambda + 2 \mu\right) + 8 \alpha_{3} \lambda \left(K \alpha_{3} + 1\right)\right) + \left(K \alpha_{3} + 1\right) \left(K \lambda \left(4 \alpha_{3}^{2} - h^{2}\right) + 8 \alpha_{3} \left(\lambda + 2 \mu\right) \left(K \alpha_{3} + 1\right)\right)\right) & 0 & - \frac{A}{h^{3} \left(K \alpha_{3} + 1\right)} \left(K \left(\alpha_{3} + 0.5 h\right) \left(K \left(4 \alpha_{3}^{2} - h^{2}\right) \left(\lambda + 2 \mu\right) + 8 \alpha_{3} \lambda \left(K \alpha_{3} + 1\right)\right) + \left(K \alpha_{3} + 1\right) \left(K \lambda \left(4 \alpha_{3}^{2} - h^{2}\right) + 8 \alpha_{3} \left(\lambda + 2 \mu\right) \left(K \alpha_{3} + 1\right)\right)\right) & 0 & \frac{A}{h^{4} \left(K \alpha_{3} + 1\right)} \left(K \left(4 \alpha_{3}^{2} - h^{2}\right) \left(K \left(4 \alpha_{3}^{2} - h^{2}\right) \left(\lambda + 2 \mu\right) + 8 \alpha_{3} \lambda \left(K \alpha_{3} + 1\right)\right) + 8 \alpha_{3} \left(K \alpha_{3} + 1\right) \left(K \lambda \left(4 \alpha_{3}^{2} - h^{2}\right) + 8 \alpha_{3} \left(\lambda + 2 \mu\right) \left(K \alpha_{3} + 1\right)\right)\right) & 0\\\frac{\mu}{h^{3} \left(K \alpha_{3} + 1\right)} \left(4 \alpha_{3}^{2} - h^{2}\right) \left(K \alpha_{3} - K \left(\alpha_{3} - 0.5 h\right) + 1\right) & 0 & \frac{\mu}{h^{3} \left(K \alpha_{3} + 1\right)} \left(4 \alpha_{3}^{2} - h^{2}\right) \left(- K \alpha_{3} + K \left(\alpha_{3} + 0.5 h\right) - 1\right) & 0 & \frac{\mu}{h^{4} \left(K \alpha_{3} + 1\right)} \left(16 K \alpha_{3}^{4} - K h^{4} + 32 \alpha_{3}^{3} - 8 \alpha_{3} h^{2}\right) & 0 & 0 & \frac{\mu \left(\alpha_{3} - 0.5 h\right) \left(4 \alpha_{3}^{2} - h^{2}\right)}{A h^{3} \left(K \alpha_{3} + 1\right)} & 0 & - \frac{\mu \left(\alpha_{3} + 0.5 h\right) \left(4 \alpha_{3}^{2} - h^{2}\right)}{A h^{3} \left(K \alpha_{3} + 1\right)} & 0 & \frac{\mu \left(4 \alpha_{3}^{2} - h^{2}\right)^{2}}{A h^{4} \left(K \alpha_{3} + 1\right)}\end{array}\right]$$
















In [15]:



    


S_in = integrate(S,(alpha3, -h/2, h/2))
S_in



    


















    
Out[15]:







$$\left[\begin{array}{cccccccccccc}- \frac{1.0}{K h^{2}} \left(0.25 A K^{2} h^{2} \mu + 1.0 A K h \mu + 1.0 A \mu\right) \log{\left (- 0.5 K h^{3} + 1.0 h^{2} \right )} + \frac{1.0}{K h^{2}} \left(0.25 A K^{2} h^{2} \mu + 1.0 A K h \mu + 1.0 A \mu\right) \log{\left (0.5 K h^{3} + 1.0 h^{2} \right )} & 0 & - \frac{1.0}{K h^{2}} \left(0.25 A K^{2} h^{2} \mu - 1.0 A \mu\right) \log{\left (- 0.5 K h^{3} + 1.0 h^{2} \right )} + \frac{1.0}{K h^{2}} \left(0.25 A K^{2} h^{2} \mu - 1.0 A \mu\right) \log{\left (0.5 K h^{3} + 1.0 h^{2} \right )} & 0 & - \frac{1.0 A \mu}{K^{2} h^{3}} \left(0.5 K h - 1.0\right) \left(0.5 K h + 1.0\right)^{2} \log{\left (- 0.5 K h + 1.0 \right )} + \frac{1.0 A \mu}{K^{2} h^{3}} \left(0.5 K h - 1.0\right) \left(0.5 K h + 1.0\right)^{2} \log{\left (0.5 K h + 1.0 \right )} + \frac{1.0}{K h^{2}} \left(2.0 A K h \mu + 4.0 A \mu\right) & 0 & 0 & \frac{1.0}{K h} \left(0.5 K h \mu + 1.0 \mu\right) + \frac{1.0 \mu}{K^{2} h^{2}} \left(0.25 K h + 0.5\right)^{2} \log{\left (- 0.5 K h + 1.0 \right )} - \frac{1.0 \mu}{K^{2} h^{2}} \left(0.25 K h + 0.5\right)^{2} \log{\left (0.5 K h + 1.0 \right )} & 0 & - \frac{1.0}{K h} \left(0.5 K h \mu + 1.0 \mu\right) + \frac{1.0 \mu}{K^{2} h^{2}} \left(0.25 K h - 0.5\right) \left(0.25 K h + 0.5\right) \log{\left (- 0.5 K h + 1.0 \right )} - \frac{1.0 \mu}{K^{2} h^{2}} \left(0.25 K h - 0.5\right) \left(0.25 K h + 0.5\right) \log{\left (0.5 K h + 1.0 \right )} & 0 & - \frac{1.0}{K^{2} h^{2}} \left(2.0 K h \mu + 4.0 \mu\right) + \frac{1.0 \mu}{K^{3} h^{3}} \left(0.5 K h - 1.0\right) \left(0.5 K h + 1.0\right)^{2} \log{\left (- 0.5 K h + 1.0 \right )} - \frac{1.0 \mu}{K^{3} h^{3}} \left(0.5 K h - 1.0\right) \left(0.5 K h + 1.0\right)^{2} \log{\left (0.5 K h + 1.0 \right )}\\0 & - \frac{1.0}{A K^{2} h} \left(1.0 K h \lambda + 2.0 K h \mu + 1.0 \lambda + 2.0 \mu\right) - \frac{1.0 \left(0.25 K h + 0.5\right)^{2}}{A K^{3} h^{2}} \left(0.25 \lambda + 0.5 \mu\right) \log{\left (- 0.5 K h + 1.0 \right )} + \frac{1.0 \left(0.25 K h + 0.5\right)^{2}}{A K^{3} h^{2}} \left(0.25 \lambda + 0.5 \mu\right) \log{\left (0.5 K h + 1.0 \right )} & 0 & \frac{1.0 \left(1.0 \lambda + 2.0 \mu\right)}{A K^{2} h} - \frac{1.0}{A K^{3} h^{2}} \left(0.25 \lambda + 0.5 \mu\right) \left(0.25 K h - 0.5\right) \left(0.25 K h + 0.5\right) \log{\left (- 0.5 K h + 1.0 \right )} + \frac{1.0}{A K^{3} h^{2}} \left(0.25 \lambda + 0.5 \mu\right) \left(0.25 K h - 0.5\right) \left(0.25 K h + 0.5\right) \log{\left (0.5 K h + 1.0 \right )} & 0 & \frac{0.25}{A K} \left(1.33333333333333 \lambda + 2.66666666666667 \mu\right) - \frac{1.0}{A K^{3} h^{2}} \left(1.0 K^{2} h^{2} \lambda + 2.0 K^{2} h^{2} \mu - 2.0 K h \lambda - 4.0 K h \mu - 4.0 \lambda - 8.0 \mu\right) - \frac{1.0 \left(0.5 K h + 1.0\right)^{2}}{A K^{4} h^{3}} \left(0.5 \lambda + 1.0 \mu\right) \left(0.5 K h - 1.0\right) \log{\left (- 0.5 K h + 1.0 \right )} + \frac{1.0 \left(0.5 K h + 1.0\right)^{2}}{A K^{4} h^{3}} \left(0.5 \lambda + 1.0 \mu\right) \left(0.5 K h - 1.0\right) \log{\left (0.5 K h + 1.0 \right )} & - \frac{1.0}{K h} \left(1.5 K h \lambda + 2.0 K h \mu + 1.0 \lambda + 2.0 \mu\right) - \frac{1.0}{K^{2} h^{2}} \left(0.25 \lambda + 0.5 \mu\right) \left(0.25 K h + 0.5\right)^{2} \log{\left (- 0.5 K h + 1.0 \right )} + \frac{1.0}{K^{2} h^{2}} \left(0.25 \lambda + 0.5 \mu\right) \left(0.25 K h + 0.5\right)^{2} \log{\left (0.5 K h + 1.0 \right )} & 0 & \frac{1.0}{K h} \left(0.5 K h \lambda + 1.0 \lambda + 2.0 \mu\right) - \frac{1.0}{K^{2} h^{2}} \left(0.25 \lambda + 0.5 \mu\right) \left(0.25 K h - 0.5\right) \left(0.25 K h + 0.5\right) \log{\left (- 0.5 K h + 1.0 \right )} + \frac{1.0}{K^{2} h^{2}} \left(0.25 \lambda + 0.5 \mu\right) \left(0.25 K h - 0.5\right) \left(0.25 K h + 0.5\right) \log{\left (0.5 K h + 1.0 \right )} & 0 & 1.0 \lambda + 0.666666666666667 \mu - \frac{1.0}{K^{2} h^{2}} \left(1.0 K^{2} h^{2} \lambda + 2.0 K^{2} h^{2} \mu - 2.0 K h \lambda - 4.0 K h \mu - 4.0 \lambda - 8.0 \mu\right) - \frac{1.0}{K^{3} h^{3}} \left(0.5 \lambda + 1.0 \mu\right) \left(0.5 K h - 1.0\right) \left(0.5 K h + 1.0\right)^{2} \log{\left (- 0.5 K h + 1.0 \right )} + \frac{1.0}{K^{3} h^{3}} \left(0.5 \lambda + 1.0 \mu\right) \left(0.5 K h - 1.0\right) \left(0.5 K h + 1.0\right)^{2} \log{\left (0.5 K h + 1.0 \right )} & 0\\- \frac{1.0}{K h^{2}} \left(0.25 A K^{2} h^{2} \mu - 1.0 A \mu\right) \log{\left (- 0.5 K h^{3} + 1.0 h^{2} \right )} + \frac{1.0}{K h^{2}} \left(0.25 A K^{2} h^{2} \mu - 1.0 A \mu\right) \log{\left (0.5 K h^{3} + 1.0 h^{2} \right )} & 0 & - \frac{1.0}{K h^{2}} \left(0.25 A K^{2} h^{2} \mu - 1.0 A K h \mu + 1.0 A \mu\right) \log{\left (- 0.5 K h^{3} + 1.0 h^{2} \right )} + \frac{1.0}{K h^{2}} \left(0.25 A K^{2} h^{2} \mu - 1.0 A K h \mu + 1.0 A \mu\right) \log{\left (0.5 K h^{3} + 1.0 h^{2} \right )} & 0 & - \frac{1.0 A \mu}{K^{2} h^{3}} \left(0.5 K h - 1.0\right)^{2} \left(0.5 K h + 1.0\right) \log{\left (- 0.5 K h + 1.0 \right )} + \frac{1.0 A \mu}{K^{2} h^{3}} \left(0.5 K h - 1.0\right)^{2} \left(0.5 K h + 1.0\right) \log{\left (0.5 K h + 1.0 \right )} + \frac{1.0}{K h^{2}} \left(2.0 A K h \mu - 4.0 A \mu\right) & 0 & 0 & \frac{1.0}{K h} \left(0.5 K h \mu - 1.0 \mu\right) + \frac{1.0 \mu}{K^{2} h^{2}} \left(0.25 K h - 0.5\right) \left(0.25 K h + 0.5\right) \log{\left (- 0.5 K h + 1.0 \right )} - \frac{1.0 \mu}{K^{2} h^{2}} \left(0.25 K h - 0.5\right) \left(0.25 K h + 0.5\right) \log{\left (0.5 K h + 1.0 \right )} & 0 & - \frac{1.0}{K h} \left(0.5 K h \mu - 1.0 \mu\right) + \frac{1.0 \mu}{K^{2} h^{2}} \left(0.25 K h - 0.5\right)^{2} \log{\left (- 0.5 K h + 1.0 \right )} - \frac{1.0 \mu}{K^{2} h^{2}} \left(0.25 K h - 0.5\right)^{2} \log{\left (0.5 K h + 1.0 \right )} & 0 & - \frac{1.0}{K^{2} h^{2}} \left(2.0 K h \mu - 4.0 \mu\right) + \frac{1.0 \mu}{K^{3} h^{3}} \left(0.5 K h - 1.0\right)^{2} \left(0.5 K h + 1.0\right) \log{\left (- 0.5 K h + 1.0 \right )} - \frac{1.0 \mu}{K^{3} h^{3}} \left(0.5 K h - 1.0\right)^{2} \left(0.5 K h + 1.0\right) \log{\left (0.5 K h + 1.0 \right )}\\0 & \frac{1.0 \left(1.0 \lambda + 2.0 \mu\right)}{A K^{2} h} - \frac{1.0}{A K^{3} h^{2}} \left(0.25 \lambda + 0.5 \mu\right) \left(0.25 K h - 0.5\right) \left(0.25 K h + 0.5\right) \log{\left (- 0.5 K h + 1.0 \right )} + \frac{1.0}{A K^{3} h^{2}} \left(0.25 \lambda + 0.5 \mu\right) \left(0.25 K h - 0.5\right) \left(0.25 K h + 0.5\right) \log{\left (0.5 K h + 1.0 \right )} & 0 & \frac{1.0}{A K^{2} h} \left(1.0 K h \lambda + 2.0 K h \mu - 1.0 \lambda - 2.0 \mu\right) - \frac{1.0 \left(0.25 K h - 0.5\right)^{2}}{A K^{3} h^{2}} \left(0.25 \lambda + 0.5 \mu\right) \log{\left (- 0.5 K h + 1.0 \right )} + \frac{1.0 \left(0.25 K h - 0.5\right)^{2}}{A K^{3} h^{2}} \left(0.25 \lambda + 0.5 \mu\right) \log{\left (0.5 K h + 1.0 \right )} & 0 & - \frac{0.25}{A K} \left(1.33333333333333 \lambda + 2.66666666666667 \mu\right) + \frac{1.0}{A K^{3} h^{2}} \left(1.0 K^{2} h^{2} \lambda + 2.0 K^{2} h^{2} \mu + 2.0 K h \lambda + 4.0 K h \mu - 4.0 \lambda - 8.0 \mu\right) - \frac{1.0 \left(0.5 K h - 1.0\right)^{2}}{A K^{4} h^{3}} \left(0.5 \lambda + 1.0 \mu\right) \left(0.5 K h + 1.0\right) \log{\left (- 0.5 K h + 1.0 \right )} + \frac{1.0 \left(0.5 K h - 1.0\right)^{2}}{A K^{4} h^{3}} \left(0.5 \lambda + 1.0 \mu\right) \left(0.5 K h + 1.0\right) \log{\left (0.5 K h + 1.0 \right )} & - \frac{1.0}{K h} \left(0.5 K h \lambda - 1.0 \lambda - 2.0 \mu\right) - \frac{1.0}{K^{2} h^{2}} \left(0.25 \lambda + 0.5 \mu\right) \left(0.25 K h - 0.5\right) \left(0.25 K h + 0.5\right) \log{\left (- 0.5 K h + 1.0 \right )} + \frac{1.0}{K^{2} h^{2}} \left(0.25 \lambda + 0.5 \mu\right) \left(0.25 K h - 0.5\right) \left(0.25 K h + 0.5\right) \log{\left (0.5 K h + 1.0 \right )} & 0 & \frac{1.0}{K h} \left(1.5 K h \lambda + 2.0 K h \mu - 1.0 \lambda - 2.0 \mu\right) - \frac{1.0}{K^{2} h^{2}} \left(0.25 \lambda + 0.5 \mu\right) \left(0.25 K h - 0.5\right)^{2} \log{\left (- 0.5 K h + 1.0 \right )} + \frac{1.0}{K^{2} h^{2}} \left(0.25 \lambda + 0.5 \mu\right) \left(0.25 K h - 0.5\right)^{2} \log{\left (0.5 K h + 1.0 \right )} & 0 & - 1.0 \lambda - 0.666666666666667 \mu + \frac{1.0}{K^{2} h^{2}} \left(1.0 K^{2} h^{2} \lambda + 2.0 K^{2} h^{2} \mu + 2.0 K h \lambda + 4.0 K h \mu - 4.0 \lambda - 8.0 \mu\right) - \frac{1.0}{K^{3} h^{3}} \left(0.5 \lambda + 1.0 \mu\right) \left(0.5 K h - 1.0\right)^{2} \left(0.5 K h + 1.0\right) \log{\left (- 0.5 K h + 1.0 \right )} + \frac{1.0}{K^{3} h^{3}} \left(0.5 \lambda + 1.0 \mu\right) \left(0.5 K h - 1.0\right)^{2} \left(0.5 K h + 1.0\right) \log{\left (0.5 K h + 1.0 \right )} & 0\\- \frac{1.0 A \mu}{K^{2} h^{3}} \left(0.5 K h - 1.0\right) \left(0.5 K h + 1.0\right)^{2} \log{\left (- 0.5 K h + 1.0 \right )} + \frac{1.0 A \mu}{K^{2} h^{3}} \left(0.5 K h - 1.0\right) \left(0.5 K h + 1.0\right)^{2} \log{\left (0.5 K h + 1.0 \right )} + \frac{1.0}{K h^{2}} \left(2.0 A K h \mu + 4.0 A \mu\right) & 0 & - \frac{1.0 A \mu}{K^{2} h^{3}} \left(0.5 K h - 1.0\right)^{2} \left(0.5 K h + 1.0\right) \log{\left (- 0.5 K h + 1.0 \right )} + \frac{1.0 A \mu}{K^{2} h^{3}} \left(0.5 K h - 1.0\right)^{2} \left(0.5 K h + 1.0\right) \log{\left (0.5 K h + 1.0 \right )} + \frac{1.0}{K h^{2}} \left(2.0 A K h \mu - 4.0 A \mu\right) & 0 & \frac{4 A}{h} \mu - \frac{A \mu}{K^{3} h^{4}} \left(K h - 2\right)^{2} \left(K h + 2\right)^{2} \log{\left (- \frac{K h}{2} + 1 \right )} + \frac{A \mu}{K^{3} h^{4}} \left(K h - 2\right)^{2} \left(K h + 2\right)^{2} \log{\left (\frac{K h}{2} + 1 \right )} + \frac{1}{K^{2} h^{3}} \left(8 A K^{2} h^{2} \mu - 16 A \mu\right) & 0 & 0 & 0.333333333333333 \mu + \frac{1.0}{K^{2} h^{2}} \left(1.0 K^{2} h^{2} \mu - 2.0 K h \mu - 4.0 \mu\right) + \frac{1.0 \mu}{K^{3} h^{3}} \left(0.5 K h - 1.0\right) \left(0.5 K h + 1.0\right)^{2} \log{\left (- 0.5 K h + 1.0 \right )} - \frac{1.0 \mu}{K^{3} h^{3}} \left(0.5 K h - 1.0\right) \left(0.5 K h + 1.0\right)^{2} \log{\left (0.5 K h + 1.0 \right )} & 0 & - 0.333333333333333 \mu - \frac{1.0}{K^{2} h^{2}} \left(1.0 K^{2} h^{2} \mu + 2.0 K h \mu - 4.0 \mu\right) + \frac{1.0 \mu}{K^{3} h^{3}} \left(0.5 K h - 1.0\right)^{2} \left(0.5 K h + 1.0\right) \log{\left (- 0.5 K h + 1.0 \right )} - \frac{1.0 \mu}{K^{3} h^{3}} \left(0.5 K h - 1.0\right)^{2} \left(0.5 K h + 1.0\right) \log{\left (0.5 K h + 1.0 \right )} & 0 & \frac{4 \mu}{3 K h} - \frac{1}{K^{3} h^{3}} \left(8 K^{2} h^{2} \mu - 16 \mu\right) + \frac{\mu}{K^{4} h^{4}} \left(K h - 2\right)^{2} \left(K h + 2\right)^{2} \log{\left (- \frac{K h}{2} + 1 \right )} - \frac{\mu}{K^{4} h^{4}} \left(K h - 2\right)^{2} \left(K h + 2\right)^{2} \log{\left (\frac{K h}{2} + 1 \right )}\\0 & \frac{0.25}{A K} \left(1.33333333333333 \lambda + 2.66666666666667 \mu\right) - \frac{1.0}{A K^{3} h^{2}} \left(1.0 K^{2} h^{2} \lambda + 2.0 K^{2} h^{2} \mu - 2.0 K h \lambda - 4.0 K h \mu - 4.0 \lambda - 8.0 \mu\right) - \frac{1.0 \left(0.5 K h + 1.0\right)^{2}}{A K^{4} h^{3}} \left(0.5 \lambda + 1.0 \mu\right) \left(0.5 K h - 1.0\right) \log{\left (- 0.5 K h + 1.0 \right )} + \frac{1.0 \left(0.5 K h + 1.0\right)^{2}}{A K^{4} h^{3}} \left(0.5 \lambda + 1.0 \mu\right) \left(0.5 K h - 1.0\right) \log{\left (0.5 K h + 1.0 \right )} & 0 & - \frac{0.25}{A K} \left(1.33333333333333 \lambda + 2.66666666666667 \mu\right) + \frac{1.0}{A K^{3} h^{2}} \left(1.0 K^{2} h^{2} \lambda + 2.0 K^{2} h^{2} \mu + 2.0 K h \lambda + 4.0 K h \mu - 4.0 \lambda - 8.0 \mu\right) - \frac{1.0 \left(0.5 K h - 1.0\right)^{2}}{A K^{4} h^{3}} \left(0.5 \lambda + 1.0 \mu\right) \left(0.5 K h + 1.0\right) \log{\left (- 0.5 K h + 1.0 \right )} + \frac{1.0 \left(0.5 K h - 1.0\right)^{2}}{A K^{4} h^{3}} \left(0.5 \lambda + 1.0 \mu\right) \left(0.5 K h + 1.0\right) \log{\left (0.5 K h + 1.0 \right )} & 0 & - \frac{16 \lambda + 32 \mu}{12 A K^{2} h} + \frac{1}{A K^{4} h^{3}} \left(8 K^{2} h^{2} \lambda + 16 K^{2} h^{2} \mu - 16 \lambda - 32 \mu\right) - \frac{\left(K h - 2\right)^{2}}{A K^{5} h^{4}} \left(\lambda + 2 \mu\right) \left(K h + 2\right)^{2} \log{\left (- \frac{K h}{2} + 1 \right )} + \frac{\left(K h - 2\right)^{2}}{A K^{5} h^{4}} \left(\lambda + 2 \mu\right) \left(K h + 2\right)^{2} \log{\left (\frac{K h}{2} + 1 \right )} & 0.666666666666667 \lambda + 0.666666666666667 \mu - \frac{1.0}{K^{2} h^{2}} \left(2.0 K^{2} h^{2} \lambda + 2.0 K^{2} h^{2} \mu - 2.0 K h \lambda - 4.0 K h \mu - 4.0 \lambda - 8.0 \mu\right) - \frac{1.0}{K^{3} h^{3}} \left(0.5 \lambda + 1.0 \mu\right) \left(0.5 K h - 1.0\right) \left(0.5 K h + 1.0\right)^{2} \log{\left (- 0.5 K h + 1.0 \right )} + \frac{1.0}{K^{3} h^{3}} \left(0.5 \lambda + 1.0 \mu\right) \left(0.5 K h - 1.0\right) \left(0.5 K h + 1.0\right)^{2} \log{\left (0.5 K h + 1.0 \right )} & 0 & - 0.666666666666667 \lambda - 0.666666666666667 \mu + \frac{1.0}{K^{2} h^{2}} \left(2.0 K^{2} h^{2} \lambda + 2.0 K^{2} h^{2} \mu + 2.0 K h \lambda + 4.0 K h \mu - 4.0 \lambda - 8.0 \mu\right) - \frac{1.0}{K^{3} h^{3}} \left(0.5 \lambda + 1.0 \mu\right) \left(0.5 K h - 1.0\right)^{2} \left(0.5 K h + 1.0\right) \log{\left (- 0.5 K h + 1.0 \right )} + \frac{1.0}{K^{3} h^{3}} \left(0.5 \lambda + 1.0 \mu\right) \left(0.5 K h - 1.0\right)^{2} \left(0.5 K h + 1.0\right) \log{\left (0.5 K h + 1.0 \right )} & 0 & - \frac{16 \lambda + 32 \mu}{12 K h} + \frac{1}{K^{3} h^{3}} \left(8 K^{2} h^{2} \lambda + 16 K^{2} h^{2} \mu - 16 \lambda - 32 \mu\right) - \frac{\left(K h - 2\right)^{2}}{K^{4} h^{4}} \left(\lambda + 2 \mu\right) \left(K h + 2\right)^{2} \log{\left (- \frac{K h}{2} + 1 \right )} + \frac{\left(K h - 2\right)^{2}}{K^{4} h^{4}} \left(\lambda + 2 \mu\right) \left(K h + 2\right)^{2} \log{\left (\frac{K h}{2} + 1 \right )} & 0\\0 & - \frac{1.0}{K h} \left(1.5 K h \lambda + 2.0 K h \mu + 1.0 \lambda + 2.0 \mu\right) - \frac{1.0}{K^{2} h^{2}} \left(0.25 \lambda + 0.5 \mu\right) \left(0.25 K h + 0.5\right)^{2} \log{\left (- 0.5 K h + 1.0 \right )} + \frac{1.0}{K^{2} h^{2}} \left(0.25 \lambda + 0.5 \mu\right) \left(0.25 K h + 0.5\right)^{2} \log{\left (0.5 K h + 1.0 \right )} & 0 & - \frac{1.0}{K h} \left(0.5 K h \lambda - 1.0 \lambda - 2.0 \mu\right) - \frac{1.0}{K^{2} h^{2}} \left(0.25 \lambda + 0.5 \mu\right) \left(0.25 K h - 0.5\right) \left(0.25 K h + 0.5\right) \log{\left (- 0.5 K h + 1.0 \right )} + \frac{1.0}{K^{2} h^{2}} \left(0.25 \lambda + 0.5 \mu\right) \left(0.25 K h - 0.5\right) \left(0.25 K h + 0.5\right) \log{\left (0.5 K h + 1.0 \right )} & 0 & 0.666666666666667 \lambda + 0.666666666666667 \mu - \frac{1.0}{K^{2} h^{2}} \left(2.0 K^{2} h^{2} \lambda + 2.0 K^{2} h^{2} \mu - 2.0 K h \lambda - 4.0 K h \mu - 4.0 \lambda - 8.0 \mu\right) - \frac{1.0}{K^{3} h^{3}} \left(0.5 \lambda + 1.0 \mu\right) \left(0.5 K h - 1.0\right) \left(0.5 K h + 1.0\right)^{2} \log{\left (- 0.5 K h + 1.0 \right )} + \frac{1.0}{K^{3} h^{3}} \left(0.5 \lambda + 1.0 \mu\right) \left(0.5 K h - 1.0\right) \left(0.5 K h + 1.0\right)^{2} \log{\left (0.5 K h + 1.0 \right )} & - 2.0 A K \lambda - 2.0 A K \mu - \frac{1.0 A}{K h^{2}} \left(0.25 \lambda + 0.5 \mu\right) \left(0.25 K h + 0.5\right)^{2} \log{\left (- 0.5 K h + 1.0 \right )} + \frac{1.0 A}{K h^{2}} \left(0.25 \lambda + 0.5 \mu\right) \left(0.25 K h + 0.5\right)^{2} \log{\left (0.5 K h + 1.0 \right )} & 0 & - \frac{1.0 A}{K h^{2}} \left(0.25 \lambda + 0.5 \mu\right) \left(0.25 K h - 0.5\right) \left(0.25 K h + 0.5\right) \log{\left (- 0.5 K h + 1.0 \right )} + \frac{1.0 A}{K h^{2}} \left(0.25 \lambda + 0.5 \mu\right) \left(0.25 K h - 0.5\right) \left(0.25 K h + 0.5\right) \log{\left (0.5 K h + 1.0 \right )} & 0 & 2.0 A K \lambda + 2.0 A K \mu - \frac{1.0 A}{K^{2} h^{3}} \left(0.5 \lambda + 1.0 \mu\right) \left(0.5 K h - 1.0\right) \left(0.5 K h + 1.0\right)^{2} \log{\left (- 0.5 K h + 1.0 \right )} + \frac{1.0 A}{K^{2} h^{3}} \left(0.5 \lambda + 1.0 \mu\right) \left(0.5 K h - 1.0\right) \left(0.5 K h + 1.0\right)^{2} \log{\left (0.5 K h + 1.0 \right )} - \frac{1.0}{K h^{2}} \left(2.0 A K^{2} h^{2} \lambda + 2.0 A K^{2} h^{2} \mu - 2.0 A K h \lambda - 4.0 A K h \mu - 4.0 A \lambda - 8.0 A \mu\right) & 0\\\frac{1.0}{K h} \left(0.5 K h \mu + 1.0 \mu\right) + \frac{1.0 \mu}{K^{2} h^{2}} \left(0.25 K h + 0.5\right)^{2} \log{\left (- 0.5 K h + 1.0 \right )} - \frac{1.0 \mu}{K^{2} h^{2}} \left(0.25 K h + 0.5\right)^{2} \log{\left (0.5 K h + 1.0 \right )} & 0 & \frac{1.0}{K h} \left(0.5 K h \mu - 1.0 \mu\right) + \frac{1.0 \mu}{K^{2} h^{2}} \left(0.25 K h - 0.5\right) \left(0.25 K h + 0.5\right) \log{\left (- 0.5 K h + 1.0 \right )} - \frac{1.0 \mu}{K^{2} h^{2}} \left(0.25 K h - 0.5\right) \left(0.25 K h + 0.5\right) \log{\left (0.5 K h + 1.0 \right )} & 0 & 0.333333333333333 \mu + \frac{1.0}{K^{2} h^{2}} \left(1.0 K^{2} h^{2} \mu - 2.0 K h \mu - 4.0 \mu\right) + \frac{1.0 \mu}{K^{3} h^{3}} \left(0.5 K h - 1.0\right) \left(0.5 K h + 1.0\right)^{2} \log{\left (- 0.5 K h + 1.0 \right )} - \frac{1.0 \mu}{K^{3} h^{3}} \left(0.5 K h - 1.0\right) \left(0.5 K h + 1.0\right)^{2} \log{\left (0.5 K h + 1.0 \right )} & 0 & 0 & - \frac{1.0}{A K^{2} h} \left(1.0 K h \mu + 1.0 \mu\right) - \frac{1.0 \mu}{A K^{3} h^{2}} \left(0.25 K h + 0.5\right)^{2} \log{\left (- 0.5 K h + 1.0 \right )} + \frac{1.0 \mu}{A K^{3} h^{2}} \left(0.25 K h + 0.5\right)^{2} \log{\left (0.5 K h + 1.0 \right )} & 0 & \frac{1.0 \mu}{A K^{2} h} - \frac{1.0 \mu}{A K^{3} h^{2}} \left(0.25 K h - 0.5\right) \left(0.25 K h + 0.5\right) \log{\left (- 0.5 K h + 1.0 \right )} + \frac{1.0 \mu}{A K^{3} h^{2}} \left(0.25 K h - 0.5\right) \left(0.25 K h + 0.5\right) \log{\left (0.5 K h + 1.0 \right )} & 0 & \frac{0.333333333333333 \mu}{A K} - \frac{1.0}{A K^{3} h^{2}} \left(1.0 K^{2} h^{2} \mu - 2.0 K h \mu - 4.0 \mu\right) - \frac{1.0 \mu}{A K^{4} h^{3}} \left(0.5 K h - 1.0\right) \left(0.5 K h + 1.0\right)^{2} \log{\left (- 0.5 K h + 1.0 \right )} + \frac{1.0 \mu}{A K^{4} h^{3}} \left(0.5 K h - 1.0\right) \left(0.5 K h + 1.0\right)^{2} \log{\left (0.5 K h + 1.0 \right )}\\0 & \frac{1.0}{K h} \left(0.5 K h \lambda + 1.0 \lambda + 2.0 \mu\right) - \frac{1.0}{K^{2} h^{2}} \left(0.25 \lambda + 0.5 \mu\right) \left(0.25 K h - 0.5\right) \left(0.25 K h + 0.5\right) \log{\left (- 0.5 K h + 1.0 \right )} + \frac{1.0}{K^{2} h^{2}} \left(0.25 \lambda + 0.5 \mu\right) \left(0.25 K h - 0.5\right) \left(0.25 K h + 0.5\right) \log{\left (0.5 K h + 1.0 \right )} & 0 & \frac{1.0}{K h} \left(1.5 K h \lambda + 2.0 K h \mu - 1.0 \lambda - 2.0 \mu\right) - \frac{1.0}{K^{2} h^{2}} \left(0.25 \lambda + 0.5 \mu\right) \left(0.25 K h - 0.5\right)^{2} \log{\left (- 0.5 K h + 1.0 \right )} + \frac{1.0}{K^{2} h^{2}} \left(0.25 \lambda + 0.5 \mu\right) \left(0.25 K h - 0.5\right)^{2} \log{\left (0.5 K h + 1.0 \right )} & 0 & - 0.666666666666667 \lambda - 0.666666666666667 \mu + \frac{1.0}{K^{2} h^{2}} \left(2.0 K^{2} h^{2} \lambda + 2.0 K^{2} h^{2} \mu + 2.0 K h \lambda + 4.0 K h \mu - 4.0 \lambda - 8.0 \mu\right) - \frac{1.0}{K^{3} h^{3}} \left(0.5 \lambda + 1.0 \mu\right) \left(0.5 K h - 1.0\right)^{2} \left(0.5 K h + 1.0\right) \log{\left (- 0.5 K h + 1.0 \right )} + \frac{1.0}{K^{3} h^{3}} \left(0.5 \lambda + 1.0 \mu\right) \left(0.5 K h - 1.0\right)^{2} \left(0.5 K h + 1.0\right) \log{\left (0.5 K h + 1.0 \right )} & - \frac{1.0 A}{K h^{2}} \left(0.25 \lambda + 0.5 \mu\right) \left(0.25 K h - 0.5\right) \left(0.25 K h + 0.5\right) \log{\left (- 0.5 K h + 1.0 \right )} + \frac{1.0 A}{K h^{2}} \left(0.25 \lambda + 0.5 \mu\right) \left(0.25 K h - 0.5\right) \left(0.25 K h + 0.5\right) \log{\left (0.5 K h + 1.0 \right )} & 0 & 2.0 A K \lambda + 2.0 A K \mu - \frac{1.0 A}{K h^{2}} \left(0.25 \lambda + 0.5 \mu\right) \left(0.25 K h - 0.5\right)^{2} \log{\left (- 0.5 K h + 1.0 \right )} + \frac{1.0 A}{K h^{2}} \left(0.25 \lambda + 0.5 \mu\right) \left(0.25 K h - 0.5\right)^{2} \log{\left (0.5 K h + 1.0 \right )} & 0 & - 2.0 A K \lambda - 2.0 A K \mu - \frac{1.0 A}{K^{2} h^{3}} \left(0.5 \lambda + 1.0 \mu\right) \left(0.5 K h - 1.0\right)^{2} \left(0.5 K h + 1.0\right) \log{\left (- 0.5 K h + 1.0 \right )} + \frac{1.0 A}{K^{2} h^{3}} \left(0.5 \lambda + 1.0 \mu\right) \left(0.5 K h - 1.0\right)^{2} \left(0.5 K h + 1.0\right) \log{\left (0.5 K h + 1.0 \right )} + \frac{1.0}{K h^{2}} \left(2.0 A K^{2} h^{2} \lambda + 2.0 A K^{2} h^{2} \mu + 2.0 A K h \lambda + 4.0 A K h \mu - 4.0 A \lambda - 8.0 A \mu\right) & 0\\- \frac{1.0}{K h} \left(0.5 K h \mu + 1.0 \mu\right) + \frac{1.0 \mu}{K^{2} h^{2}} \left(0.25 K h - 0.5\right) \left(0.25 K h + 0.5\right) \log{\left (- 0.5 K h + 1.0 \right )} - \frac{1.0 \mu}{K^{2} h^{2}} \left(0.25 K h - 0.5\right) \left(0.25 K h + 0.5\right) \log{\left (0.5 K h + 1.0 \right )} & 0 & - \frac{1.0}{K h} \left(0.5 K h \mu - 1.0 \mu\right) + \frac{1.0 \mu}{K^{2} h^{2}} \left(0.25 K h - 0.5\right)^{2} \log{\left (- 0.5 K h + 1.0 \right )} - \frac{1.0 \mu}{K^{2} h^{2}} \left(0.25 K h - 0.5\right)^{2} \log{\left (0.5 K h + 1.0 \right )} & 0 & - 0.333333333333333 \mu - \frac{1.0}{K^{2} h^{2}} \left(1.0 K^{2} h^{2} \mu + 2.0 K h \mu - 4.0 \mu\right) + \frac{1.0 \mu}{K^{3} h^{3}} \left(0.5 K h - 1.0\right)^{2} \left(0.5 K h + 1.0\right) \log{\left (- 0.5 K h + 1.0 \right )} - \frac{1.0 \mu}{K^{3} h^{3}} \left(0.5 K h - 1.0\right)^{2} \left(0.5 K h + 1.0\right) \log{\left (0.5 K h + 1.0 \right )} & 0 & 0 & \frac{1.0 \mu}{A K^{2} h} - \frac{1.0 \mu}{A K^{3} h^{2}} \left(0.25 K h - 0.5\right) \left(0.25 K h + 0.5\right) \log{\left (- 0.5 K h + 1.0 \right )} + \frac{1.0 \mu}{A K^{3} h^{2}} \left(0.25 K h - 0.5\right) \left(0.25 K h + 0.5\right) \log{\left (0.5 K h + 1.0 \right )} & 0 & \frac{1.0}{A K^{2} h} \left(1.0 K h \mu - 1.0 \mu\right) - \frac{1.0 \mu}{A K^{3} h^{2}} \left(0.25 K h - 0.5\right)^{2} \log{\left (- 0.5 K h + 1.0 \right )} + \frac{1.0 \mu}{A K^{3} h^{2}} \left(0.25 K h - 0.5\right)^{2} \log{\left (0.5 K h + 1.0 \right )} & 0 & - \frac{0.333333333333333 \mu}{A K} + \frac{1.0}{A K^{3} h^{2}} \left(1.0 K^{2} h^{2} \mu + 2.0 K h \mu - 4.0 \mu\right) - \frac{1.0 \mu}{A K^{4} h^{3}} \left(0.5 K h - 1.0\right)^{2} \left(0.5 K h + 1.0\right) \log{\left (- 0.5 K h + 1.0 \right )} + \frac{1.0 \mu}{A K^{4} h^{3}} \left(0.5 K h - 1.0\right)^{2} \left(0.5 K h + 1.0\right) \log{\left (0.5 K h + 1.0 \right )}\\0 & 1.0 \lambda + 0.666666666666667 \mu - \frac{1.0}{K^{2} h^{2}} \left(1.0 K^{2} h^{2} \lambda + 2.0 K^{2} h^{2} \mu - 2.0 K h \lambda - 4.0 K h \mu - 4.0 \lambda - 8.0 \mu\right) - \frac{1.0}{K^{3} h^{3}} \left(0.5 \lambda + 1.0 \mu\right) \left(0.5 K h - 1.0\right) \left(0.5 K h + 1.0\right)^{2} \log{\left (- 0.5 K h + 1.0 \right )} + \frac{1.0}{K^{3} h^{3}} \left(0.5 \lambda + 1.0 \mu\right) \left(0.5 K h - 1.0\right) \left(0.5 K h + 1.0\right)^{2} \log{\left (0.5 K h + 1.0 \right )} & 0 & - 1.0 \lambda - 0.666666666666667 \mu + \frac{1.0}{K^{2} h^{2}} \left(1.0 K^{2} h^{2} \lambda + 2.0 K^{2} h^{2} \mu + 2.0 K h \lambda + 4.0 K h \mu - 4.0 \lambda - 8.0 \mu\right) - \frac{1.0}{K^{3} h^{3}} \left(0.5 \lambda + 1.0 \mu\right) \left(0.5 K h - 1.0\right)^{2} \left(0.5 K h + 1.0\right) \log{\left (- 0.5 K h + 1.0 \right )} + \frac{1.0}{K^{3} h^{3}} \left(0.5 \lambda + 1.0 \mu\right) \left(0.5 K h - 1.0\right)^{2} \left(0.5 K h + 1.0\right) \log{\left (0.5 K h + 1.0 \right )} & 0 & - \frac{16 \lambda + 32 \mu}{12 K h} + \frac{1}{K^{3} h^{3}} \left(8 K^{2} h^{2} \lambda + 16 K^{2} h^{2} \mu - 16 \lambda - 32 \mu\right) - \frac{\left(K h - 2\right)^{2}}{K^{4} h^{4}} \left(\lambda + 2 \mu\right) \left(K h + 2\right)^{2} \log{\left (- \frac{K h}{2} + 1 \right )} + \frac{\left(K h - 2\right)^{2}}{K^{4} h^{4}} \left(\lambda + 2 \mu\right) \left(K h + 2\right)^{2} \log{\left (\frac{K h}{2} + 1 \right )} & 2.0 A K \lambda + 2.0 A K \mu - \frac{1.0 A}{K^{2} h^{3}} \left(0.5 \lambda + 1.0 \mu\right) \left(0.5 K h - 1.0\right) \left(0.5 K h + 1.0\right)^{2} \log{\left (- 0.5 K h + 1.0 \right )} + \frac{1.0 A}{K^{2} h^{3}} \left(0.5 \lambda + 1.0 \mu\right) \left(0.5 K h - 1.0\right) \left(0.5 K h + 1.0\right)^{2} \log{\left (0.5 K h + 1.0 \right )} - \frac{1.0}{K h^{2}} \left(2.0 A K^{2} h^{2} \lambda + 2.0 A K^{2} h^{2} \mu - 2.0 A K h \lambda - 4.0 A K h \mu - 4.0 A \lambda - 8.0 A \mu\right) & 0 & - 2.0 A K \lambda - 2.0 A K \mu - \frac{1.0 A}{K^{2} h^{3}} \left(0.5 \lambda + 1.0 \mu\right) \left(0.5 K h - 1.0\right)^{2} \left(0.5 K h + 1.0\right) \log{\left (- 0.5 K h + 1.0 \right )} + \frac{1.0 A}{K^{2} h^{3}} \left(0.5 \lambda + 1.0 \mu\right) \left(0.5 K h - 1.0\right)^{2} \left(0.5 K h + 1.0\right) \log{\left (0.5 K h + 1.0 \right )} + \frac{1.0}{K h^{2}} \left(2.0 A K^{2} h^{2} \lambda + 2.0 A K^{2} h^{2} \mu + 2.0 A K h \lambda + 4.0 A K h \mu - 4.0 A \lambda - 8.0 A \mu\right) & 0 & - \frac{A}{K^{3} h^{4}} \left(\lambda + 2 \mu\right) \left(K h - 2\right)^{2} \left(K h + 2\right)^{2} \log{\left (- \frac{K h}{2} + 1 \right )} + \frac{A}{K^{3} h^{4}} \left(\lambda + 2 \mu\right) \left(K h - 2\right)^{2} \left(K h + 2\right)^{2} \log{\left (\frac{K h}{2} + 1 \right )} + \frac{1}{4 h} \left(16 A \lambda + 32 A \mu\right) + \frac{1}{K^{2} h^{3}} \left(8 A K^{2} h^{2} \lambda + 16 A K^{2} h^{2} \mu - 16 A \lambda - 32 A \mu\right) & 0\\- \frac{1.0}{K^{2} h^{2}} \left(2.0 K h \mu + 4.0 \mu\right) + \frac{1.0 \mu}{K^{3} h^{3}} \left(0.5 K h - 1.0\right) \left(0.5 K h + 1.0\right)^{2} \log{\left (- 0.5 K h + 1.0 \right )} - \frac{1.0 \mu}{K^{3} h^{3}} \left(0.5 K h - 1.0\right) \left(0.5 K h + 1.0\right)^{2} \log{\left (0.5 K h + 1.0 \right )} & 0 & - \frac{1.0}{K^{2} h^{2}} \left(2.0 K h \mu - 4.0 \mu\right) + \frac{1.0 \mu}{K^{3} h^{3}} \left(0.5 K h - 1.0\right)^{2} \left(0.5 K h + 1.0\right) \log{\left (- 0.5 K h + 1.0 \right )} - \frac{1.0 \mu}{K^{3} h^{3}} \left(0.5 K h - 1.0\right)^{2} \left(0.5 K h + 1.0\right) \log{\left (0.5 K h + 1.0 \right )} & 0 & \frac{4 \mu}{3 K h} - \frac{1}{K^{3} h^{3}} \left(8 K^{2} h^{2} \mu - 16 \mu\right) + \frac{\mu}{K^{4} h^{4}} \left(K h - 2\right)^{2} \left(K h + 2\right)^{2} \log{\left (- \frac{K h}{2} + 1 \right )} - \frac{\mu}{K^{4} h^{4}} \left(K h - 2\right)^{2} \left(K h + 2\right)^{2} \log{\left (\frac{K h}{2} + 1 \right )} & 0 & 0 & \frac{0.333333333333333 \mu}{A K} - \frac{1.0}{A K^{3} h^{2}} \left(1.0 K^{2} h^{2} \mu - 2.0 K h \mu - 4.0 \mu\right) - \frac{1.0 \mu}{A K^{4} h^{3}} \left(0.5 K h - 1.0\right) \left(0.5 K h + 1.0\right)^{2} \log{\left (- 0.5 K h + 1.0 \right )} + \frac{1.0 \mu}{A K^{4} h^{3}} \left(0.5 K h - 1.0\right) \left(0.5 K h + 1.0\right)^{2} \log{\left (0.5 K h + 1.0 \right )} & 0 & - \frac{0.333333333333333 \mu}{A K} + \frac{1.0}{A K^{3} h^{2}} \left(1.0 K^{2} h^{2} \mu + 2.0 K h \mu - 4.0 \mu\right) - \frac{1.0 \mu}{A K^{4} h^{3}} \left(0.5 K h - 1.0\right)^{2} \left(0.5 K h + 1.0\right) \log{\left (- 0.5 K h + 1.0 \right )} + \frac{1.0 \mu}{A K^{4} h^{3}} \left(0.5 K h - 1.0\right)^{2} \left(0.5 K h + 1.0\right) \log{\left (0.5 K h + 1.0 \right )} & 0 & - \frac{4 \mu}{3 A K^{2} h} + \frac{1}{A K^{4} h^{3}} \left(8 K^{2} h^{2} \mu - 16 \mu\right) - \frac{\mu \left(K h - 2\right)^{2}}{A K^{5} h^{4}} \left(K h + 2\right)^{2} \log{\left (- \frac{K h}{2} + 1 \right )} + \frac{\mu \left(K h - 2\right)^{2}}{A K^{5} h^{4}} \left(K h + 2\right)^{2} \log{\left (\frac{K h}{2} + 1 \right )}\end{array}\right]$$
















In [12]:



    


M=Matrix([[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]])
M=L.T*M*L*A*(1+alpha3*K)
M



    


















    
Out[12]:







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
















In [13]:



    


M_in = integrate(M,(alpha3, -h/2, h/2))
M_in



    


















    
Out[13]:







$$\left[\begin{array}{cccccccccccc}0.25 A h \rho + 0.0833333333333333 h \left(- 1.0 A K h \rho + 1.0 A \rho\right) & 0 & 0.166666666666667 A h \rho & 0 & 0.05 A K h^{2} \rho + 0.5 A h \rho + 0.0833333333333333 h \left(- 1.0 A K h \rho - 2.0 A \rho\right) & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0.166666666666667 A h \rho & 0 & 0.25 A h \rho + 0.0833333333333333 h \left(1.0 A K h \rho + 1.0 A \rho\right) & 0 & - 0.05 A K h^{2} \rho + 0.5 A h \rho + 0.0833333333333333 h \left(1.0 A K h \rho - 2.0 A \rho\right) & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0.05 A K h^{2} \rho + 0.5 A h \rho + 0.0833333333333333 h \left(- 1.0 A K h \rho - 2.0 A \rho\right) & 0 & - 0.05 A K h^{2} \rho + 0.5 A h \rho + 0.0833333333333333 h \left(1.0 A K h \rho - 2.0 A \rho\right) & 0 & \frac{8 A}{15} h \rho & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0.25 A h \rho + 0.0833333333333333 h \left(- 1.0 A K h \rho + 1.0 A \rho\right) & 0 & 0.166666666666667 A h \rho & 0 & 0.05 A K h^{2} \rho + 0.5 A h \rho + 0.0833333333333333 h \left(- 1.0 A K h \rho - 2.0 A \rho\right) & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0.166666666666667 A h \rho & 0 & 0.25 A h \rho + 0.0833333333333333 h \left(1.0 A K h \rho + 1.0 A \rho\right) & 0 & - 0.05 A K h^{2} \rho + 0.5 A h \rho + 0.0833333333333333 h \left(1.0 A K h \rho - 2.0 A \rho\right) & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0.05 A K h^{2} \rho + 0.5 A h \rho + 0.0833333333333333 h \left(- 1.0 A K h \rho - 2.0 A \rho\right) & 0 & - 0.05 A K h^{2} \rho + 0.5 A h \rho + 0.0833333333333333 h \left(1.0 A K h \rho - 2.0 A \rho\right) & 0 & \frac{8 A}{15} h \rho & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\end{array}\right]$$

Cartesian coordinates





In [19]:



    


import fem.geometry as g
import fem.model as m
import fem.material as mat
import fem.shellsquared.shellsolver as s
import fem.shellsquared.mesh1D as me
import plot

stiffness_matrix_func = lambdify([A, K, mu, la, h], S_in, "numpy")
mass_matrix_func = lambdify([A, K, rho, h], M_in, "numpy")


def stiffness_matrix(material, geometry, x1, x2, x3):
    A,K = geometry.get_A_and_K(x1,x2,x3)
    return stiffness_matrix_func(A, K, material.mu(), material.lam(), thickness)

def mass_matrix(material, geometry, x1, x2, x3):
    A,K = geometry.get_A_and_K(x1,x2,x3)
    return mass_matrix_func(A, K, material.rho, thickness)



def generate_layers(thickness, layers_count, material):
    layer_top = thickness / 2
    layer_thickness = thickness / layers_count
    layers = set()
    for i in range(layers_count):
        layer = m.Layer(layer_top - layer_thickness, layer_top, material, i)
        layers.add(layer)
        layer_top -= layer_thickness
    return layers


def solve(geometry, thickness, linear, N_width):
    layers_count = 1
    layers = generate_layers(thickness, layers_count, mat.IsotropicMaterial.steel())
    model = m.Model(geometry, layers, m.Model.FIXED_BOTTOM_LEFT_RIGHT_POINTS)
    mesh = me.Mesh1D.generate(width, layers, N_width, m.Model.FIXED_BOTTOM_LEFT_RIGHT_POINTS)
    lam, vec = s.solve(model, mesh, stiffness_matrix, mass_matrix)
    
    return lam, vec, mesh, geometry



width = 2
curvature = 0.8
thickness = 0.05

corrugation_amplitude = 0.05
corrugation_frequency = 20

geometry = g.General(width, curvature, corrugation_amplitude, corrugation_frequency)

N_width = 100

lam, vec, mesh, geometry = solve(geometry, thickness, False, N_width)

print(lam)
results = s.convert_to_results(lam, vec, mesh, geometry, thickness)

results_index = 0
    
plot.plot_init_and_deformed_geometry_in_cartesian(results[results_index], 0, width, -thickness / 2, thickness / 2, 0, geometry.to_cartesian_coordinates)
to_print = 20
if (len(results) < to_print):
    to_print = len(results)
    
for i in range(to_print):
    print(results[i].rad_per_sec_to_Hz(results[i].freq))



    


















    






[ -3.19394768e+19  -3.19394768e+19  -2.92549885e+19  -2.92549885e+19
  -1.49897253e+19  -1.49897253e+19  -7.82162307e+18  -7.82162307e+18
  -7.16422574e+18  -7.16422574e+18  -3.67074805e+18  -3.67074805e+18
  -3.45934489e+18  -3.45934489e+18  -9.54711720e+17  -9.54711720e+17
  -8.47127549e+17  -8.47127549e+17  -6.11412596e+17  -6.11410641e+17
  -4.06673195e+17  -4.06673195e+17  -2.33729917e+17  -2.33729917e+17
  -1.49667149e+17  -1.49666671e+17  -9.95767780e+16  -9.95767780e+16
  -3.43768468e+16  -3.43768468e+16  -2.47352299e+16  -2.47337056e+16
  -1.86969583e+16  -1.86969583e+16  -8.41472800e+15  -8.41472800e+15
  -7.55092666e+15  -7.55092666e+15  -6.03989167e+15  -6.03952038e+15
  -5.56963198e+15  -5.56963198e+15  -5.32074452e+15  -5.32074235e+15
  -4.59037869e+15  -4.59037869e+15  -2.79136102e+15  -2.79136102e+15
  -2.31680117e+15  -2.31680117e+15  -2.24837710e+15  -2.24837710e+15
  -1.84480149e+15  -1.84480149e+15  -1.80409809e+15  -1.80409809e+15
  -1.36855863e+15  -1.36855863e+15  -1.34086376e+15  -1.33321285e+15
  -1.31669812e+15  -1.31668706e+15  -9.97099222e+14  -9.97099198e+14
  -8.51229465e+14  -8.51229465e+14  -8.25145904e+14  -8.25145904e+14
  -7.31604675e+14  -7.31604675e+14  -6.89008916e+14  -6.89008916e+14
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   9.92776217e+07   1.50987666e+08   1.98382008e+08   2.77084358e+08
   2.96223436e+08   3.44349575e+08   3.77413275e+08   5.72490493e+08
   8.24384103e+08   9.06558800e+08   1.02455931e+09   1.13968011e+09
   1.31688840e+09   1.52663400e+09   1.94057592e+09   2.52906573e+09
   3.08458792e+09   3.20237105e+09   3.56440343e+09   3.61805635e+09
   4.74838135e+09   4.76191018e+09   6.17704301e+09   6.43159203e+09
   9.15312381e+09   9.21184112e+09   1.19684458e+10   1.27483170e+10
   1.37127353e+10   1.38082250e+10   1.58389130e+10   1.63550324e+10
   1.71270946e+10   1.72007926e+10   1.77078406e+10   1.92460968e+10
   2.03636990e+10   2.07281842e+10   2.21994374e+10   2.26727651e+10
   2.29096953e+10   2.34723019e+10   2.56870760e+10   2.72149445e+10
   2.72834538e+10   2.77410427e+10   2.86463709e+10   2.94838776e+10
   3.09109268e+10   3.17549109e+10   3.33269539e+10   3.37195329e+10
   3.45592353e+10   3.48379146e+10   3.80876520e+10   3.81159022e+10
   3.89093903e+10   4.12993633e+10   4.27373152e+10   4.55723247e+10
   4.58808753e+10   4.65667835e+10   4.90038429e+10   4.96294047e+10
   5.07361960e+10   5.09817177e+10   5.11387946e+10   5.17279475e+10
   5.51214713e+10   5.53795834e+10   5.61491905e+10   5.67307153e+10
   5.83329320e+10   6.04357615e+10   6.16374677e+10   6.34349635e+10
   6.64851043e+10   6.73036685e+10   6.92535365e+10   6.96616764e+10
   7.10460925e+10   7.79084523e+10   8.04421031e+10   8.05779942e+10
   8.41465379e+10   8.42326861e+10   8.72816197e+10   8.84904690e+10
   9.22772671e+10   9.59039899e+10   9.80439071e+10   9.92627275e+10
   9.94193313e+10   1.03282037e+11   1.03480792e+11   1.03533202e+11
   1.05739695e+11   1.07830161e+11   1.08837664e+11   1.11594544e+11
   1.11651611e+11   1.15841278e+11   1.19026977e+11   1.19794243e+11
   1.22036358e+11   1.22518161e+11   1.22754458e+11   1.22993339e+11
   1.32470078e+11   1.38047850e+11   1.39828297e+11   1.40526342e+11
   1.42567150e+11   1.46403285e+11   1.48032826e+11   1.48068469e+11
   1.52257728e+11   1.60930986e+11   1.61055992e+11   1.63870497e+11
   1.63971472e+11   1.65493779e+11   1.65508521e+11   1.66367949e+11
   1.67622760e+11   1.67657097e+11   1.70211209e+11   1.70361141e+11
   1.81223918e+11   1.81870808e+11   1.92681236e+11   1.92812620e+11
   2.00252758e+11   2.05973234e+11   2.05995714e+11   2.16845455e+11
   2.17096481e+11   2.25489490e+11   2.25663728e+11   2.26208957e+11
   2.26213457e+11   2.26251199e+11   2.30571694e+11   2.34264138e+11
   2.51685240e+11   2.51685363e+11   2.53785988e+11   2.53980701e+11
   2.79288099e+11   2.79726336e+11   2.90072803e+11   2.90107430e+11
   3.01455410e+11   3.56292671e+11   3.56304153e+11   3.59269344e+11
   3.59303380e+11   3.75448003e+11   3.78200254e+11   4.09421698e+11
   4.09812400e+11   4.18239277e+11   4.18257761e+11   4.26984990e+11
   5.55436590e+11   5.55513881e+11   5.64883383e+11   5.65138246e+11
   5.69544974e+11   5.74346088e+11   5.74658909e+11   5.81778917e+11
   5.84357425e+11   6.85072042e+11   6.85072045e+11   7.58289442e+11
   7.58290509e+11   7.59785455e+11   7.59785455e+11   7.60758160e+11
   8.45549143e+11   8.45549143e+11   8.50220652e+11   8.50247660e+11
   9.55216647e+11   9.55216649e+11   1.63162945e+12   1.63182932e+12
   1.70504750e+12   1.71108805e+12   2.48272683e+12   2.48273563e+12
   2.55465699e+12   2.55465985e+12   3.10033841e+12   3.22869970e+12
   3.23481609e+12   3.28430557e+12   3.28979565e+12   4.21741526e+12
   4.27803869e+12   4.35522332e+12   4.36852841e+12   5.50596065e+12
   5.51340240e+12   5.94055242e+12   6.01358340e+12   6.01475578e+12
   6.34413871e+12   6.34414285e+12   7.54289743e+12   7.54757337e+12
   7.97094837e+12   7.97098965e+12   8.89953857e+12   8.91235564e+12
   9.32457686e+12   9.62527212e+12   9.73524160e+12   9.75675700e+12
   1.06635502e+13   1.13614223e+13   1.13953522e+13   1.13953910e+13
   1.15078213e+13   1.15080340e+13   1.25887897e+13   1.29306516e+13
   1.29306607e+13   1.39306319e+13   1.55384610e+13   1.55384611e+13
   1.61947358e+13   1.61947487e+13   1.66114691e+13   1.66116266e+13
   1.72080850e+13   1.72580831e+13   1.79315835e+13   1.79315835e+13
   2.29973508e+13   2.30273040e+13   2.53395389e+13   2.53395669e+13
   2.56069397e+13   2.56069397e+13   2.82118122e+13   2.82118122e+13
   2.91816089e+13   3.06116643e+13   3.06119455e+13   3.07688612e+13
   3.07688648e+13   3.13148552e+13   3.13148563e+13   3.27880866e+13
   3.54006110e+13   3.54006110e+13   3.84252843e+13   4.07291232e+13
   4.07291232e+13   4.58237721e+13   4.59743353e+13   4.59746384e+13
   4.60581103e+13   5.11264440e+13   5.13886662e+13   5.39195318e+13
   5.39195318e+13   5.89936359e+13   6.07020650e+13   6.07020860e+13
   6.49054508e+13   6.49054508e+13   7.01396615e+13   7.01396618e+13
   7.86221783e+13   7.86221783e+13   8.02817053e+13   8.02955486e+13
   9.20634889e+13   9.20664236e+13   9.49992456e+13   9.49993783e+13
   1.04265405e+14   1.04265405e+14   1.08998380e+14   1.08998381e+14
   1.20375797e+14   1.20375797e+14   1.25306273e+14   1.25306273e+14
   1.26955304e+14   1.26955304e+14   1.41237925e+14   1.45294281e+14
   1.45295209e+14   1.55321064e+14   1.60574689e+14   1.60574689e+14
   2.09642441e+14   2.09700182e+14   2.23531022e+14   2.23531022e+14
   2.59307656e+14   2.59312998e+14   2.60280808e+14   2.60280809e+14
   2.93463646e+14   2.93463646e+14   3.82792142e+14   3.82792142e+14
   3.93303115e+14   3.93303115e+14   4.14610899e+14   4.14610899e+14
   4.64403404e+14   4.64403404e+14   4.90869098e+14   4.90869098e+14
   4.95450962e+14   4.95450962e+14   5.84555505e+14   5.84555505e+14
   8.49848151e+14   8.52348491e+14   1.00285837e+15   1.00285842e+15
   1.05957758e+15   1.05957758e+15   1.18708027e+15   1.18708027e+15
   1.46414844e+15   1.46414844e+15   1.54811232e+15   1.54811232e+15
   1.59126899e+15   1.59126899e+15   1.98665909e+15   1.98665909e+15
   3.84949401e+15   3.84949401e+15   4.03324911e+15   4.03324954e+15
   4.30182457e+15   4.30182457e+15   5.21996432e+15   5.22025091e+15
   5.98898910e+15   5.98898910e+15   7.33817694e+15   7.33817694e+15
   1.56509309e+16   1.56509309e+16   2.13725538e+16   2.13737225e+16
   2.99505800e+16   2.99505800e+16   9.42885512e+16   9.42885512e+16
   1.38354201e+17   1.38354637e+17   2.19620066e+17   2.19620066e+17
   3.84957892e+17   3.84957892e+17   5.64608364e+17   5.64610145e+17
   8.07681030e+17   8.07681030e+17   8.96476985e+17   8.96476985e+17
   3.29779901e+18   3.29779901e+18   3.67181509e+18   3.67181509e+18
   7.30283822e+18   7.30283822e+18   7.56264223e+18   7.56264223e+18
   1.49923790e+19   1.49923790e+19   2.98187885e+19   2.98187885e+19
   3.08796342e+19   3.08796342e+19]










    






..\fem\shellsquared\shellsolver.py:55: RuntimeWarning: invalid value encountered in sqrt
  freq = np.sqrt(eigenvalues[i])










    

















    






nan
nan
nan
nan
nan
nan
nan
nan
nan
nan
nan
nan
nan
nan
nan
nan
nan
nan
nan
nan

Content source: tarashor/vibrations

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