CoCalc -- Li_Lj_coeffs.sagews

Project: PM2.5 dynamics and characterization

Path: PM_2_5/2D_wave_equation/worksheets / Li_Lj_coeffs.sagews

Views: ¹⁰²¹

%typeset_mode False

var('xi, eta, x, y')

xi_LGL = [-1.0,           \
          -0.87174014851, \
          -0.591700181433,\
          -0.209299217902,\
          0.209299217902, \
          0.591700181433, \
          0.87174014851,  \
          1.0]

eta_LGL = [-1.0,           \
           -0.87174014851, \
           -0.591700181433,\
           -0.209299217902,\
           0.209299217902, \
           0.591700181433, \
           0.87174014851,  \
           1.0]

(xi, eta, x, y)


L_xi_0 =  ((xi - (xi_LGL[1])) / (xi_LGL[0] - xi_LGL[1]))\
        * ((xi - (xi_LGL[2])) / (xi_LGL[0] - xi_LGL[2]))\
        * ((xi - (xi_LGL[3])) / (xi_LGL[0] - xi_LGL[3]))\
        * ((xi - (xi_LGL[4])) / (xi_LGL[0] - xi_LGL[4]))\
        * ((xi - (xi_LGL[5])) / (xi_LGL[0] - xi_LGL[5]))\
        * ((xi - (xi_LGL[6])) / (xi_LGL[0] - xi_LGL[6]))\
        * ((xi - (xi_LGL[7])) / (xi_LGL[0] - xi_LGL[7]))

L_xi_1 =  ((xi - (xi_LGL[0])) / (xi_LGL[1] - xi_LGL[0]))\
        * ((xi - (xi_LGL[2])) / (xi_LGL[1] - xi_LGL[2]))\
        * ((xi - (xi_LGL[3])) / (xi_LGL[1] - xi_LGL[3]))\
        * ((xi - (xi_LGL[4])) / (xi_LGL[1] - xi_LGL[4]))\
        * ((xi - (xi_LGL[5])) / (xi_LGL[1] - xi_LGL[5]))\
        * ((xi - (xi_LGL[6])) / (xi_LGL[1] - xi_LGL[6]))\
        * ((xi - (xi_LGL[7])) / (xi_LGL[1] - xi_LGL[7]))
        
L_xi_2 =  ((xi - (xi_LGL[0])) / (xi_LGL[2] - xi_LGL[0])) \
        * ((xi - (xi_LGL[1])) / (xi_LGL[2] - xi_LGL[1])) \
        * ((xi - (xi_LGL[3])) / (xi_LGL[2] - xi_LGL[3])) \
        * ((xi - (xi_LGL[4])) / (xi_LGL[2] - xi_LGL[4])) \
        * ((xi - (xi_LGL[5])) / (xi_LGL[2] - xi_LGL[5])) \
        * ((xi - (xi_LGL[6])) / (xi_LGL[2] - xi_LGL[6])) \
        * ((xi - (xi_LGL[7])) / (xi_LGL[2] - xi_LGL[7]))

L_xi_3 =  ((xi - (xi_LGL[0])) / (xi_LGL[3] - xi_LGL[0])) \
        * ((xi - (xi_LGL[1])) / (xi_LGL[3] - xi_LGL[1])) \
        * ((xi - (xi_LGL[2])) / (xi_LGL[3] - xi_LGL[2])) \
        * ((xi - (xi_LGL[4])) / (xi_LGL[3] - xi_LGL[4])) \
        * ((xi - (xi_LGL[5])) / (xi_LGL[3] - xi_LGL[5])) \
        * ((xi - (xi_LGL[6])) / (xi_LGL[3] - xi_LGL[6])) \
        * ((xi - (xi_LGL[7])) / (xi_LGL[3] - xi_LGL[7]))
        
L_xi_4 =  ((xi - (xi_LGL[0])) / (xi_LGL[4] - xi_LGL[0])) \
        * ((xi - (xi_LGL[1])) / (xi_LGL[4] - xi_LGL[1])) \
        * ((xi - (xi_LGL[2])) / (xi_LGL[4] - xi_LGL[2])) \
        * ((xi - (xi_LGL[3])) / (xi_LGL[4] - xi_LGL[3])) \
        * ((xi - (xi_LGL[5])) / (xi_LGL[4] - xi_LGL[5])) \
        * ((xi - (xi_LGL[6])) / (xi_LGL[4] - xi_LGL[6])) \
        * ((xi - (xi_LGL[7])) / (xi_LGL[4] - xi_LGL[7]))


L_xi_5 =  ((xi - (xi_LGL[0])) / (xi_LGL[5] - xi_LGL[0])) \
        * ((xi - (xi_LGL[1])) / (xi_LGL[5] - xi_LGL[1])) \
        * ((xi - (xi_LGL[2])) / (xi_LGL[5] - xi_LGL[2])) \
        * ((xi - (xi_LGL[3])) / (xi_LGL[5] - xi_LGL[3])) \
        * ((xi - (xi_LGL[4])) / (xi_LGL[5] - xi_LGL[4])) \
        * ((xi - (xi_LGL[6])) / (xi_LGL[5] - xi_LGL[6])) \
        * ((xi - (xi_LGL[7])) / (xi_LGL[5] - xi_LGL[7]))
        
L_xi_6 =  ((xi - (xi_LGL[0])) / (xi_LGL[6] - xi_LGL[0])) \
        * ((xi - (xi_LGL[1])) / (xi_LGL[6] - xi_LGL[1])) \
        * ((xi - (xi_LGL[2])) / (xi_LGL[6] - xi_LGL[2])) \
        * ((xi - (xi_LGL[3])) / (xi_LGL[6] - xi_LGL[3])) \
        * ((xi - (xi_LGL[4])) / (xi_LGL[6] - xi_LGL[4])) \
        * ((xi - (xi_LGL[5])) / (xi_LGL[6] - xi_LGL[5])) \
        * ((xi - (xi_LGL[7])) / (xi_LGL[6] - xi_LGL[7]))
        

L_xi_7 =  ((xi - (xi_LGL[0])) / (xi_LGL[7] - xi_LGL[0])) \
        * ((xi - (xi_LGL[1])) / (xi_LGL[7] - xi_LGL[1])) \
        * ((xi - (xi_LGL[2])) / (xi_LGL[7] - xi_LGL[2])) \
        * ((xi - (xi_LGL[3])) / (xi_LGL[7] - xi_LGL[3])) \
        * ((xi - (xi_LGL[4])) / (xi_LGL[7] - xi_LGL[4])) \
        * ((xi - (xi_LGL[5])) / (xi_LGL[7] - xi_LGL[5])) \
        * ((xi - (xi_LGL[6])) / (xi_LGL[7] - xi_LGL[6]))


######################################################################

L_eta_0 = ((eta - (eta_LGL[1])) / (eta_LGL[0] - eta_LGL[1]))\
        * ((eta - (eta_LGL[2])) / (eta_LGL[0] - eta_LGL[2]))\
        * ((eta - (eta_LGL[3])) / (eta_LGL[0] - eta_LGL[3]))\
        * ((eta - (eta_LGL[4])) / (eta_LGL[0] - eta_LGL[4]))\
        * ((eta - (eta_LGL[5])) / (eta_LGL[0] - eta_LGL[5]))\
        * ((eta - (eta_LGL[6])) / (eta_LGL[0] - eta_LGL[6]))\
        * ((eta - (eta_LGL[7])) / (eta_LGL[0] - eta_LGL[7]))

L_eta_1 = ((eta - (eta_LGL[0])) / (eta_LGL[1] - eta_LGL[0]))\
        * ((eta - (eta_LGL[2])) / (eta_LGL[1] - eta_LGL[2]))\
        * ((eta - (eta_LGL[3])) / (eta_LGL[1] - eta_LGL[3]))\
        * ((eta - (eta_LGL[4])) / (eta_LGL[1] - eta_LGL[4]))\
        * ((eta - (eta_LGL[5])) / (eta_LGL[1] - eta_LGL[5]))\
        * ((eta - (eta_LGL[6])) / (eta_LGL[1] - eta_LGL[6]))\
        * ((eta - (eta_LGL[7])) / (eta_LGL[1] - eta_LGL[7]))
        
L_eta_2 = ((eta - (eta_LGL[0])) / (eta_LGL[2] - eta_LGL[0])) \
        * ((eta - (eta_LGL[1])) / (eta_LGL[2] - eta_LGL[1])) \
        * ((eta - (eta_LGL[3])) / (eta_LGL[2] - eta_LGL[3])) \
        * ((eta - (eta_LGL[4])) / (eta_LGL[2] - eta_LGL[4])) \
        * ((eta - (eta_LGL[5])) / (eta_LGL[2] - eta_LGL[5])) \
        * ((eta - (eta_LGL[6])) / (eta_LGL[2] - eta_LGL[6])) \
        * ((eta - (eta_LGL[7])) / (eta_LGL[2] - eta_LGL[7]))

L_eta_3 = ((eta - (eta_LGL[0])) / (eta_LGL[3] - eta_LGL[0])) \
        * ((eta - (eta_LGL[1])) / (eta_LGL[3] - eta_LGL[1])) \
        * ((eta - (eta_LGL[2])) / (eta_LGL[3] - eta_LGL[2])) \
        * ((eta - (eta_LGL[4])) / (eta_LGL[3] - eta_LGL[4])) \
        * ((eta - (eta_LGL[5])) / (eta_LGL[3] - eta_LGL[5])) \
        * ((eta - (eta_LGL[6])) / (eta_LGL[3] - eta_LGL[6])) \
        * ((eta - (eta_LGL[7])) / (eta_LGL[3] - eta_LGL[7]))
        
L_eta_4 = ((eta- (eta_LGL[0])) / (eta_LGL[4] - eta_LGL[0])) \
        * ((eta- (eta_LGL[1])) / (eta_LGL[4] - eta_LGL[1])) \
        * ((eta- (eta_LGL[2])) / (eta_LGL[4] - eta_LGL[2])) \
        * ((eta- (eta_LGL[3])) / (eta_LGL[4] - eta_LGL[3])) \
        * ((eta- (eta_LGL[5])) / (eta_LGL[4] - eta_LGL[5])) \
        * ((eta- (eta_LGL[6])) / (eta_LGL[4] - eta_LGL[6])) \
        * ((eta- (eta_LGL[7])) / (eta_LGL[4] - eta_LGL[7]))


L_eta_5 = ((eta - (eta_LGL[0])) / (eta_LGL[5] - eta_LGL[0])) \
        * ((eta - (eta_LGL[1])) / (eta_LGL[5] - eta_LGL[1])) \
        * ((eta - (eta_LGL[2])) / (eta_LGL[5] - eta_LGL[2])) \
        * ((eta - (eta_LGL[3])) / (eta_LGL[5] - eta_LGL[3])) \
        * ((eta - (eta_LGL[4])) / (eta_LGL[5] - eta_LGL[4])) \
        * ((eta - (eta_LGL[6])) / (eta_LGL[5] - eta_LGL[6])) \
        * ((eta - (eta_LGL[7])) / (eta_LGL[5] - eta_LGL[7]))
        
L_eta_6 = ((eta - (eta_LGL[0])) / (eta_LGL[6] - eta_LGL[0])) \
        * ((eta - (eta_LGL[1])) / (eta_LGL[6] - eta_LGL[1])) \
        * ((eta - (eta_LGL[2])) / (eta_LGL[6] - eta_LGL[2])) \
        * ((eta - (eta_LGL[3])) / (eta_LGL[6] - eta_LGL[3])) \
        * ((eta - (eta_LGL[4])) / (eta_LGL[6] - eta_LGL[4])) \
        * ((eta - (eta_LGL[5])) / (eta_LGL[6] - eta_LGL[5])) \
        * ((eta - (eta_LGL[7])) / (eta_LGL[6] - eta_LGL[7]))
        

L_eta_7 = ((eta - (eta_LGL[0])) / (eta_LGL[7] - eta_LGL[0])) \
        * ((eta - (eta_LGL[1])) / (eta_LGL[7] - eta_LGL[1])) \
        * ((eta - (eta_LGL[2])) / (eta_LGL[7] - eta_LGL[2])) \
        * ((eta - (eta_LGL[3])) / (eta_LGL[7] - eta_LGL[3])) \
        * ((eta - (eta_LGL[4])) / (eta_LGL[7] - eta_LGL[4])) \
        * ((eta - (eta_LGL[5])) / (eta_LGL[7] - eta_LGL[5])) \
        * ((eta - (eta_LGL[6])) / (eta_LGL[7] - eta_LGL[6]))

u = e^(-xi^2 / 0.4^2)
u
c_x = 1.

e^(-6.25000000000000*xi^2)

poly1 = xi^3 + 2 * xi^2 + 3 * xi + 4
poly2 = -2*eta^3 + 4*eta^2 + 7*eta + 9
poly_3 = (poly1*poly2)
poly_3(-1, -1)

16

poly1 = 5*xi^3 + -2*xi^2 + -4.7211* xi + 2
poly2 = -1*eta^3 + 0*eta^2 - 9.1124*eta + 7
expand(poly1*poly2)

-5*eta^3*xi^3 + 2*eta^3*xi^2 + 4.72110000000000*eta^3*xi - 45.5620000000000*eta*xi^3 - 2*eta^3 + 18.2248000000000*eta*xi^2 + 35*xi^3 + 43.0205516400000*eta*xi - 14*xi^2 - 18.2248000000000*eta - 33.0477000000000*xi + 14

expand((L_xi_3)* (L_eta_4))

-129.727857225405*eta^7*xi^7 + 27.1519390573796*eta^7*xi^6 - 27.1519390573797*eta^6*xi^7 + 273.730966722451*eta^7*xi^5 + 5.68287960923199*eta^6*xi^6 + 273.730966722451*eta^5*xi^7 - 57.2916772505673*eta^7*xi^4 + 57.2916772505669*eta^6*xi^5 - 57.2916772505680*eta^5*xi^6 + 57.2916772505673*eta^4*xi^7 - 178.518337439857*eta^7*xi^3 - 11.9911032408375*eta^6*xi^4 - 577.583286622990*eta^5*xi^5 - 11.9911032408381*eta^4*xi^6 - 178.518337439857*eta^3*xi^7 + 37.3637484073274*eta^7*xi^2 - 37.3637484073272*eta^6*xi^3 + 120.887730163458*eta^5*xi^4 - 120.887730163459*eta^4*xi^5 + 37.3637484073272*eta^3*xi^6 - 37.3637484073274*eta^2*xi^7 + 34.5152279428116*eta^7*xi + 7.82020331954072*eta^6*xi^2 + 376.680831166362*eta^5*xi^3 + 25.3017073771593*eta^4*xi^4 + 376.680831166362*eta^3*xi^5 + 7.82020331954070*eta^2*xi^6 + 34.5152279428116*eta*xi^7 - 7.22401021413973*eta^7 + 7.22401021413968*eta^6*xi - 78.8390033617950*eta^5*xi^2 + 78.8390033617947*eta^4*xi^3 - 78.8390033617954*eta^3*xi^4 + 78.8390033617948*eta^2*xi^5 - 7.22401021413972*eta*xi^6 + 7.22401021413973*xi^7 - 1.51197968793551*eta^6 - 72.8285112658236*eta^5*xi - 16.5009417437969*eta^4*xi^2 - 245.658854496594*eta^3*xi^3 - 16.5009417437970*eta^2*xi^4 - 72.8285112658235*eta*xi^5 - 1.51197968793550*xi^6 + 15.2429504489039*eta^5 - 15.2429504489039*eta^4*xi + 51.4162061168383*eta^3*xi^2 - 51.4162061168385*eta^2*xi^3 + 15.2429504489038*eta*xi^4 - 15.2429504489039*xi^5 + 3.19033760747451*eta^4 + 47.4963607700889*eta^3*xi + 10.7613717277423*eta^2*xi^2 + 47.4963607700889*eta*xi^3 + 3.19033760747451*xi^4 - 9.94095116237084*eta^3 + 9.94095116237085*eta^2*xi - 9.94095116237085*eta*xi^2 + 9.94095116237084*xi^3 - 2.08063330348620*eta^2 - 9.18307744707700*eta*xi - 2.08063330348620*xi^2 + 1.92201092760671*eta - 1.92201092760671*xi + 0.402275383947182