volume_integral_N_8_1element.sagews

%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 / 0.6^2)
u
c_x = 1.

e^(-2.77777777777778*xi^2)

F_x = c_x * u
F_x(-0.871740148510000)

print((1. * diff(L_xi_0, xi) * L_eta_0 * F_x)(xi=0.5, eta=0.5))

0.121126584269306 -0.00141390530006489

print integrate(integrate(1. * diff(L_xi_0, xi) * L_eta_0 * F_x, xi, -1, 1), eta, -1, 1).n()
print integrate(integrate(1. * diff(L_xi_0, xi) * L_eta_1 * F_x, xi, -1, 1), eta, -1, 1).n()
print integrate(integrate(1. * diff(L_xi_0, xi) * L_eta_2 * F_x, xi, -1, 1), eta, -1, 1).n()
print integrate(integrate(1. * diff(L_xi_0, xi) * L_eta_3 * F_x, xi, -1, 1), eta, -1, 1).n()
print integrate(integrate(1. * diff(L_xi_0, xi) * L_eta_4 * F_x, xi, -1, 1), eta, -1, 1).n()
print integrate(integrate(1. * diff(L_xi_0, xi) * L_eta_5 * F_x, xi, -1, 1), eta, -1, 1).n()
print integrate(integrate(1. * diff(L_xi_0, xi) * L_eta_6 * F_x, xi, -1, 1), eta, -1, 1).n()
print integrate(integrate(1. * diff(L_xi_0, xi) * L_eta_7 * F_x, xi, -1, 1), eta, -1, 1).n()

print integrate(integrate(1. * diff(L_xi_1, xi) * L_eta_0 * F_x, xi, -1, 1), eta, -1, 1).n()
print integrate(integrate(1. * diff(L_xi_1, xi) * L_eta_1 * F_x, xi, -1, 1), eta, -1, 1).n()
print integrate(integrate(1. * diff(L_xi_1, xi) * L_eta_2 * F_x, xi, -1, 1), eta, -1, 1).n()
print integrate(integrate(1. * diff(L_xi_1, xi) * L_eta_3 * F_x, xi, -1, 1), eta, -1, 1).n()
print integrate(integrate(1. * diff(L_xi_1, xi) * L_eta_4 * F_x, xi, -1, 1), eta, -1, 1).n()
print integrate(integrate(1. * diff(L_xi_1, xi) * L_eta_5 * F_x, xi, -1, 1), eta, -1, 1).n()
print integrate(integrate(1. * diff(L_xi_1, xi) * L_eta_6 * F_x, xi, -1, 1), eta, -1, 1).n()
print integrate(integrate(1. * diff(L_xi_1, xi) * L_eta_7 * F_x, xi, -1, 1), eta, -1, 1).n()

print integrate(integrate(1. * diff(L_xi_2, xi) * L_eta_0 * F_x, xi, -1, 1), eta, -1, 1).n()
print integrate(integrate(1. * diff(L_xi_2, xi) * L_eta_1 * F_x, xi, -1, 1), eta, -1, 1).n()
print integrate(integrate(1. * diff(L_xi_2, xi) * L_eta_2 * F_x, xi, -1, 1), eta, -1, 1).n()
print integrate(integrate(1. * diff(L_xi_2, xi) * L_eta_3 * F_x, xi, -1, 1), eta, -1, 1).n()
print integrate(integrate(1. * diff(L_xi_2, xi) * L_eta_4 * F_x, xi, -1, 1), eta, -1, 1).n()
print integrate(integrate(1. * diff(L_xi_2, xi) * L_eta_5 * F_x, xi, -1, 1), eta, -1, 1).n()
print integrate(integrate(1. * diff(L_xi_2, xi) * L_eta_6 * F_x, xi, -1, 1), eta, -1, 1).n()
print integrate(integrate(1. * diff(L_xi_2, xi) * L_eta_7 * F_x, xi, -1, 1), eta, -1, 1).n()

print integrate(integrate(1. * diff(L_xi_3, xi) * L_eta_0 * F_x, xi, -1, 1), eta, -1, 1).n()
print integrate(integrate(1. * diff(L_xi_3, xi) * L_eta_1 * F_x, xi, -1, 1), eta, -1, 1).n()
print integrate(integrate(1. * diff(L_xi_3, xi) * L_eta_2 * F_x, xi, -1, 1), eta, -1, 1).n()
print integrate(integrate(1. * diff(L_xi_3, xi) * L_eta_3 * F_x, xi, -1, 1), eta, -1, 1).n()
print integrate(integrate(1. * diff(L_xi_3, xi) * L_eta_4 * F_x, xi, -1, 1), eta, -1, 1).n()
print integrate(integrate(1. * diff(L_xi_3, xi) * L_eta_5 * F_x, xi, -1, 1), eta, -1, 1).n()
print integrate(integrate(1. * diff(L_xi_3, xi) * L_eta_6 * F_x, xi, -1, 1), eta, -1, 1).n()
print integrate(integrate(1. * diff(L_xi_3, xi) * L_eta_7 * F_x, xi, -1, 1), eta, -1, 1).n()

-0.00274861591012480 -0.0162160597496870 -0.0262532272764426 -0.0317433425476952 -0.0317433425476952 -0.0262532272764426 -0.0162160597496870 -0.00274861591012480 -0.00416938629442680 -0.0245982048713678 -0.0398236238055907 -0.0481516012732554 -0.0481516012732554 -0.0398236238055907 -0.0245982048713678 -0.00416938629442680 -0.0155553552218063 -0.0917722147991800 -0.148575970363694 -0.179646405828473 -0.179646405828473 -0.148575970363694 -0.0917722147991800 -0.0155553552218063 -0.0145886180005939 -0.0860687374658498 -0.139342242256541 -0.168481706296060 -0.168481706296060 -0.139342242256541 -0.0860687374658498 -0.0145886180005939

print integrate(integrate(1. * diff(L_xi_4, xi) * L_eta_0 * F_x, xi, -1, 1), eta, -1, 1).n()
print integrate(integrate(1. * diff(L_xi_4, xi) * L_eta_1 * F_x, xi, -1, 1), eta, -1, 1).n()
print integrate(integrate(1. * diff(L_xi_4, xi) * L_eta_2 * F_x, xi, -1, 1), eta, -1, 1).n()
print integrate(integrate(1. * diff(L_xi_4, xi) * L_eta_3 * F_x, xi, -1, 1), eta, -1, 1).n()
print integrate(integrate(1. * diff(L_xi_4, xi) * L_eta_4 * F_x, xi, -1, 1), eta, -1, 1).n()
print integrate(integrate(1. * diff(L_xi_4, xi) * L_eta_5 * F_x, xi, -1, 1), eta, -1, 1).n()
print integrate(integrate(1. * diff(L_xi_4, xi) * L_eta_6 * F_x, xi, -1, 1), eta, -1, 1).n()
print integrate(integrate(1. * diff(L_xi_4, xi) * L_eta_7 * F_x, xi, -1, 1), eta, -1, 1).n()

print integrate(integrate(1. * diff(L_xi_5, xi) * L_eta_0 * F_x, xi, -1, 1), eta, -1, 1).n()
print integrate(integrate(1. * diff(L_xi_5, xi) * L_eta_1 * F_x, xi, -1, 1), eta, -1, 1).n()
print integrate(integrate(1. * diff(L_xi_5, xi) * L_eta_2 * F_x, xi, -1, 1), eta, -1, 1).n()
print integrate(integrate(1. * diff(L_xi_5, xi) * L_eta_3 * F_x, xi, -1, 1), eta, -1, 1).n()
print integrate(integrate(1. * diff(L_xi_5, xi) * L_eta_4 * F_x, xi, -1, 1), eta, -1, 1).n()
print integrate(integrate(1. * diff(L_xi_5, xi) * L_eta_5 * F_x, xi, -1, 1), eta, -1, 1).n()
print integrate(integrate(1. * diff(L_xi_5, xi) * L_eta_6 * F_x, xi, -1, 1), eta, -1, 1).n()
print integrate(integrate(1. * diff(L_xi_5, xi) * L_eta_7 * F_x, xi, -1, 1), eta, -1, 1).n()


print integrate(integrate(1. * diff(L_xi_6, xi) * L_eta_0 * F_x, xi, -1, 1), eta, -1, 1).n()
print integrate(integrate(1. * diff(L_xi_6, xi) * L_eta_1 * F_x, xi, -1, 1), eta, -1, 1).n()
print integrate(integrate(1. * diff(L_xi_6, xi) * L_eta_2 * F_x, xi, -1, 1), eta, -1, 1).n()
print integrate(integrate(1. * diff(L_xi_6, xi) * L_eta_3 * F_x, xi, -1, 1), eta, -1, 1).n()
print integrate(integrate(1. * diff(L_xi_6, xi) * L_eta_4 * F_x, xi, -1, 1), eta, -1, 1).n()
print integrate(integrate(1. * diff(L_xi_6, xi) * L_eta_5 * F_x, xi, -1, 1), eta, -1, 1).n()
print integrate(integrate(1. * diff(L_xi_6, xi) * L_eta_6 * F_x, xi, -1, 1), eta, -1, 1).n()
print integrate(integrate(1. * diff(L_xi_6, xi) * L_eta_7 * F_x, xi, -1, 1), eta, -1, 1).n()


print integrate(integrate(1. * diff(L_xi_7, xi) * L_eta_0 * F_x, xi, -1, 1), eta, -1, 1).n()
print integrate(integrate(1. * diff(L_xi_7, xi) * L_eta_1 * F_x, xi, -1, 1), eta, -1, 1).n()
print integrate(integrate(1. * diff(L_xi_7, xi) * L_eta_2 * F_x, xi, -1, 1), eta, -1, 1).n()
print integrate(integrate(1. * diff(L_xi_7, xi) * L_eta_3 * F_x, xi, -1, 1), eta, -1, 1).n()
print integrate(integrate(1. * diff(L_xi_7, xi) * L_eta_4 * F_x, xi, -1, 1), eta, -1, 1).n()
print integrate(integrate(1. * diff(L_xi_7, xi) * L_eta_5 * F_x, xi, -1, 1), eta, -1, 1).n()
print integrate(integrate(1. * diff(L_xi_7, xi) * L_eta_6 * F_x, xi, -1, 1), eta, -1, 1).n()
print integrate(integrate(1. * diff(L_xi_7, xi) * L_eta_7 * F_x, xi, -1, 1), eta, -1, 1).n()

0.0145886180005938 0.0860687374658495 0.139342242256540 0.168481706296059 0.168481706296059 0.139342242256540 0.0860687374658495 0.0145886180005938 0.0155553552218063 0.0917722147991798 0.148575970363693 0.179646405828473 0.179646405828473 0.148575970363693 0.0917722147991798 0.0155553552218063 0.00416938629442683 0.0245982048713680 0.0398236238055910 0.0481516012732558 0.0481516012732558 0.0398236238055910 0.0245982048713680 0.00416938629442683 0.00274861591012480 0.0162160597496870 0.0262532272764425 0.0317433425476952 0.0317433425476952 0.0262532272764425 0.0162160597496870 0.00274861591012480

integrate(diff(L_xi_0, xi) * e**(-xi**2/0.6**2), xi, -1, 1).n()

-0.0769612454839493

### Ignore after this line

print integrate(integrate(1. * diff(L_xi_0, xi) *  e**(-xi**2/0.6**2), xi, -1, 1), eta, -1, 1).n()

-0.153922490967899

u = e^(-xi^2 / 0.6^2)
print(numerical_integral(1. * e**(-xi**2/0.6**2), -1, 1)) 

(1.0438808902215202, 1.1589405992901371e-14)

-0.07955470515481 + 0.0769612454839494

-0.00259345967086060

for i in range(1):
    element_limits = [-1 + 0.2 * i, -0.8 + 0.2 * i]
    x_element = sum(element_limits) / 2 + xi * (element_limits[1] - element_limits[0]) / 2
    print(x_element)
    flux = e ** (- (x_element ** 2) / (0.4 ** 2))
    print(numerical_integral(diff(L_xi_0) * flux, -1, 1)[0])

0.100000000000000*xi - 0.900000000000000 -0.00201663487667