我一直在使用来自“from scipy.optimize import root”的根函数来解决需要两个方程 f(x,y) 和 g(x,y) 的其他问题,到目前为止我到现在还没发现什么阻碍 整个主题是势流,这个特定问题是关于涡流+表面上的稳定速度, 下一个代码是关于查找点 P (Xp, YP) 的坐标,其中速度为零,表面上有涡旋(涡旋强度 = -550),并且该涡旋位于 a 的左侧墙。 U : 稳定速度 cv : 涡流强度 h:涡流与表面之间的距离
import numpy as np
from scipy.optimize import root
from math import pi
cv = -550.0
U = 10.0
h = 18.0
'''
denom1 = (X + h) ** 2 + Y ** 2
denom2 = (X - h) ** 2 + Y ** 2
###########################################
# f(x,y)
###########################################
f_1_a1 = - cv * Y / denom1
f_1_a2 = cv * Y / denom2
# f(x, y)
f_1 = f_1_a1 + f_1_a2
dfx_1 = (- cv * Y) * ((-1) * (2) * (X + h)) / (denom1 ** 2)
dfx_2 = (cv * Y) * ((-1) * (2) * (X - h)) / (denom2 ** 2)
# df_x
d_f_1_x = dfx_1 + dfx_2
dfy_1 = (- cv) / denom1
dfy_2 = (- cv * Y) * (- 1) * (2) * (Y) /(denom1 ** 2)
dfy_3 = (cv) / denom2
dfy_4 = (cv * Y) * (-1) * (2 * Y) /(denom2 ** 2)
# df_y
d_f_1_y = dfy_1 + dfy_2 + dfy_3 + dfy_4
###########################################
# g(x,y)
###########################################
g_a1 = - U
g_a2 = cv * (X + h) / denom1
g_a3 = - cv * (X - h) / denom2
# g(x, y)
f_2 = g_a1 + g_a2 + g_a3
dgx_1 = cv / denom1
dgx_2 = cv * (X + h) * (-1) * (2) * (X + h) / (denom1 ** 2)
dgx_3 = (- cv) / denom2
dgx_4 = (- cv) * (X - h) * (-1) * (2) * (X - h) / denom2
dgx = dgx_1 + dgx_2 + dgx_3 + dgx_3 + dgx_4
# dg_x
d_f_2_x = dgx
dgy_1 = cv * (X + h) * (-1) * (2) * (Y) / (denom1 ** 2)
dgy_2 = (- cv) * (X - h) * (-1) * (2 * Y) / (denom2 ** 2)
dgy = dgy_1 + dgy_2
# dg_y
d_f_2_y = dgy
'''
def Proof(X, Y):
denom1 = (X + h) ** 2 + Y ** 2
denom2 = (X - h) ** 2 + Y ** 2
###########################################
# f(x,y)
###########################################
f_1_a1 = - cv * Y / denom1
f_1_a2 = cv * Y / denom2
# f(x, y)
f_1 = f_1_a1 + f_1_a2
dfx_1 = (- cv * Y) * ((-1) * (2) * (X + h)) / (denom1 ** 2)
dfx_2 = (cv * Y) * ((-1) * (2) * (X - h)) / (denom2 ** 2)
# df_x
d_f_1_x = dfx_1 + dfx_2
dfy_1 = (- cv) / denom1
dfy_2 = (- cv * Y) * (- 1) * (2) * (Y) /(denom1 ** 2)
dfy_3 = (cv) / denom2
dfy_4 = (cv * Y) * (-1) * (2 * Y) /(denom2 ** 2)
# df_y
d_f_1_y = dfy_1 + dfy_2 + dfy_3 + dfy_4
###########################################
# g(x,y)
###########################################
g_a1 = - U
g_a2 = cv * (X + h) / denom1
g_a3 = - cv * (X - h) / denom2
# g(x, y)
f_2 = g_a1 + g_a2 + g_a3
dgx_1 = cv / denom1
dgx_2 = cv * (X + h) * (-1) * (2) * (X + h) / (denom1 ** 2)
dgx_3 = (- cv) / denom2
dgx_4 = (- cv) * (X - h) * (-1) * (2) * (X - h) / denom2
dgx = dgx_1 + dgx_2 + dgx_3 + dgx_3 + dgx_4
# dg_x
d_f_2_x = dgx
dgy_1 = cv * (X + h) * (-1) * (2) * (Y) / (denom1 ** 2)
dgy_2 = (- cv) * (X - h) * (-1) * (2 * Y) / (denom2 ** 2)
dgy = dgy_1 + dgy_2
# dg_y
d_f_2_y = dgy
print "The values of u and v are:"
print f_1
print f_2
print "The derivates are:"
print dgx, dgy
print d_f_1_x, d_f_1_y
def fun_imp1(x):
X = x[0]
Y = x[1]
denom1 = (X + h) ** 2 + Y ** 2
denom2 = (X - h) ** 2 + Y ** 2
###########################################
# f(x,y)
###########################################
f_1_a1 = - cv * Y / denom1
f_1_a2 = cv * Y / denom2
# f(x, y)
f_1 = f_1_a1 + f_1_a2
dfx_1 = (- cv * Y) * ((-1) * (2) * (X + h)) / (denom1 ** 2)
dfx_2 = (cv * Y) * ((-1) * (2) * (X - h)) / (denom2 ** 2)
# df_x
d_f_1_x = dfx_1 + dfx_2
dfy_1 = (- cv) / denom1
dfy_2 = (- cv * Y) * (- 1) * (2) * (Y) /(denom1 ** 2)
dfy_3 = (cv) / denom2
dfy_4 = (cv * Y) * (-1) * (2 * Y) /(denom2 ** 2)
# df_y
d_f_1_y = dfy_1 + dfy_2 + dfy_3 + dfy_4
###########################################
# g(x,y)
###########################################
g_a1 = - U
g_a2 = cv * (X + h) / denom1
g_a3 = - cv * (X - h) / denom2
# g(x, y)
f_2 = g_a1 + g_a2 + g_a3
dgx_1 = cv / denom1
dgx_2 = cv * (X + h) * (-1) * (2) * (X + h) / (denom1 ** 2)
dgx_3 = (- cv) / denom2
dgx_4 = (- cv) * (X - h) * (-1) * (2) * (X - h) / denom2
dgx = dgx_1 + dgx_2 + dgx_3 + dgx_3 + dgx_4
# dg_x
d_f_2_x = dgx
dgy_1 = cv * (X + h) * (-1) * (2) * (Y) / (denom1 ** 2)
dgy_2 = (- cv) * (X - h) * (-1) * (2 * Y) / (denom2 ** 2)
dgy = dgy_1 + dgy_2
# dg_y
d_f_2_y = dgy
a_1 = f_1
a_2 = f_2
b_1 = d_f_1_x
b_2 = d_f_1_y
c_1 = d_f_2_x
c_2 = d_f_2_y
f = [ a_1,
a_2]
df = np.array([[b_1, b_2],
[c_1, c_2]])
return f, df
sol = root(fun_imp1, [ 1, 1], jac = True, method = 'lm')
print "x = ", sol.x
print "x0 =", sol.x[1]
print "y0 =", sol.x[0]
x_1 = sol.x[0]
x_2 = sol.x[1]
Proof(x_1, x_2)
程序发现,只有速度的一个分量为零。 一开始我以为是衍生品的问题,但没发现有什么问题。我的一个 friend 曾经说过,有时当涡流强度太高(例如超过 150)时,它可能会表现出不同的方式。
添加信息:
这是流线图:
使用此代码后:
import numpy as np
import matplotlib.pyplot as plt
vortex_height = 18.0
h = vortex_height
vortex_intensity = -550.0
cv = vortex_intensity
permanent_speed = 10
U1 = permanent_speed
Y, X = np.mgrid[-21:21:100j, -21:21:100j]
U = (- cv * Y) / ((X + h)**2 + (Y ** 2)) + (cv * Y) / ((X - h)**2 + (Y ** 2))
V = - U1 + (cv * (X + h)) / ((X + h)**2 + (Y ** 2)) - (cv * (X - h)) / ((X - h)**2 + (Y ** 2))
speed = np.sqrt(U*U + V*V)
plt.streamplot(X, Y, U, V, color=U, linewidth=2, cmap=plt.cm.autumn)
plt.colorbar()
plt.savefig("stream_plot.png")
plt.show()
我通过该程序得到的结果是:
>>>
x = [ 1.32580109e-01 3.98170636e+02]
x0 = 398.170635755
y0 = 0.132580109151
The values of u and v are:
-8.2830922107e-05
-10.1246349802
The derivates are:
-2.20709329055 0.000624761030349
-0.000624761030349 6.22388943399e-07
>>>
u 和 v 应该在哪里:
u = 0.0
v = 0.0
而不是:
u = -8.2830922107e-05(这个可以接受) v = -10.1246349802(这是绝对错误的)
当我将其更改为“hybr”时
sol = root(fun_imp1, [ 1, 1], jac = True, method = 'hybr')
我明白了:
>>>
C:\Python27\lib\site-packages\scipy\optimize\minpack.py:221: RuntimeWarning: The iteration is not making good progress, as measured by the
improvement from the last ten iterations.
warnings.warn(msg, RuntimeWarning)
x = [ -4.81817071e+02 1.96057929e+06]
x0 = 1960579.2949
y0 = -481.817070593
The values of u and v are:
2.53176901102e-12
-10.0000000052
The derivates are:
-7.14899730857e-05 5.25462578799e-15
-5.25462578799e-15 -3.87401132188e-18
>>>
我曾经得到过类似的东西,但我不太记得了,我认为在其他情况下是因为手工推导函数的错误,在当前的问题中我没有跟踪任何错误方面。
最佳答案
您使用method='lm'
,根据文档,它仅在最小二乘意义上求解方程。使用method="hybr"
,您会得到sol.success == False
。
很可能,您的雅可比行列式不正确,因为使用 jac=False
找到了根。
编辑:你的雅可比矩阵似乎是错误的,至少:
x = np.array([3, 3.])
dx = np.array([1.3, 0.3])
eps = 1e-5
dx = 1e-5 * dx / np.linalg.norm(dx)
df_num = (np.array(fun_imp1(x + dx/2)[0]) - np.array(fun_imp1(x - dx/2)[0])) / eps
df_cmp = fun_imp1(x)[1].dot(dx)/eps
print df_num
print df_cmp
打印
[-1.43834392 -0.69055079]
[ -1.43834392 -1024.60208799]
始终检查雅可比行列式与数值微分非常有用。
关于optimization - 使用 root 优化后,结果不是他们应该的,我们在Stack Overflow上找到一个类似的问题: https://stackoverflow.com/questions/19461602/