问题
This is a very general question as I feel that my errors are resulting from some misunderstanding of how scipy.solve_bvp works. I have a function def that takes an array of 12 numbers and returns a list of the system of differential equations for a given time, with shape (2,6). I will have a one dimensional array of length n for my timesteps and then an array yof input values with shape (12,n). My code aims to take simulate the motion of earth and mars over a 1000 day period subject to boundary conditions; at t=0 positions = rpast (the corresponding velocities are returned by the function find_vel_past()), the positions and velocities at t=1000 are given by rs and vs respectively. My code is at the bottom with the two functions I'm trying to solve above:
from datetime import datetime
import matplotlib.pyplot as plt
%matplotlib inline
import numpy as np
from scipy import integrate
from scipy import signal
G = 6.67408e-11 # m^3 s^-1 kg^-2
AU = 149.597e9 # m
Mearth = 5.9721986e24 # kg
Mmars = 6.41693e23 # kg
Msun = 1.988435e30 # kg
day2sec = 3600 * 24 # seconds in one day
rs = [[-4.8957151e10, -1.4359284e11, 501896.65], # Earth
[-1.1742901e11, 2.1375285e11, 7.3558899e9]] # Mars (units of m)
vs = [[27712., -9730., -0.64148], # Earth
[-20333., -9601., 300.34]] # Mars (units of m/s)
# positions of the planets at (2019/6/2)-1000 days
rspast = [[1.44109e11, -4.45267e10, -509142.], # Earth
[1.11393e11, -1.77611e11, -6.45385e9]] # Mars
def motions(t, y):
rx1,ry1,rz1, rx2,ry2,rz2, vx1,vy1,vz1, vx2,vy2,vz2 = y
drx1 = vx1
dry1 = vy1
drz1 = vz1
drx2 = vx2
dry2 = vy2
drz2 = vz2
GMmars = G*Mmars
GMearth = G*Mearth
GMsun = G*Msun
rx12 = rx1 - rx2
ry12 = ry1 - ry2
rz12 = rz1 - rz2
xyz12 = np.power(np.power(rx12,2) + np.power(ry12,2) + np.power(rz12,2), 1.5)
xyz1 = np.power(np.power(rx1, 2) + np.power(ry1, 2) + np.power(rz1, 2), 1.5)
xyz2 = np.power(np.power(rx2, 2) + np.power(ry2, 2) + np.power(rz2, 2), 1.5)
dvx1 = -GMmars * rx12 / xyz12 - GMsun * rx1 / xyz1
dvy1 = -GMmars * ry12 / xyz12 - GMsun * ry1 / xyz1
dvz1 = -GMmars * rz12 / xyz12 - GMsun * rz1 / xyz1
dvx2 = GMearth * rx12 / xyz12 - GMsun * rx2 / xyz2
dvy2 = GMearth * ry12 / xyz12 - GMsun * ry2 / xyz2
dvz2 = GMearth * rz12 / xyz12 - GMsun * rz2 / xyz2
return np.array([drx1,dry1,drz1, drx2,dry2,drz2,
dvx1,dvy1,dvz1, dvx2,dvy2,dvz2])
def find_vel_past():
daynum=1000
ts=np.linspace(0,-daynum*day2sec,daynum)
angles=np.zeros([daynum,2])
trange =(ts[0],ts[-1])
fi=np.ndarray.flatten(np.array(rs+vs))
sol= integrate.solve_ivp(earth_mars_motion,trange,fi,t_eval=ts, max_step=3*day2sec,dense_output=True)
return(sol.y[0:6][:,-1])
##return an array of six velocities at this time
def estimate_errors_improved():
daynum=1000
##generating np arrays for bouundary conditions
a=np.ndarray.flatten(np.array(find_vel_past()))
rpast=np.ndarray.flatten(np.array(rspast))
acond=np.concatenate([rpast,a])
bcond=np.ndarray.flatten(np.array(rs+vs))
t=np.linspace(0,daynum*day2sec,daynum)
y=np.zeros(([12,daynum]))
y[:,0]=acond
def bc(ya,yb):
x=yb-bcond
return np.array(x)
sol = integrate.solve_bvp(earth_mars_motion1,bc,t,y,verbose=2)
data1=np.transpose(sol.sol(t))
angles=np.zeros(daynum)
for i in range(daynum):
angles[i]=angle_between_planets(np.transpose(sol.sol(t)[:,0]))
x = t/day2sec
plt.plot(x,angles)
plt.show()
estimate_errors_improved()
I think that the reason my code isnt working is due to some error in the shapes of arrays that are being passed. I would be very grateful if someone could tell me where I am going wrong so I can fix things.
The output for sol.sol(t) I'm getting is:
Iteration Max residual Max BC residual Total nodes Nodes added
Singular Jacobian encountered when solving the collocation system on iteration 1.
Maximum relative residual: nan
Maximum boundary residual: 2.14e+11
[[ 1.44109e+11 0.00000e+00 0.00000e+00 ... 0.00000e+00 0.00000e+00
0.00000e+00]
[-4.45267e+10 0.00000e+00 0.00000e+00 ... 0.00000e+00 0.00000e+00
0.00000e+00]
[-5.09142e+05 0.00000e+00 0.00000e+00 ... 0.00000e+00 0.00000e+00
0.00000e+00]
...
[ nan nan nan ... nan nan
nan]
[ nan nan nan ... nan nan
nan]
[ nan nan nan ... nan nan
nan]]
回答1:
a few problems I think. Firstly the only reason for trying to 'run back' to the -1000 day point as far as I can see would be to obtain a good y estimate to pass to solve_bvp.
to do this simply reverse the initial velocities and run a similation to +1000 days. once you have done this flip the resulting sol.y arrays and they should serve as a good estimate for solve_bvp.
Next, you dont actually need vel past, boundary conditions of the initial position and the t=0 velocity will do perfectly.
This brings us to the next problem, your Boundary condition function looks mistaken.
it should look something like this.
\\
def bc(ya, yb):
return np.array([ya[0]-1.44109e11,ya[1] +4.45267e10,ya[2]+509142.,ya[3]-1.11393e11,ya[4]+1.77611e11,ya[5]-6.45385e9,yb[6]-27712.,
yb[7]+9730.,yb[8]+0.64148,yb[9]+20333.,yb[10]+9601.,yb[11]-300.34])
\\
Final note: you will most likley have to increase the number of nodes in the solve_bvp problem to
hope this helps
来源：https://stackoverflow.com/questions/58792981/solving-a-bvp-using-scipy-solve-bvp-where-the-function-returns-an-array