Laplace Equation, Non-Concentric Spheres

[gparker267]

1. Homework Statement

General solution for eccentric spheres, smaller sphere (radius, b) completely embedded within larger sphere of radius c. The centers of both spheres lie on z-axis, distance a, apart (note: c>b+a). Problem is symmetric, so consider θ=[0,∏], r=[0,c]. The inner sphere is grounded while the outer sphere is held at a potential f(c,θ).


2. Homework Equations
General solution (see Morse & Feshbach, 1953) is given in coordinates of larger sphere as:

V(r,θ)=Ʃ1((r/a)^s×Ʃ2(A_n×(a/b)^n×[(-1)^(n-s)n!/s!(n-s)!])-(a/r)^(s+1)×Ʃ3(A_n×(b/a)^(n+1)×[s!/n!(s-n)!]))P_s(cosθ)

V(c,θ)=f(c,θ), θ=[0,∏], r=[0,c]

Limits of summations
Ʃ1, s=0 to s=s_max (can use s_max=50)
Ʃ2, n=s to n=n_max (can use n_max=50)
Ʃ3, n=0 to s=s

3. The Attempt at a Solution
I have expanded the boundary function, f(c,θ) as a series of Legendre polynomials and computed the unknown coefficients A_n. For the special case when the offset (a) is zero,the solution behaves well (and converges nicely to prescribed boundary conditions, BCs). However, in cases where the offset is greater than zero, the solution diverges in the region where radii r<a. Any ideas on what might be going wrong?

 

[jvicens]

I would suggest the following steps to troubleshoot and improve the solution for this problem:

1. Check for errors in the equations: Double check all equations and make sure they are correctly written and accounted for. It is possible that a small error in one of the equations could lead to the divergent behavior.

2. Test for convergence: Run the solution for different values of s_max and n_max and see if the solution converges to the prescribed boundary conditions as these values increase. If the solution does not converge, it could indicate an issue with the equations or the numerical method being used.

3. Consider different boundary conditions: Instead of using a general function for f(c,θ), try using specific boundary conditions such as a constant potential or a linear potential. This can help identify if the issue lies in the boundary function or the general solution itself.

4. Explore other numerical methods: It is possible that the numerical method being used is not suitable for this particular problem. Consider using other methods such as finite element or finite difference methods to see if they provide a more accurate and stable solution.

5. Consult with other experts: Reach out to other scientists or experts in the field for their insights and suggestions. They may have encountered similar problems and can offer valuable advice on how to improve the solution.

In conclusion, troubleshooting and improving a solution for a complex problem like this requires a systematic approach and a lot of patience. By carefully examining the equations and testing different options, it is possible to find a more accurate and stable solution.