How Can the Magnetic Vector Potential be Numerically Solved in the X-Y Plane?

[merro]

hi everybody
i want to solve the wave equation of the magnetic vector potential numerically in x-y plane grid,
curl curl A= µ J
anyone can help me

thanks in advance

 

[Bill_K]

Well first you should use the identity "curl squared = grad div - del squared": ∇ x (∇ x A) = ∇ ∇·A - ∇² A. Then choose a gauge ∇·A = 0. You're left with Laplace's equation, which can be solved numerically.

 

[merro]

thanks, but i have another question , the reusltant equation will be ∇^2 A=-µ J, and A is a vector, not a scalar, this one is not Laplace , is Poisson, or

 

[merro]

because the A is a vector then must be put in links of the grid, not on the nodes, or?

 

[Born2bwire]

thanks, but i have another question , the reusltant equation will be ∇^2 A=-µ J, and A is a vector, not a scalar, this one is not Laplace , is Poisson, or

But J is a vector too. Solve the vector components separately in Cartesian coordinates and you have three Laplacian equations.

 

[merro]

do you mean that
∇^2 Ax=Jx,
∇^2 Ay=Jy,
∇^2 Az=Jz,
and Ax, Ay, Az lies on the grid nodes

 

[Born2bwire]

do you mean that
∇^2 Ax=Jx,
∇^2 Ay=Jy,
∇^2 Az=Jz,
and Ax, Ay, Az lies on the grid nodes

As you wish, though there are a few ways to setup the grid for electromagnetics depending on the exact numerical procedure. For example, a common grid for FDTD, the Yee grid, will offset the electric and magnetic fields from the grid points. That may also be appropriate here if you wish to do a second order finite difference approach.

 

[merro]

please , can you show me some examples to discrization of the magentic field potential in x-y-z plan , if anyone has a paper of chapter of book can help me. and which methods could be stable numerically