topic Hi Ryan, in Intel® oneAPI Math Kernel Library

Electrostatic Boundary Value Problem with 3-D Boundary

Ryan_H_1 — Thu, 06 Apr 2017 19:59:29 GMT

Hello,

Does anyone know if it's possible (and how) to solve a boundary value problem in Cartesian coordinates with a 3-D boundary on one face (let's say the z=0 plane). For example, I want to solve Laplace's equation in the domain 0<x<L, 0<y<L, 0<z<L, with Dirichlet BC's on all but the z=0 plane, and have several (let's say two) cylinders of length L_cyl protruding into the domain that also have Dirichlet BC's on their surfaces. Is that possible in the MKL Poisson Library?

Thank you,

Ryan

Hi Ryan,

Zhen_Z_Intel — Tue, 18 Apr 2017 01:21:34 GMT

Hi Ryan,

Did you mean the f(x,y,L) = G(x,y,z) != f(x,y,0) ? If so, I am afraid it should not be satisfied with Dirichlet boundary condition. While you are using Poisson solver, your boundary must be satisfied with one of conditions.If data on bd_a_z not equal to bd_b_z, you probably have to make sure the Neumann boundary condition, there differential equation along exterior normal direction is same that -df(x,y,z)/da_z =df(x,y,z)/db_z.

Best regards,
Fiona

Hi Fiona,

Ryan_H_1 — Tue, 18 Apr 2017 14:58:36 GMT

Hi Fiona,

Thank you very much for your response. Sorry that my description wasn't very clear.

Basically what I want to do is use the Poisson solver on a problem where one of the sides of the computation volume in Cartesian coordinates isn't a plane. More specifically, I'm trying to solve an electrostatic boundary value problem for a complex set of electrodes. Let's imagine that the computation volume is 0<={x,y,z}<=1 for reference. Essentially, I want to set up standard Dirichlet or Periodic BC's for bd_ax, bd_bx, bd_ay, bd_by, and bd_bz. Then, I want to set up a plane within the computation volume, let's say at z=0.5, that has a constant voltage, V_0, applied to it (Dirichlet condition). However, I want the plane to have a circular hole of radius R in the center of it. From this hole, I want to extend a cylinder down to the z=0 plane, which I want to bias to, say -V_0 (Dirichlet condition on the cylinder also). On the z=0 plane, I'd just use a Periodic BC (I guess...). So, there would be essentially a two part problem: 1.) solving the BVP in the top half of the computational volume within a rectangular prism from 0.5 <=z <=1, and 2.) then solving the BVP within the cylinder. The issue is that I don't know before hand what the potential (u(x,y,z=0.5)), or it's derivatives, are on the z=0.5 plane are equal to within the radius of the cylinder (i.e., for x^2+y^2<=R). Otherwise, I could just solve the two problems independently, and tie the results together in my simulations.

Does that make sense? If you'd like a conceptual illustration, I can provide you with one. A similar system is also described in this article from the guys at UC San Diego http://positrons.ucsd.edu/papers/TCB15.pdf , and a conceptual illustration is shown on page 4.

Thank you again for your response,

Ryan H