I am trying to solve a Laplace problem with a point grounding.
I am imposing periodic boundary conditions on top and bottom.
I expect one-dimensional which is consistent with the following.
cond1 = AA( 0) == 0;
cond2 = DA(d/2) == 0;
ode = diff(diff(AA,x),x) == -1
Az = dsolve(ode,[cond1,cond2]));
The following is the NGSolve Python script… Please give me an advise.
from netgen.occ import *
from numpy import *
d = 0.005
h = 0.005
xp = [-d/2, d/2, d/2,-d/2]
yp = [-h/2,-h/2, h/2, h/2]
face = MoveTo(xp[0], yp[0]).LineTo(xp[1], yp[1]).LineTo(xp[2], yp[2]).LineTo(xp[3], yp[3]).Close().Face()
face.faces.name = "air"
face.edges.Nearest((-d/2, 0, 0)).name = "left"
face.edges.Nearest(( d/2, 0, 0)).name = "right"
face.edges.Nearest(( 0,-h/2, 0)).name = "top"
face.edges.Nearest(( 0, h/2, 0)).name = "bottom"
face.vertices.Nearest((-d/2, -h/2, 0)).name = "gnd"
face.edges.Min(Y).Identify(face.edges.Max(Y), "periodicy", IdentificationType.PERIODIC)
from ngsolve import *
from ngsolve import Mesh
from ngsolve.webgui import Draw
geo = OCCGeometry(face, dim=2)
mesh = Mesh(geo.GenerateMesh(maxh=h/10, quad_dominated=True) )
Draw(mesh)
fes = H1(mesh, order=2, dirichlet='gnd')
u = fes.TrialFunction()
v = fes.TestFunction()
a = BilinearForm(fes)
a += grad(u)*grad(v)*dx
c = Preconditioner(a, type="bddc")
a.Assemble()
f = LinearForm(fes)
f += v*dx
f.Assemble()
gfAz = GridFunction(fes)
solvers.CG(sol=gfAz.vec, rhs=f.vec, mat=a.mat, pre=c.mat, printrates='', tol=1e-16, maxsteps=10000)
Draw(gfAz, mesh)