[Getdp] Von Neumann misconception?

Lars Rindorf lrf at teknologisk.dk
Tue Aug 9 08:32:42 CEST 2011


Hi John

The short answer is that from you brief description I cannot see what is wrong.

According to Gauss' theorem the flux through a closed surface equals the sources inside the surface. So I guess you cannot fix both charge and flux in your problem.

KR Lars

Fra: John_V [mailto:jvillar.john at gmail.com]
Sendt: 6. august 2011 20:12
Til: Lars Rindorf
Cc: getdp at geuz.org
Emne: Re: [Getdp] Von Neumann misconception?

Thanks Lars. If I understand you, you're telling me that my answer A was the right one. (Eimposed was a very simple function in my case. It was constant.) But in my solution, the field at UpperEndCap is much different from Eimposed. Why is that?

I am suspecting the real answer is B. This suspicion is, however, based only on observation. I don't know whether this makes sense in terms of how the program is supposed to operate. Domain_Ele does not exclude UpperEndCap. That is, the nodes on UpperEndCap (a surface) are also nodes of Domain_Ele, which is the entire mesh. Does that offer a way to explain why the field is not what I expected?

John
On Thu, Aug 4, 2011 at 2:23 AM, Lars Rindorf <lrf at teknologisk.dk<mailto:lrf at teknologisk.dk>> wrote:
Hi John

The Neumann (von Neumann is a different mathematician) boundary condition fixes the 'influx' grad(V) normal to the boundary, and it arises when grad(V) = -E is known and can be replaced by a function, such as Eimposed. A dirichlet boundary condition, e.g. V=0, fixes the magnitude.

Be aware that in the formulation the 'Eimposed' is the normal component of the incoming field/flux and it is thus a scalar not a vector.

KR Lars

Fra: getdp-bounces at ace20.montefiore.ulg.ac.be<mailto:getdp-bounces at ace20.montefiore.ulg.ac.be> [mailto:getdp-bounces at ace20.montefiore.ulg.ac.be<mailto:getdp-bounces at ace20.montefiore.ulg.ac.be>] På vegne af John_V
Sendt: 3. august 2011 14:21
Til: getdp at geuz.org<mailto:getdp at geuz.org>
Emne: [Getdp] Von Neumann misconception?

Consider an electrostatic problem in a cylindrical volume, bottom end cap (surface) constrained to V=0, source terms in some volume elements, and top end cap with a von Neumann constraint implemented as shown in the formulation below. Does the 3rd Galerkin term (the von Neumann term)
a) Require that the electric field through the upper end cap be equal to Eimposed?
b) Require that the electric field through the upper end cap be equal to the sum: field produced by charges in SourceDomain + Eimposed?
c) Something else?

Formulation {
  { Name Electrostatics_v; Type FemEquation;
    Quantity {
      { Name v; Type Local; NameOfSpace Hgrad_v_Ele; }
    }
    Equation {
      Galerkin { [ epsr[] * Dof{d v} , {d v} ]; In Domain_Ele;
                 Jacobian Vol; Integration GradGrad; }
      Galerkin { [ -q[]*chargeUnit/eps0/ElementVol[] , {v} ]; In SourceDomain;
                 Jacobian Vol; Integration GradGrad; }
      Galerkin { [ Eimposed , {v} ]; In UpperEndCap;
                 Jacobian Sur; Integration GradGrad; }
    }
  }
}

The mesh is a cylindrical volume oriented along z. Domain_Ele refers to all volume elements of my mesh. SourceDomain is a set of volume elements in which there are charges. (These are near the lower end cap of the cylindrical mesh.) The 3rd term was meant to impose my von Neumann condition. When I wrote it I thought I was doing (a) above, but the result suggests otherwise.

John

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