[Getdp] Von Neumann misconception?

Lars Rindorf lrf at teknologisk.dk
Thu Aug 4 08:23:56 CEST 2011

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] På vegne af John_V
Sendt: 3. august 2011 14:21
Til: 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.

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