# Implementation of a variational formulation with boundary terms

Dear ONELAB users,

We encounter difficulties in the implementation of a variational formulation with boundary terms in GetDP. We want to solve an eigenmode problem for a cylindrical waveguide. The structure is invariant in θ, thus we can restrict the study to a 2D cell (see the attached png file) with an axisymmetric axis (Oy).

- A quasi-periodic boundary condition is enforced on both vertical walls Γ1 and Γ2 :

(𝛾∈ℝ, the case of an evanescent mode)

- A Dirichlet condition is imposed on the bottom wall Γd,
- A Neumann condition is imposed on the top wall ΓN.

The field perpendicular to the structure is calculated. The **Form1P** space is therefore used.
The variational formulation of the problem is written as:

It is known that in the cylindrical PEC case (invariance following Oy):

When solving the weak formulation in GetDP, the direct implementation of walls terms using : `Galerkin {[ Vector [0,1,0] /\ Dof {d e}, {e}]}`

does not give appropriate results (the term seems to have no contribution in the matrix, it is as if they do not exist). On the other hand, when we replace by in the formulation: `Galerkin {[ -gam [] * Dof {e}, {e}]}`

we get the correct eigenvalue that we were looking for (f_0=2.53 GHz for γ=15) and when we export the matrix, we see the contribution of the additional terms.

A comparison of the plots of and in PostOperation shows that they appear to be identical (when solving with the " " boundary term).

The simple. geo and .pro considered for this simulation are attached.

Is there something wrong in our implementation of the boundary terms ?