Neumann BC (05-bc-neumann)¶

Let us recall the domain and boundary markers from the previous example:

We will solve the same equation

(1) $-\mbox{div}(\lambda \nabla u) - C_{src} = 0$

but now it will be equipped with a Neumann (heat flux) boundary condition

(2) $\lambda \frac{\partial u}{\partial n} = C_{flux}$

on the “Outer” boundary. Here $C_{flux}$ is a constant prescribed heat flux value. Recall that $C_{flux} = 0$ means a perfectly insulated (adiabatic) wall.

On the boundary parts “Bottom”, “Left” and “Inner” we keep the nonconstant Dirichlet condition from example P01/04-bc-dirichlet:

$u(x, y) = Ax + By + C.$

Since the solution on the “Outer” boundary is unknown, the test functions are nonzero there, and thus the weak formulation is

(3) $\int_\Omega \lambda \nabla u \cdot \nabla v \;\mbox{d\bfx} - \int_{\Gamma_{Outer}} \lambda \frac{\partial u}{\partial n}v \;\mbox{dS} - \int_\Omega C_{src} v\;\mbox{d\bfx} = 0.$

Using the Neumann condition (2), we obtain

(4) $\int_\Omega \lambda \nabla u \cdot \nabla v \;\mbox{d\bfx} - \int_{\Gamma_{Outer}} C_{flux} v \;\mbox{dS} - \int_\Omega C_{src} v\;\mbox{d\bfx} = 0.$

This weak formulation contains all integrals from the example 04-bc-dirichlet plus one new surface integral

$- \int_{\Gamma_{Outer}} C_{flux} v \;\mbox{dS}$

Using a default surface vector form¶

This integral can be added using the default H1-form DefaultVectorFormSurf. The header of the custom form CustomWeakFormPoissonNeumann is defined in definitions.h:

class CustomWeakFormPoissonNeumann : public WeakForm<double>
{
public:
  CustomWeakFormPoissonNeumann(std::string mat_al, Hermes1DFunction<double>* lambda_al,
                               std::string mat_cu, Hermes1DFunction<double>* lambda_cu,
                               Hermes2DFunction<double>* vol_src_term, std::string bdy_heat_flux,
                               Hermes2DFunction<double>* surf_src_term);
};

and its constructor in definitions.cpp:

CustomWeakFormPoissonNeumann::CustomWeakFormPoissonNeumann(std::string mat_al, Hermes1DFunction<double>* lambda_al,
                                                           std::string mat_cu, Hermes1DFunction<double>* lambda_cu,
                                                           Hermes2DFunction<double>* vol_src_term, std::string bdy_heat_flux,
                                                           Hermes2DFunction<double>* surf_src_term) : WeakForm<double>(1)
{
  // Jacobian forms - volumetric.
  add_matrix_form(new DefaultJacobianDiffusion<double>(0, 0, mat_al, lambda_al));
  add_matrix_form(new DefaultJacobianDiffusion<double>(0, 0, mat_cu, lambda_cu));

  // Residual forms - volumetric.
  add_vector_form(new DefaultResidualDiffusion<double>(0, mat_al, lambda_al));
  add_vector_form(new DefaultResidualDiffusion<double>(0, mat_cu, lambda_cu));
  add_vector_form(new DefaultVectorFormVol<double>(0, HERMES_ANY, vol_src_term));

  // Residual forms - surface.
  add_vector_form_surf(new DefaultVectorFormSurf<double>(0, bdy_heat_flux, surf_src_term));
};

Sample results¶

The output for the parameters $C_{src} = 3000$ , $\lambda_{Al} = 236$ , $\lambda_{Cu} = 386$ , $A = 1$ , $B = 1$ , $C = 20$ and $C_{flux} = 0$ is shown below: