Axisymmetric Problems (09-axisym)¶

Model problem¶

We solve stationary heat transfer in a hollow cylindrical object shown in the following schematic picture:

The symmetry axis of the object is aligned with the y-axis. The object stands on a hot plate

$u = T_{bottom} \ \ \ \mbox{on}\ \Gamma_{bottom}$

where $\Gamma_{bottom}$ denotes its bottom face. On the rest of the boundary we prescribe a radiation (Newton) condition

$-\lambda \frac{\partial u}{\partial n} = \alpha (u - T_{ext}).$

Here $\lambda$ is the thermal conductivity of the material, $\alpha$ the heat transfer coefficient between the object and the air, and $T_{ext}$ the exterior air temperature.

Using default weak forms in axisymmetric mode¶

All default weak forms provided by Hermes can be used for 2D planar problems, 3D problems that are symmetric about the x-axis, and 3D problems that are symmetric about the y-axis. The mode is set via the optional parameter GeomType in the constructor of the default form. Thus the user can think in terms of the planar formulation of the problem, without having to bother with the axisymmetric forms of the differential operators.

The planar form of the stationary heat transfer equation is

$-\mbox{div}(\lambda \, \mbox{grad}\, u) = 0.$

Hermes provides DefaultJacobianDiffusion and DefaultResidualDiffusion for the diffusion operator

$-\mbox{div}(\lambda \, \mbox{grad}\, u).$

For their headers we refer to the Doxygen documentation.

Custom weak forms¶

The weak formulation is custom because of the Newton boundary condition (see definitions.h and definitions.cpp).

Sample results¶

Results for the values $T_{bottom} = 100$ , $T_{ext} = 0$ , $\lambda = 386$ and $\alpha = 20$ are shown below. We start with the stationary temperature distribution: