FEATool has been designed to be able to perform Octave and Matlab multiphysics simulations in all spatial dimensions (1D, 2D, and 3D). However, running full 3D simulations often requires a lot of computational resources, in the form of memory and simulation time, and it is therefore desirable to find simplifications to reduce simulations to two or even one dimension if possible.
Problems which feature cylindrical and rotationally symmetric geometries and solutions can be reduced to 2D through an axisymmetric coordinate transformation (sometimes also referred to as 2.5D). A symmetry axis, usually r=0, is taken as reference around which the coordinates, gradient, and divergence operators are transformed. In this way the governing equations will be reduced to two dimensions while having a rotationally symmetric three dimensional solution.
For example, a typical 2D convection and diffusion equation for the unknown c can be rewritten
c' - D*(cx_x + cy_y) + (u*cx_t + v*cy_t) = R
r*c' - r*D*(cr_r + cz_z) + r*(u*cr_t + v*cz_t) = r*R
D is the diffusion coefficient,
v the convection velocities, and
R reaction coefficient. In this case it is sufficient to simply replace
z, and multiply the whole equation with
r. Note that the underscore
t for example in
cx_t indicates that the coefficient is multiplied with a test function and treated implicitly in the finite element context (the time derivative is indicated with a single quote as in the first term
More complex equations are treated analogously, for example the axisymmetric Navier-Stokes equations in the FEATool syntax reduce to
-r*miu_ns*(2*ur_r + uz_z + vr_z) + r*rho_ns*(u*ur_t + v*uz_t) + r*p_r + 2*miu_ns*u/r - p = 0 -r*miu_ns*( vr_r + uz_r + 2*vz_z) + r*rho_ns*(u*vr_t + v*vz_t) + r*p_z = 0 u_t + r*ur_t + r*vz_t = 0
In addition to modifying the equations an appropriate boundary condition for the symmetry boundary must be chosen such as a symmetry Neumann condition or slip condition for the Navier-Stokes equations.
To perform axisymmetric modeling one can either use the custom equation physics mode to define PDEs from scratch, or the edit equation GUI feature by clicking on the edit eqn button in the Equation Settings dialog box. This will show the full equations which can be edited directly as shown in the image below.
Moreover, as usual it is also possible to perform Matlab script modeling on the command line. A tutorial example modeling axisymmetric fluid flow can be found in the FEATool documentation
In addition, three other axisymmetric model m-script examples can be found in the examples directory of the FEATool distribution.