Weakly-compressible Navier-Stokes and Stokes elements automatic differentiation

ELEMENT DESCRIPTION:

Current directory contains the required files for the Automatic Differenctiation (AD) of both the _"symbolic_stokes"_ and _"weakly_compressible_navier_stokes"_ elements of the FluidDynamicsApplication. These elements implement a Stokes and a weakly-compressible Navier-Stokes formulation for both 2D and 3D cases.

SYMBOLIC GENERATOR SETTINGS:

• Dimension to compute: This symbolic generator is valid for both 2D and 3D cases. The element has been implemented using a template argument for the problem dimension, so it is advised to set the dim_to_compute flag as "_Both_". In this case the generated .cpp file will contain both 2D and 3D implementations.
• Formulation: As mentioned, the symbolic generator is valid for both Stokes and Navier-Stokes formulations. To switch the formulation set the "_formulation_" variable to either "_Stokes_" or "_NavierStokes_"
• Linearisation settings: "_FullNR_" considers the convective velocity as _"v-vmesh"_, implying that v is taken into account in the differenctiation of the LHS and RHS. On the contrary "_Picard_" option (a.k.a. QuasiNR) defines the convective velocity as "a", so it is considered as a constant in the differenctiation of the LHS and RHS. Note that this option is only meaningful in the Navier-Stokes formulation as there is no convective term in the Stokes case.
• Divide by rho: If divide_by_rho flag is set to "_True_" the mass conservation equation is divided by the density for the sake of having a better conditioned system of equations. It is therefore advised to keep it activated.
• Artificial compressiblity: If set to "_True_", the time derivative of the density is introduced in the mass conservation equation and rearanged to the pressure time derivative by using the simple state equation $$\partial p/\partial\rho = c^{2}$$, being "_c_" the fluid speed of sound. This adds some extra terms to the usual Navier-Stokes equations that are intended to act as a soft artificial compressibility. Such artificial compressibility is controlled by the value of "_c_", meaning that the artificial compressibility terms vanishes as "_c_" tends to infinite. So far this option is only available in the Navier-Stokes formulation. Although the generator is already prepared to include it in the Stokes element, slight modifications would be required in the header and source template files.
• Darcy term: If set to "_True_" a darcy term is included in the momentum conservation equation. Such Darcy term is governed by a resistance constant (1/L^2) retrieved from the elemental database.

INSTRUCTIONS

Run:

python generate_weakly_compressible_navier_stokes_element.py

Then, depending on the selected formulation, a file "_symbolic_stokes.cpp_" or "_weakly_compressible_navier_stokes.cpp_" is automatically generated. Such file muest be copied within the "_custom_elements_" folder of the FluidDynamicsApplication. The corresponding header file ("symbolic_stokes.h" or "weakly_compressible_navier_stokes.h") is already stored in such folder.