A natural application in the framework of shape optimization is the aerodynamic design of an airfoil. An object defined by a domain with boundary immersed in a fluid will experience a net force given by
where is the stress tensor, is the fluid pressure, is the kinematic viscosity, is the strain rate tensor, and is the fluid velocity vector field.
The force exerted by the fluid on the airfoil parallel to the fluid velocity at infinity is referred to as the drag force, whereas the force in perpendicular direction is often named the lift.
We look for the optimal shape in order to minimize the drag, as this force produces energy losses, and at the same time maximize the lift for a fixed airfoil volume. This can be expressed by means of the functional
where and are weighting factors for drag and lift forces, respectively, and with . The state variables are subject to a set of constraints in the fluid domain with boundary , namely the steady Navier-Stokes equations,
and to the volume constraint
The adjoint problem reads as
The directional derivative of the functional is given by
and the cost function descreases by choosing the normal displacement to the controlled boundary as
for a sufficiently small value of .
The remaining Lagrange multiplier associated to the volume constraint is computed in order to ensure the volume conservation,
In summary, the shape optimization iteration is as follows:
- Solve the primal problem.
- Solve the adjoint problem with the previously computed state variables.
- Compute the Lagrange multiplier for the volume constraint.
- Compute the displacement field.
- Update the displacement field of the previous iteration.
- Perform the mesh motion.
The solver must be compiled in the terminal. It is advisable to first clean previous compilations with
and then use
OpenFOAM C++ library must be installed in order to compile the code.
The OpenFOAM distribution provided by the OpenFOAM Foundation was used.
Running a Case
The above shape optimization process has been implemented in the open-source C++ library OpenFOAM. The already existing solver adjointShapeOptimizationFoam is a topological optimization routine that relies on a porosity variable and on the calculation of volume sensitivities to determine which regions of the domain must be blocked to the fluid passage in order to minimize a known functional. We have coded a new solver shapeOptimizationFoam that takes some ideas from the aforementioned one, but which performs a shape optimization iteration instead by computing shape sensitivities.
The solver has been tested in the minimization of the the cost function
The adjoint problem reads as
In order to run the solver move to the case folder shapeOptimizationFoamCase and type in the command line
The solver has been run for three different Reynolds numbers and the cost functional value has been normalized with
where and are the cylinder diameter and thickness, respectively.
It might be needed to use
sed -i -e 's/\r$//' filename
in order to be able to execute
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