At the core of this technique is a system of adjoint equations that holds sensitivity information, which connects local discretization errors to the error in a quantity of interest. This sensitivity information can be obtained by direct solution of the adjoint equations, but this is a non-trivial task. In fact, some would say that it is impossible. In 2001, at a Nasa Ames lecture series, this was also what I heard as a young PhD student, eager to share my new results that contradicted that claim.
Guided by a posteriori estimation of the error in wind load prediction, the adaptive algorithm constructs local error indicators that decide what cells in the mesh must be refined to optimize the mesh.
These are some of the challenges met when to solve the adjoint Navier-Stokes equations:
- The adjoint equations are linearized at a computed approximation of the primal (usual) Navier-Stokes solution, which introduces a linearization error. Since turbulent flow is a chaotic system, this linearization error could potentially grow exponentially.
- In the adjoint equations the direction of time is reversed. Thus the full time series of the primal solution must be computed and stored to solve the adjoint equations. This represents a huge amount of data for a large scale CFD problem.
- A so called discrete adjoint approach (solely based on the discretized form of the adjoint equations) often fails, since the stability of the discrete primal system does not translate to stability of the discrete adjoint system.
- Turbulence models make the adjoint equations much more complicated, by introducing additional variables and sensitivities.
Many years have passed since I completed that PhD thesis, but still today adaptive mesh refinement is often assumed to imply heuristic error indicators, such as solution gradients. My message is that there is also a less known family of adaptive algorithms that are based on a posteriori error estimation, where the mesh optimization directly connects to the target of the simulation (the wind load of a building, for example). The resulting adaptive algorithms have been shown to be highly efficient. But of course, there are still many exciting challenges to face in this area. Coupled problems, design optimization, and the mathematical foundation are but a few examples.
_For an automated cloud service for adaptive algorithms in CFD, register for free here.
_For a review of research in the area, see Encyclopedia of Computational Mechanics - John Wiley & Sons Ltd.
_For an open source implementation, see FEniCS-HPC.