Optimization and Computational Fluid Dynamics - Department of ...

126 Nicolas R. Gauger
Spalart-Allmaras-Edwards one-equation, and the Wilcox k–ω two-equation
turbulence models.
For both solvers, FLOWer as well as TAU, first activities are launched for
the automated generation of discrete adjoint solvers by the use of AD tools.
For the FLOWer code, the AD tool TAF [11] is used while ADOL-C [14] is
used for the TAU code.
5.6 Automatic Differentiation Applied to an Entire
Design Chain
For the optimizations presented in this section, we used the following tools
covering the four steps of the design chain:
For the surface deformation, the tool defgeo has been used. This tool has
been implemented in order to compute deformations based on Hicks-Henne
as well as cosine functions.
The grid deformation within our optimization chain is done by a tool
named meshdefo. This tool uses a public domain linear equations solver to
compute the above mentioned coefficients of the interpolation.
DLR flow solver TAUij is used to compute the flow around the deformed
airfoil. TAUij is a quasi 2D version of TAUijk [17], which again is based on
a cell centered developer version of the DLR TAU code [29]. TAUij solves
the quasi 2D Euler equations. For the spatial discretization, the MAPS+
[26] scheme is used. To achieve second order accuracy, gradients are used to
reconstruct the values of variables at the cell faces. A slip wall and a far-field
boundary condition are applied. For time integration, a Runge-Kutta scheme
is used. To accelerate the convergence, local time stepping, explicit residual
smoothing and a multigrid method are used. The code TAUij is written in
C and comprises approximately 6,000 lines of code distributed over several
files.
To compute the difference vectors of the original to the transformed shape
geometry, another program named difgeo had to be implemented.
The chain to compute the cost function value is illustrated in Fig. 5.4.
This entire optimization chain has been differentiated by the use of ADOL-
C, which operates in reverse (adjoint) mode [10, 27]. This differentiated chain
can be written as
∂CD ∂CD ∂m ∂dx ∂x
= · · ·
∂X ∂m ∂dx ∂x ∂X .
Note that the first term on the right side corresponds to the differentiation of
TAUij, the second term to the differentiation of meshdefo, thethirdtermto
the differentiation of difgeo and the last term to the differentiation of defgeo.
Since difgeo computes only the differences dx = x−xs and xs is a static initial
surface, its corresponding factor becomes the unit matrix and therefore