Optimization and Computational Fluid Dynamics - Department of ...

124 Nicolas R. Gauger
The main advantage of the AD method over the above two mentioned
methods is that gradients can be computed with the best possible accuracy.
In forward mode the gradient computation speed is dependent on the number
of design variables which can be expensive if the evaluation trace for the cost
function is long. In reverse mode the gradient computation is independent on
any input and therefore is very efficient when numerous design variables are
needed.
There are mainly two possible implementations of AD methods. The first
is the so-called source to source which means that the primal evaluation trace
is transferred into a differentiated evaluation trace. The second is based on
operator overloading which only yields an executable.
However, the problem in reverse mode for both implementation strategies
is that primal computation informations have to be recomputed and/or stored
to compute backwards again. A fair amount of memory is thus essential for
the chosen approach.
For more information on multiple output and input variables including
vector valued functions, please refer to [13].
5.5 Adjoint Flow Solvers
5.5.1 Continuous Adjoint Flow Solvers
Within the MEGAFLOW project [22], an adjoint solver following the continuous
adjoint formulation has been developed and widely validated for the
block-structured flow solver FLOWer [7, 8]. The adjoint solver, which was
implemented by hand, can deal with the boundary conditions for drag, lift
Table 5.3 Reverse differentiated evaluation trace
1 v−1 = x1 =1.5
2 v0 = x2 =0.5
3 v1 = v−1/v0 =1.5/0.5=3.0
4 v2 =sin(v1)=sin(3.0) = 0.1411
5 v3 = v1 + v2 =3.0+0.1411 = 3.1411
6 y = v3 =3.1411
7 v3 = y =1.0
8 v1 = v3 =1.0
9 v2 = v3 =1.0
10 v1 = v1 + v2 cos(v1)=1.0+1.0cos(3.0) = 0.01
11 v0 = −v1v1/v0 = −0.01 × 3.0/0.5=−0.06
12 v−1 = v1/v0 =0.01/0.5=0.02
13 x2 = v0 = −0.06
14 x1 = v−1 =0.02