AIAA JOURNAL
Vol. 59, No. 7, July 2021
Regularization for Adjoint-Based Unsteady Aerodynamic
Optimization Using Windowing Techniques
Steffen Schotthöfer,∗ Beckett Y. Zhou,† Tim Albring,‡ and Nicolas R. Gauger§
Technical University of Kaiserslautern, 67663 Kaiserslautern, Germany
Downloaded by SOUTH UNIVERSITY OF SCIENCE AND on December 12, 2021 | http://arc.aiaa.org | DOI: 10.2514/1.J059983
https://doi.org/10.2514/1.J059983
Unsteady aerodynamic shape optimization presents new challenges in terms of sensitivity analysis of timedependent objective functions. In this work, we consider periodic unsteady flows governed by the unsteady
Reynolds-averaged Navier–Stokes (URANS) equations. Hence, the resulting output functions acting as objective
or constraint functions of the optimization are themselves periodic with unknown period length, which may depend on
the design parameter of said optimization. Sensitivity analysis on the time average of a function with these properties
turns out to be difficult. Therefore, we explore methods to regularize the time average of such a function with the
so-called windowing approach. Furthermore, we embed these regularizers into the discrete adjoint solver for the
URANS equations of the multiphysics and optimization software SU2. Finally, we exhibit a comparison study between
the classical nonregularized optimization procedure and the ones enhanced with regularizers of different smoothness,
and we show that the latter result in a more robust optimization.
Nomenclature
CD
Cf
CL
d
Gn
g
Hn
h
J
JM
Jw
L
M
m
n
nd
p
q
R
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
Re
R
=
=
s
=
T
=
t
=
drag coefficient
skin-friction coefficient
lift coefficient
number of spatial dimension of the flow domain
direct (primal) fixed point iteration at time step n
scalar output of a dynamical system
adjoint fixed point iteration at time step n
period normalized scalar output of a dynamical system
period average of a scalar function
finite time average of a scalar function
windowed finite time average of a scalar function
Lagrangian function
finite time span
number of grid points of the discretized flow domain
current physical time step
dimension of design space
current fictitious time step
order of convergence of a windowed time average
residual of the spatially discretized unsteady Reynoldsaveraged Navier–Stokes equations
Reynolds number
extended residual of the unsteady Reynolds-averaged
Navier–Stokes equations discretized in space and time
physical time normalized with regard to the period length
T or a given time frame M
(design dependent) period length of a periodic scalar
output
physical time
u
=
u0
un
u n
w
X ad
Δt
Δτ
σ
τ
=
=
=
=
=
=
=
=
=
spatially discretized and time dependent solution of the
unsteady Reynolds-averaged Navier–Stokes equations
initial condition of the flow
flow solution at physical time step n
adjoint flow solution (Lagrange multiplier) at time step n
windowing function
set of admissible designs
physical time step
fictitious time step
system or design variable
fictitious time
Subscripts
f
tr
=
=
final (for example, final time)
transient
I.
A
Introduction
ERODYNAMIC shape optimization has been a subject of
active research for the last two decades [1–5]. The goal of the
design process usually consists of optimizing one or multiple aerodynamic coefficients such as drag or lift of the object in focus to
improve its flow properties. Since the advent of adjoint based methods [6,7], for which the computational cost is independent of the
number of design variables, it is possible to address many applicable
large-scale problems such as aerodynamic and aerostructural optimizations of complete aircraft configurations.
Challenges of the optimization problem also arise from the nature of
the equations used to model the fluid flow. Traditionally, the underlying
problem is considered to be in a steady state, which is a reasonable
assumption for many flow conditions. However, many fluid flows in
industry are naturally unsteady and turbulent. Such flows occur for
example in aerodynamic and aeroacoustic design of rotorcraft and wind
turbines or active flow control for high-lift devices [1]. In comparison to
steady-state aerodynamic shape optimization, the unsteady counterpart
has only recently attracted more attention [1,3,5,8–10]. This is due to the
fact that large amounts of solution data and computational time are
required to solve the unsteady adjoint equation. Nevertheless, more
research is being carried out in this area because, on the one hand, timeaccurate numerical methods have been further developed and, on the
other, the computing power of high-performance computing centers has
increased.
For flow conditions in which large-scale vortex structures are
formed (which is the case, for example, in the wake of bluff bodies
as an airfoil with a high angle of attack), scale-resolving methods are
required. The reason for this is that turbulence models are not
designed for detached flows [11]. Scale-resolving methods like
Presented as Paper 2020-3130 at the AIAA Aviation 2020 Forum, Virtual
Event, June 15–19, 2020; received 12 July 2020; revision received 24
November 2020; accepted for publication 30 November 2020; published
online 24 February 2021. Copyright © 2020 by The Authors. Published
by the American Institute of Aeronautics and Astronautics, Inc., with permission. All requests for copying and permission to reprint should be
submitted to CCC at www.copyright.com; employ the eISSN 1533-385X to
initiate your request. See also AIAA Rights and Permissions www.aiaa.org/
randp.
*Ph.D. Candidate, Chair for Scientific Computing, Bldg. 34, Paul-EhrlichStrasse; steffen.schotthoefer@kit.edu. Student Member AIAA.
†
Research Scientist, Chair for Scientific Computing, Bldg. 34, PaulEhrlich-Strasse; yuxiang.zhou@scicomp.uni-kl.de. Member AIAA.
‡
Ph.D. Candidate, Chair for Scientific Computing, Bldg. 34, Paul-EhrlichStrasse; tim.albring@scicomp.uni-kl.de.
§
Professor, Chair for Scientific Computing, Bldg. 34, Paul-Ehrlich-Strasse;
nicolas.gauger@scicomp.uni-kl.de. Associate Fellow AIAA.
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SCHOTTHÖFER ET AL.
large-eddy simulation, detached-eddy simulation, and direct numerical simulation resolve some part of the chaotic dynamics of turbulent
fluid flows. Chaotic systems give rise to new challenges for sensitivity analysis, which is one aspect in the solution process of the shape
optimization problem. There are different approaches to cope with
the fact that in chaotic systems, meaningful time-averaged sensitivities cannot be directly computed. On the one hand, finite difference
methods produce wrong results; on the other hand, a forward or
adjoint mode of automatic differentiation is able to differentiate the
solver correctly. However, due to the chaotic nature of the system, the
sensitivities are by orders of magnitude too big to be useful [8,12]. In
the search for efficient algorithms for a meaningful sensitivity analysis, least-squares shadowing approaches were considered [12–14],
for example. Both scale-resolving methods and sensitivity analysis of
chaotic systems are computationally very expensive.
Unsteady Reynolds-averaged Navier–Stokes (URANS) methods
are therefore often used to calculate large-scale unsteady turbulent
flows. In URANS simulations, the increased dissipation induced by
turbulence models partially dampens the chaotic behavior of the
system. In many cases, URANS-governed flows exhibit limit-cycle
oscillations, which are the focus of this work. Here, a meaningful
objective function for the shape optimization is a windowed temporal
average of an aerodynamic coefficient over the length of a period of
the limit cycle. A property of unsteady flows that exhibit limit-cycle
oscillations is that not only the shape of the limit cycles but also the
length of the period depend on the design parameters. Understanding
sensitivity analysis for limit-cycle oscillations in URANS-governed
flows is an important first step to tackle the problem of sensitivity
analysis for chaotic systems.
The goal of this paper is to apply methods of Krakos et al. [15], who
use a windowing smoothing approach originating from the field of
signal processing to regularize the time-averaged output function, in
the context of aerodynamic shape optimization. Furthermore, we aim
to explore the behavior and efficiency of windowed shape optimization
in different unsteady flow contexts to determine use cases that profit
from the windowing approach. Additionally, we present new insights
into the way windowed time-averaged sensitivities behave in systems
with high-Reynolds-number flows whose instantaneous sensitivities
exhibit exponentially increasing amplitudes. We embed the regularized
objective function in the discrete optimization framework of the multiphysics computational fluid dynamics (CFD) suite SU2 [16], where
we use automatic (or algorithmic) differentiation (AD), replacing the
labor-intensive and error-prone manual differentiation of the discretized equations. Another advantage of AD-based adjoints is the fact
that AD leads to accurate to machine-precision results by construction
because they do not incur any roundoff or truncation error. Furthermore, although turbulence models are not analytically differentiable,
they are still algorithmically differentiable. Therefore, the frozen turbulence assumption, which is typically used in many URANS-based
adjoint formulations, is eliminated. The AD-based discrete adjoint
framework inherits the same convergence properties as the primal flow
solver, and thus yields a robust method to compute adjoints.
The remainder of this paper is structured as follows. In Sec. II, the
challenges in sensitivity analysis in the presence of limit-cycle oscillations are discussed, and we review the windowing framework by
Krakos et al. [15] to overcome these challenges. Section III describes
the discrete primal and adjoint flow solver of SU2. Furthermore, the
windowing regularization approach is embedded in the discrete adjoint
solver, and we show convergence properties of the adjusted adjoint
solver. In Sec. IV, numerical results are presented, where we validate
the convergence properties of the windowed time-average objective
functions and we showcase where the traditional approach fails. In
addition, we validate the consistency of the primal (tangent) AD mode
and the adjoint AD mode for computing sensitivities of windowed
time-averaged objective functions. We showcase the advantages of
windowed sensitivity computations in the case of high-dimensional
surface sensitivities compared to the traditional approach. Here, we
display time-averaged surface sensitivities of the NACA0012 airfoil
calculated with different windows and pin down critical zones. Finally,
we perform a series of shape optimization procedures on the
NACA0012 airfoil using different windows and compare the results
with regard to our previous findings. In Sec. V, we present numerical
results for a pitching airfoil to illustrate a test case, where the period
length is not dependent on the design parameters.
To our knowledge, this is the first application of the windowing
approach to unsteady optimization.
II.
Computing Sensitivities of Limit-Cycle Oscillations
The sensitivity computation of a given objective function such as
the time-averaged drag or lift of an airfoil is an essential part of any
optimization attempt. If one employs the URANS equations for the
flow simulation that acts as the optimization constraint, we arrive at a
spatial discretization of the form
d
u Ru 0;
dt
ut 0 u0
t ∈ 0; tf (1)
where Ru is the residual obtained by a finite volume approach for
solving the turbulent unsteady compressible Navier–Stokes equations, possibly with an additional turbulence model. The vector ut
denotes the solution of the URANS equations on each grid point of
the computational domain, and u0 are the freestream conditions that
display the initial values of the flow.
This approach, called the method of lines, can be seen as a coupled
system of ordinary differential equations that govern a dynamical
system. We define the scalar instantaneous output g ∈ R of the
dynamical system, which might be the time-dependent drag CD or
lift coefficient CL with additional dependence on a system parameter
σ ∈ Rnd . In many applications, this dynamical system exhibits periodic behavior after some transient phase, which is assumed to be
exceeded at t ttr . It makes sense to compute the mean value over a
period Tσ as the scalar output of the dynamical system
Jσ 1
Tσ
Z
ttr Tσ
gt; σ dt
(2)
ttr
that later acts as the objective function of an optimal control problem.
In the following paragraphs, we shift the timescale by ttr for the sake
of easier notation.
Sensitivity computation of a limit-cycle oscillator imposes several
challenges, namely, that one often does not know the exact duration
of a period T and that the period length is dependent on the system
parameter σ, which might be an initial condition, a boundary condition, a design variable, or an additional parameter. A common
approach to compute the time average without prior knowledge of
Tσ is to average over a fixed time M
JM σ; M 1
M
Z
M
gt; σ dt
(3)
0
The corresponding sensitivity of J and JM with respect to σ is
Z Tσ
Z Tσ
d
d
1
1
d
gt;σdt Jσ gt;σdt
dσ
dσ Tσ 0
Tσ 0 dσ
(4)
d
1
J σ; M dσ M
M
Z
M
0
d
gt; σ dt
dσ
(5)
Note that we cannot simply interchange differentiation and integration in Eq. (4) due to dependence of T on σ. The hope is to reduce
the error in jJM − Jj, respectively,
d
JM − d J
dσ
dσ as M grows, and so the question at hand is whether
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SCHOTTHÖFER ET AL.
lim JM J
M→∞
and (more importantly)
lim
d
JM M→∞ dσ
d
J
dσ
To answer this question in detail, Krakos et al. [15] introduced a
more general notation of a weighted time average
Downloaded by SOUTH UNIVERSITY OF SCIENCE AND on December 12, 2021 | http://arc.aiaa.org | DOI: 10.2514/1.J059983
J w σ; M 1
M
Z
M
w
0
t
gt; σ dt
M
(6)
that we call a windowed time average. We have that ws ∈ Cl R; R
is a windowing function if it satisfies
Z1
wsds 1
(7)
ws 0; s ∈
= 0; 1;
0
Furthermore, l denotes the differentiability; and we state that l −1 denotes a piecewise continuous function. It is easy to see that the
characteristic function of the open interval 0; 1; i.e., 10;1 is a
feasible C−1 window, which we call a square window. In this case,
the definitions of J w and JM coincide. Since the windowing function
w is independent of σ, the windowed time-averaged sensitivity is
given by
d
1
J σ; M dσ w
M
Z
M
w
0
t d
gt; σ dt
M dσ
(8)
In their work [15], Krakos et al. were able to prove that the
convergence of Jw and d∕dσJw depends on the differentiability
of the windowing function. We recall Krakos et al.’s Theorem 1 for
the sake of completeness. To this end, we define the substitution
st t
;
T
hs; σ gt; σ
(9)
Note that hs; σ has a period length of one.
Theorem 1: The windowed time average Jw and the windowed
time-averaged sensitivity d∕dσJw computed with a window ws ∈
Cl R; R spanning k M∕T periods converge to J with order
jJ w σ; M − Jσj ≤ khk∞ Ok−q d
Jw σ; M − d Jσ ≤ k∂σ hk∞ Ok−q dσ
dσ
1 dT −q−1 T dσ k∂s hk∞ Ok
1
(10)
(11)
where
8
1;
>
>
<
q l 1;
>
>
:
l 2;
l −1;
l ≥ 0; leven;
its transient phase and exhibits periodic behavior. Then, we use a
sufficiently smooth window function and compute the windowed
time average in Eq. (6) over a finite time span M starting at ttr . We
achieve convergence as M → ∞ with a convergence speed given by
the smoothness of the window as described by Theorem 1. Similarly,
we get the algorithm for the long-time window-averaged sensitivity
by first calculating the instantaneous sensitivity until the transient
phase has passed and then computing the windowed time-averaged
sensitivity of the objective function in Eq. (8). To compute the
instantaneous sensitivity, we use the automatic differentiation capabilities of SU2 with both forward and adjoint modes.
(12)
l > 0; lodd:
For f ∈ C∞ R; R, we get an exponential rate of convergence.
As a result, we get that the difference between the actual timeaveraged sensitivity d∕dσJ and the approximation using a squarewindow weighting d∕dσJ M is at most j1∕TdT∕dσjO1. This
implies very slow convergence or none at all. Krakos et al. [15] developed two methods for approximating J, namely, long-time windowing and short-time windowing. We use the long-time windowing
approach, which has the advantage that it does not need a priori
knowledge of the real period length Tσ and directly uses Theorem 1.
The idea of long-time windowing is the following: First, one computes the solution of the dynamical system of Eq. (1) until it exceeds
III.
Optimization Framework
The simulation of the compressible URANS equations in the
high-performance fluid dynamic solver SU2 [4,8,16,17] is generally
done by establishing the method of lines; i.e., one first performs the
spatial discretization of convective and viscous fluxes and arrives at
the coupled system of ordinary differential equations given by Eq. (1).
The spatial discretization is implemented using a finite volume scheme
on a vertex-based median-dual grid with several numerical fluxes
available, e.g., Jameson–Schmidt–Turkel (JST) [18], Roe [19], and
advection upstream splitting method (AUSM) [20]. We solve the
resulting ordinary differential equation (ODE) system with a dual
time-stepping method. Using a second-order backward differentiation
formula (BDF) method for time discretization, we first approximate
the solution u to Eq. (1) by
R un ;un−1 ;un−2 ;σ
3 n
2
1 n−2
u Run − un−1 u 0; n 1;:::;nf
2Δt
Δt
2Δt
(13)
The solution vector of each time step un ∈ Rd2m represents the
d 2 conservative variables of the URANS equations in d spatial
dimensions at m grid points of the computational domain. Note that un
is additionally dependent on the variable(s) of a turbulence model
(e.g., Spalart–Allmaras [21]), which is omitted for the sake of simplicity in the notation of this work. The residual R (and therefore R ) is
via boundary conditions directly and via the solution un indirectly,
dependent on the design of the considered structure σ ∈ Rnd . The
design in this work is a freeform deformation (FFD)–box parametrization of an airfoil using nd control points. Therefore, the extended
residual R is a mapping given by
R :Rd2m × Rd2m × Rd2m × Rnd → Rd2m
(14)
The initial condition for the BDF method is given by the freestream
values of the flowfield. The flow at initial time has no physical meaning and needs a transient phase that is assumed to exceed at time step
ntr , where it exhibits physically meaningful behavior. We further
assume that the system exhibits a limit-cycle oscillation after ntr time
steps. This assumption is reasonable in many cases; however, it should
be stressed that the duration of the transient phase may be dependent
on the design parameter σ, and the value of ntr should be chosen big
enough.
A dual time step τ is used to converge the residual R to zero. This
can be done by employing an implicit Euler method for the (fictitious
time) system of differential equations
∂τ un R un 0
(15)
yielding
unp1 −unp ΔτR unp1 ;un−1 ;un−2 0; p 1;2;3; :::
(16)
Using a linearization of R around unp , given by
unp1 − unp Δτ R unp ∂u R jnp unp1 − unp 0;
p 1;2;3;:::
(17)
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SCHOTTHÖFER ET AL.
the scheme can be written in the form of a fixed-point iteration
Gn :Rd2m × Rd2m × Rd2m × Rnd → Rd2m
(18)
that marches unp to a fixed point un , which can be seen as the steadystate solution of the fictitious time ODE given by Eq. (15). The
iterator for each physical time step n 1; : : : ; nf reads as
unp1
G unp ; un−1 ; un−2 ; σ ;
n
p 1; 2; 3; : : :
(19)
where un−1 and un−2 denote the fixed points of the previous physical
time steps. To set up the discrete optimal control problem, we
approximate the continuous output in Eq. (6) by a midpoint rule
1
t − ttr
gt; σ dt
Jw σ; M w
M ttr
M
nf
X
1
n − ntr
w
≈
gun σ; σ
nf − ntr nn
nf − ntr
Downloaded by SOUTH UNIVERSITY OF SCIENCE AND on December 12, 2021 | http://arc.aiaa.org | DOI: 10.2514/1.J059983
Z
ttr M
(20)
where we abbreviate gt; ut; σ at time tn nΔt by gun ; σ. The
time span to take the average is given by M nf − ntr Δt tf − tr . Note that taking the limit M → ∞ in the long-time windowing approach is equivalent to taking the limit nf → ∞ in Eq. (20). In
the following, we also denote the discrete approximation by J w. We
can now formulate the discrete optimal control problem
σ∈X ad
Jw nf
X
1
n − ntr
w
gun σ; σ
nf − ntr nntr
nf − ntr
subject to un σ Gn un ;un−1 ;un−2 ;σ ;
n 1;:::;nf
∂u n L 0;
(21a)
(21b)
The set of admissible designs is given by Xad ⊂ Rnd , which is open,
bounded, and nonempty.
The idea to solve the optimization problem is to evaluate the
objective function with a flow solution corresponding to an initial
design σ 0 . We then compute a descent direction of the objective
function, which is given by the total derivative of J w with respect
to σ, −d∕dσJ w , that is then used in an update step using, e.g., a
quasi-Newton method or a line search method. The implementation
of the SU2 solver for discrete optimization problems with partial
differential equation (PDE) constraints is done via the adjoint
approach, which we want to adjust with respect to the weighted
objective function Jw . Here, the idea is to avoid the computation of
the computationally expensive terms dun ∕dσ in Eq. (22) and appear
when applying the chain rule to d∕dσJw ; see Refs. [4,6,7]:
nf
X
d 1
dun
∂un Jw
Jw u ;: : :; unf σ;σ ∂σ Jw u1 ;: : :; unf ;σ dσ
dσ
n1
(25)
n 1; : : : ; nf
(26)
Adjoint equations:
∂un L 0;
Design equation:
We therefore introduce the Lagrangian corresponding to the optimal control problem given by Eq. (21). This is a function
(23)
(27)
The state equations are already given by Eq. (21b). The discrete
adjoint equations, which are used to efficiently compute the total
derivative of the objective function Jw with respect to σ, are obtained
by computing the partial derivatives ∂un L for all n 1; : : : ; nf . The
adjoint equations of the KKT system, given by Eq. (26), yield the
expression
u n ∂un Gn T u n ∂un Gn1 T u n1 ∂un Gn2 T u n2
T
1
n − ntr
1fn≥ntr g
w
∂un gun ; σ
nf − ntr
nf − ntr
(28)
for all n nf ; : : : ; 1. The characteristic function 1fn≥ntr g indicates
that the seeding of the objective function is only performed from
time step ntr on. We remark that since the window function w is only
dependent on the current time step n, the adjustment to traditional
adjoint state equations [8] is marginal. This eases the integration
into existing high-performance solvers like SU2. Analogous to the
direct equations, we may write Eq. (28) as a set of fixed-point
iterations H n :
u np1 Hn u np ; u n1 ; u n2 ; σ
∂un Gn T u np ∂un Gn1 T u n1 ∂un Gn2 T u n2
T
1
n − ntr
n
n
1fn≥ntr g
w
∂ gu ; σ
nf − ntr
nf − ntr u
(29)
This adjoint fixed-point iteration has to converge for each physical time step n, starting from final time nf and marching backward
in time like the continuous adjoint equation formulated in Ref. [1].
The adjoint states u n1 and u n2 denote the converged adjoint fixed
points of the previous time steps n 2 and n 1. By the Banach
fixed-point theorem, we have convergence of this fixed-point iteration if it is a contraction. The norm of the derivative of the iterator
H nu with respect to its argument u n can be written as
(22)
L:Rnf d2m × Rnf d2m × Rnd → R
n 1; : : : ; nf
∂σ L 0
tr
min
where u n denotes the adjoint state vector at time step n. We
obtain the following Karush–Kuhn–Tucker (KKT) system under
the assumption of differentiability of L with respect to σ, un , and
u n for all n 1; : : : ; nf , which is given by the construction of the
fixed-point iteration:
State equations
k∂u n H n k k∂un Gn T k k∂un Gn k
(30)
Convergence of the adjoint iterator therefore depends only on the
convergence of the primal iterator and is independent of the chosen
windowing function. We have
k∂un Gn T k < 1
defined as
in a suitable norm if the direct (pseudotime) iteration is near
convergence to a steady-state solution. Finally, the design equation
is given by the partial derivative of the Lagrangian with respect to
the design σ,
Lu1 ; : : : ; unf T ; u 1 ; : : : ; u nf T ; σ
nf
X
1
n − ntr
w
gun ; σ
nf − ntr
nf − ntr nntr
nf
X
n0
u n T Gn un ; un−1 ; un−2 ; σ − un ∂σ L (24)
nf
X
n0
1fn≥ntr g
1
n − ntr
w
∂ Jun u n T ∂σ Gn
nf − ntr σ
nf − ntr
(31)
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Downloaded by SOUTH UNIVERSITY OF SCIENCE AND on December 12, 2021 | http://arc.aiaa.org | DOI: 10.2514/1.J059983
IV.
Periodic Detached Flow Around the NACA0012
Airfoil
In the following, we present numerical results of an aerodynamic
shape optimization performed in the test case of the NACA0012 airfoil.
All computations are done on the Regionales Hochschulrechenzentrum Kaiserslautern (RHRK) high-performance computing center in
Kaiserslautern. The presented methods are published in the opensource CFD suite SU2, version 7.0.1 “Blackbird” [16].
The test case at hand is a subsonic unsteady turbulent flow around
the NACA0012 airfoil. The mesh is a quadrilateral O grid that wraps
around the NACA0012 airfoil. It consists of 27,125 elements, of
which 217 are wall-boundary elements and 217 are far-field boundary elements. We use the URANS solver of the SU2 CFD suite,
where we employ the JST [18] scheme for the Navier–Stokes fluxes,
whereas the turbulent viscosity is calculated using the Spalart–Allmaras turbulence model [21] with a scalar upwind scheme. A FFD–
box parametrization is used to model the airfoil surface with a total of
242 control points, which act as design variables for the sensitivity
computations. In particular, for the comparison between the windows, a single FFD design variable is chosen. The flow configuration
we considered is as follows: 1) Mach number of 0.3, 2) angle of attack
of 17.0 deg, 3) freestream temperature of 293.0 K, and Reynolds
numbers of 103 and 106 .
The simulation is computed with a time step of Δt 0.0005 s for
several final times tf . Naturally, the discrete objective function is
computed with the same time step.
The sensitivity dCD ∕dσ of the drag coefficient CD with respect to
the design variables is computed with the help of the built-in automatic differentiation support of SU2 based on the software package
CoDiPack [22]. We use two modes of AD. The first is the primal
(forward) mode, which represents a tangent vector evaluation. The
computational effort scales with the number of design variables.
The second, more important mode is the adjoint mode of AD,
which is independent of the number of design variables. Here, the
sensitivities are computed using a combination of the Lagrangian
framework in Eqs. (29) and (31) and several adjoint AD techniques.
In each time step, we have to compute the fixed points of these
iterators. All appearing derivatives in these equations (e.g., ∂un Gn )
are computed using the adjoint AD mode. We avoid the memoryinefficient black-box fashion adjoint AD and instead make use of
several advanced AD techniques for the inner iterations, which are
described in detail and discussed with respect to memory and computational costs in Refs. [4,22]. The so-called checkpointing method
is used connect the fixed point of each time step to finally compute the
sensitivity of the unsteady flow. For a detailed discussion of this
method and its computational and memory costs, we refer to
Refs. [3,9]. In Sec. IV.A, the forward AD mode is used; however,
in Sec. IV.B, we show consistency of both approaches in our test case,
and thus the results of Sec. IV.A are valid for the more practical
adjoint mode. The angle of attack of 17 deg results in a detached
primal flow, as can be seen in Fig. 1. In the wake of the airfoil, a vortex
street can be seen that illustrates the oscillating character of the flow,
which is exhibited after the transient phase. We can see the transient
phase in Fig. 2 as well as the periodic behavior of CD in its limit-cycle
oscillation. It should be noted that the sensitivity d∕dσCD exhibits
periodic behavior in the same time frame with the same period length
a) t1 = 1/3T( )
as CD . However, the amplitude of its limit-cycle oscillation changes
in time. Therefore, it is worthwhile to consider first the long-time
behavior of the drag sensitivity with different Reynolds numbers,
which is displayed in Fig. 3.
We compare the aforementioned flow configuration with two
choices of Reynolds numbers: Re 103 and Re 106 . Both flows
are detached and exhibit periodic behavior after some transient phase.
Considering the drag sensitivities, we can see in Figs. 3a and 3b that
the amplitude of d∕dσCD grows linearly in the case of Re 103 .
However, in the case of Re 106 , the growth in the amplitude is
exponential, as can be seen in Figs. 3c and 3d. We experience faster
growth in amplitude for the case of Re 106 since higher Reynolds
numbers result in a flowfield that is far more sensitive to changes in the
flow configurations, which is (in our case) the design σ of the airfoil.
Taking Theorem 1 into account, we expect good convergence
behavior for the case of Re 103 and worse behavior for the case
of Re 106 . This is due to the fact that the upper bound for the
difference
d
Jw σ; M − d Jσ
dσ
dσ
scales with k∂s CD k∞ and
1 dT T dσ k∂σ CD k∞
1
If
d
CD dσ
∞
grows exponentially by the chain rule, k∂s CD k∞ or
1 dT T dσ k∂σ CD k∞
1
must grow exponentially as well, which can affect the convergence
behavior of the windowed time-averaged sensitivity. However, in our
test case, the exponential growth of the amplitude dominates only
after some time (see Figs. 3b and 3d), and so it is still worth considering the windowed sensitivities.
A. Validation of the Windowing Approach
In the following, we validate the embedding of the windowing
approach in the sensitivity analysis that is necessary for aerodynamic
shape optimization. We apply several windows to the time-averaged
lift coefficient CD :
Jw σ; tf − ttr 1
tf − ttr
Z
tf
ttr
w
t − ttr
C t; σ dt
tf − ttr D
(32)
Analogously, we consider the discrete windowed time-averaged
lift coefficient
c) t3 = T( )
b) t2 = 2/3T( )
6
Fig. 1 Velocities of a flow around NACA0012 airfoil with Re 10 and a 17 deg angle of attack at different time points of a period. The period length is
Tσ ≈ 31Δt.
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SCHOTTHÖFER ET AL.
Hann-square window:
2
whsq s 1 − cos2πs2 ;
3
s ∈ 0; 1
(36)
Bump window:
wbmp s 1
−1
;
exp
A
s − s2
s ∈ 0; 1
(37)
where
Z
1
A
exp
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0
Fig. 2 CD σ;nΔt and d∕dσCD σ;nΔt over n with Re 106 ; the
window for the adjoint run spans iterations 500 to 1200.
Jw σ; nf − ntr nf
X
1
n − ntr
w
C nΔt; σ
nf − ntr D
nf − ntr nntr
(33)
We compare several windowing functions, which were suggested
by Krakos et al. [15], with different orders of differentiability,
namely,
Square window:
wsq s 10;1 s;
s ∈ 0; 1
(34)
Hann window:
wh s 1 − cos2πs;
a) Linear scale, Re = 103
c) Linear scale, Re = 106
Fig. 3
s ∈ 0; 1
(35)
−1
ds
s − s2
(38)
The different windowing functions can be seen in Fig. 4. Considering the order of differentiability, we get the following orders of
convergence q for the windowed time average and order qs for the
windowed time-averaged sensitivity:
wsq ∈ C−1 ⇒ q 1 and qs 0
wh ∈ C1 ⇒ q 3 and qs 2
whsq ∈ C3 ⇒ q 5 and qs 4
wbmp ∈ C∞ ⇒ q ∞ and qs ∞, i.e., the exponential rate of
convergence.
It is reasonable to start the windowed time averaging at iteration
ntr 500, where the transient phase of the system has already passed
and the limit-cycle oscillator has been reached. One period of CD
consists of approximately 31 time steps in both configurations.
We start the analysis of our results with the windowed drag
coefficient for both Reynolds cases Re 103 and Re 106 since
the results are similar. Figure 5a (respectively, Fig. 6a) shows
CD σ; nΔt over time step n with the previously described flow
configuration as well as the windowed time averages over the end
time step n. Hence, the time span to average CD is M n − 500Δt.
We can observe that all high-order windows converge within a short
b) Logarithmic scale, Re = 103
d) Logarithmic scale, Re = 106
Long-time behavior of d∕dσCD σ;nΔt over n; higher Reynolds numbers lead to exponential growths in the period amplitude of the sensitivity.
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SCHOTTHÖFER ET AL.
Fig. 4 Square-, Hann-, Hann-square-, and bump windows over s.
amount of periods. The bump window needs approximately k 5
periods to level off, and the Hann windowed average oscillates for
a some more periods before converging to the same value as the
bump-windowed average. We can see that the amplitude of the square
window also reduces, but at a much slower rate than the higher-order
windows. Note that the windowed time averages for final times of
a) CD ( , n t) and Jw ( , n – ntr) up to time step 1200
Fig. 5
n ≤ 531 (i.e., where the time span is less than one period) are not
meaningful.
Now, we consider the long-time behavior of different windowed time
averages for final times of n 104 , which are equal to k 300 periods
in Fig. 5b (respectively, Fig. 6b). We can observe several effects. First,
none of the higher-order windows show any oscillatory behavior,
whereas the square window still oscillates for very high iteration
numbers. Second, we can see an asymptotic convergence of all windows as the iteration number increases. Both effects are expected since
Theorem 1 postulates convergence in Ok1 for the square window,
where k is the number of periods passed within the averaged time span.
Furthermore, we can see in Fig. 7a (respectively, Fig. 7b) that the mean
value of the period shifts slightly upward over time. In Fig. 5b (respectively, Fig. 6b), we can see that all windowed time averages follow this
trend. Indeed, we have observed in Fig. 7 that the shift in the mean value
stops at n 4500. We can see that higher-order windows have less
problems adapting to this trend in their long-time behavior since they
weight function values at (normalized) times near to s 0 and s 1
much less than the square window, as can be seen in Fig. 4. More
generally speaking, the decreased weighting of values near to s 0 and
s 1 leads to a decreased sensitivity with respect to the behavior of the
instantaneous output in these regions. As a result, higher-order windowed averages output useful values; although, in this example, there
exists an additional trend, i.e., the function is not exactly periodic. In
analogy to Fig. 6, the sensitivity of CD with respect to the design
variable σ as well as the corresponding windowed time averages are
displayed in Fig. 8 with Re 103 and in Fig. 9 with Re 106 . Let us
first analyze the results of the lower-Reynolds-number case. The results
completely validate Theorem 1 since the higher-order windowed
b) Jw ( , n – ntr) up to time step 10,000
CD σ;nΔt and Jw σ;n − ntr with different windows over n, ntr 500, and Re 103 .
a) CD ( , nΔ t) and Jw ( , n – ntr) up to time step 1200
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b) Jw ( , n – ntr) up to time step 10,000
Fig. 6 CD σ;nΔt and Jw σ;n − ntr with different windows over n, ntr 500 and Re 106 .
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SCHOTTHÖFER ET AL.
a) Re = 103
b) Re = 106
Fig. 7 CD σ;nΔt long-time behavior with different Reynolds numbers. Note the slight shift upward in time of the period mean in both cases, which stops
at approximately n 4500.
a)
d
d
CD (
nΔ t ) and dd
(
n − ntr ) up to time step 1200
b)
d J
w
d
(
−
) up to time step 5000
Fig. 8 d∕dσCD σ;nΔt and d∕dσJw σ;n − ntr with different windows over n, ntr 500 and Re 103 .
Fig. 9 d∕dσCD σ;nΔt and d∕dσJw σ;n − ntr with different windows over n, ntr 500 and Re 106 .
time-averaged sensitivities quickly converge (see Fig. 8b), and the
square window leads to an oscillatory time average with a nondecreasing amplitude. Theorem 1 postulates a convergence rate for the squarewindowed time average in O1. This is in line with the findings of
Krakos et al. [15], who analyzed the convergence behavior of the
windows.
Now let us consider the flow with Re 106 . Considering
Fig. 9a, we can observe again that the higher-order window-averaged
sensitivities do not show any oscillatory behavior after the first few
periods. The square window first reduces its amplitude, but this is
solely due to the fact that the amplitude of the instantaneous sensitivity
reduces up to n 1050. We can see in Fig. 9b that the amplitude of the
square window increases again with increasing iteration number n.
This reflects the findings of Theorem 1, which postulates no convergence for the square-windowed time-averaged sensitivity. Now, we
consider the long-time behavior of the higher-order windows in Fig. 9b.
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SCHOTTHÖFER ET AL.
We can observe a divergent behavior for all windows as time increases.
The divergence rate is slow for the higher-order windows but higher for
the square window. This is a result of the previously discussed feature
of higher-order windows to be less sensitive to values of the instantaneous output near to s 0 and s 1. The windowed time-averaged
sensitivities assume smaller and smaller values. Considering the corresponding (instantaneous) sensitivities in Fig. 3c, we can see that the
minimum of d∕dσCD (i.e., the lower part of the envelope of this
signal) grows much faster to −∞ than the bounding maxima of
d∕dσCD (i.e., the upper part of the envelope) grows to ∞. Thus,
the effective mean value of a period also shifts downward, which is
reflected by the divergent behavior of the windowed time-averaged
sensitivities. However, it should be stressed that Theorem 1 does not
give clear convergence bounds in this case, as discussed in Sec. IV,
since the exponential growth of the amplitude of the sensitivity implies
exponential growth of
d
CD dσ ∞
Also interesting is the time frame between n 3500 and n 4000
time steps, where the square-windowed time-averaged sensitivity
changes its sign due to its oscillation. This can lead to massive errors
in the sensitivity computation. In the context of aerodynamic shape
optimization, this may result in wrong signs or magnitude of the
components of the sensitivity vector, i.e., a nondiminishing error in
the descent direction of the design process. In such cases, a higherorder window that does not oscillate yields much more meaningful
results, which can be used in a shape optimization process.
B. Verification of the Adjoint Solver
The adjoint flow solver of SU2 is verified using the NACA0012
airfoil with the same flow conditions as given earlier in this paper with
Re 103 and Re 106 . To compare adjoint and primal computations,
we choose a fixed final time step of nf 1200 and a fixed starting time
of ntr 500. We therefore average over 700 time steps, which correspond to approximately 22.5 periods. We only change the choice of the
windowing function to ensure a fair comparison to the traditional
approach and between the windows. Table 1 shows the values of the
sensitivity with respect to one design variable, computed with forward
and adjoint modes. By forward mode, we refer to the primal (tangent)
computation of the derivatives using the forward AD mode. By adjoint
mode, we refer to the Lagrangian method derived in Sec. III in combination with the adjoint AD mode. The adjoint derivative calculation is
performed using an iterative scheme that uses the fixed-point structure of
the flow solver, i.e., Eqs. (29) and (31). Therefore, consistency of adjoint
and primal sensitivities depends on the convergence of the adjoint
iteration in Eq. (29), which itself depends on the convergence of the
primal iterator in Eq. (19). The adjoint flow solver with a square window
has been verified up to machine precision in the work of Zhou et al. [3].
Table 1 shows that in this test case, the direct and adjoint solvers match
up to five digits of accuracy.
Table 1 Sensitivity of the drag coefficient CD with respect to a single
FFD–box design variable, computed with different differentiation
techniques and different windows
Reynolds number
106
106
106
106
103
103
103
103
Windowing
function
Square
Hann
Hann-square
Bump
Square
Hann
Hann-square
Bump
Adjoint mode
Forward mode
7.69759919 10−3
8.35141629 10−3
8.36581390 10−3
8.36382254 10−3
6.55959994 10−4
6.60942188 10−4
6.61312049 10−4
6.61556835 10−4
7.69759176 10−3
8.35144493 10−3
8.36588240 10−3
8.36381374 10−3
6.55957658 10−4
6.60948670 10−4
6.61315461 10−4
6.61555351 10−4
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C. Comparison of Surface Sensitivities Using Different Windows
In the following, we consider the surface sensitivities of CD in the
setting of the NACA0012 airfoil. We choose surface sensitivities over
sensitivities with respect to the design variables since they give more
intuitive insights into how the wing is deformed in the design process.
In this test case, there are 217 surface points on the airfoil. We compare surface sensitivities calculated with the bump and square windows with different windowing time spans. The square window acts
as a benchmark and represents the case where no specific windowing
function is chosen, and therefore the traditional approach.
As discussed in Sec. IV.A and shown in Fig. 9a, the squarewindowed time average first reduces and then increases its amplitude,
which distorts the convergence behavior of the square window.
Therefore, we choose ntr 1503 instead of ntr 500. We average
over 700 and 709 time steps up to a final iteration of nf1 2203
(respectively, nf2 2212). The difference of nine time steps is
approximately 29% of a period length. Figure 10 shows bump- and
square-windowed time-averaged surface sensitivities with different
final times nf . The blue symbols represent the sensitivities on the
suction side, whereas the red symbols represent those on the pressure
side of the airfoil. The sensitivities are plotted on their respective
x-axis locations on the airfoil; i.e., for every point on the x axis, there
are four sensitivities plotted. For reference, the NACA0012 airfoil is
plotted on the right-hand-side y axis.
In Fig. 10a, we can see bump- and square-windowed time-averaged surface sensitivities at the final time nf1 . In both methods, the
sensitivities at the suction side near the leading edge are the highest,
followed by the sensitivities on both sides near the trailing edge. This
is expected, since the flow detaches at the leading edge and forms big
vortices at the trailing edge, as can be seen in Fig. 1. Note that the
suction-side sensitivities near the leading edge are too big in magnitude to be displayed in Fig. 10. The absolute difference between both
methods is the highest near the leading edge at the suction side
followed by a region at 20% airfoil length on the pressure side, where
the square-window values show a deviation from the bump window
values. Except for this region, the relative difference of the sensitivities ranges from 5 to 20%.
Next, we consider Fig. 10b, where the bump window values are
shown for different final times nf1 and nf2 . We can see that even at the
high-sensitivity regions at the trailing and leading edges, the bumpwindowed time-averaged sensitivities barely differ. This shows that
the bump window is quite robust with respect to the choice of time
span to average. Contrary to this, we can see in Fig. 10c that the
square-windowed time-averaged sensitivities change clearly over the
span of nf2 − nf1 9 time steps. In the highly sensitive area at 20%
airfoil length on the pressure side, the changes in the square-window
values are big compared to the changes in the bump-window values in
Fig. 10b. The relative difference of the sensitivity vectors in the
Euclidean norm computed with bump and square windows at nf1 is
9.2%, which displays a significant change in the descent direction of
the optimization procedure. We can even observe a sign change at five
surface points when comparing the two windows. The relative difference between the bump- windowed sensitivity vectors at nf1 and nf2
is 0.84% in comparison to the relative difference between the squarewindowed sensitivity vectors at both times, which comes to 9.1%.
On the one hand, typically one does not know the exact period
length of the objective function, and therefore is forced to choose an
arbitrary time span to average the windowed sensitivity; on the other
hand, a relatively small change in the chosen time span leads to a
significant error in the square-windowed time-averaged sensitivity.
This further reinforces the thesis that higher-order windows, such as
the bump window, lead to a more robust sensitivity calculation, and
therefore to increased robustness in the shape optimization with
respect to the choice of the averaging time span.
D. Comparison of Optimization Results Using Different Windows
In this subsection, we evaluate the results of an aerodynamic shape
optimization run performed by SU2 with different choices of windowing functions as objective function regularizers. We consider the
optimization problem
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SCHOTTHÖFER ET AL.
a) Bump and square window at
1
b) Bump window at
1
and
2
c) Square window at
1
and
2
Fig. 10 Drag surface sensitivities at Re 106 and ntr 1503 with different end times of nf1 2203 and nf2 2212. Airfoil surface in black for
reference.
min Jw σ;nf −ntr σ∈X ad
nf
X
1
n−ntr
C nΔt;σ
w
nf −ntr nn
nf −ntr D
SLSQP optimization algorithm by Kraft [23], which then computes a
new design for the next design iteration.
tr
subject to Gun ;σ un ; ∀ n 1;:::;nf
nf
X
1
n−ntr
Cw σ;nf −ntr w
C nΔt;σ ≥ 0.96;
nf −ntr nn
nf −ntr L
tr
Xad −0.05;0.05242 ∈ R242
(39)
Here, nd 242 and Xad is cuboid in Rnd . Gun ; σ is the fixed-point
form of the URANS solver, which we eliminate formally by the
Lagrangian in Eq. (24). We input the obtained sensitivities to the
1. Test Case: Re 106
We use the windowed time-averaged lift coefficient as an inequality constraint that is greater than or equal to 0.96, which is the
approximate value of the windowed time-averaged lift for the baseline configuration. We start the windowed time average at iteration
ntr 1500 and average up to nf 2200. This corresponds to
the configuration in the last subsection. The flow configuration is
described in the beginning of this section with a Reynolds number of
Re 106 . We multiply the computed gradients with a relaxation
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SCHOTTHÖFER ET AL.
factor of 0.1 such that the step sizes proposed by the optimizer are not
too big for the admissible set Xad .
We compare the optimization results of optimization procedures
computed with the different windowing functions introduced earlier
in this paper. Figure 11 compares the flow around the baseline
geometry described by the NACA0012 airfoil to the results of the
shape optimization process, where once the traditional square window and once the Hann-square window are used. All flowfield
snapshots are taken at a time step of n 1618, i.e., past the transient
phase of the flow around the base geometry. Whereas the flow
optimized using the square window, depicted in Fig. 11b, is unsteady
and periodic; the design optimized with the Hann-square window
exhibits a stationary flow, as seen in Fig. 11c. The other high-order
windows, Hann and bump, exhibit a stationary flow as well. In
Fig. 11d, we can see the skin-friction coefficient Cf in the x direction
over the suction side of each displayed airfoil. Note that the value of
Cf is plotted at a time step of n 1618; however, the point of flow
separation has a relative change of less than 5% for the baseline
design and less than 1% for the optimized designs over the course of a
period. In the baseline NACA0012 airfoil, the flow detaches itself
almost directly at the leading edge (i.e., at 2% airfoil length), as can be
seen in the green graph of the Cf plot in Fig. 11d. The flow around the
design optimized with the traditional square window detaches itself
further back on the airfoil at 20% airfoil length. Lastly, we can see that
the flow around the Hann-square-window optimized design separates
at 30% airfoil length, which is a 50% improvement compared to the
square window. The further back on the wing the flow separates, the
lower is the drag induced by the separated flow. Intuitively, this hints
that the optimization using higher-order windows is more effective
than the optimization using a square window.
A more detailed report of the different optimization runs is given in
Fig. 12. Here, we can quantify the intuitive result of Fig. 11. The
figure displays values of the optimization objective Jw and the
optimization constraint Cw , as well as the bound of the inequality
constraint. Both lift Cw and the lower bound of the lift constraint
Cw 0.96 are measured on the right-hand-side y axis of Fig. 12a.
Note, however, that the axis of Fig. 12a is scaled by a factor of two
a) Flow around the initial NACA0012 airfoil
c) Flow around optimized design using Hann-square
windowing
2527
since the values of J w and Cw oscillate much more in the squarewindow run compared to the other optimization runs. We can see that
the square-windowed optimization struggles to satisfy the inequality
constraint while minimizing the objective Jw . Each time the optimizer reduces Jw , it oversteps the lift constraint Cw ≥ 0.94, which
results in an infeasible design. The subsequent design then increases
both Cw and Jw but reduces Jw a bit compared to the first design. This
results in a very slow optimization procedure compared to the ones
where higher-order windows are used.
Contrary to the square-window run, the design process of a higherorder window produces designs that do not repeat themselves in such
a pattern; see Figs. 12b–12d. After two infeasible designs, the optimizer manages to reduce the windowed time-averaged drag to 30% of
the baseline value while keeping the design feasible on most optimization iterations. These effects are explainable by considering the
surface sensitivities in Sec. IV.C. We have seen that the surface
sensitivities computed with the square window changes when the
time frame over which the sensitivity was averaged was shifted.
During the shape optimization process, we change the airfoil design,
and hence the limit cycle changes as well. This may result in a shifted
phase, shape, or period length of the new limit cycle compared to the
baseline limit cycle or a combination of these effects; see Ref. [24]. We
have seen that a higher-order window mitigates the effect of a shifted
limit cycle, which is equivalent to shifting the averaging time frame
better than the square window; i.e., it is more robust with respect to a
shifted limit cycle. Another aspect of the increased oscillation in the
square-window design process, compared to the higher-order design
processes, is given by the time-dependent sensitivities computed with
different windows, shown in Fig. 9b. As discussed in Sec. IV.A, the
design variable sensitivities computed using a square window oscillate,
depending on the length of the averaging time; in this high-Reynoldsnumber test case, the amplitude even increases. Higher-order windows,
however, dampen their oscillatory behavior as time increases.
Whereas a changed shape of a period is not important for any
windowing, including the square window, a change in the period
length affects convergence speed of the windowed averaging since a
windowed time average converges, depending on the number of
completed periods in the given time frame M.
b) Flow around optimized design using square-windowing
d) Skin-friction coefficient Cf in the x direction
over x direction of the suction side of all (designs airfoil
length is normalized to one)
Fig. 11 Comparison of baseline design and optimized designs using different windows after 15 design iterations. Displayed are isosurfaces of Mach
number of each control volume at time step n 1618 as well as corresponding Cf plot over their suction sides.
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SCHOTTHÖFER ET AL.
a) Square windowing
b) Hann windowing
c) Hann-square windowing
d) Bump windowing
Fig. 12 Shape optimization performed by SLSQP with different windows used. Both (feasible and infeasible) design steps are displayed. Re 106 .
2. Test Case: Re 103
We briefly consider the test case at a lower Reynolds number. We
look again at the optimization problem in Eq. (39) but at Re 103 ,
ntr 500, and nf 684. The lift constraint Cw is set to be greater or
equal to 0.76, which is the windowed time-averaged lift of the baseline geometry. All other flow properties are set to the same values as in
the aforementioned test case.
We have chosen a smaller time frame of 168 time steps, which
corresponds to approximately five periods. We can see in Fig. 8 that
after 168 time steps, the square-windowed time average still oscillates; whereas higher-order windowed time averages are already
leveled. Therefore, we expect worse results in the case of a squarewindowed optimization compared to its higher-order counterparts,
which should perform comparably well. Indeed, we can see in
Fig. 13a that the square-windowed lift constraint Cw , which is
colored red, oscillates for the first few design iterations and stays
infeasible (i.e., below the lift constraint) for twice as much than the
higher-order windowed lift constraints. The last infeasible squarewindowed lift appears at iteration 8, whereas the last infeasible Hannwindowed value appears at iteration 6 (respectively, 4) in the case of
Hann-square and bump windowing.
The windowed time-averaged drag coefficient Jw , which is the
optimization objective, does not depend much on the chosen windowing function as opposed to the results of the Re 106 test case.
Furthermore, the oscillating behavior of the square-windowed
designs is not as extreme as the square-windowed designs of the
Re 106 test case. This can be explained by considering the amplitude of both the instantaneous output as well as the windowed time
averages of both test cases. The amplitudes of the Re 103 test case
at iteration 684 are an order of magnitude smaller than the corresponding results at Re 106 at nf 2200. Hence, the SLSQP
optimizer can handle the square-windowed time-averaged sensitivities of the Re 103 test case better than the ones of the Re 106
test case. For completeness, we note that the optimization procedure
as well as all other computations are conducted on 16 cores of an Intel
Xeon SP 6126 processor for each test case. The total time cost for one
design iteration including direct and two adjoint computations as well
as the mesh deformation and postprocessing is approximately 15 h.
V.
Periodically Pitching Airfoil
In this test case, we consider a NACA64A010 airfoil, modeled by a
grid consisting of 12,897 quadrilaterals and 23,205 triangles. The
airfoil surface consists of 250 wall-boundary elements and, at the far
field, we have 68 elements. We compute sensitivities with respect to
50 design variables, which model the airfoil surface using Hicks–
Henne functions. The flow is computed with the URANS solver of
SU2, where we use the Roe method for the Navier–Stokes fluxes and
the upwind method for the transport equation in the Spalart–Allmaras
turbulence model. The angle of attack in this configuration is relatively low at 5 deg, and we achieve a periodic flow by tilting the wing
around its longitudinal axis. Therefore, the airfoil increases or
decreases its angle of attack. Here, we use a pitching amplitude of
3 deg and an angular frequency of 53.3491 rad∕s. The pitching is
modeled by mesh movement using an arbitrary Lagrange–Eulerian
(ALE) formulation, as explained in Ref. [16]. The remaining freestream parameters are given as 1) a Mach number of 0.35, 2) a
freestream temperature of 88.15 K, 3) a Reynolds number of 106 ,
and 4) a Reynolds length of 1.0 (length of the airfoil).
The simulation is computed with a time step of Δt 0.001636 s,
which is equal to 72 time steps per pitching period.
In the following, we study windowed time averages and windowed
time averaged sensitivities of the periodically pitching airfoil.
Here, we display first the time-dependent drag and lift coefficients
as well as their sensitivities in Fig. 14. We have a transient phase of
approximately 130 time steps; after it, both drag CD and lift CL
oscillate with a period amplitude of approximately 0.19 in the case of
CD (respectively, 0.01) in the case of CL . We can see that, in contrast
to Fig. 3 (which displays the NACA0012 test case at Re 103 ), the
sensitivities of neither drag nor lift grow in amplitude. Their amplitudes are 0.1 (respectively, 0.5). The effect of a nonincreasing amplitude of the sensitivity can be explained by the fact that the period
SCHOTTHÖFER ET AL.
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a) Square-windowing
c) Hann-square-windowing
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b) Hann-windowing
d) Bump-windowing
Fig. 13 Shape optimization performed by SLSQP with different windows used. Both (feasible and infeasible) design steps are displayed. Re 103 .
Fig. 14 Drag and lift, and their sensitivities of the pitching airfoil over iteration number.
length is only dependent on the angular frequency, which is independent of the design parameters.
We can see in Fig. 15 the windowed time averages and their
sensitivities. We start the windowed time average at ntr 130, where
the transient phase has passed. Both the windowed time averages and
the windowed time-averaged sensitivities converge for all higherorder windows after a couple of time steps. Furthermore, we can see
that the square window performs almost equally well in the case of the
time-averaged sensitivity as the time average. The reason for this is
that the period length is independent of the design parameter. This
means that in Theorem 1 we can use the convergence boundary given
by Eq. (10) rather than Eq. (11) for the time-averaged sensitivities
since the second equality in Eq. (4) holds in this case.
Figure 16a compares the square-windowed time averages of the
NACA0012 test case computed at Re 103 to the current
NACA64A10 test case. Figure 16b compares the corresponding
square-windowed time-averaged sensitivities. The timescale of both
figures is given by k Δtnf − ntr ∕T, where T is the period
length of the limit cycle of the specific test case. The curves display
the relative difference of the windowed time averages (respectively,
their sensitivities) to their corresponding limit values. Since a computation of the analytic value Jσ is not possible, use the bumpwindowed value for a very high value of k as an approximation. One
can see that the square window in the NACA64A10 test case has
approximately the same slope as its sensitivity, which confirms the
aforementioned consideration. Note that the displayed value of the
slope is corrected by the ratio of scales of the x and y axes. The slope
of the windowed time average in the NACA0012 case asymptotically
matches the counterpart of the NACA64A10 case. However, the
square-windowed time-averaged sensitivity of the NACA64A10 case
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SCHOTTHÖFER ET AL.
a) Windowed time average
b) Windowed time-averaged sensitivity
Fig. 15 Jw σ;n − ntr and d∕dσJw σ;n − ntr with different windows over n, ntr 130.
a) Windowed time average
b) Windowed time-averaged sensitivity
Fig. 16 Relative differences of square windows to their limits, where k nf − ntr ∕T. We display the NACA0012 airfoil at Re 103 and the
NACA64A10 airfoil.
has a much steeper slope than its NACA0012 counterpart. This validates the aforementioned consideration. With this test case, we have
seen an example for which the windowing approach is not absolutely
necessary. However, usage of higher-order windows still speeds up the
convergence rate of the time-averaged output and the time-averaged
sensitivity.
VI.
Conclusions
In this paper, the unsteady aerodynamic optimization framework of
SU2 was combined with the windowing approach to obtain meaningful and robust design sensitivities. First, the windowed time average
was defined as an output to the direct flow solver. Then, the long-time
windowing approach was embedded in the discrete adjoint solver of
SU2 using a Lagrangian method. The discrete adjoint directly inherits
the convergence properties of the primal flow solver.
Both the primal and adjoint solvers with windowed output functions have been applied in the NACA0012 and NACA64A010 test
cases, which display an unsteady turbulent detached flow that exhibits
limit-cycle oscillations. The superiority of the windowing approach
using high-order windows for sensitivity analysis of a period-averaged
quantity was shown and compared to the traditional nonwindowed
approach. Whereas nonwindowed (or square-windowed) averaged
sensitivities tend to oscillate even for long-time averages, high-order
windows quickly converge. An increased robustness of computed
sensitivities and a resulting more efficient optimization procedure
using high-order windows for the sensitivity calculation, as compared
to the traditional approach, was demonstrated by taking the example of
the NACA0012 airfoil as a two-dimensional test case. Furthermore,
high-order windowing does not increase the computational cost of
the sensitivity calculation. The windowing approach was successfully
implemented in optimization that was governed by URANS flows,
which have a clear periodicity with a single frequency due to the fact
that only one turbulent scale is resolved. However, more realistic
simulations of detached flows require scale-resolving methods [11]
that result in a range of different frequencies in the time-dependent
output functions. Future work will investigate if it is possible to apply
windowed optimization in a meaningful way for these methods.
Acknowledgments
The computational resources provided by the Regionales Hochschulrechenzentrum Kaiserslautern (RHRK) high-performance computing
center via the Elwetritsch high-performance cluster at the Technical
University of Kaiserslautern is gratefully acknowledged.
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J. Larsson
Associate Editor