Linköping University
IEI - Department of Management and Engineering
Division of Applied Thermodynamics and Fluid Mechanics
Master Thesis 2019|LIU-IEI-TEK-A–19/03403-SE
CFD-Method for 3D Aerodynamic
Adjoint Simulations
For External Automotive Aerodynamics
Marcus Lång
Academic supervisor: Roland Gårdhagen
Industrial supervisor: Mattias Olander
Examiner: Matts Karlsson
Linköping University
SE-581 83 Linköping, Sweden
+4613-28 10 00, www.liu.se
Abstract
Today’s rules and regulations regarding emissions from combustion vehicles are very strict
and the travel range per tank and/or charge, especially for electric vehicles, is a crucial
factor which will always be considered by the customers. Hence, automotive manufacturers strive to boost fuel and battery economy. This can, to a great extent, be done by
improving the aerodynamics of the vehicle for lower drag. The conventional CFD process
for aerodynamic development is relatively time consuming and there is rarely enough time
or resources to find the optimal design in all regions of the vehicle. Hence, the adjoint
solver was investigated to make the aerodynamic development process more efficient by
providing sensitivities of the geometry with respect to drag force.
The adjoint solver was investigated both through a literature review as well as by performing CFD and adjoint simulations. The CFD and adjoint simulations were performed using
Fluent 2019 R1 and the realizable k−ε turbulence model. It was found that it is important
to monitor surface sensitivities during the solution in addition to the adjoint residuals to
assess convergence of the adjoint simulation. It is also recommended to analyse regions
of high residuals in the domain to ensure that they are far away from the surface(s) of
interest. Investigations regarding different stabilization schemes as well different meshes
for the adjoint solver were performed.
It was concluded that the residual minimization scheme (RMS) is the preferred stabilization scheme. It was found that a coarser mesh can be used to reduce localized transient
behaviour if the adjoint solver has trouble converging. It was found that a simplified model
of a fully detailed car geometry is necessary to reduce the complexity and the resolution
of the mesh to be able to use the RMS and to avoid local instabilities. A proposed CFD
and adjoint procedure with guidelines and recommendation was developed.
Keywords: Discrete adjoint method, shape sensitivity, gradient-based optimisation, Fluent, vehicle aerodynamics, CFD, stabilization scheme, residual minimization, adjoint convergence assessment
i
Acknowledgements
First of all, I would like to acknowledge CEVT and the CAE Energy department for giving
me the opportunity to do my master’s thesis together with them. I would like to thank
my industrial supervisor at CEVT, Mattias Olander, as well as my academic supervisor,
Roland Gårdhagen, for their support and guidance during the project. I would also like to
express my appreciation to Samuel Gabriel at CEVT who also assisted me with solutions
and answers to various problems and questions that arose. Last but not least, I want to
thank Aditya Naronikar, my friend and study partner from Linköping University, whom I
spent uncountable hours with to acquire the required knowledge to master a thesis within
this subject.
Marcus Lång, Gothenburg, June 2019
ii
Nomenclature
Abbreviations and Acronyms
Abbreviation
ACN
AMG
AWA
CAE
CEVT
CFD
CFL
DNS
FRCS
MP
MRF
NOD
PID
RANS
RMS
RKE
SSM
UDM
URF
Meaning
Artificial Compressibility Number
Algebraic Multigrid
Area-Weighted-Average
Computer Aided Engineering
China Euro Vehicle Technology
Computational Fluid Dynamics
Courant–Friedrichs–Lewy
Direct Numerical Simulation
Flow Rate Courant Scaling
Motion Point
Moving Reference Frame
Normal Optimal Displacement
Property Identification
Reynolds-Averaged Navier-Stokes
Residual Minimization Scheme
Realizable k-ε
Shape Sensitivity Magnitude
User Defined Memory
Under-Relaxation Factor
Latin Symbols
Symbol
A
a
B
CD
CL
Cp
c
FD
k
L
M
p
Description
Projected frontal area
Speed of sound
Additive constant
Coefficient of drag
Coefficient of lift
Pressure coefficient
Control variable
Drag force
Turbulent kinetic energy
Length of the vehicle
Mach number
Pressure
iii
Units
2
m
−1 ms
[−]
[−]
[−]
[−]
[−]
[N ]
2 −2 m s
[m]
[−]
[P a]
Symbol
Rij
Re
Sij
t
Uτ
u
u+
y
y+
Description
Reynolds stresses
Reynolds number
Strain rate
Time
Friction velocity
Velocity
Dimensionless velocity
Wall distance
Dimensionless wall distance
Units
[P a]
[−]
−1 s
[s]
−1 ms
−1 ms
[−]
[m]
[−]
Description
Lagrange multiplier
Difference between two values
Variation
Kronecker delta
Dissipation of turbulence energy
von Karman’s constant
Dynamic viscosity
Density
Kinematic viscosity
Turbulent kinematic viscosity
Viscous stress tensor
Shear stress at wall
Units
[−]
[−]
[−]
[−]
2 −3 m s
[−]
kgm−1 s−1
kgm−3
2 −1 m s
2 −1 m s
[P a]
[P a]
Greek Symbols
Symbol
α
∆
δ
δij
ε
κ
µ
ρ
ν
νt
τij
τw
Subscripts and superscripts
Abbreviation
i,j,k
∞
Meaning
x,y,z components
Freestream value
iv
Contents
Abstract
i
Acknowledgements
ii
Nomenclature
1 Introduction
1.1 Background . . . .
1.2 Problem Statement
1.3 Purpose . . . . . .
1.4 Objectives . . . . .
1.5 Delimitations . . .
iii
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1
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2 Theory
2.1 Fluid Dynamics . . . . . . . . . . . . . .
2.1.1 Aerodynamic Drag . . . . . . . .
2.1.2 Assumptions and Simplifications
2.2 Computational Fluid Dynamics . . . . .
2.2.1 The Governing Equations . . . .
2.2.2 Turbulence Modelling . . . . . .
2.2.3 Realizable k − ε . . . . . . . . .
2.2.4 Near Wall Flow . . . . . . . . . .
2.3 Adjoint Solver . . . . . . . . . . . . . . .
2.3.1 Discrete Adjoint Method . . . . .
2.3.2 Solution Methods . . . . . . . . .
2.3.3 Solution Controls . . . . . . . . .
2.3.4 Adjoint Literature Review . . . .
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4
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. 5
. 5
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. 6
. 7
. 7
. 9
. 9
. 11
. 11
. 13
3 Method
3.1 Model Setup . . . . . . . . . . . . . . .
3.1.1 Geometry and Mesh . . . . . . .
3.1.2 Solver Setup . . . . . . . . . . .
3.1.3 Boundary Conditions . . . . . .
3.1.4 Computational Domain . . . . .
3.1.5 Convergence Assessment . . . . .
3.2 Adjoint Setup . . . . . . . . . . . . . . .
3.2.1 Adjoint Convergence Assessment
3.2.2 Adjoint Solver Setup . . . . . . .
3.3 Post-processing . . . . . . . . . . . . . .
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15
15
15
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22
25
4 Results
27
4.1 Simplified Car Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.1.1 Baseline Mesh Settings . . . . . . . . . . . . . . . . . . . . . . . . . . 27
v
Contents
4.1.2 Stabilization Schemes and Meshes . . .
4.1.3 Varying Number of Modes using RMS .
Full Car Model . . . . . . . . . . . . . . . . . .
Case Study - Single Part Adjoint Optimisation
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33
39
43
47
5 Discussion
5.1 Potential & Applicability of the Adjoint Solver
5.2 Adjoint Method . . . . . . . . . . . . . . . . . .
5.2.1 Convergence Assessment . . . . . . . . .
5.2.2 Stabilization Schemes and meshes . . .
5.2.3 Robustness . . . . . . . . . . . . . . . .
5.2.4 Single Part Adjoint Optimisation . . . .
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50
50
51
52
52
55
56
4.2
4.3
6 Conclusion
58
7 Future Work
59
References
60
Appendices
A1
A Adjoint Procedure
A1
vi
1
1.1
Introduction
Background
This master thesis was performed together with China Euro Vehicle Technology AB
(CEVT) in Gothenburg, Sweden. The project was conducted within the CAE Energy
department in the field of aerodynamics. CEVT is an innovation centre for the future
vehicles of Geely automotive group. CEVT’s research and technology development involve all different kinds of aspects in the automotive industry such as power transmission
and drivelines, chassis, aerodynamics and the vehicles’ visual design as well as the total
architecture of a car.
Today, the rules and regulations regarding emissions from combustion vehicles are very
strict and the travel range per tank and/or charge, especially for electric vehicles, is
a crucial factor which will always be considered by the customers. Hence, automotive
manufacturers strive to boost fuel and battery economy. This can, to a great extent,
be done by improving the aerodynamics of the vehicle for lower drag. However, the
external design of the car can be varied in almost an infinite amount of designs which
makes experimental testing expensive and time-consuming. First of all, new parts would
have to be manufactured for each design. In addition, experimental measuring equipment
and facilities are required and need to be run and maintained. Instead, different designs
are simulated using Computational Fluid Dynamics (CFD) for faster and cheaper design
iterations.
The conventional CFD process is still relatively time-consuming due to large models with
many details. As of now, each simulation requires several thousand CPU hours. Hence,
it can take many days to evaluate several designs. However, a primal flow field from a
CFD simulation coupled with an adjoint solver could provide very valuable insight of the
sensitivity of the geometry to a specified observable, which is drag force in this case. This
allows the engineer to evaluate the sensitivity map to locate the most crucial regions of
the vehicle and make modifications accordingly to reduce drag. It also allows the engineer
to easily find regions of the car which has a great impact on the drag which might not
have been thought of before. The adjoint method is relatively new within the field of CFD
but has already been applied to both internal and external flow problems [1, 2, 3, 4, 5, 6].
The adjoint solver has evolved during the last couple of years with increased stability and
has shown great potential [1].
1
1 Introduction
1.2
Problem Statement
Generally in the automotive industry, the aerodynamic development is performed through
an iterative process where the engineer investigates different concepts one by one through
3-dimensional aerodynamic simulations using CFD. The result from the simulation for
each concept is analysed quantitatively through drag and load deltas, and qualitatively
from looking at the pressure and velocity of the flow around the vehicle. The engineer then
changes the concept based on the analysis. The conventional CFD process for external
aerodynamics at CEVT is illustrated in fig. 1. This process can be very time-consuming
and there is rarely enough time or resources to find the optimal design in all regions of
the vehicle.
Figure 1: Typical aerodynamic development process using CFD
1.3
Purpose
To make the aerodynamic development process more efficient, an adjoint solver can be
used to provide information about the sensitivity of the geometry in relation to the flow
around the vehicle. For example, the adjoint solver can calculate the sensitivity of the drag
depending on the geometrical shape of the vehicle. A reliable adjoint optimisation method
would enable the engineers to increase the performance of the car while maintaining or even
reducing the computational time and resources required for each design loop compared to
the current CFD method. Hence, the adjoint method will be investigated and a method
with guidelines of how to use it will be developed.
1.4
Objectives
The objectives of this study are to:
• Investigate the potential of Fluents adjoint solver and evaluate its applicability to
external aerodynamic flow around a ground vehicle
• Develop a CFD-method with guidelines to run external 3D adjoint simulations for
aerodynamic analysis using Fluent’s adjoint solver to achieve information regarding
the sensitivities of the vehicle with respect to drag force
2
1 Introduction
1.5
Delimitations
The delimitations of the study are:
• The time frame of the project is limited to the spring of 2019 (middle of January beginning of June 2019)
• Computational resources are limited to the available cluster at CEVT
• A car model of type SUV A will be the geometry of focus in this study
• Only consider drag as the objective parameter in the adjoint solver
• Simulation and solver settings are set to be similar to CEVT’s standard procedure
(however, the robustness of the developed method will be tested and possibly tweaked
using other settings as well)
• Available software packages for aerodynamic simulations and adjoint calculations
are the standard software used at CEVT which are ANSA, Harpoon, Fluent and
Ensight
• No experimental validation was performed
3
2
Theory
The theory chapter contains information regarding important concepts and equations utilized in the study. The fundamental physical principles and relevant equations to fluid
dynamics, aerodynamics and adjoint optimisation are presented in the following sections.
2.1
2.1.1
Fluid Dynamics
Aerodynamic Drag
Aerodynamics is the study of fluid flow around a body and the forces acting on the
body. An aerodynamic analysis is an essential part of the development of a road vehicle,
both for stability and handling of the car as well as for the fuel economy, as mentioned
previously. This study focused on the drag force of the car but all the forces and moments
are important to take into consideration when designing a vehicle [7]. The drag force, or
air resistance, is the force acting opposite to the direction of motion of the car. The drag
force of the car (or any geometrical object for that matter) is defined by [8]:
FD = 0.5CD Aρu2∞
(1)
Where A, ρ, u∞ and CD are the projected frontal area, the density of the fluid, the velocity
of the car and the coefficient of drag, respectively. The drag coefficient is a dimensionless
quantity which depends on the geometrical shape of the car. The coefficient of drag can
be altered by adjusting the external geometry of the car since the velocity and the density
are constant parameters in this study. The value CD A is usually presented instead of just
CD or the drag force as a comparison between designs, as shown in eq. (2).
CD A =
FD
0.5ρu2∞
(2)
The aerodynamic drag force for a vehicle can be decomposed into two types of drag which
are the profile drag and the induced drag. The profile drag can further be divided into
two components which are the form drag (also called pressure drag) and viscous drag
(also called skin friction drag). The form drag is caused due to the pressure distribution
over the vehicle. In general, cars have a high-pressure region in front of the car and a
low-pressure region behind the car caused by flow separation. The pressure differences
create a resultant force which is the pressure drag. The viscous drag is caused by the
wall shear stresses between the surfaces of the car and the fluid. The induced drag (also
called trailing vortex drag) is a 3-dimensional type of drag caused by lift forces. The
pressure differences in certain regions of the vehicle equalize and creates a trailing vortex
that propagates in the streamwise direction. Typical regions where this can be expected
are at the A-pillars and at the back of the vehicle around the C-pillars.[8, 9]
4
2 Theory
2.1.2
Assumptions and Simplifications
Several assumptions were made to simplify the problem in hand. The fluid was considered to be an incompressible Newtonian fluid. Additionally, external body forces such as
gravitational, electric or magnetic forces were neglected. The flow field was assumed to be
of a steady nature and hence, the flow field variation with respect to time was neglected.
It is commonly known that air could be considered incompressible, without losing significant accuracy in calculations if the Mach number is less than 0.3 [8]. At ground level, the
speed of sound is roughly equal to 340 m s−1 (the speed of sound varies with temperature
but the variation is not significant for this study) [10]. Due to the low velocity of commercial passenger cars, which in this study was set to be 100 km h−1 , the Mach number
was found to be much less than 0.3, see eq. (3). The flow can still be considered incompressible even if local velocities around the vehicle increases three times the freestream
velocity, which is not expected to be reached.
M=
u∞
100
=
≈ 0.08
a
3.6 ∗ 340
(3)
Where u∞ is the velocity of the vehicle and a is the speed of sound. The flow was assumed
to be incompressible and the temperature was considered to be constant implies that the
energy equation can be excluded, i.e. an isothermal process.
2.2
2.2.1
Computational Fluid Dynamics
The Governing Equations
Fluid flow can be described based on three fundamental physical principles which are
conservation of mass, Newton’s second law and the first law of thermodynamics [11].
These physical principles are commonly known as the continuity equation, Navier-Stokes
momentum equations (in x, y, and z-directions) and the energy equation and forms the
base of CFD. Considering the assumptions mentioned earlier, the simplified continuity as
well as the momentum equation are shown in eq. (4) and (5), respectively [12].
ρ
∂ui
=0
∂xi
(4)
∂
∂p
∂τij
(ui uj ) = −
+
∂xi
∂xi
∂xj
(5)
Where ρ is the density, u is the velocity component in ’i, j, k’-directions, p is the pressure
and τij is the viscous stress tensor. The viscous stress tensor is defined as shown in eq. (6)
[13].
1
τij = 2ρν Sij − Skk δij
3
5
(6)
2 Theory
Where ν is the kinematic viscosity, δ is the Kronecker delta and Sij is the strain rate which
is defined in eq. (7) [13].
Sij =
2.2.2
1 ∂ui
∂uj +
2 ∂xj
∂xi
(7)
Turbulence Modelling
Turbulent flow is an irregular, disorganised and chaotic three-dimensional behaviour of the
flow in time and space [11, 14]. The Navier-Stokes equations can be solved using a direct
numerical simulation (DNS) which resolves all the turbulent scales of the flow. However,
a DNS requires an extreme amount of computational resources which makes it unfeasible
to apply for industrial flow problems with today’s resources. Instead, the turbulence of
the flow can be modeled and there are several different methods to model turbulent flow.
The most common method, which is usually a sufficiently accurate and computationally
viable method with today’s computing resources, is the Reynolds-Average Navier-Stokes
(RANS) approach. The RANS approach is based on Reynolds decomposition, where the
flow quantities are decomposed into one mean term (φ̄) and one fluctuating term (φ0 ), as
shown in eq. (8) [11, 14]. Where φ is any scalar quantity of the flow.
φ = φ̄ + φ0
(8)
By replacing the scalars with the mean and the fluctuating term in the governing equations,
an average form of the equations are obtained, see eq. (9).
∂
1 ∂ p̄
∂
∂ ūi
0 0
(ūi u¯j ) = −
+
− ρui uj
ν
∂xi
ρ ∂xi ∂xj
∂xj
!
(9)
This leads to an undefined system of equations since there are more unknown variables
than equations, which is known as ’the closure problem’ [11]. One of the most common
approach is to make use of the Boussinesq hypothesis which states that the Reynolds
stresses (Rij ) are proportional to the mean rates of deformation by introducing a term
called eddy viscosity or turbulent viscosity (νt ), see Reynolds stress tensor in eq. (10) [11,
13].
0
0
Rij = −ρui uj = νt
∂ u¯j
∂ ūi
+
∂xj
∂xi
!
2
− ρkδij
3
(10)
Where k is the turbulent kinetic energy which is defined as:
1 02
k = ui
2
6
(11)
2 Theory
2.2.3
Realizable k − ε
The utilized turbulence model was the Realizable k-ε (RKE) model which is a two-equation
model. The model includes one transport equation for turbulent kinetic energy (k) and
another transport equation for the dissipation rate (ε). RKE is an improved variation of
the standard k-ε model and it has two main differences which are [13, 15]:
• A new formulation of the eddy viscosity
• A modified transport equation for the dissipation rate
The standard k- overpredicts the eddy viscosity in large separation regions and hence,
produce non-realizable results in these situations which is why the RKE model was developed [15, 16]. The eddy viscosity formulation in the standard k-ε model contains a
constant. This was modified in the RKE model such that the eddy viscosity contains a
variable Cµ instead, see eq. (12) [13, 15].
νt = Cµ
k2
ε
(12)
The variable Cµ varies with the mean rotation rate, the strain rate tensor, the angular
velocity of the rotation as well as the turbulence quantities k and ε as shown in eq. (13).
Cµ =
1
√
A0 + AS
k
Sij Sij +Ω̃ij Ω̃ij
ε
(13)
Where A0 is a constant equal to 4.04 and AS is a parameter which is a function of the
strain rate. Ω̃ij is a function of the angular velocity, dissipation rate and the mean rotation
rate [15].
The modified transport equation for the dissipation rate was derived from an exact equation for the transport of the mean-square vorticity fluctuation as explained by Tsan-Hising
Shih et al. in [15].
2.2.4
Near Wall Flow
A boundary layer will form near any no-slip surface where the flow is attached. The
gradients in the boundary layer wall are very strong and the viscous effects are large.
The near wall region can be divided into three different layers. First, the viscous sublayer,
where the molecular viscosity is dominant and the flow has a laminar behaviour. Secondly,
the buffer layer, where the molecular viscosity and eddy viscosity are of similar magnitude.
Thirdly, the logarithmic layer, where the turbulent effects are dominant.[13, 17]
A variable, y + , is introduced as a dimensionless variable which defines the distance from
the wall and is based on wall shear stress, see eq. (14).
y+ =
7
Uτ y
ν
(14)
2 Theory
Where Uτ is the friction velocity and is defined as shown in eq. (15).
Uτ =
r
τw
ρ
(15)
Where τw is the shear stress at the surface. The near wall mesh resolution can be adjusted
to capture the viscous sublayer, if physics permits. A y + of approximately 1 is required
to properly resolve the viscous sublayer [11]. However, it would require a very fine mesh
in the near-wall region which would increase the computational time significantly. Instead
of resolving the sublayer, a wall function approach can be used to bridge the effects using
the law of the wall. The wall function approach requires the first cell to be located in the
logarithmic layer, which corresponds to a y + of 30-500.[11, 13] The region in between the
viscous sublayer and the logarithmic layer is the buffer layer (5 < y + < 30) as mentioned
previously. The velocity profile using the linear law as well as the log-law are visualized in
fig. 2 together with data from a DNS simulation, where u+ is the dimensionless velocity.
The DNS data are taken from a DNS simulation with a fully developed turbulent channel
flow with a Re of 4200 from [18].
25
u+
20
Linear law
Log-Law
DNS
15
10
5
0
1
5
10
y+
30
70
500
Figure 2: Velocity profile in the near wall regions for turbulent flow to visualize the linear-law
and the log-law to DNS data. DNS data was taken from [18]
The linear law is a linear relationship as seen in eq. (16) while the log-law refers to a
logarithmic relationship between u+ and y + , which are shown in eq. (17).[11, 12]
u+ = y +
u+ =
1
ln(y + ) + B
κ
(16)
(17)
κ is called the von Karman’s constant and is usually set to a value of 0.40-0.41 while
B is called the Additive constant and is usually approximately 5.1-5.5 depending on the
wall roughness. These constants are found from experimental data and are valid for all
turbulent high Reynolds number flows past relatively smooth walls.[11, 12]
8
2 Theory
2.3
Adjoint Solver
The adjoint solver is a gradient-based optimisation tool which provides sensitivities of the
surface of the geometry and other parameters with respect to an objective. There are
two types of adjoint solvers, the discrete solver and the continuous solver. The discrete
adjoint solver was implemented by Fluent since they believe that it provides more useful
sensitivity data, compared to the continuous approach, for their users [16, 19]. All theory,
methods and controls shown in the following subsections are equations and settings utilized
or available as user-options in Fluent 2019 R1.
2.3.1
Discrete Adjoint Method
Fluent’s adjoint solver is based on two main assumptions which are that the turbulence is
frozen and that the primal flow field is steady [16, 19]. Hence, the adjoint solver does not
take turbulent effects into consideration with respect to a change in geometry and flow
field. This is a drawback of the adjoint solver, especially in regions where turbulent effects
are significant such as in regions where the flow is separated.
A previous CFD simulation of the flow field is required since the adjoint approach makes
use of the flow variables to calculate the sensitivities. This implies that the primal flow
field needs to be steady for the adjoint solver to produce accurate and reliable results [20].
Hence, it is important that the RANS solution does not oscillate such that the adjoint
solver would produce different results depending on when the primal solution is stopped,
unless the oscillations take place in an region that is not of interest.
The discrete adjoint solver is discretized from the equations used in the flow field solver and
then linearized. Hence, the discrete adjoint equation is directly depending on the type of
scheme used to solve the flow equations [21]. This basic concept of Fluent’s adjoint solver
is explained below. The adjoint objective (d) is a function of both the state of the flow
(φ) and the control variables (c) as shown in eq. (18). The control variables are the points
which defines the surfaces of the car.
d = d φ(c), c
(18)
The governing equations can be written as in eq. (19) in compact form.
R φ(c), c = 0
(19)
The optimisation formulation is presented below, where the goal is to minimise the objective with the flow equations as the constraint.
minimize
d
subject to R = 0
The observation variable will change δd with a variation in both flow state and control
variable in a linear fashion as stated in eq. (20). The residual variation (δR) should remain
zero when the flow field is changed, see eq. (21).
9
2 Theory
∂d
∂d
δφ +
δc
∂φ
∂c
(20)
∂R
∂R
δφ +
δc = 0
∂φ
∂c
(21)
δd =
δR =
This results in a constrained optimisation problem. However, the problem can be converted
to an unconstrained formulation by introducing Lagrange multiplier (α). Hence, eq. (22)
is obtained from eq. (20) and 21 by introducing α.
δd =
∂d
∂R
∂R
∂d
δφ +
δc − α
δφ +
δc =
∂φ
∂c
∂φ
∂c
∂d
∂R
−α
δφ +
∂φ
∂φ
∂d
∂R
−α
δc
∂c
∂c
(22)
The value of α should be chosen such that the variations of the flow variables are eliminated
which requires the following relation (eq. (23)) to be solved:
∂d
∂R
=α
∂φ
∂φ
(23)
This results in an explicit formulation between the objective variable and the control
variables which is shown in eq. (24).
δd =
∂d
∂R
−α
δc
∂c
∂c
(24)
Equation (23) is the so-called adjoint equation and eq. (24) is the adjoint sensitivity
equation. Since the equations are linear they can be solved in one step. However, that
would put very high requirements on the computational resources in terms of both CPU
and RAM, especially when simulating larger models. In practice, the adjoint equations
are solved using an iterative procedure. First, the adjoint equations are computed. The
adjoint residuals from the computation are used to determine a correction to the adjoint
solution using algebraic multigrid (AMG) iterations (inner iterations). The correction is
added and the adjoint variables are updated. Then, the adjoint equations are computed
again until convergence is reached.[19, 22]
The derivation of the equations used in the discrete adjoint approach is based on the theory in ANSYS Fluent’s Advanced Add-On Modules documentation [19], ANSYS Fluent’s
User’s Guide [16] and ANSYS Adjoint Solver Best Practice Guide [22]. It is suggested to
go through ANSYS documentation if the reader would like even more details or additional
information regarding the discrete adjoint solver.
10
2 Theory
2.3.2
Solution Methods
The default schemes in Fluent’s adjoint module for the adjoint solution are standard for
pressure, first order upwind for momentum and Green-Gauss cell-based for the spatial
gradient discretization. These schemes are used as default due to their superior stability
compared to the second order schemes. However, if the same scheme is used in both the
primal RANS flow field calculation and in the adjoint solution, the most accurate discrete
derivative prediction is achieved [16, 23]. Using different schemes for the two solvers
can lead to a deterioration of the adjoint results even if a second order adjoint scheme
would reduce the truncation error [22, 24]. The difference can be observed by comparing
the observable in Fluent’s report definition to the value of the observable in the adjoint
module. All schemes that are implemented for the spatial discretization in the Fluent
solver are not available in the adjoint solver. Hence, Fluent implemented a feature which
makes it possible to achieve the ’best match’ between the flow field schemes and the adjoint
solution schemes. However, as mentioned, the first order schemes are recommended for
convergence stability [19].
In general, the adjoint calculation is much less stable compared to the primal solver and
the accuracy of the primal flow field calculation is of greater importance to achieve valuable
adjoint results since the primal flow results is used as input to the adjoint solver. Hence,
a combination with first order primal and second order adjoint schemes would not make
sense and would not necessarily result in a more accurate prediction either.
2.3.3
Solution Controls
This section contains information regarding the available settings in Fluent’s adjoint module including Fluent’s recommendations.
Solution-Based Controls Initialization
The ’solution-based controls initialization’ selects the advancement controls (which are
explained in detail below) based on the state of the flow for the initialization [19]. If
disabled, the solver starts with default values or the values specified by the user.
Auto-Adjust Controls
The ’auto-adjust controls’ is a feature which is implemented to improve convergence by automatically adjusting the advancements settings, within certain reasonable limits, during
the calculation.
Stabilized Scheme
Since the adjoint solver is based on a linear assumption, it can not damp any instabilities
that may grow during the solution. Hence, a stabilization scheme has to be used to limit
and damp any instabilities that might appear. Is it possible to run the adjoint solver
without the use of any stabilization scheme but it is highly recommended by Fluent to
use a stabilization scheme in problems with complex geometries, large separation regions
11
2 Theory
and/or large Reynolds number flows as well as models with high cell count (> 30 million)
[19, 25]. A stabilization scheme is also necessary to reduce oscillations and instabilities in
the adjoint solver when the cell count is very large [1].
Four different stabilization settings were available which are the modal, spatial, dissipation
and residual minimization scheme (RMS). However, the modal and spatial schemes are
not recommended for any flow case according to Fluent and were deprecated in Fluent
2019 R1 [26]. Hence, the dissipation scheme or the RMS are recommended for all complex
flow problems. Information about the dissipation and the RMS scheme follows below:
• The dissipation scheme introduces non-linear dampening to damp the growth of
instabilities. It has the capability to handle both weak and strong instabilities.
Another advantage is that the computational resources in terms of memory and
time are comparatively low. It is important to select the order of dampening to be
one order larger than the pressure and momentum scheme to reduce its effect on the
adjoint solution. The dissipation scheme is the second most recent developed adjoint
stabilization scheme and is one of the most effective schemes according to ANSYS
[1]. If the dissipation scheme is used but the solution advancement oscillates or
diverges, ANSYS recommends to increase the dampening factor by 10 and decrease
the spatial dampening order to 1.[22]
• The RMS is the most recent development and was implemented in Fluent 19.2 and
is based on a Krylov subspace using the generalized minimum residual method [16].
The memory requirements can be very high, depending on the number of modes
(which is user defined), but the overall residuals are guaranteed to decrease, which
makes it the most reliable and effective scheme [26]. The modes correspond to
the number of vectors in the subspace span of the Krylov sequence [27]. A larger
subspace span is beneficial for the residuals to decrease. The additional required
RAM for the RMS is directly proportional to the number of modes and to the cell
count of the model. Fluent recommend to initially run the adjoint solver without
any stabilization scheme or with the dissipation scheme before activating the RMS.
The residual minimization scheme should be activated when the solution has been
run for a number of iterations or when the adjoint solution starts to diverge [16].
Apply Preconditioning
If preconditioning is enabled, the following options are available:
• Courant Number (CFL), controls the stability and rate of the solution advancement.
A larger CFL value would decrease the computational time while increasing the risk
of stabilization issues [19]. The CFL number has the most influence on the convergence stability and rate among all available preconditioning and under-relaxation
settings. A typical CFL number is in the range of 1-5 for external automotive aerodynamics. However, cases with local instabilities might require a CFL < 1.[22]
• Artificial Compressibility Number (ACN), controls the adjoint continuity residual in
the inner iterations [16][28]. While it is possible to set the ACN above one, Fluent
recommend values in the range of 0.01 ≤ ACN ≤ 1 [19]. If the residuals during the
12
2 Theory
solution advancement seem to ’bounce’ it can be an indication of a too large ACN
value.[22]
• Flow Rate Courant Scaling (FRCS), is a scaling value that needs to be greater
than zero [19]. A higher value of the FRCS corresponds to a faster computation of
the inner AMG iterations. However, a too large value could cause instability and
divergence issues.
Preconditioning of the problem is when the problem is adapted to a form that is more suitable for numerical solving methods and, as mentioned previously, Fluent’s adjoint solver
was using iterative numerical methods. Hence, it is suitable for more complex problems
such as for most cases involving turbulent flow. If the preconditioning is disabled, the
CFL number is automatically set to an infinite value which can cause solver instabilities.
The preconditioning parameters explained above can be used to adjust the stability and
convergence rate of the inner iterations in the AMG solver [19]. If the inner iterations
in the AMG solver diverge or fail to converge within a reasonable number of iterations
specified by the user, a reduction in either CFL or FRCS may be required. Alternatively,
the ACN could be increased to reduce possible instabilities. However, a more aggressive
inner convergence of the AMG loop will generally result in an improved convergence of
the outer loop, as long as the inner AMG loop converges.
Under-Relaxation Factors
The under-relaxation factors (URF) are applicable for the adjoint momentum, continuity
and local flow rate equations. Any value between zero and one can be specified whereas
a higher value implies a more aggressive progression of the computation which could lead
to instabilities or, otherwise, a faster convergence rate. It is worth to mention that the
preconditioning settings have a much higher influence on the advancement stability than
the URFs [29].
Algebraic Multigrid
The AMG is, as mentioned previously, used to determine a correction of the adjoint
solution. The AMG algorithm has two settings which are ’Tolerance’ and ’Maximum
Iterations’, where the user can specify the desired convergence criteria as well as a set
number of maximum iterations for the inner AMG loop. A tolerance value greater than
0.1 is considered to be an aggressive setting while a value lower than 0.1 is considered to
be a conservative value. It is also recommended to keep the inner iterations between 3-5
and to decrease all residuals by a factor of 2 within each inner iteration to achieve a stable
outer convergence.[22]
2.3.4
Adjoint Literature Review
CFD adjoint solvers have been evaluated in several studies and are known to have stability
issues in various flow related problems. However, the stability of the solver has been
improved year by year with every new Fluent release and the adjoint solver has shown
great potential during the last couple of years where car models have been analysed and
13
2 Theory
improved using the adjoint method [1, 5, 6, 30, 31, 32]. This section aims to point out
recommendations regarding the mesh and the adjoint settings to avoid instabilities during
the solving procedure that have already been found in other studies.
First off, it was recommended to disable the ’Solution-based control initialization’ since it
caused instabilities in the solver when solving an external aerodynamics adjoint problem.
Additionally, the dissipation scheme was recommended to be used as the stabilization
scheme with default settings to achieve a steady convergence. However, it is worth to
mention that these conclusions were found in a study that was performed before the RMS
scheme was introduced as an option in Fluent’s adjoint solver.[32] In addition, ANSYS
recommended to always run the adjoint solver using the first order scheme for the momentum equations and then apply the second order scheme in the end of the design iterations
to increase the accuracy. However, other studies found it difficult to reach convergence
using the second order scheme, depending on the case [29, 30, 33, 23]. Hence, ANSYS
recommended to use first order schemes and claimed that it was usually ’accurate enough’
to provide useful sensitivity information [16]. The adjoint solver is highly dependant on
the convergence quality of the primal solution in order to obtain reliable adjoint results,
else it will most likely show non-physical results [3].
Another study found that the adjoint solver has to run using first order schemes with
a coarse mesh where the cell skewness is lower than 0.8 [30]. A study regarding the
application of adjoint methods on drag reduction of cars states that a high quality mesh is
required, if the geometry is complex, for the adjoint solver compared to the conventional
CFD solver [4]. ANSYS recommend to use a coarse mesh, both to reduce computational
time and to improve convergence of the solver [1, 25].
It is worth to note that most of these previous studies, where the adjoint method has been
applied to road vehicles, have been simulated with a simplified model of the car. That is
different configurations of the car e.g. car models without front and tail lights, suspension,
floor details, door gaps and without engine bay flow etc.[1, 5, 6, 30, 31, 32] Additionally,
none of the previous studies presented a clear methodology with a workflow or settings
that can be followed and used to evaluate other geometries. The RMS was not found to
be utilized in any previous study either which was most probably due to the recent release
of the RMS stabilization scheme in Fluent at the time when this study was performed.
14
3
Method
The method chapter contains information about the primal model setup as well as the
adjoint setup.
3.1
Model Setup
The geometry, primal solver settings, boundary conditions, computational domain size
and primal convergence criteria are presented in this section. It was considered important
to keep the simulation setup similar to CEVT’s procedures for external aerodynamics
regarding the solver settings, computational domain and meshing strategy to ensure that
the adjoint investigation and results were applicable to their current methods.
3.1.1
Geometry and Mesh
The analysed vehicle geometry was an SUV A which is shown in fig. 3. It is a detailed
geometry and included all details of the car such as suspension, brakes, engine bay area,
cooling fan, heat exchangers, exhaust system and floor details etc. A simplified model of
this geometry was also analysed to reduce the computational costs while investigating the
adjoint solver and is explained in further detail later in the report, see fig. 4 on page 17.
15
3 Method
Figure 3: SUV A geometry
The geometry was discretized in Harpoon, which is a hex-dominant mesher. The maximum
element size in the domain was set to 320 mm while the minimum size was set to 1.25 mm.
The large elements were placed near the outer boundaries of the domain while the smallest
elements were placed near the car. Several refinement boxes were created around the
vehicle in critical regions where large gradients were expected and to properly capture the
wakes around the vehicle. One large refinement box was created around the vehicle with
an extension behind the vehicle to capture the wake. Two smaller refinement boxes were
created at the rear of the vehicle with an even finer grid density. Additionally, refinement
boxes were created under the car as well as around and behind the mirrors, which is
presented in the following sections. Three prism layers were created on the surfaces of
the car which resulted in a y + around 30-70 in this case. The use of the refinement boxes
with small elements near the car together with the size of the outer prism layer ensures a
smooth transition from the prisms to the hex-elements.
The harpoon meshes were created using scripts to ensure mesh consistency, such that the
mesh procedure and location of grid refinements were the same every time a new mesh
was created, to avoid user errors. The primary setting for the user is to change a base
level mesh size, which all the sizings in the script were related to. Hence, the whole mesh
is coarsened/refined if the base level is changed.
16
3 Method
Simplified Car Geometry
The simplified model included a flat underbody and closed grilles which meant that the
engine bay flow, suspension, brakes and the fan were not included. Additionally, details
such as wipers and Moving Reference Frames (MRF) regions for the wheels where not
included either. These simplifications allowed for less complexity which resulted in more
stable solution advancements and faster computations of the model’s primal solution as
well as the adjoint solutions. Hence, more adjoint simulations were run in a shorter period
of time which allowed several different adjoint solver settings to be tested quicker.
The whole geometry of the car was divided into several different surface regions, called
Property Identification (PID), in the pre-processing software ANSA. This was important
to be able to refer to the individual PID regions and hence, facilitate the meshing procedure
while also having the possibility to monitor scalars/sensitivities on each PID during the
solution of the flow field and the adjoint. Also, depending on the used software, the use
of different PID’s makes it possible to morph just a certain surface or a certain set of
surfaces. The PID regions are shown in fig. 4, note that several colours look very similar
but are of different nuance.
Figure 4: The surfaces of the car were divided into different PID regions, where each specific
color is a separate PID (note that several colours look similar but are of different nuance)
Each PID was named with a certain mesh number which defined the level of mesh refinement at that specific surface. A cross-section view of the mesh at the symmetry plane near
the vehicle can be seen in fig. 5. A few of the refinement boxes are also visible in fig. 5.
The total cell count for the simplified geometry mesh was around 60 million elements.
17
3 Method
Figure 5: Mid-cross-section cut to visualize the mesh and the refinement boxes. Important to
note that this figure does not show the whole flow domain
The mesh quality is of great importance when running CFD simulations. A good mesh
quality will decrease the simulation time by increasing the rate of convergence and the
stability of the solver. ANSYS mesh recommendations for Fluent’s conventional flow solver
and the adjoint solver are to keep the maximum skewness less than or equal to 0.95 while
the minimum orthogonal quality should be kept larger than or equal to 0.05 [16, 34]. The
mesh recommendations are guidelines and the solver can reach convergence with worse
quality but the adjoint solver is more sensitive to poor orthogonal quality compared to
the primal flow solver [34, 35]. The mesh quality metrics from the simplified car geometry
mesh are shown in table 1. It was observed that the mesh quality metrics were at the
recommended limits for Fluent’s solvers. An ideal element has a skewness of 0, an aspect
ratio of 1 and minimum orthogonal quality of 1, which in the case of a hexahedron element
would be a cube [16].
Table 1: Mesh quality metrics for the baseline simplified car geometry mesh
Mesh Quality Metrics
Value
Maximum Skewness
Maximum Aspect Ratio
Minimum Orthogonal Quality
0.95
105
0.055
In addition, two more meshes were created. One without prism layers, to achieve a better
quality mesh since all of the elements with high skewness were found to be in the prism
layer of the mesh, and one coarser mesh. The mesh quality metrics from the mesh without
prism layers and the coarse mesh can be seen in table 2 and 3, respectively. The mesh
without prisms had much better quality compared to the previously created mesh and had
a cell count of 50 million. The coarse mesh was of poor quality and had a total cell count
of 30 million, which is roughly half compared to the baseline mesh. It is important to note
that mesh quality is taken from Fluent and the skewness can be one since the values are
automatically rounded.
18
3 Method
Table 2: Mesh quality metrics for the simplified car mesh without prisms
Mesh Quality Metrics
Value
Maximum Skewness
Maximum Aspect Ratio
Minimum Orthogonal Quality
0.73
69
0.083
Table 3: Mesh quality metrics for the coarse mesh
Mesh Quality Metrics
Value
Maximum Skewness
Maximum Aspect Ratio
Minimum Orthogonal Quality
1
163
0.051
Full Car Geometry
The full car geometry included all the details that CEVT includes in their standard aerodynamic analysis such as open front, engine bay area, porous zones for heat exchangers,
underbody details and MRF-regions for the wheels and for the fan etc. Same meshing
procedure used for the simplified car geometry was applied to the detailed model as well.
The mesh had a cell count of around 160 million cells and the cross section of the mesh for
the full car geometry are presented in fig. 6. The mesh quality metrics are shown in table 4
and it can be seen that the mesh is not within Fluent’s mesh quality recommendations.
Figure 6: Cross-section cut to visualize the mesh and the refinement boxes. Important to note
that this figure does not show the whole flow domain
19
3 Method
Table 4: Mesh quality metrics for the full car geometry
3.1.2
Mesh Quality Metrics
Value
Maximum Skewness
Maximum Aspect Ratio
Minimum Orthogonal Quality
1
264
0.025
Solver Setup
Fluent, version 2019 R1, which utilizes the finite volume method, was used as the CFD
solver. The turbulence model was set to RKE with enhanced wall treatment as the nearwall treatment for convergence stability reasons. The pressure-based solver was used since
the flow was assumed incompressible. The coupled algorithm was used as the pressurevelocity coupling. The main advantages of the coupled algorithm are that it is more
robust and efficient in terms of convergence rate [16]. The spatial gradient discretization
was set to the Green-Gauss Cell-based method which was a more robust scheme since
it is first order accurate and it has a comparatively low computing cost. The standard
pressure scheme was utilized and the first order upwind scheme was used for the turbulent
kinetic energy and the turbulent dissipation rate. For stability reasons, the momentum
equations were initially solved using the first order upwind scheme. After 1000 iterations,
the second order upwind scheme was applied and the solver was set to run to convergence.
The convergence assessment for the primal flow field is explained in section 3.1.5 on page
21.
3.1.3
Boundary Conditions
An inlet velocity of 27.78 m s−1 was specified normal to the inlet surface with a turbulent
intensity of 0.1% and a viscosity ratio of 200. The outlet was set to a pressure outlet with
a gauge pressure of 0 Pa. All the surfaces of the car were set to no-slip walls. The ground
under the car was set to a moving no-slip wall with the same velocity and direction as the
inlet velocity as shown in fig. 7. The other three far-field walls of the domain surrounding
the car were set to symmetry boundaries. Fluent assumes that the flux and gradients of
all quantities are zero at a symmetry boundary, which is also the case for a free slip wall
boundary [16].
Additional boundary conditions for the full geometry model were MRF regions, rotational
velocity of the tires as well as porous zones for the radiators and other porous regions.
All the rims had a specified fluid region each with an applied rotational velocity around
the wheel hub centre axis to match the free stream velocity of 27.78 m s−1 . Also, the tire
wall surfaces were set to a moving wall with the same rotational velocity as for the MRF
regions in the rims. Another MRF region was specified for the fan in the engine bay with
a rotational velocity that corresponds to its operational angular velocity at the specified
travelling velocity of the vehicle. There are several heat exchangers placed inside the grille
in the engine bay area. Each of these heat exchangers had its own fluid zone and each fluid
zone was set to be a porous zone with a specified viscous and inertial resistance based on
data from the hardware supplier of each radiator.
20
3 Method
Figure 7: Visualization of the domain and the boundary conditions (domain and vehicle not to
scale)
The fluid was set to air with the following properties; the density (ρ) and the dynamic
viscosity (µ) were set to 1.205 kg m−3 and 1.805 × 10−5 Pa s, respectively. This resulted in
a Reynolds number of approximately 2 700 000 which is defined as shown in eq. (25) [11].
Where the projected frontal area (A) of the SUV model was approximately 2.5 m2 .
√
ρu∞ A
Re =
µ
3.1.4
(25)
Computational Domain
The computational domain around the vehicle was calculated based on the dimensions
of the vehicle. The domain was ensured to be large enough to not affect the outcome of
the results. The domain was much larger behind to car to make sure that the wake was
captured properly.
3.1.5
Convergence Assessment
Since the adjoint solver requires a well-converged primal solution to provide reliable results
it was important to ensure that the mesh quality and convergence criteria did not have
a significant effect on the stability or the results of the final solution. The solution was
deemed converged only when two conditions were fulfilled. Firstly, once the residuals
were leveled out for 500 iterations at a sufficiently low value, usually when the continuity
equation and the residuals for the transport equations for the turbulent quantities were
less than 1E-3 and the momentum equations less than 1E-5. Secondly, when negligible
variations in CL and CD were obtained in the last 500 iterations. Alternatively, if the
solution reached a continuously oscillating state with a standard deviation of less than or
equal to 0.003 for both CL and CD .
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3 Method
3.2
Adjoint Setup
The adjoint set-up in Fluent is explained for the different cases in this section together
with information about adjoint convergence assessment. Settings according to the recommendations in the literature review and Fluent guides were implemented together with
tweaks of the solution control values to make the solution advancement more stable.
3.2.1
Adjoint Convergence Assessment
There are two ways to assess the convergence of the adjoint solver. First of all, the
adjoint residuals that are printed and shown during the solution advancement as in the
conventional flow solver. However, it was important to not blindly look at these adjoint
residuals. Depending on the selected observable, which in this study was set to be the
drag force, an adjoint ’wake’ might be present in front of the vehicle due to a backward
calculation of the adjoint field. This leads to a region of several cells with higher residuals
which makes the global adjoint residuals misleading. This is not necessarily an issue since
the wake is most probably relatively far away from the vehicle and surface of interest but
should be treated with caution. Other areas where higher residuals can be expected, if
a hex-dominant mesh is used, are at the edges of the refinement boxes due to the rapid
change of element size. Hence, it was recommended to make sure that the adjoint residuals
level out and to pay attention to the residuals around the surfaces of interest. Therefore,
it was decided to monitor the sensitivities of the important surfaces. The adjoint residuals
in the domain from the last iteration can be saved by activating 5 User Defined Memory
(UDM) locations and store the adjoint residuals in each memory location. The residuals
can then be visualized using the cell registers in Fluent.
Secondly, at the time of this study, no built-in function to monitor convergence of the
adjoint solver, like Fluent’s report monitors for the primal flow field, exists. The adjoint
solver is located in another module within Fluent and hence, the monitor features are not
available during the adjoint solution process. Instead, a separate scheme script was used
(mainly developed by ANSYS [24] but modified and adapted in this study to fit each case)
and implemented in the journals to write adjoint surface data at every desired iteration
to a file. Another script was written in Python to clean, read and plot the data. This
surface sensitivity data was used as an additional check to assess convergence of the adjoint
solution, i.e the solution was converged when negligible differences in the sensitivities were
obtained during a sufficiently large number of iterations depending on the stabilization
scheme utilized. However, the use of the scheme script for monitoring sensitivities during
the solution increased the computational time significantly and stored a huge amount of
’trash’ data files in addition to the intended file. Hence, the monitor scripts were not
always used to store data every iteration but rather when it was necessary, such as, at the
end of the adjoint simulation and with a larger output interval than every iteration.
3.2.2
Adjoint Solver Setup
Both the simplified and the full car model were divided into different regions to be able to
write surface data to assess convergence. Several surfaces at different locations of the car
were added to the adjoint monitor scripts. The location of each surface on the simplified car
are shown in fig. 8 (the corresponding surfaces were monitored in the full car simulations).
22
3 Method
The simulations were initially set to write sensitivity data, using the script mentioned
in the previous section to assess convergence, every iteration which greatly increased the
simulation time, especially for the dissipation case. In further simulations where the
dissipation stabilization scheme was used it was specified to write surface sensitivity data
every 25th iteration instead. The computational time for the RMS was not as affected by
the output intensity since the sub-iterations took the most time. The full car simulations
with the RMS would require an approximated 3 TB additional RAM which is not feasible
with today’s computational resources and it would require several hundreds of thousand
CPU hours. The monitored surfaces for the full car simulations with the dissipation
scheme were only exported during the last 2 000 iterations and with a write interval of 50
iterations to reduce the simulation time.
Figure 8: Surfaces and their respective names that were monitored during the adjoint solution to
assess convergence
The solution methods for the adjoint gradient, pressure and momentum discretization
were, as recommended, set to Green-Gauss Cell-Based, standard and first order upwind,
respectively. Hence, the gradient and the pressure scheme were the same in both the
primal flow solver and the adjoint solver. The solution-based control initialization and
the auto-adjust controls were disabled. Stabilization schemes were used and are explained
later in this subsection. The preconditioning was enabled for all cases and the specific
settings were very conservative due to the complex geometries (see table 5).
23
3 Method
Table 5: Advancement control settings
Advancement Control
Value
CFL
ACN
FRCS
URF Adjoint Momentum
URF Adjoint Continuity
URF Adjoint Local Flow Rate
Tolerance
Maximum Iterations
0.2
0.01
0.2
0.2
0.2
0.2
0.0001
30
Dissipation Scheme
The dissipation scheme was activated with the settings shown in table 6. It is worth to
note that the first order damping was used with a larger damping factor than default. The
simulations with the dissipation scheme were set to run for 5 000 iterations.
Table 6: Stabilization scheme settings for the dissipation scheme
Dissipation Scheme Setting
Value
Damping Factor
Damping Relaxation
Damping Order
Suppression
10
0.9
1
Enabled
Residual Minimization Scheme
The residual minimization scheme utilized the settings shown in table 7. The number
of elements using the baseline mesh together with 800 modes was found to required a
massive amount of RAM and cannot be increased due to the computational limitations.
The predicted additional memory required for the RMS, with these settings and the 60
million mesh was 1.1 TB. The RMS demands a huge amount of RAM and would not
be feasible for larger models with today’s resources if the number of modes is kept at the
same number. Larger models with more details will most likely have more local instabilities
which would require even more modes to stabilize the solution. Simulations with 100, 200
and 400 modes were also investigated using the simplified model and the baseline mesh.
Due to instabilities and rapidly growing residuals without any stabilization scheme, it
was chosen to initially run the adjoint solver with the dissipation scheme and the settings showed in table 6. The RMS was then activated after 100 iterations and set to run
for another 30 iterations. Each of these 30 iterations was set to run with a number of
sub-iterations, one for each mode minus the recycled modes, which makes the RMS computational demanding in relation to both time and hardware requirements. The number
of recycled modes were always kept 0.25 times the number of modes.
24
3 Method
Table 7: Stabilized Scheme Settings for the RMS
3.3
RMS Setting
Value
Number of Modes
Number of Recycled Modes
AMG Iterations
800 (100,200,400)
200 (25,50,100)
6
Post-processing
The pressure coefficient (Cp ) from the primal flow field simulations was presented together
with the adjoint results to be able to easily interpret and judge if the adjoint results seemed
physically reasonable. The adjoint residuals and the surface sensitivity monitor plots were
presented for each case to assess stability and convergence of the adjoint solver.
There are several different ways to visualize the results computed from an adjoint solution
that could be used to manually locate and improve the shape of the car or to automatically
morph the geometry to achieve lower drag. The adjoint solution data visualized in this
study is the log10 shape sensitivity magnitude (SSM) and the normal optimal displacement
(NOD).
The SSM highlights regions of the geometry where a small change in the geometry results
in a large change in the objective. A higher SSM value indicates a more sensitive region.
Due to a large range of the SSM values, the log10 (SSM ) was presented instead to make
it easier to analyse and interpret the results. This allows the engineer to find regions
of the car where very small geometry changes can be made to decrease the drag of the
vehicle.[16] While looking at the SSM results, it is important to remember that even the
surfaces where the SSM are low can have a significant impact on the drag. However, that
requires a small change over a larger area or a large change over a small area. However, it
is not possible to extract information regarding the magnitude of the surface displacement
from the SSM.
The NOD shows how much and in which direction the surfaces should be displaced to
achieve, in this case, lower drag. A positive value indicates a normal displacement of the
surface into the flow domain, such that the surface is pulled outwards. In the same way, a
negative value indicates that the surface should be pushed inwards to reduce the drag.[16]
It is important to note that the NOD is not calculated during the adjoint solution but
instead calculated afterward using computed adjoint sensitivity data utilizing Fluent’s
’Design-Tool’ in the adjoint module [16]. Hence, it should be treated with caution but can
be very informative and of great value.
There are options available in the design tool to set the desired drag reduction to a certain
percentage or to optimally decrease the drag which should result in an accurate prediction
of the objective. The optimum decrease was chosen because of the possible misprediction
that can occur if a too large percentage value is set. This possible misprediction can be
attributed to the non-linearities of larger geometry changes in relation to the flow field.
The geometry change was done in Fluent without any restrictions to verify the adjoint
data.
The settings utilized in the design tool to calculate the NOD for each case are shown in
25
3 Method
table 8. The selected zones to be modified for the car were all the surfaces of the car while
only the surfaces of the side view mirrors were selected in the case study. The case study
is a part of this study where the adjoint results of only the side view mirrors are analysed
separately. The same surfaces were selected as the regions to create a design space around
the object. The design space is then filled with a number of points called motion points
(MP). The number of MPs in X, Y and Z direction utilized to calculate the NOD is case
dependent. It is important that the number of MPs are less than the number of mesh
nodes in each direction. That is to avoid rough elements in the surfaces since the adjoint
data is not always perfectly smooth which can induce separation and hence, increase the
drag instead of reducing it. However, a too smooth NOD might not always be desired
since smaller details and shapes can easily be missed out. This is a trade-off which has to
be decided by the user to achieve the desired information about the surface(s) of interest.
When the NOD was calculated in this study a coarser set of MPs were used for the car
to get an overview of the results. However, when the side view mirrors were analysed
separately, a much finer set of MPs were used to achieve a more detailed NOD map of the
mirror surfaces.
Table 8: Settings utilized in the design tool to calculate NOD for the car as a whole and for the
side view mirrors separately
Setting
Car
Side View Mirrors
Freeform Scale Factor
X Motion Points
Y Motion Points
Z Motion Points
Freeform Max Iterations
Freeform Constraint Relaxation
Freeform Preconditioning
Freeform Parameter Relaxation
0.1
25
25
25
250
0.1
4
0.6
1
15
80
15
500
0.1
20
0.5
26
4
4.1
Results
Simplified Car Model
The simplified car geometry was run with all possible combinations of the different meshes
and stabilization schemes explained previously and are presented in the following sections.
4.1.1
Baseline Mesh Settings
The pressure coefficient on the surfaces of the vehicle from the primal solution using the
baseline mesh is shown in fig. 9. As expected, high-pressure regions were observed in front
of the car, at the windshield and at the mirrors. It was also observed that relatively high
pressure was present at the leading edge of the B and C-pillars. However, the pressure in
front of the car was much higher.
Figure 9: Pressure coefficient (Cp ) on the surfaces of the simplified vehicle
Figure 10 shows the regions of the domain with an isosurface with a total pressure coefficient of zero which is colored by the velocity magnitude. It was observed, and expected,
that a large wake is present behind the vehicle. There are also wake regions behind the
side view mirrors, around the A and C-pillars and at the wheels.
27
4 Results
Figure 10: Isosurface with a total pressure coefficient of zero to visualize wake regions, colored
by the velocity magnitude (the bottom view is from the back of the vehicle)
The residuals from the adjoint simulations utilizing the dissipation scheme and the RMS
as the stabilization schemes are shown in fig. 11a and 11b, respectively. It was observed
that the residuals in the RMS case dropped very rapidly once the RMS was activated and
stabilized quickly after just a few iterations. The residuals using the dissipation scheme
were oscillating and required around 5 000 iterations before the residuals leveled out.
The adjoint solutions using the dissipation scheme and the RMS took 13 000 and 23
300 CPU hours to run, respectively. However, since the adjoint solution was difficult to
converge, the simulations were run using very conservative settings. Additionally, the
RMS simulation was run 30 iteration even though it converged at around 15-20 iterations
as seen in fig. 11b. Both these simulations were set to write sensitivity data every iteration
which greatly increased the simulation time, as mentioned in section 3.2.2.
28
4 Results
101
Adj-Con
Adj-xMom
Adj-yMom
Adj-zMom
Adj-Flowrate
10−1
100
Residuals
Residuals
100
10−2
10−3
10−1
Adj-Con
Adj-xMom
Adj-yMom
Adj-zMom
Adj-Flowrate
10−2
10−3
10−4
0
1000 2000 3000 4000 5000
Iterations
(a) Residuals utilizing the dissipation scheme
0
20
40
60 80 100 120
Iterations
(b) Residuals utilizing the RMS
Figure 11: Adjoint residuals using different stabilization schemes
The region of the domain where the highest residuals were observed are presented in
fig. 12. The adjoint residual for the momentum equation in the x-direction at a cross
section plane in the middle of the domain are shown. However, it is worth to mention that
all residuals were high in the same region. It was observed that the so-called ’adjoint wake’
was present in front of the vehicle, as previously explained to be expected in section 3.2.1.
This region with higher residuals was observed to be present in all adjoint simulations
using the simplified geometry. From fig. 12 it can also be observed that slightly higher
residuals were found at the transition between the large refinement box and the far field
mesh.
Figure 12: Residual (adj-xMom) contour of a cross section plane simulated with the dissipation
scheme (the whole domain is not visible in this figure)
The area-weighted-average (AWA) log10 SSM monitored during the adjoint solutions utilizing the RMS and the dissipation scheme are presented in fig. 13 and 14, respectively.
The location of the monitored surfaces was shown in the method chapter in fig. 8. It
was observed that the sensitivities leveled out quickly once the RMS was activated. The
29
4 Results
7
2500 5000
Iterations
Headlights
6
5
0
2500 5000
Iterations
5
0
7
2500 5000
Iterations
Rear Spoiler
6
5
0
2500 5000
Iterations
6
5
0
7
2500 5000
Iterations
Tail Lights
6
5
0
2500 5000
Iterations
AWA log10 SSM
0
6
Bumper
7
AWA log10 SSM
5
7
AWA log10 SSM
6
A-Pillars
AWA log10 SSM
Side View Mirrors
AWA log10 SSM
7
AWA log10 SSM
AWA log10 SSM
AWA log10 SSM
sensitivities using the dissipation scheme were initially oscillating and several surface sensitivities still showed an increasing/decreasing advancement behaviour even after 5000
iterations. However, the sensitivities at a few of the monitored surfaces were found to
level out, such as for the A-pillars, bumper, headlights etc.
Roof
7
6
5
0
2500 5000
Iterations
Rear Body
0
2500 5000
Iterations
7
6
5
Figure 13: Surface sensitivity monitors using the dissipation scheme
7
6
5
0
65 100
Iterations
5
0
7
65 100
Iterations
Rear Spoiler
6
5
0
65 100
Iterations
6
5
0
7
65 100
Iterations
Tail Lights
6
5
0
65 100
Iterations
AWA log10 SSM
65 100
Iterations
Headlights
Bumper
7
AWA log10 SSM
0
6
AWA log10 SSM
5
7
AWA log10 SSM
6
AWA log10 SSM
7
A-Pillars
AWA log10 SSM
AWA log10 SSM
AWA log10 SSM
Side View Mirrors
Roof
7
6
5
0
65 100
Iterations
Rear Body
0
65 100
Iterations
7
6
5
Figure 14: Surface sensitivity monitors using the RMS (the RMS scheme was activated at
iteration 100)
The log10 SSM for both stabilization schemes are presented in fig. 15. It was observed that
the most sensitive regions for both cases were the same in general. These regions were
the A-pillars, the edge between the bumper and the hood as well as the pillars around the
30
4 Results
windows (B and C-pillars) and the rear spoiler. The largest difference between the two
cases was seen around the tail lights of the vehicle.
Figure 15: log10 SSM for the simplified car geometry using different stabilization schemes
The NOD as well as the actual geometry change are presented in fig. 16 and 17, respectively. The NOD was observed to be similar for the two stabilization schemes. The NOD
results are reasonable from a physical point of view compared to the Cp in fig. 9. The
adjoint results proposed that the bumper should be pushed forward to reduce the size of
the high-pressure region by making the front sharper while decreasing the size of the rear
part of the car to reduce the size of the wake. It was also observed that a part of the side
view mirrors was pushed forward while the rest of the mirror was displaced inwards, to
achieve a more aerodynamic shape, shaping it towards a droplet. The actual geometry was
obtained through mesh morphing in Fluent based on the NOD results. As seen in fig. 17,
the whole car was morphed (including the tires and windshield) which is unrealistic but
allowed for the adjoint results to be verified.
31
4 Results
Figure 16: NOD for the simplified car geometry using different stabilization schemes
Figure 17: Actual geometry change after the adjoint optimisation. Red indicates that the
surface was displaced outwards and blue indicates that the surface was displaced inwards or that
the geometry retained its shape from the baseline
The predicted value of CD A in Fluent compared to the actual verified values are presented
in table 9. It was observed that the prediction was off by around 3.5 % for the RMS and
2.3 % for the dissipation case. The CD A decreased roughly 4 % compared to the baseline
32
4 Results
case for the RMS while the dissipation scheme resulted in a CD A decrease of 4.87 %.
Table 9: Actual and predicted values of ∆CD A for the two stabilization schemes
4.1.2
Case
Actual ∆CD A [%]
Predicted ∆CD A [%]
Baseline
RMS
Dissipation Scheme
-4.05
-4.87
-7.60
-7.21
Stabilization Schemes and Meshes
The results from the coarse mesh and the mesh without prisms are presented in this
section. The primal flow solutions obtained from these two cases compared to the primal
baseline mesh case showed a relatively small difference. The ∆CD A for the other meshes
was found to be within 3.5 % to the baseline case, see table 10. It was also observed from
fig. 18 that the pressure fields of the vehicles were similar in both cases. This comparison
was performed to ensure that the other meshes primal flow fields and values were similar
to the baseline mesh flow field. If too large differences would have been seen already in
the primal flow field, the adjoint results would be very inaccurate.
Table 10: ∆CD A (%) comparison for the different meshes
Case
∆CD A [%]
Baseline
Without prisms
Coarser mesh
-3.5
-1.5
Figure 18: Cp contours on the surface of the car using different meshes
33
4 Results
Using the Dissipation Scheme
The adjoint solver was run utilizing the mesh without prism layers and the coarser mesh
with the dissipation scheme. The residuals for both cases are presented in fig. 19a and 19b.
It was observed that the residuals were much more stable and leveled out 1 500 iterations
earlier in both cases compared to the baseline mesh case. The mesh without prism layers
and the coarse mesh took 8 600 and 4 300 CPU hours to run, respectively.
Adj-Con
Adj-xMom
Adj-yMom
Adj-zMom
Adj-Flowrate
Residuals
100
10−1
10−2
10−3
101
Adj-Con
Adj-xMom
Adj-yMom
Adj-zMom
Adj-Flowrate
100
Residuals
101
10−4
10−1
10−2
10−3
10−4
0
1000 2000 3000 4000 5000
Iterations
0
(a) No prism mesh
1000 2000 3000 4000 5000
Iterations
(b) Coarse mesh
Figure 19: Adjoint residuals using different mesh setups
The monitored surface sensitivities from the two adjoint simulations are shown in fig. 20. It
was observed that the surface sensitivity between the two cases differs in both advancement
behaviour and quantity. It was also observed that the coarser mesh model experienced
fewer oscillations for the side view mirrors, bumper and the tail lights compared to the
finer meshes. All the monitored surface sensitivities are not converged such as the rear
body for the no prism mesh and roof for the coarser mesh.
34
4 Results
0
6
5
5
2500 5000
Iterations
Headlights
0
2500 5000
Iterations
0
7
AWA log10 SSM
AWA log10 SSM
7
6
5
5
2500 5000
Iterations
Rear Spoiler
0
AWA log10 SSM
6
6
0
7
2500 5000
Iterations
2500 5000
Iterations
Tail Lights
6
5
0
2500 5000
Iterations
Roof
7
6
5
0
2500 5000
Iterations
Rear Body
0
2500 5000
Iterations
7
AWA log10 SSM
5
Baseline
Coarse
No Prism
7
AWA log10 SSM
6
Bumper
7
AWA log10 SSM
7
A-Pillars
AWA log10 SSM
AWA log10 SSM
Side View Mirrors
6
5
Figure 20: Surface sensitivity monitors using the dissipation scheme and different meshes. The
legend in the first subplot is valid for every subplot in this figure
The log10 SSM was observed to be slightly larger and more pronounced using the coarse
mesh compared the mesh without prisms, see fig. 21. It was observed that the largest
differences were at the C-pillars and around the tail lights.
Figure 21: log10 SSM for the simplified car geometry simulated with different meshes using the
dissipation scheme
35
4 Results
The NOD comparison between the different meshes are shown in fig. 22. It was observed
that the NOD results of no prism mesh were slightly asymmetric. However, both the Cp
and the log10 SSM were observed to be symmetric over the vehicle. It can also be seen
that the front should be displaced outwards for both cases, as observed for the baseline
mesh case as well. The NOD at the rear spoiler is showing the opposite to one another,
where the no prism mesh shows the best correlations with the baseline mesh.
Figure 22: NOD of the car using different meshes simulated with the dissipation scheme
Residual Minimization Scheme
The residuals from the adjoint simulation utilizing the RMS for the mesh without prism
and the coarser mesh are shown in fig. 23a and 23b, respectively. The former took around
21 000 CPU hours to simulate while the latter required around 8 600 CPU hours to finish.
It was observed that the residuals drop rapidly once the RMS scheme was activated (at
iteration 100) and levels out quickly afterward.
36
4 Results
100
10−1
Adj-Con
Adj-xMom
Adj-yMom
Adj-zMom
Adj-Flowrate
10−2
Residuals
Residuals
100
10−3
10−4
Adj-Con
Adj-xMom
Adj-yMom
Adj-zMom
Adj-Flowrate
10−1
10−2
10−3
0
20
40
60 80 100 120
Iterations
0
(a) No prism mesh
20
40
60 80 100 120
Iterations
(b) Coarse mesh
Figure 23: Adjoint residuals using different mesh setups
The monitored surface sensitivities for the two meshes are presented in fig. 24. It was
observed that the sensitivities from the coarse mesh were leveled out while most of the
obtained sensitivities from the no prism mesh were found to increase or decrease. The
behaviour and magnitude from the coarse mesh correlates well with the monitored surface
sensitivities from the baseline mesh.
0
6
5
5
65 100
Iterations
Headlights
0
65 100
Iterations
0
7
AWA log10 SSM
AWA log10 SSM
7
6
5
0
6
5
65 100
Iterations
Rear Spoiler
AWA log10 SSM
6
0
7
5
65 100
Iterations
65 100
Iterations
Tail Lights
6
0
Roof
7
6
5
0
65 100
Iterations
Rear Body
0
65 100
Iterations
7
AWA log10 SSM
5
Baseline
Coarse
No Prism
AWA log10 SSM
6
7
Bumper
7
AWA log10 SSM
7
A-Pillars
AWA log10 SSM
AWA log10 SSM
Side View Mirrors
65 100
Iterations
6
5
Figure 24: Surface sensitivity monitors using the RMS and different meshes. The legend in the
first subplot is valid for every subplot in this figure
The log10 SSM contours are presented in fig. 25 and it was observed that the sensitivity
37
4 Results
similar similar for the baseline mesh and the coarse mesh. It was also observed that the
sensitivity for the mesh without prism was lower on the bumper, hood and roof compared
to the baseline case.
Figure 25: log10 SSM for the simplified car geometry simulated with different meshes using the
RMS
The NOD was observed to be very similar in the front of the vehicle, as seen in fig. 26. The
rear of the vehicle differed slightly whereas the case without prisms wanted to displace
more of the rear outwards in a circular like fashion.
38
4 Results
Figure 26: NOD of the car using different meshes simulated with RMS
4.1.3
Varying Number of Modes using RMS
Three additional simulations were run with the baseline mesh to investigate if less number
of modes can be used to decrease the computational time and demands. The three runs
utilized 100, 200 and 400 modes and the computational time and requirement for these
simulations are compared to the baseline simulation with 800 modes in table 11. As
mentioned in the theory, the computational requirement is proportional to the number of
modes. However, it was observed that the computational time was not proportional to
the number of modes but rather have an exponential relation.
Table 11: Comparison of the computational time and the RAM required for different number of
modes using the baseline mesh
Number of Modes
100
200
400
800
Modes
Modes
Modes
Modes
RAM [GB]
Computational Time [CPU Hours]
140
280
560
1 100
1 300
2 800
7 000
23 000
The residuals from the iteration where the RMS was activated for the 100, 200, 400 and
800 mode simulations are presented in fig. 27a, 27b, 27c and 27d, respectively. It was
observed that residuals leveled out in all cases. However, the residuals were found to be
lower when simulations were run with a larger number of modes.
39
4 Results
100 Modes
101
Adj-Con
Adj-xMom
Adj-yMom
Adj-zMom
Adj-Flowrate
10−1
100
Residuals
Residuals
100
200 Modes
101
10−2
10−3
10−1
10−2
10−3
10−4
100 105 110 115 120 125 130
Iterations
10−4
100
115
120
125
130
125
130
(b) RMS using 200 modes
400 Modes
800 Modes
101
100
Residuals
100
Residuals
110
Iterations
(a) RMS using 100 modes
101
105
10−1
10−2
10−3
10−1
10−2
10−3
10−4
100
105
110
115
120
125
10−4
100
130
Iterations
105
110
115
120
Iterations
(c) RMS using 400 modes
(d) RMS using 800 modes
Figure 27: Adjoint residuals utilizing the RMS scheme with varying number of modes with the
baseline mesh
The monitored surface sensitivities and the log10 SSM from the simulations with a varying
number of modes are shown in fig. 28 and 29, respectively. It was observed that the surface
sensitivities were very similar. More modes resulted in more stabilized sensitivities for most
of the monitored surface. It was also observed, that the 400 mode data was found to result
in a closer match to the 800 mode data compared to the 100 and 200 mode data.
40
4 Results
AWA log10 SSM
6.5
6.0
5.5
100
115 130
Iterations
6.5
6.0
5.5
100
5.5
5.0
100
115 130
Iterations
Rear Spoiler
6.5
115 130
Iterations
Bumper
5.5
100
115 130
Iterations
5.5
5.0
100
115 130
Iterations
Tail Lights
6.0
Roof
6.0
AWA log10 SSM
AWA log10 SSM
6.0
7.0
6.5
100
115 130
Iterations
Headlights
A-Pillars
6.5
AWA log10 SSM
5
100
100 Modes
200 Modes
400 Modes
800 Modes
AWA log10 SSM
6
7.5
AWA log10 SSM
Side View Mirrors
AWA log10 SSM
AWA log10 SSM
7
115 130
Iterations
Rear Body
6.0
5.5
100
115 130
Iterations
Figure 28: Monitored surface sensitivities for each adjoint run with varying number of modes.
The legend in the first subplot is valid for every subplot in this figure
41
4 Results
Figure 29: log10 SSM on the surface of the car for different amount of modes
It was observed from fig. 30 that the largest differences between the NOD on the surfaces
of the car were at the rear of the car. The simulation with 100 modes proposed a larger
displacement of the geometry in the rear of the car compared to the 800 mode case. The
200 mode case was found to be in better correlation with the 800 mode case and proposed
a displacement that better fit the 800 mode case.
42
4 Results
Figure 30: NOD of the surface of the car for different amount of modes
4.2
Full Car Model
The Cp on the surfaces of the full car is presented in fig. 31. It was observed that a high
pressure-region exists at the bumper of the car. However, this region is smaller compared
to the simplified model since air flows through the upper and lower grilles and into the
engine bay. A few elements stick out from the middle of the hood in the low-pressure
region which was caused by the mesher but did not affect the results significantly since
they were few and very small.
43
4 Results
Figure 31: Pressure coefficient on the surfaces of the full car model
An isosurface of the total pressure coefficient set to a value of zero is presented in fig. 32
and colored by the velocity magnitude. It can be seen that a large and unsymmetrical
wake are present in the rear of the vehicle. It was also observed that there were larger
regions of zero total pressure behind the mirrors and at the A and C-pillars, just as in the
simulations with the simplified model.
Figure 32: Isosurface with a total pressure coefficient of zero to visualize wake regions, colored
by the velocity magnitude (the bottom view is from the back of the vehicle)
The residuals and the monitored surface sensitivities are shown in fig. 33 and 34, respectively. The adjoint residuals were observed to oscillate and the simulation required around
52 000 CPU hours to reach 5000 iterations. The surface sensitivities did not level out and
most of the monitored surfaces had values decreasing/increasing instead of stabilizing.
44
4 Results
101
Adj-Con
Adj-xMom
Adj-yMom
Adj-zMom
Adj-Flowrate
Residuals
100
10−1
10−2
10−3
10−4
3000
4000
Iterations
5000
4.5
3000 4000 5000
Iterations
5.5
3000 4000 5000
Iterations
Rear Spoiler
7
6
5
3000 4000 5000
Iterations
Bumper
6
5
3000 4000 5000
Iterations
Tail Lights
7
6
5
3000 4000 5000
Iterations
AWA log10 SSM
6.5
7
AWA log10 SSM
5.5
A-Pillars
AWA log10 SSM
6
3000 4000 5000
Iterations
Head Lights
6.5
7.5
AWA log10 SSM
7
AWA log10 SSM
Side View Mirrors
8
AWA log10 SSM
AWA log10 SSM
AWA log10 SSM
Figure 33: Adjoint residuals of the last 2000 iterations from the full car simulation
7
Roof
6
5
3000 4000 5000
Iterations
Rear Body
6.5
5.5
4.5
3000 4000 5000
Iterations
Figure 34: Monitored surface sensitivities from the full car simulation
The region of high residuals was observed in front of the vehicle and is shown in fig. 35.
The region extends all the way from the inlet to the bumper of the vehicle.
45
4 Results
Figure 35: Residual (adj-xMom) contour at the mid plane
The log10 SSM on the surfaces of the full car is presented in fig. 36. It was observed that
most sensitive regions were found at the A-pillars and the tail lights as well as most of the
sharp and small corners and edges.
Figure 36: log10 SSM on the surfaces of the full car
The NOD of the full car model is presented in fig. 37. It was observed that the NOD in
the front of the vehicle was asymmetric.
46
4 Results
Figure 37: NOD of the surfaces of the full car
4.3
Case Study - Single Part Adjoint Optimisation
The side view mirrors were analysed in more detail to verify the applicability of the
adjoint method. The NOD of the side view mirrors, from the coarse mesh case that was
run with the RMS, was analysed independently with an informative variable range of the
legend to facilitate the visualization and interpretation of the NOD. The monitored surface
sensitivity for the side view mirrors during the adjoint solution was found to oscillate with
small variations each iteration. The NOD of the side view mirror is presented in fig. 38.
Figure 38: NOD of the side view mirror
The geometry of the mirrors was adjusted in accordance with the NOD shown above.
For simplicity, the geometry change was performed by morphing the mesh in Fluent as
done previously with the simplified car model but this time only with the side view mirrors. The morphing was performed without any constraints and with a relatively large
displacement allowance (scaled 1:1 with the NOD values) which resulted in a maximum
allowed displacement of 14 mm at the blue regions on the outside side of the mirrors.
The geometry change is shown in fig. 39, where the baseline and the optimised mirror
are overlapped. The red mirror is the optimised shape while the transparent grey is the
47
4 Results
baseline geometry. It was observed that the projected frontal area of the optimised side
view mirrors is smaller than the baseline geometry. However, the projected frontal area of
the two optimised mirrors were only 1.5 % less compared to the baseline design. Hence,
the projected frontal area of the whole vehicle remains more or less the same since the
mirrors are fractions of the total projected frontal area of the car.
Figure 39: Geometry change between the baseline shape (colored in transparent grey) and the
optimised shape (colored in red)
The ∆CD A of the whole car decreased significantly and is presented in table 12. The drag
force on the mirrors decreased 22.75 % compared to the baseline design.
Table 12: Actual and predicted values of ∆CD A for the baseline car compared to the baseline
car with morphed mirrors
Case
Actual ∆CD A [%]
Predicted ∆CD A [%]
Baseline Car
Car with Optimised Mirrors
-0.62
-0.83
A comparison of the static pressure coefficient and an isosurface with a total pressure
coefficient of zero for both the baseline geometry and the optimised design are presented
in fig. 40 and 41, respectively. It was observed that the pressure region in front of the mirror
was larger with the baseline geometry compared to the optimised design. Additionally, it
was observed from fig. 41 that the region of the wake was smaller for the optimised design
compared to the baseline design.
48
4 Results
Figure 40: Comparison of the pressure coefficient on the side view mirror between the baseline
and optimised shape
Figure 41: Comparison of isosurface with a total pressure coefficient of zero around the side view
mirror between the baseline and optimised shape colored by the velocity magnitude
49
5
5.1
Discussion
Potential & Applicability of the Adjoint Solver
The potential of the adjoint solver was investigated both by performing a literature study
as well as CFD and adjoint simulations. The adjoint results in the literature review give
the impression that it has great potential and is widely applicable to different simplified
geometries for external aerodynamic flow. However, as mentioned, the details regarding
the adjoint method in these studies were very vague and it would not be possible to recreate any simulation setup. Hence, several adjoint simulations were run with different
settings to further examine the potential and applicability of the adjoint solver.
It was found that running adjoint simulations were more difficult than initially thought,
especially after going through the previous studies which make the adjoint simulation
procedures look very straight forward. However, it was found that Fluent’s adjoint solver
is applicable to external aerodynamic simulations to a certain extent and is discussed
further in section 5.2 and its following subsections.
The adjoint solver was found to provide sensitivities of the surfaces as well as NOD recommendations. This is valuable information which can be used to optimise the car in
unexpected locations and directions. The adjoint optimisation method is preferred over
other optimisation methods such as design of experiment since there are no specific parameters and/or dimensions of the shapes but rather free-form surfaces. The biggest
limitations of the adjoint solver is the frozen turbulence assumption and the fact that the
adjoint equation is linearized, as explained in the theory chapter. It is important to take
these factors into consideration when working with the adjoint solver. The adjoint solver
cannot predict sensitivities and design changes which are affected by turbulence, such as in
separated flow regions, as accurate as in attached flow regions. The adjoint solver together
with the design tool cannot predict too large design changes due to the linearization of
the adjoint equation. Hence, it is important to make small changes to the geometry after
each adjoint solution to ensure an improvement of the observable. Two other factors that
need to be taken into consideration when using the adjoint tool is the visual design of the
vehicle as well as manufacturability and cost of the parts. However, these factors always
have to be considered and might be more important to keep in mind if the geometry is
morphed rather then manually changed.
The linear nature of the adjoint equation is one of the main reasons for the convergence
problems that were experienced. Any instabilities introduced during the solution due to
either physical (flow separation or regions with localized transient behavior) or numerical
(poor mesh quality or discretization) will grow exponentially without any stabilization
scheme and is difficult to converge. It can be compared to the governing equations which
are non-linear and can limit the growth of instabilities which results in a smoother convergence. This is the reason why the convergence of the primal flow field is more stable and
easier to converge than the adjoint equation. The instabilities were however limited by
the use of stabilization schemes which are discussed in section 5.2. Two other factors that
50
5 Discussion
were found, from both the literature review and the simulations in this study, to have a
considerable effect on the adjoint convergence were the mesh quality and mesh resolution,
which is further discussed in section 5.2 and its following subsections.
5.2
Adjoint Method
Several adjoint simulations were performed to find settings that resulted in both converged
and diverged solutions. The diverged solutions were not presented but will be discussed
briefly in this paragraph to make the reader aware of some of the adjoint solvers limitations in regards to settings. The solution-based control-initialization and the auto-adjust
controls features seemed to be useful tools as explained in the theory but were found
to diverge the solution, which was also found in the literature review. The auto-adjust
controls feature decreased the ACN continuously until the solution diverged. As seen in
the results from the dissipation scheme (see e.g. fig. 11a), the residuals bounce which
can be an indication of a too large ACN, as mentioned in the theory chapter. Hence, the
auto-adjust control correctly decreased the ACN but to a level where the solution blew
up.
The adjoint solver had to be run with first order discretization, as recommended in the
theory and literature review, due to instabilities of second order schemes. The schemes
were set to be the same in the adjoint solver as used in the primal solver as far as possible.
However, the momentum discretization had to be set to first order even though the second
order was used in the primal flow field simulation. This results in a slight distortion of
the adjoint results. However, the value of the observable in the adjoint solver was always
compared to the value of the observable through Fluent’s report definition to make sure it
was sufficiently accurate. The difference was never more than 5 % throughout the study
(usually around 1-2 %) and experiences from Oliver Zühlke et al [22] and ANSYS [23]
have shown that reasonable and useful results are still possible to achieve with different
discretization schemes. Even if the exact same schemes are used, it is not certain that the
values from the adjoint and report definition are exactly the same since the evaluation of
the observable is not identical as explained and showed in [33, 36].
When coupling the adjoint solver to an aerodynamic automotive analysis it is important
that the user has experience in CFD or preferably automotive aerodynamics without
blindly trusting the results from the adjoint solver. The only observable in this study
was set to be the drag force which implied that the adjoint solver did not take lift into
consideration at all. Hence, it is important that the user has experience or knowledge
about how a certain design change would change the lift forces and balance of the car. In
addition, it is always recommended to run a verification simulation of the adjoint prediction
to realise the actual performance increase/decrease before any design decisions are taken
based on only the sensitivities or the NOD.
As mentioned in the theory the adjoint solver is based on one primal flow field. Hence, it
is crucial that this primal flow field does not oscillate for the adjoint to produce reliable
results. If it does, different results might be obtained depending on when the primal
solution is stopped. If the primal solution oscillates, it is considered important to locate
the region(s) of transient behaviour to judge if it can have an impact on the surfaces of
interest for the adjoint optimisation. The primal solutions in this study had, as mentioned,
a much lower standard deviation of CL and CD than 0.003 and was not considered to affect
51
5 Discussion
the adjoint solution. However, it is important to keep this in mind when using the adjoint
solver in the future for other cases.
5.2.1
Convergence Assessment
The convergence assessment of the adjoint simulations were found to be one of the most
difficult parts of the study and has not been presented, discussed or investigated in the
previous studies that were reviewed. This made the author question the trustworthiness
of these studies. The adjoint residuals give a good indication of the solution advancement
but can be misleading. The use of a monitor script increases the information about the
solution advancement both during and after the simulation and makes it easier to track
the sensitivities on surfaces of interest. As mentioned and shown previously (see fig. 12),
an adjoint wake can be present in front of the vehicle due to a backward calculation of the
adjoint field. The adjoint wake can be resolved with a finer grid but that would increase
number of elements and the computational time significantly and is not necessary. Instead,
the regions of high residuals can be located using UDM’s to make sure that they are away
from the surface(s) of interest, the one(s) that are about to be re-designed. It was also
considered important to look at the adjoint result and compare it to the primal solution
and use engineering judgment to deem if the adjoint results are reasonable. As a side note,
if any geometry is about to be simulated with the intention of a following adjoint simulation
it is important to split the PIDs of the surfaces according to the users preferences prior
to the primal simulation so that these surfaces of interest can be monitored (this cannot
be changed in Fluent afterward).
Since reasonable results were obtained from several adjoint simulations in this study, it
proves that the adjoint residuals do not always have to be very low to provide useful
information. Also, even though all surface sensitivities were not fully leveled out but
oscillating with small variations, it was still possible to adjust the geometry accordingly to
the NOD and obtain less drag. If the sensitivities are oscillating, the most optimal design
change cannot be predicted but it will still most probably be an improvement compared
to the baseline geometry.
5.2.2
Stabilization Schemes and meshes
Stabilization Schemes
A stabilization scheme was required due to the complex geometries and large separation
regions to avoid exponential instability growth caused by the linear nature of the adjoint
equation as previously explained. The two schemes investigated in this study were the
dissipation scheme and the RMS together with different types of meshes. The RMS
scheme was found to be very effective and rapidly reduced the residuals (see e.g. fig. 11b).
However, the RMS was very computationally expensive in terms of RAM and was relatively
time-consuming depending on the size of the mesh and the number of modes. The number
of modes is directly related to the convergence of the solution and it was found that 400
modes were sufficient and resulted in roughly the same results as the case with 800 modes
for the simplified case with the baseline mesh as seen from fig. 28, 29 and 30. By lowering
the number of modes from 800 to 400 the computational time decreased by a factor of 3 and
the required RAM by a factor of 2. The number of modes is expected to be geometry and
52
5 Discussion
mesh dependent and has to be changed depending on the case. Different geometries and
meshes will result in different regions of physical instabilities (separation and localized
transient behaviour). However, the results from this study provide an indication and
background for a reasonable initial guess of the number of modes for other similar cases.
It is recommended that the adjoint simulation runs initially with the dissipation scheme,
as done in this study with 100 iterations, before activating the RMS. It is not necessary
to run exactly 100 iterations but a number of iterations is required for the RMS to work
effectively. The most important thing is that the adjoint solver does not diverge rapidly
during these initial iterations. If it does, the RMS should be activated as soon as possible,
as explained in the theory chapter [16].
The dissipation scheme had to be run using first order damping since the second order
damping was found to result in a diverged solution. This was an issue since, as mentioned in the theory chapter, it is important to run the adjoint solver with a second order
dampening scheme to avoid distortion of the adjoint solution. However, due to the instabilities, the order of dampening had to be set to the same order as the pressure and
momentum scheme which resulted in a higher influence on the solution. The adjoint simulations utilizing the dissipation scheme still seemed to result in a converged solution for
the simplified model if used with first order dampening. However, several of the monitored
surface sensitivities were found to have an increasing/decreasing advancement behaviour
or were found to oscillate compared to the monitored sensitivities using the RMS which
were much more stable.
The dissipation scheme utilized with the baseline mesh was initially run with 5 000 iterations as shown in Figure 11a and 13. However, it was later run for an additional 10
000 iterations to see if the increasing/decreasing advancement behaviour of the sensitivities kept increasing/decreasing or if they eventually leveled out. The adjoint momentum
residuals were found to remain stable while the continuity adjoint residual were found to
oscillate below 1E-4. The adjoint flow rate residual were found to slowly increase but
was still lower then the adjoint momentum residuals. All monitored sensitivities that previously were increasing/decreasing did not level out during these 15 000 iterations but
started oscillating randomly instead. A few of the monitored surfaces that seemed to have
leveled out after 5 000 iterations were found to oscillate during the additional iterations.
This is another reason for why the RMS is preferred over the dissipation scheme with a
first order dampening. This is probably also the case for the adjoint simulations with the
dissipation scheme with the no prism and coarse mesh but they were not run for additional iterations due to the long simulation time. However, it can be seen from fig. 20
when looking at the roof sensitivity for the coarse mesh that it seems to be stable after a
while but later, after around 2 500 iterations, started to show an unsteady behaviour.
Mesh
The mesh investigations were initiated due to all the instabilities and convergence issues
that occurred in the beginning of this study and since other studies (see literature review)
found that a coarse mesh can help improve the convergence while maintaining sufficient
accuracy [1, 25]. However, the previous studies have only utilized the dissipation scheme
without any further information regarding specific settings. The simulations with the
coarse mesh was faster due to the lower element count and it was found that the convergence was much smoother for the simulations with the dissipation scheme (see fig. 19a
and 19b) compared to the baseline (see fig. 11a and 11b). The smoother convergence can
53
5 Discussion
be attributed to less small regions with localized transient behavior. However, the results
from the coarse mesh using the dissipation scheme did not correlate well with the baseline
mesh. On the other hand, the RMS results with the coarse mesh showed a good correlation with the baseline. Hence, having a reduced mesh resolution can greatly help the
adjoint convergence since the calculation is more diffused but it is important to be careful
to avoid mispredictions. A coarse mesh with fewer instabilities allows for fewer modes if
the RMS is utilized which results in even faster computations. It seems reasonable to use
a coarser mesh together with the RMS as the stabilization scheme but the required mesh
resolution is expected to be very case dependent (just as the mesh for primal flow fields
for different cases) and has to be decided by the user for the specific case.
The mesh without prism layers was run with the same near wall treatment as the other two
cases. The first cell height without prism layers was much larger but still with a y + within
the logarithmic layer. Due to the differences in the mesh and mesh quality, compared
to the baseline, different results were obtained even though the same settings were used
(as seen in fig. 21, 22, 25 and 26 as well as in the sensitivity monitors in fig. 20 and
24). The results were expected to be more similar and can possibly be attributed to the
adjoint solver’s sensitivity of the input. The most important finding was not necessarily
the results in relation to the baseline but rather how different the convergence was with
a mesh of improved quality. Even though the convergence of this simulation was much
smoother oscillations were still present.
It is important to remember that the adjoint solver is based on the results of the primal
flow field. Hence, the most important thing to consider is to have a mesh that results
in a sufficiently accurate primal flow field. Secondly, it is highly recommended to have a
mesh of as good quality as possible but not at the cost of accuracy by removing prism
layers. Thirdly, a coarse mesh can be used together with the RMS depending on the
case but it is suggested to avoid coarse meshes for the dissipation scheme. The mesh
recommendations from Fluent is only recommendations and both the primal flow solver
and the adjoint solver can reach convergence even with a quality that is worse. However,
it is as mentioned preferred to have a mesh of as good quality as possible.
Full Car Model
If the adjoint initialization is performed without fully compatible primal settings two types
of messages can appear. The first type of message is a warning message about settings that
are not supported such as porous zones. However, the adjoint solver can still initialize and
run by automatically changing the settings to corresponding supported settings during the
calculation. As a result, the adjoint field might be inaccurate around these regions and the
sensitivities should be treated and evaluated carefully [16]. The second type of message
is an error message which completely stops the initialization. The immediate stop occurs
due to reasons such as the use of unsupported boundary conditions or turbulence models.
Both of these messages appeared prior to the adjoint simulation of the full geometry.
The warning messages appeared due to the porous zones of the heat exchangers inside
the grille. The error message was due to the use of ’radiator’ boundary conditions of
preamble surfaces in the domain which were used to measure mass flow rate and pressure.
These ’radiator’ boundaries are not supported and were changed to an ’interior’ type of
boundary condition which still is a 100 % preamble boundary. The next flow simulation
can either be run again with the original setup by reverting the interior boundaries to
54
5 Discussion
radiator boundaries to allow for a fair comparison between the two designs. The radiator
boundary conditions and the interior boundary condition are both preamble boundaries
in this study since the solver is run without the energy equation. However, they were
found to produce slightly different results when two cases were run with exactly the same
settings except for these preamble boundaries. Hence, it is important to use the same
boundary conditions in both the primal flow field and the verified simulation. This is
not an issue if the geometry is changed manually outside of Fluent and set to be re-mesh
and re-solved. However, if the geometry is morphed in Fluent and re-run directly, it is
important to revert to the original settings. The post processing was performed in Ensight
and the solution values at the interior surfaces cannot be exported to Ensight which was
the reason for the usage of the radiator boundary in the first place [37]. From the adjoint
solver’s point of view, data from these preamble surfaces are not of interest and mass flow
rate and pressure at the surfaces can still be obtained from Fluent. Hence, if a primal
simulation is run with the intention of a following adjoint simulation, it is recommended
to run them both using the interior preamble boundary.
The full geometry model had to be run with the dissipation scheme since it was found to
be too computationally expensive in terms of RAM to run it with the RMS. The residuals
were very unstable and it was found that after 5 000 iterations, which took approximately
52 000 CPU hours, that the residuals were still oscillating (see fig. 33). In addition, most
of the monitored sensitivities did not level out (see fig. 34). An adjoint wake was found
to be present in front of the vehicle as in the simplified case. However, the wake in the
full model was larger and much closer to the vehicle as seen in fig. 35. High residuals were
found at the grille and the front bumper of the car. The cause of the is most likely due to
two reasons. The first reason is that the grille was open and the engine bay was included
in the simulation which increases the number of local instabilities. The second reason is
the use of porous media. As mentioned above, a warning message appears when porous
media is used in the adjoint solver. This decreases the quality of the results and could be a
reason for the higher residuals. Both of these reasons together with the fact that the car is
unsymmetrical inside the engine bay can be attributed to the non-symmetric result of the
NOD in the front of the full car (see fig. 37). Another fact that is important to remember
is that the NOD was not calculated during the adjoint solution but in the ’Design-Tool’
which can produce inaccurate results depending on the input. However, that was not
likely to be the reason in this case. A possible third reason can be that the maximum
skewness of the mesh was very close to 1 which can cause numerical errors. However, it is
worth to note that the simplified model also ran with the coarse mesh which was of equal
quality and reached convergence.
5.2.3
Robustness
The adjoint solver relies upon an accurate primal flow solution to be able to provide useful
results. Hence, as mentioned, the solver settings were kept similar to CEVT’s methods
for external aerodynamics. Several different settings such as Fluent’s pseudo transient
calculation as well as non-equilibrium wall treatment were also tested using the simplified
model to ensure that the adjoint solver was compatible and converged to reasonable results.
In addition, a simulation was also performed with an even finer mesh near the wall with
a y + ≈ 1 using the RKE model with enhanced wall treatment. The adjoint solution
advancement in all these simulation were similar to the baseline case seen in this study
and therefore not showed in the results. The results from the adjoint solver were slightly
55
5 Discussion
different compared to the baseline case since the results from the primal flow field were
different. However, not significantly and unexpectedly different which implies that the
adjoint solver can be run using y + ≈ 1 meshes as well as different near wall treatments.
These simulations with other settings were run with the RMS as the stabilization scheme
to ensure robustness of the adjoint solver.
5.2.4
Single Part Adjoint Optimisation
The single part adjoint optimisation was performed with only the side view mirrors in
focus as a typical application of the adjoint solver. The adjoint simulation was still run
for the whole car to get a flow field around the car and the mirrors as realistic as possible
to achieve accurate adjoint predictions. Hence, the adjoint results were taken from the
RMS simulation of the coarse model with 800 modes and were further analysed. This
case was chosen due to the stable monitored sensitivity of the mirrors and for the fast
computations of the coarser mesh since it was found to be a relatively close match to the
baseline case. Since a single part was analysed, it was important to change the variable
range of the legend to highlight the NOD changes as shown in fig. 38. This value of the
NOD will obviously vary from part to part and case to case depending on the surface(s)
in focus.
The geometry was changed according to the results from the NOD contour and the case
was re-run to verify that the adjoint results and prediction was correct and gave an improvement of the baseline design. As mentioned, the geometry was morphed in Fluent and
there were several reasons for this. First of all, it was much easier to adjust the geometry
in accordance with the NOD using a morpher since it is based on the NOD results. Secondly, using a more or less identical mesh allowed for a fair comparison between the two
designs by reducing mesh induced differences of the results. Thirdly, the whole process
was rapid and the simulation of the morphed geometry continued from the previous primal
flow field which significantly decreased the simulation time. The time required to morph
the mirrors and re-run the flow field for the whole car was approximately one-fifth of the
time required to re-mesh and run the simulation from the beginning again. Note that
this is excluding the time required to do the actual geometry change manually in ANSA,
or any other software that allows manually geometry change, which would increase the
required time even further.
The improvement found from the mirror optimisation was significant, both by analysing
the difference in force on the mirrors themselves as well as the total CD A of the whole car
(see table 12). The proposed design change of the mirrors was partly unintuative but was
found to decrease the drag force significantly. It was found that the design changes of the
mirrors resulted in a reduced region of high-pressure region in the front of the mirrors as
well as a smaller wake behind the mirrors. It is important to highlight that these design
changes are not fully realistic since no constraints were set on the glass of the mirrors
prior to morphing. However, the glass of the mirrors are located in the wakes and were
not in contact with any attached flow. Hence, this design change is a cause of the frozen
turbulence assumption and was not expected to have a large influence of the results. If
the user would like to morph the mirrors and constrain the glass or any other surface of
the model it is recommended to split these surfaces into different PID regions to be able to
select them separately in Fluent’s design tool. However, this is not necessary if the design
is changed manually.
56
5 Discussion
The monitored surface sensitivity of the mirrors was relatively stable but with small oscillations. It was found that the adjoint optimisation was successful even though the
sensitivity was not fully steady. Hence, the adjoint tool can be used on surfaces that have
small sensitivity oscillations. The most optimal design might not be predicted/obtained
but an improvement can still be expected as long as the design is not close to a local optimum [20]. Reaching or advancing to a shape close to the local optimum requires several
adjoint optimisation loops due to the linearization that limits displacement magnitude of
the design change each iteration.
This type of optimisation, where one part/surface of the car is analysed at time, can
be applied to different parts with the use of a single adjoint simulation of the full car.
However, one calculation in the design tool has to be performed per part/surface to obtain
an informative NOD map of the part/surface.
57
6
Conclusion
Fluent’s discrete adjoint solver has been investigated and evaluated with regard to external
aerodynamic flow simulations by performing a literature study as well as CFD and adjoint
simulations. It was found that running 3D adjoint simulations around ground vehicles are
more complicated than it seems from previous studies. It was concluded that the gradientbased adjoint optimisation is a powerful tool that is applicable for external aerodynamic
analysis to a certain extent.
A CFD-method with guidelines to run external 3D adjoint simulations for aerodynamic
analysis was developed by investigating different adjoint solver settings such as advancement settings and stabilization schemes together with different meshes. It is recommended
to use the RMS stabilization scheme and preferably with a baseline mesh instead of a
coarser mesh. However, a coarse mesh can be used in some cases when the adjoint solver
has trouble converging due to localized transient behaviour. The mesh quality was found
to be of great importance and it is recommended to run the adjoint solver with a mesh of
as high quality as possible.
It was also found that the adjoint solver has to run with first order discretization schemes
for stability reasons. It was concluded that it is very important to monitor surface sensitivities during the solution in addition to the adjoint residuals to assess convergence of
the adjoint simulation. Even if the adjoint residuals are relatively high and sensitivities
slightly oscillating it is still possible to achieve an improved design, if the initial design is
not already close to the Pareto front. It is also recommended to analyse regions of high
residuals in the domain to ensure that they are far away from the surface(s) of interest.
It was found that a simplified model of a fully detailed car geometry is necessary to reduce
the complexity and the resolution of the mesh to be able to use the RMS and to avoid
local instabilities. A proposed CFD and adjoint procedure is presented in Appendix A as
a flowchart together with a list of comments and guidelines to each step.
It was also concluded that it is important that the engineer who analyse the adjoint results
have previous experience or knowledge in the field of CFD and preferably automotive aerodynamics in order to gain an advantage over the conventional aerodynamic development
process.
58
7
Future Work
The adjoint solver together with a robust, reliable and versatile mesh morphing method
can reduce the computational time per design loop further by eliminating the need for
manual re-design, re-meshing, and re-solving of the primal flow field. It would also allow
for more fair comparisons between different designs since the mesh is more or less the same
except for the morphed region(s). If implemented, it is important to consider the surface
and mesh quality after each morph to avoid rough surfaces and poor quality meshes that
can lead to increased numerical errors. It is also important that the morphing method
has functionalities to add constraints, both external and internal, to add limits of surfaces
that cannot be exceeded and to restrict the morphing from occupied spaces due packaging
of other details such as electronics in the side view mirrors as an example. However, the
process cannot be fully automated to run a certain number of design iterations due to
two reasons. Firstly, the adjoint solver only takes drag force into consideration (lift and
balance of the car are not included). Secondly, the exterior design and visual look of the
vehicle have to be approved by a designer/design team.
It is proposed that the developed method is revised in a few years if the adjoint solver
in Fluent is improved and the available computational resources most probably increased.
The adjoint solver is, as of now, continuously updated with new features, tweaks and
stabilization schemes for robustness and accuracy. Hence, it might be possible to run
more complex models once it is computationally faster and more robust.
59
References
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61
A
Adjoint Procedure
A proposed and recommended adjoint procedure for 3D external aerodynamic simulations
are presented below as a flowchart, see Figure A.1, together with explanatory steps from
1-10. It is not recommended to blindly use this flowchart guide. Instead, use it as a
compliment and summary of the results and conclusions obtained in this study.
1) Pre-processing by setting up the specific PID’s. Important to note that all surfaces
that are about to be monitored require a separate PID. All surfaces that might
be morphed and constrained in Fluent also requires a separate PID. Set up the
adjoint surface monitor scripts to match the PID’s. It is also possible to monitor
the sensitivities of the surfaces manually but that is very time-consuming. Hence, a
monitor script is highly recommended. It is also recommended that the geometry is
as simplified as possible but still representative for the full geometry.
2) Create a mesh (as good quality as possible). A coarse mesh can be used together
with the residual minimization scheme but it is recommended to run the adjoint
solver with a mesh that generates an accurate solution of the primal flow field at
first.
3) Run the primal flow field and make sure that the primal flow solution converges
and shows the expected results. Since only one primal flow field is used for the
adjoint solver it is important that the primal solver is as steady as possible for the
adjoint to produce reliable results. If large oscillations are present in the primal flow
solver, locate them and see if they are close to the surfaces of interest for the adjoint
optimisation.
4) Run the adjoint solver for 50-100 iterations using the dissipation scheme with the
settings shown in Table 5 and 6. Change to the RMS stabilization scheme with the
settings (or similar settings) found in Table 7 and run the simulation to convergence
(∼15-30 iterations). It is recommended to run everything from a journal and include
the monitor scheme script. If the monitor script is used, run at least 5 iterations
before activating it. Set the number of modes depending on the geometry and
number of elements in the mesh. Run the adjoint solver with first order schemes and
preferably with the same type of schemes that were used in the primal flow field.
5) Analyse the monitored data as well as the adjoint residuals. If the residuals and the
sensitivities of interested leveled out, move on to the design tool. If not, locate the
problematic region, if possible, by storing the residuals in UDMs and visualize them
using cell registers in Fluent or export the UDMs and visualized them in Ensight. A
more stable solution will be obtained if the number of modes is increased, the quality
of the mesh is improved, the resolution of the mesh is decreased or if the geometry
is simplified even further to avoid local instabilities. It is recommended, if possible,
to simplify the geometry since the computational time increases exponentially with
A1
A Adjoint Procedure
more modes.
6) Run the design tool to obtain NOD data for the specific surface(s) to be analysed/changed. The design tool parameters should be similar to the ones found in
Table 8. However, the scaling factor and the number of motion points have to be
changed depending on the surface(s) of interest. Export the desired data to the
post-processing tool at this point if necessary.
7) Re-design the surface(s) or morph the mesh. If the geometry is changed manually,
re-meshing is required.
8) Verify the adjoint results by running the primal flow solver with the new design.
If the geometry was morphed, continue the primal flow solver from the previous
simulation (saves a lot of time!).
9) Continue the adjoint optimisation either by analysing another part/surface of the
vehicle from the same adjoint simulation by selecting the part/surface in the design tool. Or, by running another adjoint simulation to make an additional design
iteration on an already optimised part/surface (if several design iterations are performed, the adjoint solver can also be re-run from the previous iteration to save
computational time if the geometry was morphed).
10) End of adjoint optimisation.
A2
No
A Adjoint Procedure
No
1) Pre-Processing
2) Mesh
3) Run Primal Flow
Solver
4) Run Adjoint Solver
No
5)
Residuals &
Sensitivities
ok?
Yes
6) Calculate NOD
Using Design Tool
7) Change Geometry
OR Morph Mesh
8) Run Primal Flow
Solver To Verify the
Results
9) Keep Optimising?
Yes
No
10) End of Adjoint
Optimisation
Figure A.1: Adjoint procedure presented as a flowchart
A3