- Research
- Open Access

# Adjoint methods for car aerodynamics

- Carsten Othmer
^{1}Email author

**4**:6

https://doi.org/10.1186/2190-5983-4-6

© Othmer; licensee Springer. 2014

**Received:**30 March 2013**Accepted:**5 March 2014**Published:**3 June 2014

## Abstract

The adjoint method has long been considered as the tool of choice for gradient-based optimisation in computational fluid dynamics (CFD). It is the independence of the computational cost from the number of design variables that makes it particularly attractive for problems with large design spaces. Originally developed by Lions and Pironneau in the 70’s, the adjoint method has evolved towards a standard tool within the development processes of the aeronautical industries. Its uptake in the automotive industry, however, lags behind. The first systematic applications of adjoint methods in automotive CFD have interestingly not taken place in the classical shape design arena, but in a relatively young discipline of sensitivity-based optimisation: fluid dynamic topology optimisation. While being an established concept in structure mechanics for decades already, its transfer to fluid dynamics took place just ten years ago. We demonstrate that specifically for ducted flow applications, like airducts for cabin ventilation or engine intake ports, it constitutes a very powerful tool and has matured over the last years to a level that allows its systematic usage for various automotive applications. To drive adjoint-based shape optimisation to the same degree of maturity and robustness for car applications is the subject of ongoing research collaborations between academia and the car industry. Achievements and challenges encountered during these efforts are presented.

## Keywords

- Computational Fluid Dynamics
- Topology Optimisation
- Shape Optimisation
- Drag Reduction
- Adjoint Method

## 1 Background

Due to the high computational effort associated with black-box optimisation, where the number of CFD evaluations scales roughly linearly with the number of design variables, the explorable design space is very limited: In our current practice, these methods cannot afford more than ten design parameters. An additional obstacle that inhibits the further adoption of black-box optimisation within the regular automotive development process is the necessity of a versatile but stable CAD parameterisation. For e.g. an engine intake, the setup of the parameterised model might take an experienced CAD engineer several weeks before it comes up to the stability requirements of an automatic optimisation process chain. It is these limitations that call for a radically different approach to automotive CFD optimisation - especially in view of applications like entire vehicle aerodynamics that entail shape optimisation of highly complex free-form geometries with a practically unlimited design space.

Despite being already a long-established tool in the aerospace sector, the adjoint method has only recently started to also enter the automotive development processes. This backlog is mainly due to two obstacles specific to the automotive industry.

**Obstacle #1** *Rather than on in*-*house codes*, *the car industry relies almost exclusively on commercial CFD software*.

The implementation of the adjoint method requires access to the source code - either to implement the analytically derived adjoint equations into the code (so-called continuous adjoint) or to perform a manual or automatic differentiation of the primal code (discrete adjoint, see [5] for a comparison). The sources of commercial CFD codes are, however, inaccessible for the user, and by the time we started our project to industrialise the adjoint method, the software vendors of the CFD solvers that are operational at Volkswagen declined to implement it into their codes. As a remedy, we selected the well-validated and versatile CFD toolbox OpenFOAM® [6] as our platform to develop adjoint-based optimisation methods for car applications. In addition to its open source character, OpenFOAM comes with a high-level symbolic programming style, which makes it ideally suited to implement additional partial differential equations - like the adjoint Navier-Stokes equations.

**v**and

*p*stand for velocity and pressure, respectively, and

*α*for the porosity distribution. The effective kinematic viscosity

*ν*is the sum of molecular and turbulent viscosity, and the rate of strain tensor $\mathsf{D}(\mathbf{v})$ is defined as $\mathsf{D}(\mathbf{v})=\frac{1}{2}(\mathrm{\nabla}\mathbf{v}+{(\mathrm{\nabla}\mathbf{v})}^{T})$. For a given cost function

*J*composed of contributions from the design domain boundary $\mathrm{\Gamma}=\partial \mathrm{\Omega}$ and from the interior of Ω,

**u**and

*q*denoting adjoint velocity and adjoint pressure, respectively. While the cost function contribution from the domain interior ${J}_{\mathrm{\Omega}}$ gives rise to source terms in the adjoint field equations, the surface contribution ${J}_{\mathrm{\Gamma}}$ enters the adjoint boundary conditions:

where *β* denotes the normal node displacement.

Within the still ongoing collaboration with Prof. Giannakoglou’s group at the National Technical University of Athens (NTUA) and Dr. de Villiers at Engys®, this basic implementation was subsequently enhanced towards an inclusion of an adjoint turbulence model [9], optimisation of external flows [10], sensitivities for flow control [11], adjoint wall functions [12], the inclusion of heat conduction and constraints [13] and mildly compressible flows [14]. In addition, the uptake and further development of the basic adjoint code by other researchers lead to remarkable advances [15–18], especially in the area of topology optimisation for exhaust systems [19, 20]. As a result of these development efforts, a versatile continuous adjoint code in OpenFOAM for steady-state RANS is now available.

**Obstacle #2** *The geometries of entire vehicles and even of single automotive parts are highly complex*.

The complexity of typical automotive geometries, especially of entire vehicles, which consist of an assembly of several hundred parts, necessitates the use of an automatic process for the generation of the computational mesh. Even though the sophistication of automatic meshing software is continuously increasing, the achievable mesh quality is still inferior to a hand-made mesh, especially concerning the creation of layers along the surface of the flow domain. While this quality suffices for obtaining reasonable results for primal computations, it has been found that the adjoint equation system reacts much more delicately to imperfections of the computational mesh. This is particularly the case in the presence of flow separation, which is commonly encountered in flows in and around automotive geometries.

This lack of robustness of the adjoint when running on typical industrial meshes is a problem that has been accompanying the adjoint development in OpenFOAM from the very beginning - and for certain applications it is still inhibiting the systematic use of the adjoint method in the regular computational processes. The source of this instability is the so-called ‘adjoint transpose convection’ $\mathrm{\nabla}\mathbf{u}\cdot \mathbf{v}$ - the second convection term on the left-hand side of Eqn. (4). By developing several formulations for this term, as well as explicit and implicit treatments, major steps towards increased stability were made. As it introduces a high degree of cross-coupling between the three components of the velocity, recent efforts concentrate on solving the adjoint equation system as a single block rather than in a segregated manner [21].

While those developments are very promising, the current practice of running the adjoint - at least on meshes of the typically encountered quality in the car industry environment - still requires to trade robustness against accuracy: by using limiters, relaxed adjoint boundary conditions or by damping the adjoint transpose convection in an intelligent way that does not harm the *qualitative* correctness of the sensitivities. To report the achievements as well as the challenges of this pragmatic approach to solving adjoints for car applications is the intention of this article. It is organized as follows: The concepts of topology optimisation and shape optimisation for ducted flows are introduced in the next section, along with an application of these two complementary methodologies to engine port flows. Section 3 focuses on external aerodynamics, comprising a full-vehicle validation study, an example of a component optimisation and a comparison of steady-state RANS sensitivities with those based on a time-averaged transient primal flow. The article finally closes with a comparatively young adjoint application area - car drag reduction by flow control.

## 2 Ducted flows

Geometric optimisation of ducted flows in cars, like airducts for cabin climatisation, engine air intakes or exhaust systems, are commonly subject to severe packaging constraints. This gave rise to the development and adoption of topology optimisation methods for automotive applications. After a concise retrospective of these development efforts, it will be shown that especially for ducted flows, topology optimisation is a perfect complement to the classical shape optimisation.

### 2.1 Topology optimisation

Topology optimisation is a well-established tool in computational structure mechanics [22] with widespread industrial use. In its simplest realisation, a topological optimisation starts from the available design domain filled up entirely with solid material of a certain density. In an iterative fashion, the given loads are applied, the stresses are computed all over the domain and the areas with low stresses are weakened by assigning a lower density to them. After several iterations, this method retains high-density material only in regions that are critical to fulfill the structural task, and in this way generates optimal lightweight structural designs. It delivers an un-biased design from scratch that automatically fulfills the installation space constraints.

Despite being based on the same concept of flow optimisation, these two approaches exhibit significant differences. Klimetzek’s method is a rather pragmatic solution: The local criterion he employs to identify counterproductive cells is the deviation between the actual velocity direction and a desired direction computed as polygonal traces between inlet and outlet. If this deviation exceeds a certain angle, the corresponding cells are blocked for the fluid. His criterion allows to efficiently remove unwanted recirculations from the flow and - albeit being used for other cost function as well - is therefore ideally suited to reduce the pressure drop. From the date of its original publication, Klimetzek’s method has already been applicable to three-dimensional turbulent Navier-Stokes flows and is seeing a continuously increasing number of successful applications in industrial flow optimisation.

In contrast, the method proposed by Borrvall and Petersson was restricted to two-dimensional Stokes flows between parallel plates, and the punishment of counterproductive areas is performed by locally decreasing the distance between the plates - until zero upon convergence. What inspired a series of subsequent research works is, however, the versatility of their approach that lies in the way how they identify counterproductive fluid cells: by computing actual sensitivities via the adjoint method.

In order to extend the Borrvall and Petersson method towards three-dimensional flows, the punishment via plate distance was replaced by introducing a Darcy porosity term $-\alpha \mathbf{v}$ into the governing Eqn. (1) (see [25] and [26] for a finite-element and a finite-volume implementation, respectively). The individual cell porosities allow a continuous transition between fluid ($\alpha =0$) and solid ($\alpha ={\alpha}_{\mathrm{max}}$) and act as the actual design variables of the optimisation problem. With this setting, topological sensitivity maps like those shown in Figure 3 are nothing but the derivative of the cost function w.r.t. an increase in the cell porosity *α*.

Another requirement for the industrial applicability of adjoint-based topology optimisation was its generalisation from pure Stokes flows to turbulent Navier-Stokes flows. After the principal feasibility of laminar Navier-Stokes topology optimisation was first demonstrated via Automatic Differentiation of an academic CFD code [27], the concept was implemented via a continuous adjoint also for turbulent flows [8] - albeit under the assumption of ‘frozen turbulence’, i.e. fully neglecting the variation of the turbulent flow quantities. Finally, the development of a continuous adjoint turbulence model [9] now allows to run topology optimisation for turbulent Navier-Stokes flows under full consideration of variations in the turbulence fields [13].

For the low swirl case, it can be assumed that the geometry is only driven by the swirl sensitivities where the modifications have a very small detrimental effect on the pressure loss. As a result, we see a small reduction in pressure loss along with a modest increase in swirl. Predictably, the pressure loss starts to go up as the swirl weighting is increased and the fluid looses additional energy due to the highly swirling flow and the blockages required to produce it. What is encouraging is the extent to which the swirl can be reliably increased - by more than 500% - compared to the relatively modest relative increase in pressure loss (high swirl case of Figure 7c).

For topology optimisation, it is critical that the design extracted from the optimisation process be verified with a high accuracy simulation. To this end, the blocked element description from the high swirl case was extracted and incorporated into a new CAD description of the port flow geometry (Figure 7d). It is immediately apparent that the modified design is rather complex in its detail and not something a designer would likely create through intuition or experience. Neither could these geometries easily be arrived at via a process of parameter-based stochastic optimisation. The most important point to note is that the accurately rebuilt geometry very nearly reproduces the results from the topology optimisation run. While we do not contend that this will consistently be the case, it certainly proves the efficacy of the process in this instance.

### 2.2 Shape optimisation

The topology optimisation method as introduced above, acts on the fluid domain like a sculptor on a piece of rock: It removes the unwanted portions and reduces the original geometry to its optimal shape. Since the original volume mesh is not changed during the optimisation process, the final geometry consequently consists of a subset of those cells that made up the original shape. Thus, the inherent geometrical resolution of topology optimisation corresponds to the cell size, and the final shape will inevitably have a ragged surface. Topology optimisation is therefore to be regarded as a *drafting* method. For *fine-tuning* geometries that are already close to their optimal shape, it is ideally complemented by so-called shape optimisation. Rather than on topological or volume sensitivities, shape optimisation is based on the detailed information contained in surface sensitivity maps in order to morph the shape towards a further improvement of the cost function.

A surface regularisation method that comes up to these requirements has been developed by Prof. Bletzinger and his team at the Technical University of Munich (TUM) [28, 29]. The results of applying their method to the exhaust port above are shown in Figure 8. Depending on the size of the smoothing radius and on whether morphing is allowed only in the inward direction or both inwards and outwards, different pressure drop reductions between 2% and 20% are achieved. Since this shape optimisation process is set up such that primal and adjoint calculation are closely coupled to the shape update itself (‘one-shot optimisation’), the overall cost of the optimisation amounts to not more than four primal computations.

As demonstrated above, topology optimisation and shape optimisation are complementary methods: the former for drafting, the latter for fine-tuning. Especially for the optimisation of car components that are subject to design space constraints - and in the increasingly tighter getting installation spaces within the engine compartment and the passenger cabin this is the case for the majority of components - the combination of both methods suits the requirements of automotive component design very well. Starting from the feasible installation space, they are capable of delivering a fine-tuned optimal geometry, thereby each of them making use of the full potential of the available design space.

## 3 External aerodynamics

The adjoint-based topology and shape optimisation methods described above are obviously not restricted to ducted flows but can equally well be applied to external flows. The external aerodynamics of entire vehicles is, however, a peculiar application for optimisation methods: Except for dedicated low emission cars, vehicle shapes are to a high degree driven by aesthetic considerations rather than aerodynamic performance. This has two implications: (1) Given the rather limited design freedom, external aerodynamic optimisation is more about fine-tuning. Topology optimisation is therefore not an adequate tool. (2) Since aesthetic requirements can impossibly be casted into mathematical constraints, the *automatic* optimisation of vehicle shapes forbids itself - unless restricted to small portions like spoilers, mirrors or, of course, to the underbody.

Under these circumstances, the mere computation of surface sensitivity maps - with neither a topological nor a shape update - turned out to be a tool that fits very nicely into the specific requirements of the development of external vehicle shapes: The information contained in these maps provides the designer with concrete suggestions for aerodynamic improvements, and it is in his/her hand to translate this information into an aerodynamically improved shape without compromising the aesthetics.

### 3.1 Validation of RANS shape sensitivities for external aerodynamics

*locally*- an adjoint is not needed to get this piece of information. The added-value of the adjoint, however, becomes obvious for regions 1, 3 and 4: the rear end of the car where the adjoint ‘suggests’ a spoiler, the blue area of the front wing which should be further exposed, and the wheel spoiler area in front of the front tyres. This information cannot be deduced by looking only at the primal. The effect of changing the car shape here is

*non-local*and happens further downstream of the sensitive regions themselves. It can therefore only be identified by the adjoint.

*z*-direction (i.e. up and down, Figure 14). Within a one-shot optimisation driven by a simple steepest descent algorithm the

*z*-coordinates of the morphing control points were moved towards their optimal position after roughly 20 iterations (Figure 14). The overall cost of the optimisation corresponded to less than five primal flow solutions and resulted in a drag decrease of 2%. Given the fact that the XL1 with a drag coefficient of less than 0.2 was aerodynamically nearly perfect already, this was a significant reduction. As a beneficial side-effect, lift decreased by 30%.

Rather than with free-form deformation boxes, which allow to modify shapes in a very controlled manner but are quite tedious to set up, another option of translating surface sensitivities into an improved shape are of course the node-based morphing algorithms from TUM as introduced in the previous section. Since they are capable of moving each and every node into its optimal position, they are ideally suited for the optimisation of vehicle details - like external mirrors.

### 3.2 RANS sensitivities vs. approximate time-averaged DES sensitivities

The preceding section summarised the achievements in the development of adjoint-based shape optimisation methods for external aerodynamics of vehicles. These methods rest upon steady-state RANS simulations. Meanwhile, however, the common practice of external aerodynamics computations has moved - at least in the largest part of the Volkswagen Group - from steady-state RANS to unsteady Detached-Eddy Simulations (DES, [31]). To create an adjoint to a time-varying primal like DES entails particular difficulties: Since the adjoint runs backwards in time, all primal states for the time period of interest have either to be stored or recomputed when running the adjoint. The huge amount of storage space and computational time can, in principle, be reduced to a manageable amount by clever checkpointing algorithms [32, 33], but for the typical model sizes of car external aerodynamics an adjoint to a DES run is still not feasible yet.

Our provisional solution to compute sensitivity maps for primal DES computations therefore looks as follows: We plug the time-averaged velocity **v** and pressure *p* from the primal DES computation into a RANS turbulence model in order to solve for a turbulent viscosity ${\nu}_{t}$. With these quantities - time-averaged **v** and *p* as well as RANS-${\nu}_{t}$ - we run the existing RANS adjoint code to obtain the sensitivities. A validation study similar to the one above for the XL1 revealed that we can only expect qualitative accuracy from this procedure. Quantitatively, the sensitivities are not correct. Still, the following comparison will demonstrate that these approximate DES sensitivities are useful and actually have an added value against pure RANS sensitivities.

### 3.3 Adjoint methods for flow control

Geometrical modifications, be it via shape or via topology optimisation, are not the only option of improving the fluid dynamic performance of car components or the entire vehicle. Recently there is a growing interest in qualifying active flow control devices for vehicle aerodynamics: Brunn et al. [34] demonstrated how steady blowing and oscillating jets can be successfully applied to reduce drag of the well-known Ahmed body [35]. Bideaux et al. [36] used the same bluff body shape to investigate the effect of a line of pulsed jet actuators on drag and lift, while Heinemann et al. [37] experimentally investigated continuous jets on a 1:4 model of a passenger production car.

Since flow control of entire vehicles is a rather young subject, experience for the optimal layout of jet configurations is still very limited. The questions to be answered when designing a flow control concept are: (1) What kind of jets should be applied - blowing or suction? (2) Where should the jets be positioned to have a maximum effect? (3) What is the achievable aerodynamic improvement?

To answer these questions, the adjoint method is again the tool of choice, as it allows to compute a sensitivity map of the whole car for the positioning of jets. While for shape sensitivity maps, the design variables were the normal displacements of all surface nodes, it is now the normal flow velocity ${v}_{n}$ on each surface element that constitutes the design space. In other words, the car is regarded as having a perforated surface with initially zero normal mass flux through it, and the adjoint method is then applied to compute the sensitivity of drag ${F}_{x}$ or lift ${F}_{z}$ w.r.t. changing the mass flux through each of the ‘holes’ [11].

## 4 Summary and outlook

Starting from the obstacles of introducing the adjoint method into automotive CFD, we have reported on the efforts undertaken with various partners to develop an industrially viable adjoint solver. Based on OpenFOAM, powerful optimisation methods - topology optimisation for ducted flows, and shape optimisation for ducted as well as external flows - have been developed. While topology optimisation is being used systematically already, the implementation of shape optimisation into the regular automotive development process has just started with some initial promising results. Both methods will most surely have an impact on overall vehicle performance and consumption in the future.

The adjoint development efforts have so far concentrated on steady-state phenomena. For an extension towards inherently transient flow applications, like aeroacoustics, flow control with pulsating jets and transient aerodynamics with DES, the adjoint has to become transient as well. This is a subject that is under investigation by several research groups already (e.g. [39–41]) and is currently addressed in the Marie-Curie International Training Network ‘aboutFlow’ [42]. Its industrialisation can be regarded as the next challenge in developing adjoint methods for the automotive industry.

## Additional Material

## Declarations

### Acknowledgements

The author is truly grateful to E de Villiers and his colleagues at Engys, KC Giannakoglou and his team at NTUA (above all EM Papoutsis-Kiachagias and AS Zymaris), E Stavropoulou, M Hojjat and KU Bletzinger from TUM, JD Müller (Queen Mary University London) and the FlowHead team, as well as E Skaperdas and K Haliskos from BETA CAE Systems. The support from many CFD colleagues of the Volkswagen Group in the form of continuous interest in the adjoint method, provision of resources and of valuable feedback is also gratefully acknowledged.

## Authors’ Affiliations

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