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Optimization and Engineering manuscript No. (will be inserted by the editor)

Combustion engine optimization: a multiobjective approach Stefan Jakobsson · Muhammad Saif-Ul-Hasnain · Robert Rundqvist · Fredrik Edelvik · Bj¨ orn Andersson · Michael Patriksson · Mattias Ljungqvist · Dimitri Lortet · and Johan Wallesten Received: date / Accepted: date

Abstract To simulate the physical and chemical processes inside combustion engines is possible by appropriate software and high performance computers. For combustion engines a good design is such that it combines a low fuel consumption with low emissions of soot and nitrogen oxides. These are however partly conflicting requirements. In this paper we approach this problem in a multi-criteria setting which has the advantage that it is possible to estimate the trade off between the different objectives and the decision of the optimal solution is postponed until all possibilities and limitations are known. The optimization algorithm is based on surrogate models and is here applied to optimize the design of a diesel combustion engine. Keywords Combustion engine · simulation-based optimization · multiobjective optimization · surrogate models · radial basis functions

Fredrik Edelvik Fraunhofer-Chalmers Centre Tel.: +46-31-7724246 Fax: +46-31-77260 E-mail: [email protected] Stefan Jakobsson · Muhammad Saif-Ul-Hasnain · Robert Rundqvist · Bj¨ orn Andersson Fraunhofer-Chalmers Centre Michael Patriksson Chalmers University of Technology Mattias Ljungqvist Volvo Car Corporation Dimitri Lortet · and Johan Wallesten Volvo Powertrain Corporation

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1 Introduction It is nowadays possible to simulate the physical and chemical processes inside combustion engines by appropriate software and high performance computers. These simulations can predict, for example, fuel consumption and emission of soot and nitrogen oxides. By varying the design parameters of the engine, different configurations can be simulated and their performances compared. Evidently, it is favorable for combustion engines to have a low fuel consumption as well as a low emission of soot and nitrogen oxides. These are partly conflicting requirements - for highly efficient engines we have high burning temperatures which may result in the formation of a lot of nitrogen oxides. Since the engine must work well in certain ranges of load and speed conditions, it may be necessary to take several such conditions into account in the optimization. In addition, we may for structural reasons have some constraints on the maximal pressure inside the engine during the combustion stroke. In this paper we emphasize the multiobjective character of this problem. In multiobjective optimization the goal is to find the Pareto optimal solutions, that is, all solutions for which there exist no other point which is better in all objectives[11,4]. Instead of combining all different requirements into a single goal function it is advantageous to keep the goal functions separate during the course of optimization. This makes it possible to estimate the trade off between the different objectives and the decision of the optimal solution is postponed until all possibilities and limitations are known. Mathematically the multiobjective problem can be written as min {fj (x)}N min F (x), j=1 = x∈Ω x∈Ω cL ≤ c(x) ≤ cU cL ≤ c(x) ≤ cU

(1)

where Ω is the parameter space and c the possibly vector-valued constraint function with upper and lower limit cU and cL . For convenience we let F be the vector valued function with all objectives. The performance of the engine will depend on many things and it is important to focus on the most crucial variables. Our choice of design parameters is described later. The main drawback of the multiobjective approach is that it is in general more expensive to explore a Pareto front than to find the optimum for a single objective problem. This problem gets more severe as the number of objectives increases. In many applications there is no interest to find the whole Pareto front but only a part of it. Because of legislation or other reasons we may be forced to put upper limits on the emissions and we may also put limits on the fuel consumption. By reducing the Pareto front of interest, for example, by putting constraints on the objectives, we can make the algorithm converge faster. These constraints are of a different type than upper and lower limits and geometrical constraints in the sense that they are expensive to compute. Such constraints must be handled efficiently by the optimization algorithm. The objective functions are often expensive to compute because they depend on output from full three dimensional simulations of combustion engines

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which can take several days of CPU time even on fast computers whereas more simple models may only take a couple of minutes. For the simple models where one can afford thousands of simulations, evolutionary multiobjective algorithms will often give a good approximation of the Pareto front. However, evolutionary algorithms are often implemented as general purpose optimization algorithms which make small assumptions on the objective functions, and the convergence to the true Pareto front may therefore be quite slow. In this paper we instead concentrate on methods which use surrogate models for the objective functions. A surrogate model is an approximation of the true objective function based on all data evaluated so far. In the multiobjective case a surrogate model is created for each objective function so we get a vector valued surrogate model. There are many options for surrogate models [8]. The simplest examples are response surfaces which are least square approximations with low order polynomials. Other options are, for example, neural networks, lolimot models, multivariate splines, Kriging and radial basis functions. All these approximations can be combined with transformations of both the parameter space and the objective functions themselves to yield more accurate approximations. The problem is then which approximation to use. A way to compare different types of approximations, estimate parameters and validate the surrogate models is cross validation which is the topic of Section 4. In this paper we apply a recently developed multiobjective optimization, algorithm, denoted qualSolve[7], based on approximations with radial basis functions. A key property of the algorithm is to adaptively improve these approximations in interesting areas of the parameter space. The result of the algorithm is both a set of approximately Pareto optimal solutions and also expansions of all objective functions in radial basis functions which can be used for post-processing. Because of the long simulation times it is beneficial for the optimization to run several simulations using different parameter settings simultaneously. Most optimization algorithms are serial, only one evaluation is done at a time, but if a cluster is available it is necessary that the optimization algorithm can generate a given number of new evaluation points in each iteration step to use the computer resources efficiently. To summarize this discussion in a specification of a suitable algorithm for combustion engine optimization: An efficient multiobjective algorithm for combustion engine optimization should be a two stage model based on approximation of the objective functions with surrogate model. It should be able to handle both hard and soft constraints, and allow for several simultaneous simulations. The paper is divided into seven sections. In the next section we go through some basic concepts for surrogate models. The following section deals with validation of the surrogate models through cross validation and this technique is also applied to find outliers (incorrect data) which may be the result of some error during the simulation. In Section 4 the optimization algorithm qualSolve is summarized, more details are found in the work of Jakobsson et. al. [7]. In Section 5 a case study for multiobjective optimization of a diesel combustion

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engine is described, and in Section 6 results are presented. The last section includes a discussion of the results and some conclusions.

2 Surrogate Models A surrogate model is an approximation of the objective function based on the already evaluated data. In the multiobjective case, a surrogate model is created for every objective function so in a sense we get a vector valued surrogate model. There are many ways of constructing surrogate models. The most simple examples are approximations with low order polynomials. In these cases the approximations are also called response surfaces. Other methods include neural network, Kriging and approximation with radial basis functions to name a few. Furthermore, these methods can be combined with transformations of both the parameter space and the objective space. In this work we focus on interpolation/approximation with radial basis functions combined with transformations and scaling of the input variables. In the following section we also describe how cross validation can be used for evaluating different types of surrogate models, finding suitable values for free parameters in the models and also to find outliers in the data.

2.1 Radial basis functions Radial basis functions provide a very general and flexible way of interpolation in several dimensions, even for unstructured data where it is often impossible to apply polynomial or spline interpolation. The method is a popular choice in many different areas, ranging from statistics to applications to partial differential equations, because of its good approximation properties and ease of implementation. More information on RBFs can be found in [1], [2] and [12]. Connected to each radial basis function is a native space equipped with a semi-norm uniquely determined by the basis function. The RBF interpolant can the be interpreted as the minimal norm solution (in the native space) among those functions which interpolate all data, S = arg min S˜ ˜ k )=fk S(x



By relaxing the interpolation condition to allow a small error at the data point we will get a smoother approximation of the function S(xk ) = fk + ek . We replace the problem with S = arg

min

˜ k )=fk +ek S(x

2 η S˜



+ (1 − η)kek2l2

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where η is a parameter in the range (0, 1). In the limit η → 0+ the error vector will be zero and we get the standard RBF interpolant. More details can be found in [7]. In many situation we know that a certain quantity is strictly positive. This holds for example for fuel consumption and emissions soot and NOx . In that case it may be advantageous to approximate the logarithm of the objective function by an RBF expansion and then compose it with the inverse transformation, i.e., the exponential transformation.

3 The qualSolve Algorithm D. R. Jones categorize in [8] global optimization methods into one and two stage methods. In the two stage approach the first stage is to create a surrogate model which fits the computed data. In the second step this surrogate is used to find new search points. This should be compared with the one stage approach where the new search points are found without using the surrogate model. According to Jones the potential risk with a two stage approach is that the model is trusted too much so that the optimization stops too early or the search is to local. It is therefore crucial for a two-stage algorithm that the surrogate model is of good quality, i.e., it approximates the true objective function well and that no areas are unexplored. The multiobjective optimization algorithm developed here is based on approximation of the objective functions with so called surrogate models and falls therefore into the category of two stage methods. The surrogate models used are interpolation/approximation with radial basis functions perhaps in combination with transformation of the objectives. Prior the optimization, all parameters are scaled so that their lower and upper limits are 0 and 1, which means that the optimization is carried out in the unit hyper cube with the same dimension as the number of input parameters. If there are any additional constraints these are also transferred to the hyper cube. The idea of the algorithm is to improve the quality of the surrogate models during the course of optimization, especially in the interesting areas, that is, close to the Pareto front. The improvement is measured by a quality function which is defined for the approximation. This quality function involves the uncertainty of the surrogate models approximation, depends on the values of the surrogate model and measures the improvement of the approximations in a weighted average sense. The uncertainty UX (x) of the approximation at a point x is measured by its distance to the closest already evaluated point UX (x) =

min

k=1,...,N

kx − xk k.

We measure the total uncertainty in the relevant areas by Z UX (x)ω(S(x)) dV (x). Ω

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The weighting function ω included in this measure should have high values in the interesting regions (near the Pareto front) and low values far from the Pareto front. A basic requirement on ω is that it should be monotonic in the ˜ , i.e. it is better in multiobjective sense: if a point x dominates another point x all objectives S(x) ≤ S(˜ x), then ω(S(x)) > ω(S(˜ x)). Our choice of weighting function will be specified later. To select a new point y to be evaluated we want to minimize this averaged uncertainty. This leads to the definition of the quality function which is given by Z Q(y) = (UX (x) − UX∪y (x)) ω(S(x)) dV (x). Ω

where UX∪y (x) is the uncertainty when y is added to the already evaluated points. The new evaluation point is then chosen so that it maximizes the quality function arg max Q(y). (2) y∈Ω

The choice of weight function made here is  ω(z) = exp −σdistZS∗ (z) ,

where the function distZS∗ is the smallest Euclidean distance between the point z and the Pareto front ZS∗ of the scaled surrogate model and σ is a positive real number. For high values of σ new evaluation points are likely to be close to the Pareto front (local search) of the surrogate model whereas low values yields a more global search. In order to find a balance between local and global search this number σ is varied cyclically during the course of optimization. At some instances we even choose σ = ∞ and in this case we choose the point on the Pareto front which is as far away from an already evaluated point as possible. A more detailed explanation of the algorithm can be found in the work of Jakobsson et. al. [7]. Two modifications have been made to the algorithm for this application. Firstly, in order to run several cases simultaneously we must generate several new evaluation points at the same time. We solve this problem by not updating the surrogate model between consecutive calls to the optimizer for minimization problem (3). Between each call the newly found point is added to the set X. This makes it less favourable to put an additional point in its immediate surrounding. Secondly, in many cases the designer is not interested in the whole Pareto front. Some parts of the front may not be of interest because some of the objectives are to high there. Here we have solved this problem by computing the function in the domain where the surrogate model satisfies this extra constraint. 4 Validation of surrogate models As already mentioned there are many ways to create surrogate models from data. One may then ask which is the most suitable for a certain problem.

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Cross validation offers a method to compare the different model options in an objective way. In this paper we use cross validation both to estimate free parameters in the RBF models and to find outliers in the data.

4.1 Cross validation parameter estimation Cross validation is a statistical method for evaluating and comparing different models based on their predictability. The presentation, terminology and notation here follows the work of Hjort [5]. Let DS

= data set

= all data which can be analyzed

ES = estimation set = subset of DS used for estimation TS = test set = subset of DS used for testing Normally the sets ES and TS are disjoint and their union is DS. For cross model validation and for parameter estimation the data set DS is divided into a number of different pairs (ESk , TSk ) with ESk ∩ TSk = ⊘ and ESk ∪ TSk = DS. A common approach is the leave-one-out cross validation, in this case TSk = (xk , fk ) (a single data point) and ESk = all points except (xk , fk ). Let Sk be the surrogate model based on the estimation set ESk . The cross validation measure is QCV =

N X

2

(Sk (xk ) − fk ) .

k=1

If we have several different surrogate models to choose from then the cross validation methodology gives a sort of best candidate, the one with the least cross validation measure. If the surrogate model depends on some parameter, for example the parameter η in the RBF approximation, then the cross validation measure depends on this parameter QCV (η) and we choose the parameter which minimizes the cross validation measure η = arg min QCV (η). η

There are some problems with the standard cross validation measure as it is applied here. Errors in the surrogate model can be the result of two things. Either we have not enough data to construct a reliable model or we have errors in our input data. If the data points are not evenly distributed then the model should predict points which have close neighbors better than points which are far from other evaluated points. This suggests that we should put more weight on points which have close neighbors in the cross validation measure. When the surrogate model is a composition of for example a RBF expansion and a transformation, the transformation may damp or amplify errors depending on the value of the surrogate model. Both these problems indicate that some further analysis should be made in order to be successful.

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4.2 A cross-validation method for finding outliers A problem with this kind of simulation based optimization with very long simulation times is the risk for getting corrupted data into the optimization (the optimization campaign may take several months). It may happen that the simulation software is updated or there are other changes of the computer system that may cause the simulations to produce false output. It is then important to identify these errors and rerun the corresponding cases. We developed a two-point cross validation technique to find these erroneous points. The method is as follows. Let X = {xk }N k=1 be the set of all evaluated points and define wk =

N  2 X SX\(xk S xl ) (xl ) − fl ,

k = 1, . . . , N,

l=1,l6=k

where SX\(xk S xl ) is the surrogate model constructed from all data except the two points xk and xl . This surrogate model is then evaluated at xl and compared with the data fl . We see that wk is the ordinary leave-one-out cross validation measure with the point xk excluded from the data set. The interpretation of a relatively low value of wk for a specific k is that we then can predict the other data better without this value which suggest that it might be incorrect.

5 Case study - Optimization of a diesel combustion engine 5.1 Optimization problem formulation The objectives chosen in the present work are NOx content, soot content and IMEP. NOx content is the average mass fraction of nitrogen oxides in the exhaust gasses and should be minimized. Correspondingly, soot content is the average mass fraction of soot compounds in the exhaust gasses, a property that also should be minimized. IMEP is a measure of the work output from the engine and should be maximized. These three objectives form a natural problem encountered when designing a new engine - how to minimize the emissions at a given workload. The part of engine design chosen here is the injection of the diesel spray, which is an important but also well contained part of the design cycle. The parameters to vary in the spray are span angle, nozzle tip protrusion, swirl number, injection nozzle hole diameter and later in the course of research also injection timing. In Figure 1 the significance of the first four design parameters are illustrated. The fifth and last design parameter is injection timing, which here is defined as the time at which the first burst of fuel is injected into the combustion chamber. The same injection profile, shown in Figure 2 is used throughout the optimization, but with different offset times. All input parameters with box constraints and the objective functions are summarized in Tables 1 and 2.

9 Table 1 Optimization objectives. OBJECTIVES IMEP NOx soot

Type maximize minimize minimize

DESIGN PARAMETERS Span angle Nozzle hole diameter Nozzle tip protrusion Swirl number Injection timing

Min 55◦ 0.125 mm 1.45 mm 2.0 338 CAD

Table 2 Design parameters. Max 85◦ 0.17 mm 2.0 mm 2.65 358 CAD

Fig. 1 A snapshop of a CFD-simulation at the time of injection of the main burst of fuel. Illustrated is the significance of the four spray parameters that are related to the geometry. Tip protrusion is the distance the nozzle in inserted into the geometry, span angle is the angle at which the fuel is injected, and swirl number relates the swirling flow to the radial flow in the combustion chamber at the time of injection. The nozzle hole diameter is, naturally, the hole of the nozzle injecting the fuel.

5.2 CFD simulation The simulated case represents the Volvo D5 engine. The engine speed is 2000 rpm and the operating conditions correspond to an engine torque of 122 Nm. The simulation is started when the air intake valve has closed, which happens at the beginning of the compression of the combustion chamber. The flow development is then solved for during the compression phase, injection and combustion of fuel and during the subsequent expansion of the combustion chamber. The simulation is terminated when the exhaust valve is opened at the end of the expansion, and the gas composition at this point in time is

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Fig. 2 Inlet mass flow of fuel as a function of crank angle degree.

taken to represent the composition of the engine exhaust gases. The following stroke of the piston, when combustion residuals are evacuated and new air is introduced, is not simulated. As a consequence, the state of the gas at intake valve closure has to be prescribed. Test engine data is used to estimate this initial state. The commercial flow solver Star-CD was used for the combustion simulation. The choice was natural as Volvo has previous experience of diesel combustion simulations in Star-CD, and since reactive flows with droplets and moving meshes is easily handled with the software package. A 72 degree sector of a cylinder was simulated in a 200000 hexahedral cell mesh - the 72 degrees corresponds to a fifth of the cylinder, which is natural as the diesel in this case is injected through a five-hole nozzle. The mesh has been prepared to start at the beginning of the compression stroke, counting from the start of the intake cycle the starting point is at Crank Angle Degree 240, that is, 60 degrees into the compression part of the cycle. As the combustion chamber is compressed and expanded, different versions of the mesh is used where a number of layers of computational cells are omitted in order to get a good aspect ratio of the cells even when the chamber is at its smallest. A snapshot of the mesh is shown in Figure 3.

5.3 Model details The CFD model used is k-ε with standard wall functions [3]. This is a debatable choice in a swirling and compressible flow scenario, but the CFD model also needs to be executable within a reasonable time as the number of computed simulations by necessity is rather large in a multiobjective and multiparameter optimization case like this, hence it may be motivated to use a simple but

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Fig. 3 Mesh of the 72 degree cut of the cylinder. The sections in different colors correspond to the parts of the domain solved one each processor.

reasonably fast model as the k-ε is. Moreover, being so widely used many of the drawbacks of the model are known and can be compensated for. At the end of the compression cycle a pilot spray of diesel is injected into the cylinder, this injection is evaporated and mixed by the flow and ignited by the compression. The main injection of fuel takes place around 15-20 Crank Angle Degrees later, at the beginning of the expansion stroke. The fuel injection profile is shown in Figure 2. Size distributions of the injected droplets are computed using the Huh atomization model, in which an energy spectrum and an atomization time scale computed from the turbulence quantities is used to construct the probability density function for droplet size [6]. The complex chemistry of diesel combustion is simplified using a turbulent flame-speed model. Reaction rates are determined using the Eddy Break-Up (EBU) model and the Laminar and Turbulent Characteristic Time (LTCT) model. The EBU model, originally proposed by Magnussen, Hjertager and Spalding [10], is one of the most straightforward way to tackle turbulent combustion, using a single-step global reaction. The LTCT model is based on computing local time-scales for the rate of combustion, time-scales that are formed from a laminar time-scale computed from an Ahrrenius expression and a turbulent time-scale that is computed from the turbulent quantities i.e. k and ε. At the completion of the simulation of the combustion cycle, the output parameters are computed. IMEP is computed by integrating the pressure on the piston over the cycle. The soot emissions are computed using the Mauss

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soot model [9], and the NOx emissions are computed from the transient temperature field using an extended Zeldovich model [3].

6 Results The results in this section are based on RBFs constructed from 238 calculations in the optimization loop following an initial DoE of 56 CFD cases. Each simulation takes around 24 hours to conduct on a four CPU node based on the Xeon 5160 (Woodcrest) processor. The long computation time emphasizes the need for minimizing the number of computational runs as much as possible. For a high number of objectives it is a complex task to actually visualize the solution, but equally important as the main idea of the multiobjective approach is to save the decision making for the ones who know the process at hand. In this case the number of objectives is only three, which means that the Pareto front can be visualized in three dimensions, and also in two with a bit of work. Figure 4 contains such a visualization of the Pareto surface for the three objectives - soot, NOx and IMEP. For different values of IMEP, the trade off curve between soot and NOx is shown. Each curve can be regarded as a horizontal cut through the two dimensional surface that is the true Pareto front in this case. In Figure 5, 6 and 7 part of the RBFs for IMEP, NOx and soot are shown for constant values of tip protrusion and swirl number - which were found to be the parameters of least impact on the objectives. Apart from the final results the development of the RBFs during optimization is shown through the thinner isolines. It should be noted that as the RBFs depend on five variables, the task of reconstructing them in a reliable way from as few as 250 samples is by no means straight-forward. The main idea behind the optimization scheme is to focus on the promising regions, and this means that the parts of the RBF that lie closer to a potential minimum is to be considered more trustworthy. From the RBFs shown, it is evident that the optimum for soot and IMEP are to be sought in the mid range of span angle. This is in strong contrast to the optimum in NOx which lies towards the boundaries, specifically close to the maximum span angle allowed. This trade-off is actually well known in engine engineering and is commonly referred to as the diesel dilemma. The reason is that good combustion with little soot requires and leads to high temperatures in the cylinder, but high temperatures in the cylinder also lead to a high rate of formation of NOx . In Figures 8 and 9 the difference between two different possible choices of realization is illustrated. In one case NOx is really low at the expense of bad IMEP and high soot formation. As can be seen, the temperature in that case is lower, indicating an uncompleted combustion of the fuel. In the other case, NOx is higher and soot is lower; in this case a higher temperature and the absence of residual fuel droplets indicate a more complete combustion process. In this work, however, the Pareto front rather than a specific choice of trade-off is the end result of the optimization loop. Convergence is considered

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to be reached when there is no longer any significant development of the Pareto front; it is then up to the designer of the engine/car to make the trade-off and pick the solution of choice. In Figure 10, the development of the Pareto front during the last 50 simulations is shown. Using the converged solution it is also possible to construct plots of the optimal design parameters along chosen paths on the Pareto front. In Figure 4 two such paths, taken as two different levels of IMEP, are indicated. The design parameters are then illustrated along these paths in Figures 11 and 12. It is interesting to note how the parameter set seems to be piecewise smooth and continuous, with a few jumps in the solution. As we move along the selected path the solution jumps from one set of parameters to a radically different set, which in effect is a completely different engine realization. In Figure 11 we see six or seven different such engine states appearing, and in Figure 12 we can see two. This type of information can easily be extracted in the multiobjective approach and should be judged and exploited by a skilled designer. Apart from leaving the work of judging the trade-offs outside the optimization loop, the multiobjective approach outlined in this work is also flexible in the specification of input parameters. Initially, only the four spray shape variables from Figure 1 were used as design parameters. During the simulation procedure described in this work, it was noted that injection timing was strongly influential on the objectives and it was then decided to include this as a fifth design parameter, knowing that with the concept of surrogate models the effort to start varying a fixed modeling variable as an extra input parameter is small. In this case, a few DoE runs were made to get a starting guess of the variations of the RBFs with varying injection timing, and then the optimization loop needed another 100 simulations in order to come up with improved RBFs to take injection timing into account. 6.1 Cross validation and approximation Cross validation can be used to estimate the parameter η in the RBF expansion, which determines the level of approximation. Table 3 list the optimal values of η found for the approximations. Cross validations were evaluated at the end of the optimization, 25 iterations before the end and 50 iterations before the end. The general trend is that NOx and particularly soot is more Table 3 Optimal value of η as determined by cross validation for different data sets. Data set 50 before end 25 before end full set

IMEP 0.0018 1.8557e-15 1.8557e-15

NOx 0.1628 0.2800 0.2694

soot 0.2952 0.2952 0.3162

difficult to interpolate which reflects in the larger values of η for those objectives. There is a large jump in the magnitude of the η parameter for IMEP

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when extending the data set. This indicates that one or more points were added which made approximation superfluous in the surrogate model. Figure 13 shows the Pareto front extracted from surrogate models with the η values taken from the table. This plot can be compared with Figure 10. The former figure shows generally smoother curves with less sharp bends. On a general note it can be said that the soot values in the Pareto fronts based on approximations are larger than with pure interpolation, which indicates that low soot regions might be hard to predict. This could for example be the case if low soot is a very local phenomenon in design space. Looking at the results in Table 3 it is clear that we should expect interpolation to perform well with IMEP. This can be used to check the CFD data for possibly corrupted data. For instance, a rapid variation of the l2 value of the objective will indicate that something is wrong with the interpolation. Figure 14 shows the result of a validation of the CFD data using the described cross validation technique. Noteworthy is the dip in the l2 line for IMEP, suggesting that the cases 64 and 67 may be incorrect, which they indeed were.

Fig. 4 Visualization of the Pareto Front with constraints on IMEP. The two marked curves correspond to the cases plotted in Figures 11 and 12, respectively.

7 Conclusions In this paper we have performed a multiobjective optimization of a diesel combustion engine. No quantitative comparison with different multiobjective

15 0.17

12

0.165

11.5

Hole diameter [mm]

0.16

11

0.155 10.5 0.15 10 0.145 9.5 0.14 9

0.135

8.5

0.13 0.125 55

60

65

70 75 Span angle [°]

80

85

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Fig. 5 IMEP as function of the hole diameter and span angle. The other design variables were set to their middle value. The thicker lines in the contour corresponds to the complete data set, and the other two to all but the last 42 and all but the last 84, respectively. 0.17

−2.8

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0.14

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0.135

−4.2

0.13

−4.4

0.125 55

60

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70 75 Span angle [°]

80

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Fig. 6 NOx as function of the hole diameter and span angle. The other design variables were set to their middle value. The thicker lines in the contour corresponds to the complete data set, and the other two to all but the last 42 and all but the last 84, respectively.

16 0.17

−4

0.165 −5

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0.16 0.155

−6

0.15 −7 0.145 0.14

−8

0.135 −9 0.13 0.125 55

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70 75 Span angle [°]

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Fig. 7 Soot as function of the hole diameter and span angle. The other design variables were set to their middle value. The thicker lines in the contour corresponds to the complete data set, and the other two to all but the last 42 and all but the last 84, respectively.

Fig. 8 CFD image showing temperature field, fuel droplets and fluid streamlines for a simulation case on the Pareto front where IMEP and soot are bad, but NOx is good.

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Fig. 9 CFD image showing temperature field, fuel droplets and fluid streamlines for a simulation case on the Pareto front where IMEP and soot are good, but NOx is bad. Note that there are less visible fuel droplets and generally a higher temperature in this case than in the previously shown case. This is due to the more complete combustion in this case.

Fig. 10 Evolution of the Pareto curves with constraints on IMEP. Full line: full data set. Dashed line: All but the last 25 simulations. Dotted line: All but the last 50 simulations. Surrogate models based on interpolation.

18 1 0.9

Scaled parameter value

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

0.2

0.4 0.6 Position on Pareto front

0.8

1

Fig. 11 Design parameters along the selected path of IMEP ≥ 10. 1 0.9

Scaled parameter value

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

0.2

0.4 0.6 Position on Pareto front

0.8

Fig. 12 Design parameters along the selected path of IMEP ≥ 12.

1

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Fig. 13 Evolution of the Pareto curves with constraints on IMEP. Full line: full data set. Dashed line: All but the last 25 simulations. Dotted line: All but the last 50 simulations. Surrogate models built with the value of η taken from table 3.

optimization algorithms have been made in this work, but a few qualitative comments can be made about its performance. The optimization algorithm is developed with the particular application of engine optimization in mind and is based on successive approximations of the objectives with surrogate models. The use of surrogate models gives flexibility that allows for adjustments to the problem solved during the process of optimization, and the multiobjective approach of focusing on the Pareto front gives a flexibility in a design selection - selecting a range of design values also makes it a robust method. Cross validation is a useful statistical tool in this approach, as it makes it possible to compare different models and also find outliers in the data. For the future, more work needs to be devoted to determine the convergence of the algorithm. In each iteration step we get an approximate Pareto front for the surrogate model. Based on this it might be possible to define a good convergence criterion. The algorithm also needs some modification in order to be able to handle higher dimensional problems than the ones treated in this work. Finally, a better treatment of the bounds on the objectives should be implemented in order to facilitate an investigation of a selected part of the Pareto front. Acknowledgements This work has been funded in part by the Swedish Foundation for Strategic Research (SSF) through the Gothenburg Mathematical Modeling Centre (GMMC). The work was supported by CD-adapco by providing Fraunhofer-Chalmers Centre with a research license of STAR-CD and also Ulf Engdar at CD-adapco for giving the introductory

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Fig. 14 Results from a two-point cross validation of simulation data for combustion engines.

course of the software at FCC. The computations were performed on C3SE computing resources at Chalmers University of Technology.

References 1. M. D. Buhmann, Radial basis functions, Acta numerica, 2000, Acta Numer., vol. 9, Cambridge Univ. Press, Cambridge, 2000, pp. 1–38. 2. M. D. Buhmann, Radial basis functions, Cambridge Monographs on Applied and Computational Mathematics, vol. 12, Cambridge University Press, Cambridge, 2003. 3. CD-adapco, User guide, star-cd version 3.26, 2005. 4. H. W. Hamacher and K.-H. K¨ ufer, Inverse radiation therapy planning – a multiple objective optimization approach, Discrete Applied Mathematics 118 (2002), no. 1-2, 141–161. 5. J. S. U. Hjorth, Computer intensive statistical methods, Chapman & Hall, London, 1994, Validation model selection and bootstrap. 6. K. Y. Huh and A. D. Gosman, A phenomenological model of diesel spray atomization, Proc. of the International Conference on Multiphase Flows (Sep. 1991.). 7. S. Jakobsson, M. Patriksson, J. Rudholm, and A. Wojciechowski, A method for simulation based optimization using radial basis functions, Submitted for publication, January 2008. 8. D. R. Jones, A taxonomy of global optimization methods based on response surfaces, J. Global Optim. 21 (2001), no. 4, 345–383. 9. A. Karlsson, I. Magnusson, M. Balthasar, and F. Mauss, Simulation of soot formation under diesel engine conditions using a detailed kinetic soot model, Journal of Engines, SAE-Transactions (1998), no. 981022.

21 10. B. F. Magnussen and B. H. Hjertager, On mathematical models of turbulent combustion with special emphasis on soot formation and combustion, Proc. Combustion Institute 16 (1976), 719–729. 11. K. Miettinen, Nonlinear multiobjective optimization, International Series in Operations Research & Management Science, Kluwer Academic Publishers, Boston, 1999. 12. Holger Wendland, Scattered data approximation, Cambridge Monographs on Applied and Computational Mathematics, vol. 17, Cambridge University Press, Cambridge, 2005.