Struct Multidisc Optim (2012) 46:129–136DOI 10.1007/s00158-011-0757-1

INDUSTRIAL APPLICATION

Enhanced POD projection basis with application to shapeoptimization of car engine intake port

Manyu Xiao · Piotr Breitkopf ·Rajan Filomeno Coelho · Catherine Knopf-Lenoir ·Pierre Villon

Received: 26 August 2011 / Revised: 2 December 2011 / Accepted: 26 December 2011 / Published online: 13 January 2012c© Springer-Verlag 2012

Abstract In this paper we present a rigorous method for theconstruction of enhanced Proper Orthogonal Decomposi-tion (POD) projection bases for the development of efficientReduced Order Models (ROM). The resulting ROMs areseen to exactly interpolate global quantities by design,such as the objective function(s) and nonlinear constraintsinvolved in the optimization problem, thus narrowing thesearch space by limiting the number of constraints that needto be explicitly included in the statement of the optimiza-tion problem. We decompose the basis into two subsets oforthogonal vectors, one for the representation of constraintsand the other one, in a complementary space, for the min-imization of the projection errors. An explicit algorithm ispresented for the case of linear objective functions. The pro-posed method is then implemented within a bi-level ROMand is illustrated with an application to the multi-objectiveshape optimization of a car engine intake port for two com-peting objectives: CO2 emissions and engine power. Weshow that optimization using the proposed method producesPareto dominant and realistic solutions for the flow fieldswithin the combustion chamber, providing further insightinto the flow properties.

Keywords Proper Orthogonal Decomposition ·Reduced order modeling · Intake port · Optimization

M. Xiao · P. Breitkopf (B) · C. Knopf-Lenoir · P. VillonLaboratoire Roberval, Université de Technologie de Compiègne,Compiègne, Francee-mail: piotr.breitkopf@gmail.com

R. Filomeno CoelhoBATir, Université Libre de Bruxelles, Brussels, Belgium

1 Introduction

Using high-fidelity simulation models to predict theresponse of structures for design optimization and uncer-tainty quantification often leads to an unacceptable com-putational cost, thus motivating the research of techniquesto extract features from complex physical fields using areduced number of full-order numerical experiments. TheProper Orthogonal Decomposition (Berkooz et al. 1993)is of particular interest in the optimization of engineeringproblems, where a set of scalar objective/constraint func-tions that depends on the values of design variables, isevaluated such as a surface/volume integration of a veloc-ity/stress/. . . field obtained from a finite element/volumesimulation. Literature reviewed thus far shows a large vol-ume of work recently published on improving the precisionof POD-like ROMs and on interpolation between theseROMs.

One of the research directions focuses on minimizingthe number of full-scale analyses by including informa-tion about the gradients. Weickum et al. (2009) enriched aPOD for the transient response of linear elastic structuresby using the gradients of the POD modes with respect tothe design/random parameters for robust shape optimiza-tion. In the same spirit (Carlberg and Farhat 2011; Hay et al.2010) extended the concept of POD snapshots to includederivatives of the state variables with respect to system inputparameters.

In order to avoid generating additional sampling points,an effort has been made to develop interpolating strategiesbetween the ROMs. Missoum (2008) used Lagrange inter-polation to control the relative contributions of individualmodes in order to perform a random-field-based probabilis-tic optimization of a tube impacting a rigid wall. Mathelinand Le Maitre (2010) proposed polynomial transformation

130 M. Xiao et al.

of the POD projection coefficients over a coarse time-step,for application to the 2D flow past a circular cylinder inasymptotic and transient cases. Amsallem et al. (2009) inter-polated the ROM data in a tangent space to the manifold ofsymmetric positive-definite matrices, and mapped the resultback to the manifold for the dynamic characterization of aparameterized structural model so as to evaluate its responseto a given input. Degroote et al. (2010) compared splineinterpolation of the reduced-order system matrices in theoriginal space as well as in the tangent space to the Rieman-nian manifold with Kriging interpolation of the predictedoutputs for a steady-state thermal design problem and prob-abilistic analysis via Monte Carlo simulation of an unsteadycontaminant transport problem.

Our approach in this paper targets the development ofmulti-objective/multi-disciplinary optimization techniquesusing high-quality ROMs. In this work we improve the bi-level reduced-order model strategy (Filomeno Coelho et al.2007, 2008) based on the POD of the initial data set andon kriging/RBF/MLS/. . . approximation of the projectioncoefficients. In Xiao et al. (2010) we have introduced theconstrained POD, which consists basically of the adapta-tion of POD coefficients in order to interpolate quantitiesof interest (objective/constraint functions). Here, we focuson further tailoring the POD technique in order to mod-ify the basis vectors rather than the coefficients in order tointerpolate global quantities exactly. The benefits are a bet-ter physical meaning of the adapted POD modes, a lowernumber of basis vectors and a deeper insight into the post-processed optimization results. This approach, focusing onthe improvement of the precision of the ROM by an appro-priate choice of basis vectors, may be used in local as wellas in global versions of the POD.

The paper is organized in three parts: in the first section,we revisit the standard POD, reformulated here as a mini-mization problem. This presentation allows the introductionof additional constraints, aiming to enhance the projec-tion coefficients of a standard POD basis (Xiao et al.2010). Section 2 is the central part of the paper, where thebasis vectors are considered as variables of a constrainedminimization problem and an algorithm is presented forthe explicit computation of additional modes. In the thirdsection, we present the data analysis and the results obtainedfor the multi-objective shape optimization results of a carengine intake port. We close with some concluding remarksand prospects for future work.

2 Proper Orthogonal Decomposition

Let{v(k) = (

v(k)1 , . . . , v

(k)n

)T; k = 1, . . . , M}

be a set ofsnapshots of the velocity field obtained by running a “high

fidelity” model on a representative sample of M points inthe design space, where n is the size of a snapshot andcorresponds typically to the number of degrees of free-dom associated with a finite element model. The PrincipalComponent Analysis of the data allows us to express thesnapshots in terms of an average snapshot v and the set ofbasis vectors �

v(k) = v + �α(k) (1)

The basis � is usually presented (after mean centering thedata) as a set of eigenvectors of the covariance matrix,or singular value decomposition of the data matrix. Thecoefficients α are calculated by the projection of thesesnapshots on the basis �

α(k) = ⟨v(k), �

⟩, k = 1 . . . M. (2)

The reduced order model is obtained by using only a sub-set �,m of the first m<<n modes. The snapshots are thenreproduced with the error given by

ε =

M∑

k=m+1λk

M∑

i=1λi

(3)

where we make the reasonable assumption that the numberof experiments is less than the dimensionality of the finiteelement problem, M < n.

In the current paper, we use an alternative presentation,more suitable for the derivation of constrained approxima-tion, in which the basis vectors may be interpreted as asolution of the minimization problem

� = Argmin

(M∑

k=1

∥∥∥v(k) − v(k)∥∥∥

L2(�)

)

(4)

In Xiao et al. (2010), (4) was exploited in order to intro-duce an additional constraint requiring that some functions(objective or optimization constraints) have the same valuewhether evaluated using an exact or an approximate field,thus

J(v) = J (v) (5)

at least for the snapshot sampling points. Our approach thenwas to express the approximate snapshot using a standardPOD basis (4)

v(k) = v + �γ (k) (6)

with a modified set of coefficients γ obtained by solving theminimization problem

γ (k) = Argmin�=const

(M∑

k=1

∥∥∥v(k) − v(k)∥∥∥

L2(�)

)

Enhanced POD projection basis with application to shape optimization of car engine intake port 131

subject to the set of constraints

J(

v(k))

= J(

v(k))

, k = 1 . . . M. (7)

In the case of linear constraints, the discrete form of (7) is

CT v = CT v(k) = c(k), k = 1 . . . M (8)

and γ are obtained by solving

[�T

,mM�,m �T,mC

CT �,m 0

] {γ (k)

λ

}=

{�T

,mM(v(k) − v

)

c(k) − CT v

}

(9)

with the matrix W representing both the interpolationoperator over the reference grid and the numerical inte-gration of the quantity of interest J (v), M a standardfinite element mass matrix and λ the Lagrange multipliers.This approach was successfully applied to several prob-lems using a bi-level approximation, coupling POD withkriging/RBF/Diffuse Approximation and we showed thatsatisfying the interpolation property (5) was critical ina multi-objective problem. There remain however somedifficulties, since the existence of a solution to (9) cannotbe guaranteed in a general case.

3 Computing constrained modes �

Here, we propose an alternative formulation in whichinstead of modifying the coefficients, we express theapproximate field using a modified set of orthogonalmodes �

v(k) = v + �α(k) (10)

obtained by imposing the constraints (5) on the minimiza-tion problem

� = Argmin�T �=I

(M∑

k=1

(∥∥∥v(k) − v(k)∥∥∥

L2(�)

))

(11)

and with the coefficients α obtained by projecting the snap-shots on the modified basis in the same way as in standardPOD (2)

α(k) = ⟨v(k), �

⟩, k = 1 . . . M. (12)

The main idea behind this algorithm is based on the obser-vation that the necessary and sufficient condition (NSC) forthe existence of a solution to (11) is that for any velocityfield v in the high fidelity model space, there should exist afield w in the reduced order space such that

CT v = CT w (13)

To satisfy this condition, it is sufficient for ImC to belongto the reduced space with basis �. ImC is obtained by QRfactorization of the matrix C

C = QR = [Q1 Q2

] [R1

0

]= Q1R1 (14)

where R1 is an upper triangular matrix p × p, Q1 is n × p,Q1 is n × (n − p) and Q1 and Q2 have orthogonal columns.Since R1 is invertible, ImC = ImQ1, so the the first p termsof the basis

� = [ψ1 . . . ψp ψp+1 . . . ψm

](15)

may be assimilated with Q1 and the remaining basis vectorsare to be found in the ImQ2 giving

� = [Q1 Q2] (16)

where is the matrix of first m − p eigenvectors of the

covariance matrixM∑

k=1

(v(k) − v

)TQ2QT

2

(v(k) − v

)of the

snapshots v(k) projected on Q2. Finally, the coefficients α ofthe constrained approximation v(k) = v + �α(k) are givenas in (2) by

α(k) = �T(

v(k) − v)

(17)

There is a clear interpretation of the role of these two sub-sets of the basis vectors: the first subset � = [

ψ1 . . . ψp]

is sufficient for optimization since due to the orthogonalityproperties of Q1 and Q2, the remaining m − p modes haveno contribution to the quantities of interest such as objec-tive functions and constraints. This is a key advantage, asthe size of the ROM may now be kept low! The questionthen arises, do we ever need to compute the second subsetof modes? The answer is clearly yes, as the full set of modesprovides a deeper insight into the optimized solutions per-mitting the reconstruction of detailed flow field beyond theglobal quantities involved in optimization process. On theother hand, since our approach preserves global quantitiesby design, it narrows the search space by limiting the num-ber of constraints that need to be explicitly included in theoptimization problem statement.

4 Application: intake port of an automotivediesel engine

We use the test case presented below to illustrate threefeatures of the proposed approach:

– the constrained POD has analogous convergence prop-erties to the standard POD,

132 M. Xiao et al.

Fig. 1 Six CAD variablesdefining the geometry of theduct

– the additional p modes have a simple physical interpre-tation,

– the first p basis vectors alone are sufficient for opti-mization.

We show below, that the above properties allow us tooutperform the standard POD in a multi-objective optimiza-tion case. This allows us, when projecting the results to thefull space of M modes, to get a qualitative insight into theoptimal solution.

4.1 Description of the test case

In the automotive industry, an engine must be optimizedfor the competing objectives of performance and com-pliance with environmental standards. This study focusesspecifically on the design and optimization of the intakeduct of the cylinder. The goal is to find a set of shapecompromises for the duct maximizing two objectives:

– vorticity or tumble T , related to the ability of the fluidand fuel to mix efficiently in the combustion chamber(pollution reduction);

– mass flow Q, characterizing the power of the engine.

The velocity field in the combustion chamber dependson the shape of the intake port which is driven by six designvariables y = (y1,. . . , y6)

T (Fig. 1).

T and Q are computed by integrating the velocity fieldwithin a fixed zone � chosen in the cylindrical part of thecombustion chamber⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

T = T (y) =∫

�

(x − x0) vzd�

Q = Q (y) =∫

S

vzd S(18)

where vz is the axial component of the velocity field, S is thecylinder cross section and x − x0 is the radial coordinate.

The optimization problem may be formally stated as

max (T (v (y) ) , Q (v (y))

yinf ≤ y = (y1, . . . , y6)T ≤ ysup (19)

Before discussing the solution of the optimization problem,we first begin with a description of the Design of Experi-ments and the analysis of the data in the next subsection.

4.1.1 Simulation process

The parametric CAD model was generated using STAR-Design. CFD mesh of approximately 90,000 cells wasobtained and fluid flow computation was carried out usingSTAR-CCM code. The computations were stationary, thenumber of iterations was 1,000 (enough to reach conver-gence in most cases) with pressure conditions in input and

Fig. 2 Flowchart of the processand definition of the zone ofinterest for computation of Tand Q (18)

Enhanced POD projection basis with application to shape optimization of car engine intake port 133

(a) (b) (c)

Fig. 3 POD and CPOD2 error for the velocity field (a) and for objective functions T (b) and Q (c)

output and a turbulence k-ε model. The flowchart is givenin Fig. 2 which gives also the location of the volume used tocompute both the tumble and the mass flow.

4.2 Data analysis

4.2.1 Design of experiments

The Design of Experiments (DOE) consists of a combi-nation of Latin Hypercube Sampling (LHS) points and atwo-level factorial design for a total of 146 design points.

4.2.2 Reconstruction error analysis

Figure 3 shows the relative L2 error on the velocity field vreconstructed with the standard POD and using the proposedapproach (curve labeled “CPOD2”, for “Constrained PODversion 2”). The error is only slightly higher for CPOD2 anddecreases at the same rate as for the POD. More precisely,the initial part of the CPOD2 curve is shifted by two units tothe right, corresponding to the number of additional modesp = 2. Figure 3b and c show the convergence of the relativeL2 error on T = (T1, · · · , TM )T and Q = (Q1, · · · , QM )T

against the number of POD modes used in reconstruction.As expected, for CPOD2 the error is zero.

Fig. 4 Constrained mode 1. T = 1.034, Q = 1.968*10−7

4.2.3 Comparison of traditional and constrainedPOD modes

The first p constrained modes do not depend on the veloc-ity field as they are obtained by a QR decomposition of theconstraint matrix C without considering the snapshots. Thematrix C size is r × n, where r = 2 is the number of con-straints and n is the number of nodes on the reference gridin the zone of interest. In our case, the QR decompositiongives p = 2 and the corresponding velocity fields are illus-trated in Figs. 4 and 5. We observe readily, that the firstmode (Fig. 4) corresponds to the pure rotation of the fluidwith no mass flow, while the second mode (Fig. 5) depicts auniform flow with no vorticity.

Figure 6 compares the first POD modes (left) with theirconstrained counterparts p + 1. . . m (right).

As is usual with the standard POD, the physical inter-pretation of these modes is not straightforward. However,we may make two observations here. First, there is anoverall similarity between the corresponding vector fields.When comparing the CPOD2 modes with the POD ones,the difference is mainly due to the subtraction of the puremass flow and the pure vorticity components. The secondobservation is quite obvious when looking at the underlyingmath: for the CPOD2, the mass flow and vorticity are null

Fig. 5 Constrained mode 2. T = 1.7*10−14, Q = 2.3* 10−2

134 M. Xiao et al.

POD mode 1

T = 0.806, Q = -0.138

constrained mode 3

T= -2.7 *10-14, Q = 0

POD mode 2

T = -0.406, Q = 5.0*10-3

constrained mode 4

T = 2.4*10-14, Q = 0

POD mode 3

T= -0.068, Q = 1.87*10-3

constrained mode 5

T = 1.948 *10-14, Q = 0

Fig. 6 Comparison of POD modes with their CPOD2 counterparts

for modes p + 1. . . m. So, as far as optimizing T and Q,it is sufficient to restrict oneself to the first two constrainedmodes.

4.3 Multicriteria optimization

The multicriteria optimization was performed using thegenetic algorithm MOGA (Multi-Objective Genetic Algo-rithm, Eddy and Lewis 2001) available in the environmentDAKOTA developed by Sandia Labs Dakota (2011), and areillustrated in Fig. 7a for 12 POD modes. Results using theconstrained POD with 2 modes are shown in Fig. 7b.

The crosses correspond to the points measured withthe fine model, and circles denote the points of the final

population of the genetic algorithm and are interpreted asan optimized Pareto set. The POD-based optimized Paretoset is not acceptable since it does not dominate the initialdata. The Pareto set obtained using CPOD2 is acceptable inthe sense that the final MOGA points dominate the initialdata. Another advantage of CPOD2 is that for an optimalpoint obtained here with only 2 modes, we can reconstructthe complete velocity fields using an arbitrary number ofmodes. In Table 1, the values of T and Q are given for achoice of four Pareto points used in cross-validation withfull-scale analysis (marked with asterisks in Fig. 7b) alongwith the corresponding values of the design variables.

Since the final reconstruction of the optimal designobtained with two CPOD2 modes consists of performing

Enhanced POD projection basis with application to shape optimization of car engine intake port 135

Fig. 7 Original (DOE, solidline) and optimized Pareto sets,a original POD, b constrainedPOD

(a) (b)

Table 1 Example compromisesolutions from the Pareto set Solution T/T_max Q/Q_max Y1 y2 y3 y4 y5 y6

number

1 1.0406 0.8448 0 0 0 1.0 0.2699 0.5721

2 0.7421 0.8981 0.1108 0.4635 0.2541 1.0 1.0 1.0

3 0.4682 0.9482 0.8717 1.0 0.0226 0.5343 0.0172 1.0

4 0.2029 1.0016 1.0 1.0 1.0 1.0 1.0 1.0

(a) – point 1 (b) – point 2

(c) – point 3 (d) – point 4

Fig. 8 Optimal velocity fields for four points of the Pareto set. a point 1. b point 2. c point 3. d point 4

136 M. Xiao et al.

a simple interpolation of coefficients for the optimal de-sign parameters values followed by a linear combinationof the already available modes, we may use a highervalue for m. The velocity fields reconstructed with the fullset of m = 146 modes for the four solutions are shown inFig. 8a–d.

We see that for point 1 corresponding to the lowest val-ues of the valve and short turn angles, the minimal bowlheight and a maximal port diameter allow for maximizingT by adjusting the runner length and the downdraft anglegiving a vorticity dominated flow (Fig. 8a), resulting in anenvironmentally friendly design. The power of the engineis however too low due to an insufficient mass flow. Con-versely, for the optimal point 4, corresponding to maximalvalues of all the parameters, a high value of Q is obtainedand the field (Fig. 8d) is dominated by the mass flow. Thissolution (which belongs to the Pareto set of the initial DOE)has to be discarded due to the excessive CO2 emission rate.Points 2 and point 3 (Fig. 8b, c), provide a compromisebetween the two objectives, both with a maximal downdraftangle. Once again, the increase in port diameter and runnerlength allows us to adjust the vorticity while a maximumvalue of the bowl height permits the tuning of the valveangle, short turn radius, the port diameter and the runnerlength for higher engine power.

5 Conclusions

The numerical results show that with the standard POD, thereconstruction error may lead to unreliable results, espe-cially in multi-objective optimization problems. This isbecause the basis truncation produces an error in the calcu-lation of the functionals of the velocity field. The approachproposed in this paper overcomes this limitation by modify-ing the orthogonal modes, by imposing the conservation ofintegral quantities.

Two conclusions may be drawn. First, realistic Paretosets are obtained more easily than with the original PODapproach. The second advantage is that the number ofmodes needed for optimization is reduced to the numberof objective functions: two in the case presented, with noimpact on the quality of estimation of the quantities ofinterest. Moreover, the final solutions may be reconstructedusing a high number of modes (much higher than normallyused in POD), providing a finer physical insight into theoptimized solutions.

In terms of future prospects, even though the use of a con-strained approach within local POD approximations seemsstraightforward, further research is needed to investigatededicated manifold interpolation techniques. An extensionof the approach to problems with energy-type quadraticconstraints is also currently under investigation.

Acknowledgments This work was supported by the Agence Nationalede la Recherche in the scope of OMD2 project ANR-08-COSI-007.

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