MULTILEVEL SHAPE PARAMETERIZATION FOR AERODYNAMIC …perso.unifr.ch/ales.janka//papers/desideri-janka-eccomas.pdf · 2007-11-06 · ECCOMAS 2004 P. Neittaanm¨aki, T. Rossi, S. Korotov,

European Congress on Computational Methods in Applied Sciences and EngineeringECCOMAS 2004

P. Neittaanmaki, T. Rossi, S. Korotov, E. Onate, J. Periaux, and D. Knorzer (eds.)Jyvaskyla, 24–28 July 2004

MULTILEVEL SHAPE PARAMETERIZATION FORAERODYNAMIC OPTIMIZATION – APPLICATION TO

DRAG AND NOISE REDUCTION OFTRANSONIC/SUPERSONIC BUSINESS JET

Jean-Antoine Desideri? and Ales Janka†

?INRIA Sophia Antipolisproject OPALE

2004, route des Lucioles, B.P.9306902 Sophia Antipolis, France

e-mail: [email protected]

†Ecole Polytechnique Federale de LausanneInstitut d’Analyse et Calcul Scientifique

CH-1015, Lausanne, Switzerlande-mail: [email protected]

Key words: Optimum-shape design, aerodynamics, multilevel algorithm, hierarchicalsmoothing, embedded Bezier shape parameterizations, free-form deformation

Abstract. We present the construction and report on the experimentation of a shape-optimization method applied to the aerodynamic design of a transonic or supersonic busi-ness jet. The main numerical ingredients are: a 3D unstructured-grid compressible-flowfinite-volume solver, free-form deformation approach for shape and mesh movement basedon self-adaptive and multilevel 3D tensorial Bezier polynomial representations by thedegree-elevation technique, and a general genetic or simplex optimizer. In our test-cases,we reduce the pressure drag and, for the supersonic case, a criterion of the noise source ofsupersonic bang. Our experiments demonstrate the versatility of the free-form deforma-tion approach within an unstructured grid volume discretization, and the cost-efficiency ofhierarchical optimization strategies (parameterization degree-enhancement or V-cycles).

1

Jean-Antoine Desideri and Ales Janka

1 SHAPE PARAMETERIZATION

1.1 Bezier curves and degree elevation

Consider a Bezier curve in Rd (d=2 or 3) of degree n supported by a control polygonthat connects the n + 1 control points pk ∈ Rd (k = 0, 1, ..., n), according to the followingparameterization:

x(t) =n∑

k=0

Bkn(t)pk, (1)

where Bkn(t) is the following Bernstein polynomial:

Bkn(t) = Ck

n tk (1− t)n−k, (2)

in which Ckn =

n!

k! (n− k)!, and the parameter t varies from 0 to 1. Consequently, the

derivative of order α with respect to t is given by:

x(α)(t) =n!

(n− α)!

n−α∑k=0

Bkn−α(t) ∆αpk, (3)

in which ∆ is the forward finite-difference operator (∆pk = pk+1 − pk = −−−−→pkpk+1), and∆α its α-th power. Hence in particular, the information p0, p1, p2 (or symmetricallypn, pn−1, pn−2) determine the endpoint p0 (or symmetrically pn) and the local tangentand curvature.

A very important property of the Bezier parameterization is related to the followingdegree-elevation process [3] which is the basis of our multi-level parametrization. Tointroduce this process precisely, multiply the expression of x(t) by t + (1− t) = 1 to get:

x(t) =n∑

k=0

Ckn tk+1 (1− t)n−k pk +

n∑k=0

Ckn tk (1− t)n+1−k pk

=n+1∑`=1

C`−1n t` (1− t)n+1−` p`−1 +

n∑k=0

Ckn tk (1− t)n+1−k pk

=n+1∑k=0

Ckn+1 tk (1− t)n+1−k p′

k

provided the following definitions are made:

p′

0 = p0

p′k =

Ck−1n pk−1 + Ck

n pk

Ckn+1

=k

n + 1pk−1 + (1− k

n + 1)pk (1 ≤ k ≤ n)

p′n+1 = pn

(4)

2


This means that the Bezier curve defined in (1) can be represented exactly by a higher-degree formula also of Bezier type,

x(t) =n+1∑k=0

Bkn+1(t)p

′k, (5)

with new control points p′k from (4). Hence, equations (4) and (5) define a degree-elevation

process that maintains the Bezier curve unchanged.

1.2 Conventional representations of 2D airfoils by Bezier curves

Bezier parameterizations as described above are commonly used to represent the upperand lower portions of an airfoil in 2D. The condition x0 = x1 = 0 on the abscissas of thefirst two control points p0 and p1 ensure a vertical tangent. Usually the coordinates arenormalized by the chord-length and thus one sets xn = 1, and y0 = yn = 0, see Figure 1.

Figure 1: RAE2822 airfoil, degree-16 Bezier curvefit and control polygons associated with the upper andlower surfaces

In this example, the abscissas of the intermediate control points x2, x3, ..., xn−1 havebeen adapted to minimize the following total variation

TV(y(`)

k )

=n∑

k=1

∣∣∣y(`)k − y

(`)k−1

∣∣∣ (6)

associated with the least-square approximation of the initial targetted airfoil RAE2822by the Bezier curve of given order n (here 16). This quantity (TV ) is viewed as a measure

3


of regularity of the control polygon. Numerical experiments in aerodynamic optimum-shape design [1, 5] have demonstrated that improved performance was achieved by thisadaption.

1.3 Tensorial Bezier parametrization and Free-Form Deformation

More general approach to parameterization by Bezier-like formulas is in conjunctionwith the “free-form deformation” method [4]. Let us take for a displacement of each nodex ∈ Rd of the computational domain a tensorial Bezier formula (e.g. for d=3):

x(t1, t2, t3) = x0 +n1∑

k=0

n2∑`=0

n3∑m=0

Bkn1

(t1) B`n2

(t2) Bmn3

(t3)pk,`,m , (7)

where Bkn(t) are the Bernstein polynomials as above, and x0 denotes the position of the

node x in the original, non-deformed configuration.

The free-form deformation approach is a very versatile technique consisting in repre-senting only the geometrical displacements of the entire volumic domain, instead of theoptimized surface geometry itself, see Figure 2 for illustration.

Figure 2: Illustration of a global mesh movement by the free form deformation method with a tensorialBezier parameterization in 2D, degrees n1 = 10, n2 = 3

This method has the following particular merits:

1. the parametrized surfaces to be optimized can be fairly general (e.g. a completeaircraft fuselage);

2. the update of the volumic mesh is naturally included in the procedure;

3. it inherits the differentiability and degree-elevation properties of Bezier curves.

4


Combining the free-form deformation framework with the Bezier tensorial descriptiongives us the possibility to exploit degree-elevation and to construct a hierarchy of nestedparameterization spaces.

2 MULTILEVEL ALGORITHMS FOR SHAPE OPTIMIZATION

Based on the above shape representation by free-form deformation, a shape optimiza-tion can be performed by looking for optimal values of the Bezier control points pk,`,min the formula (7).

To achieve computational efficiency, our construction is modelled on the concept ofmultilevel algorithms such as multigrid methods. In particular we construct a hier-achy of nested parameterizations by degree elevation process and we propose a MultilevelOptimum-Shape Algorithm in the following way [2].

Algorithm 2.1 Let us take for simplicity d = 1, the target (finest) parametrization ofdegree n = n1, and let us propose a two-level optimization algorithm:

1. Use the degree elevation process to construct a hierarchy of embedded spaces of Bezierparameteric curves B2 ⊂ B1, with respective degrees n(2) < n(1). Set the currentshape Γ to the initial shape Γ0.

2. Make a few iterations of a standard optimization method for searching a new shapeΓ(1) = Γ + δΓ(1), on the space B1, i.e. look for correction δΓ(1) ∈ B1.

3. Take the new shape and try to improve it by a coarser correction in the space B2.Make a few iterations of an optimization method to obtain shape Γ(2) = Γ(1) + δΓ(2),with δΓ(2) ∈ B2. Thanks to the embedding B2 ⊂ B1, the total correction δΓ(1) + δΓ(2)

still belongs to B1.

4. Set Γ← Γ(2) and repeat the whole V-cycle from the step 2, until convergence.

The whole algorithm is sketched in Figure 3 for a two-level setting, generalizations of thealgorithm to multi-level setting and d > 1 is straightforward.

3 APPLICATION TO AERODYNAMICS

This contribution is intended to demonstrate the generality of the Bezier tensorialparameterization within the free-form deformation approach, and the efficiency of theresulting multilevel optimum-shape algorithm based on degree elevation, for a noise-drag-reduction problem for supersonic and transonic business jets.

5


find δΓ(1) ∈ B1:Γ(1) ← Γ + δΓ(1),

by a few design iterations,

find δΓ(1) ∈ B1:Γ← Γ(2) + δΓ(1),


find δΓ(2) ∈ B2:Γ(2) ← Γ(1) + δΓ(2),


Figure 3: Schematics of two-level optimization V-cycle (B2 ⊂ B1 are the search spaces associated withthe coarse and fine Bezier parameterizations respectively).

3.1 Transonic business jet test case

For purpose of demonstration mostly of the versatility of the free-form deformationapproach, the shape parameterization on a Falcon 50 business-jet geometry (courtesy ofDassault Aviation) has been considered using a coarse-mesh only.

The objective of the geometrical optimization is to re-design the reactor-pylon-fuselagejoint on the rear of the aircraft, to produce a smoother pressure distribution in the regionclose to the fuselage, at a transonic flight regime (Ma=0.85). Typically, for a classicalshape description by patches (e.g. of Bezier, B-spline or NURBS type), this geometricalconfiguration would require the use of a specialized CAD software to manage the fittingof more patches together.

Figure 4: Free-form deformation control box around the reactor-pylon-fuselage configuration of Falcon 50(courtesy of Dassault Aviation), detail of pressure distribution on the fuselage, original design

The Free-Form Deformation, on the contrary, is very simple to set-up and handle.Enclosing the part of the fuselage to be optimized into a parameterization box, like in

6


Figure 5: Simplex convergence history: cost of all investigated shapes at each iteration of the Nelder-Meadsimplex method

Figure 4, one can parameterize only a much simpler object (the inside of a brick), whileenforcing, of course, that the allowed brick deformations satisfy continuity conditionsbetween the deformations of the inside of the brick and the fixed outside fuselage. Atensorial Bezier parameterization of degrees 7-7-2 is used, fixing all the control points onall the brick faces to ensure C0 continuity of deformations, at least.

The notation of Bezier degrees “7-7-2” stands for a tensorial formula (7) correspondingto degrees n1 = 7, n2 = 7 and n3 = 2 in the longitudinal, vertical direction, and spanwise

Figure 6: Pressure distribution on the fuselage of Falcon 50 (left) and modified design (right) of thereactor-pylon-fuselage part with a pressure-smoothing effect in the neighbourhood of the pylon-fuselagejoint

7


horizontal direction respectively.

By allowing the unconstrained Bezier control points to move only in spanwise direction,we obtain 36 design parameters. The cost-functional to minimize reflects the aerodynamicconstraints on non-increasing drag and non-decreasing lift, and a measure of pressurevariation on the whole fuselage, in the form

J(Γ) = JCL+ JCD

+ JP ,

where the lift and drag penalties JCLand JCD

are formulated by

JCL=

0 if dCL ≥ −10−3

104 · |dCL| if dCL < −10−3 , JCD=

0 if dCD ≤ 10−3

104 · |dCD| if dCD > 10−3 , (8)

with dCL = (CL−C0L)/C0

L, dCD = (CD−C0D)/C0

D, the index 0 denotes the original design.

The pressure variation penalty has the form:

JP =

∑T∈τh(Γ)

∫T(p− pT )2dΩ

∑T∈τh(Γ0)

∫T(p0 − p0

T )2dΩ, (9)

where τh(Γ) denotes for the set of boundary triangles of the discretized fuselage Γ, pT isthe average pressure on a triangle, and the index 0 refers to the original design.

The Nelder-Mead simplex method has been used to search for the optimal values of 36shape parameters. The convergence rate of the simplex minimization is drawn in Figure 5.

The optimization has improved lift CL by 1.72% w.r.t. the original design, whilethe drag CD has been maintained (slight increase of 0.0483% w.r.t. the original). Thepressure-variation measure JP has been reduced by 0.7%. Note, that the integrals in(9) have been taken over the whole aircraft fuselage Γ, and not only in the vicinity ofthe reactor-pylon configuration. As a result, although the attained reduction of pressure-variation might seem negligible, its important local effect in the neighbourhood of thepylon is clearly seen from the pressure distributions for original and optimal designs, asploted in Figure 6.

8


3.2 Supersonic business jet test case

One of the major issues in the design of a supersonic civil aircraft is the noise due topressure-shock propagation to the ground. The aim of the present shape optimization isto improve the wing shape of a supersonic business jet (courtesy of Dassault Aviation)so that the level of noise caused by the pressure shock is reduced. At the same time,however, the total lift CL should not decrease, total drag CD should not increase and thewing-box volume of the wing should not diminish.

ΩSB

pressure

ground

10km

Limits of computational domain

Bow wave Tail wave

Control box

Figure 7: Sonic-boom emission below the aircraft

Within the construction of a mono-criteria cost functional, we are not actually inves-tigating the pressure-shock propagation to the ground, instead, we take as a measure ofthe noise strength the integral of pressure gradient in a reference domain ΩSB below theaircraft, see Figure 7.

The aerodynamic and geometric constraints are put as penalties into the cost functionformula in the following way:

J(Γ) = JCL+ JCD

+ JSB + JV ol,

where the lift and drag penalties JCLand JCD

are formulated like in (8). The sonic-boom

9


Figure 8: Pressure distribution below the aircraft: original design (upper left), single-level optimal shape(upper right), multi-level optimal shape (bottom)

penalty has the form

JSB =

∫ΩSB

(∇p)2dΩ∫ΩSB

(∇p0)2dΩ,

with the index 0 denoting the original design. The penalty JV ol ≥ 0 expresses a relativereduction of inner wing-box volume with respect to the original configuration. In case ofan increase in volume, we take JV ol = 0.

For description of shape modifications we are using a free-form deformation frameworkwith three tensorial 3D Bezier nested parametrizations with degrees 4-1-2 (coarsest), 4-1-4(middle), 6-1-4 (finest).

10


Figure 9: Free-form deformation: curvi-linear parameterization box with Bezier control box of degree4-1-4. Control points marked by • are fixed, the resting control points might be shifted vertically

The notation of Bezier degree “6-1-4” means that we are using in the tensorial formula(7) degrees n1 = 6 in the chordwise direction, n2 = 1 in the wing-thickness direction, andn3 = 4 in the spanwise direction. Unlike the example of free-form deformation in Figures 2and 4, the parametric box is now curved and is tightly encompassing the wing surface in3D, see Figure 9. We allow control points to be displaced vertically (in wing-thicknessdirection), except for points along the leading and trailing edges and wing root, which arebeing fixed.

Let us perform two kinds of optimization. First, we take the finest parameterization(degree 6-1-4) and search the minimum directly in this single-level space by the meansof Nelder-Mead simplex algorithm. Taking into account the constrained control points,we have 40 independent non-constrained shape parameters. We perform 500 iterations ofthe simplex algorithm and plot the convergence rate of the optimization.

Second, we are repeating the same optimization task, but now using a multi-levelapproach. Starting from the coarsest parametrization 4-1-2, we perform 150 steps ofNelder-Mead simplex algorithm for 12 independent non-constrained shape parameters.Next, starting from the best shape obtained with the coarse parameterization, we makeanother 150 iterations of the simplex algorithm for slightly refined parameterization ofdegree 4-1-4 with 24 independent shape parameters, followed by another 150 iterationswith the finest parameterization 6-1-4 with 40 shape parameters.

Let us observe the speed of convergence of the two optimizations and the quality of theobtained results. Figure 10 shows the convergence histories of the two approaches. Weclearly see, that the gradually elevated approach converges not only faster (thanks to theinfluence of the first block of 150 iterations with only coarse parameterizations), but alsoto a better solution in the same target parametric space (of degree 6-1-4)! The reductionsof lift, drag and sonic-boom noise measure are summarized in Table 1.

In both cases, we observe that the wing-box volume remained unchanged (up to 0.05%

11


Figure 10: Convergence history of simplex algorithm with only finest parameterization (degree 6-1-4) vs.simplex algorithm with the degree of parameterizations gradually increased at iterations 150, 300 and450

strategy dCL [%] dCD [%] JSB[%] δ(

wing-boxvolume

)[%] optimal cost J

single-level 5.77 0.06 82.27 0.013 0.9070multi-level 8.75 -0.58 79.38 0.053 0.8909

Table 1: Summary of optimization results: target parameterization of degree 6-1-4, single-level vs. multi-level approach

of internal volume), the three-level degree-elevation process resulted in an improved lift(8.75% increase w.r.t. the original design), and a lower drag (0.58% reduction); simulta-neously, the sonic-boom noise measure JSB was reduced of more than 20%. The corre-sponding pressure fields in a planar section under the aircraft are displayed in Figure 8.

12


4 CONCLUSIONS

We have proposed to use the free-form deformation method in conjunction with nestedtensorial-Bezier parameterizations to construct a multi-scale description of the deforma-tion of 3D shapes and volumes.

Two aerodynamic optimization test-cases involving a generic business jet geometryin a transonic and a supersonic regimes have been presented. The versatility of thesetting for treating shape optimization problems involving complex 3D geometries hasbeen demonstrated.

Additionally, the concept of multi-level parameterization has revealed itself very promis-ing for shape optimization in aerodynamics in particular. These first tests show, that amulti-level strategy converges faster, and to a better solution, then optimizations witha single-level fine parameterization. Here, we have shown an example of a multi-leveloptimization with gradually enriched parametric spaces. Nevertheless, other multi-levelstrategies are possible, in particular V-cycle and full-multi-level iterations as proposedin [2].

We have been using a Nelder-Mead simplex method as an example of a general opti-mization method. However, the multi-level approach is not limited to simplex algorithms.One can easily concieve an algorithm using different basic optimization methods on dif-ferent levels of parameterization. We think that within this class of algorithms, a worthychoice would be a combination of a global genetic algorithm on coarsest parametric leveland simplex or gradient-like methods on finer levels.

The multi-level concept can be also further improved by an auto-adaptation process ofthe parameterization, intruduced in [1, 5] for Bezier curves and 2D airfoils. Indeed, Bezierparameterization of a curve not being unique, one can choose one parameterization out ofthe class of admissible ones. The best choice of a parameterization for shape-optimizationis inherently connected to the optimization process. The idea of [1, 5] thus consists incombining a few loops of optimization with reparametrization. The role of the former isto reveal the overall trend in the evolution of the optimized shape, while the aim of thelatter is to describe accurately the optimum shape, and thus increase the potential of theminimization algorithm by self-detection of the best search space.

13


REFERENCES

[1] A. Clarich and J.-A. Desideri. Self-Adaptive Parameterization for aerodynamic opti-mum shape design, INRIA Research Report 4428, March 2002.

[2] J.-A. Desideri. Hierarchical Optimum-Shape Algorithms Using Embedded Bezier Pa-rameterizations, Finnish-French Seminar on Scientific Computing, Jyvaskyla, Fin-land, 2002, E. Heikkola and O. Pironneau Eds., CIMNE, Barcelona Publish, 2003.

[3] G. Farin. Curves and Surfaces for Computer-Aided Geometric Design – A PracticalGuide, W. Rheinboldt and D. Siewiorek Eds., Academic Press, Boston, 1990.

[4] J. A. Samareh. Multidisciplinary Aerodynamic-Structural Shape Optimization Us-ing Deformation (MASSOUD), AIAA-2000-4911. (8th AIAA/NASA/USAF/ISSMOSymposium on Multidisciplinary Analysis and Optimization, September 6-8,2000/Long Beach, CA.)

[5] Z.L. Tang and J.-A. Desideri. Towards Self-Adaptive Parameterization of BezierCurves for Airfoil Aerodynamic Design, INRIA Research Report 4572, Sept. 2002.

Acknowledgements

This study has been sponsored by the Reseau de Recherche et d’Innovation Tech-nologique (RRIT) “Recherche Aeronautique sur le Supersonique” of the French Minister-ies of Research and Transports. Ales Janka is currently partially sponsored by the SwissNational Scientific Fund FNS-20-101391/1.

14

Documents

MULTILEVEL SHAPE PARAMETERIZATION FOR AERODYNAMIC …perso.unifr.ch/ales.janka//papers/desideri-janka-eccomas.pdf · 2007-11-06 · ECCOMAS 2004 P. Neittaanm¨aki, T. Rossi, S. Korotov,