Optimal aeroacoustic shape design using approximation modeling

Transcription

1 Center for Turbulence Reserch Annul Reserch Briefs Optiml erocoustic shpe design using pproximtion modeling By Alison L. Mrsden, Meng Wng AND Petros Koumoutskos 1. Introduction Reduction of noise generted by turbulent flow pst triling edge continues to pose chllenge in mny eronuticl nd nvl pplictions. Aerocoustics problems relted to such pplictions necessitte the use of lrge-eddy simultion (LES) or direct numericl simultion in order to cpture wide rnge of turbulence scles which re the source of brodbnd noise. Much previous work hs focused on development of ccurte computtionl methods for the prediction of triling-edge noise. For instnce, erocoustic clcultions of the flow over model irfoil triling edge using LES nd erocoustic theory hve been presented by Wng & Moin (2) nd were shown to gree well with experiments. To mke the simultions more cost-effective, Wng & Moin (22) successfully employed wll models in the triling-edge flow LES, resulting in drstic reduction in computtionl cost with miniml degrdtion of the flow solutions. With the recent progress in simultion cpbilities, the focus cn now move from noise prediction to noise control. The gol of the present work is to pply shpe optimiztion nd control theory to the triling-edge flow previously studied, in order to minimize erodynmic noise. In this work pproximtion modeling techniques re pplied for shpe optimiztion, resulting in significnt noise reduction in severl cses Choice of optimiztion method One generl distinction mong optimiztion techniques is between grdient-bsed methods nd non-grdient-bsed methods. The choice of method for prticulr problem depends on fctors such s the cost of evluting the function, the level of noise in the function, nd the complexity of implementtion. Grdient-bsed methods generlly include djoint solutions nd finite-difference methods. Non-grdient-bsed methods my include pttern-serch methods, pproximtion models, response-surfce methods nd evolutionry lgorithms. In LES-bsed erocoustic shpe design, cost-function evlutions re computtionlly expensive, nd hence the efficiency of the optimiztion routine is crucil. Therefore, key considertion is cost minimiztion when choosing n optimiztion method. One of the difficulties in grdient-bsed optimiztion methods is the clcultion of the grdient of the cost function with respect to the control prmeters. The most widely-used grdient method is to solve n djoint eqution in ddition to the flow equtions, s hs been successfully demonstrted by Jmeson et l. (1998) nd Pironneu (1984). However, djoint methods re difficult to implement for time-ccurte clcultions, nd cn present dt storge issues. Additionlly, djoint solvers re not portble from one flow solver to nother. Becuse of these fctors, the method of incomplete sensitivities ws initilly chosen for the grdient clcultion. This method, suggested by Mohmmdi & Pironneu (21), ignores the effects of geometric chnges on the flow field when computing the

2 22 A. L. Mrsden, M. Wng & P. Koumoutskos grdient of surfce-bsed cost function. This mkes it simple to use, nd fr more costeffective thn solving the full djoint problem. In fct, only smll dditionl cost to the flow computtion is needed for every itertion. Exmples demonstrting the method re given by Mohmmdi & Pironneu (21). Initilly, ppliction of the method of incomplete sensitivities produced seemingly promising results, s presented by Mrsden, Wng & Mohmmdi (21). However, on further study the method ws found to brek down for severl importnt cses. A systemtic evlution of the incomplete-sensitivity method ws crried out by comprisons with the exct grdient in the cse of single control prmeter. It ws found tht the exct nd incomplete grdients do not gree with ech other; furthermore, they do not lwys hve the sme sign, s shown by Mrsden et l. (22). This finding shows tht neglecting the stte contribution to the grdient is not vlid for the problem of triling-edge noise. We thus conclude tht the incomplete sensitivities pproch is not dequte for the present ppliction. In choosing n lternte optimiztion method, we re concerned with identifying method tht hs robust convergence properties nd yet is computtionlly fesible. To this end, the method of pproximtion modeling ws chosen for explortion. Approximtion modeling ws developed for use in engineering optimiztion problems which require the use of expensive numericl codes to obtin cost function vlues. Grdient informtion for these problems is often difficult or impossible to obtin. In ddition, mny optimiztion problems hve ssocited dt sets which hve error, or cost functions which re noisy. For ll of these resons, there hs recently been considerble interest in using pproximtion modeling for optimiztion with lrge engineering simultions Introduction to pproximtion modeling Approximtion modeling is fmily of non-grdient-bsed methods which rely on model, or surrogte, functions to pproximte the ctul function. Optimiztion is performed, not on the expensive ctul function but on the model, which is chep to evlute. The use of pproximtion modeling for expensive functions hs been demonstrted by Booker et l. (1999), Serfini (1998), Chung & Alonso (22) nd others. These surrogtes cn be polynomils, in which cse the models re clled response surfce models, or interpolting functions such s splines or more dvnced functions. As n illustrtion of this method, let us ssume we wish to find minimum of the function y = f(x) within n llowble domin x min x x mx. The bsic procedure using n pproximtion-modeling technique is s follows. First, we begin with set of initil dt points x = [x 1, x 2,..., x n ] where the function vlues re known. We then fit surrogte function through these points to pproximte the ctul function. We express the surrogte function s ŷ = ˆf(x). Becuse the surrogte function is inexpensive to evlute, its minimum (within the llowble rnge of x) cn be esily found using stndrd optimiztion methods. When the minimum is found, the ctul function is evluted t this point, the surrogte fit is updted, nd the process continues itertively until convergence to minimum function vlue. Approximtion-modeling methods hve severl possible vritions. One is the choice of surrogte function. Others include the choice of initil dt distribution, nd the use of merit functions to ensure good distribution of the dt. In this work, results re presented using pproximtion-modeling methodology on the model problem of Mrsden et l. (21). Detils of the optimiztion procedure re discussed in section 3. In section 4, results re presented for one-prmeter cse, for which we compre the performnce of severl surrogte functions. In section 5 we present results of clcultion using two

3 Optiml erocoustic shpe design 23 Figure 1. Model irfoil used in shpe optimiztion. The right hlf of the upper surfce is llowed to deform. prmeters, nd discuss the effects of the initil dt set on the finl solution. Extension of the method to severl prmeters is lso discussed. Using pproximtion-modeling methods, significnt reduction in cost function is demonstrted for severl one- nd two-prmeter cses. The method is shown to be robust nd computtionlly ffordble. In ddition, the method hs uncovered new irfoil shpe which gives greter reduction in noise thn previously chieved. 2. Formultion nd cost-function definition We begin by formulting the generl optimiztion problem. Given prtil differentil eqution A(U, q, ) = defined on the domin Ω with control vribles, stte vrible U nd design prmeters q, we wish to minimize given cost function J(U, q, ). The control problem cn be stted s min {J(U, q, ) : A(U, q, ) = x Ω, b(u, q, ) = x Ω} (2.1) where b(u, q, ) is the boundry condition of the PDE. In our problem, the stte equtions re the Nvier-Stokes equtions nd the cost function is the coustic source. The ultimte gol of this work is to optimize n irfoil shpe with fully-turbulent flow t the triling edge. Becuse of the cost of LES clcultions, the optimiztion method is first implemented nd vlidted on n unstedy lminr model problem, which is the subject of the present work. The irfoil geometry for the model problem is shown in figure 1 nd is shortened version of the irfoil used in experiments of Blke (1975). The irfoil chord is 1 times its thickness, nd the right hlf of the upper surfce is llowed to deform. The flow is from left to right nd results presented in this work re t chord Reynolds number of Re = 1,. Previously, in Mrsden et l. (21), results were lso presented for Re = 2,, nd it ws shown tht the cost function ws esily reduced to zero. The focus of the present work is therefore on the higher Reynolds number. Before discussion of the optimiztion method, we outline the derivtion of the cost function for the model problem. For unstedy lminr flow pst n irfoil t low Mch number, the coustic wvelength ssocited with the vortex shedding is typiclly long reltive to the irfoil chord. Noise genertion from n cousticlly-compct surfce cn be expressed s follows, using Curle s extension to the Lighthill theory (Curle 1955), ρ M 3 4π x i x 2 Ḋi(t M x ), Ḋ i = t S n j p ij (y, t)d 2 y (2.2) where ρ is the dimensionless coustic density t fr field position x, p ij = pδ ij τ ij is the compressive stress tensor, n j is the direction cosine of the outwrd norml to the irfoil surfce S, M is the free-strem Mch number, nd y is the source-field position vector. All the vribles hve been mde dimensionless, with irfoil chord C s the length scle, free strem velocity U s velocity scle, nd C/U s the time scle. The density nd pressure re normlized by their mbient vlues. Note tht (2.2) implies the three-

4 24 A. L. Mrsden, M. Wng & P. Koumoutskos dimensionl form of Lighthill s theory, which is used here to compute the noise rdited from unit spn of two-dimensionl irfoil. The rdition is of dipole type, cused by the fluctuting lift nd drg forces. The men coustic intensity cn be obtined from (2.2), s I = M 6 16π 2 x 2 ( D 1 cos θ + D 2 sin θ) 2 (2.3) where the overbr denotes time verging, nd θ = tn 1 (x 2 /x 1 ). To minimize the totl rdited power, we need to minimize the integrted quntity 2π I(r, θ)rdθ = M 6 ( ) 2 2 D 1 + D 2. (2.4) 16π x Hence, the cost function is defined s J = ( 2 n j p 1j (y, t)d y) t 2 + S which corresponds exctly to the coustic source function. ( 2 n j p 2j (y, t)d y) t 2 (2.5) S 3. Optimiztion procedure In this section, we outline the steps in the lgorithm used to optimize the irfoil shpe. Our im is to find the minimum of the cost function defined by (2.5). The cost function, J, depends on control prmeters corresponding to the surfce deformtion. To strt the optimiztion process, the cost function is evluted for severl initil points in the prmeter spce. The subsequent steps re s follows: 1. Fit surrogte function through the set of known dt points 2. Estimte the function minimum using the surrogte function 3. Evlute the true function vlue t the estimted minimum 4. Check for convergence 5. Add new dt point to list of known points nd go bck to 1. Itertions continue in this wy until the prmeters hve converged to give finl irfoil shpe. The control prmeters re defined s follows. Ech prmeter corresponds to deformtion point on the irfoil surfce which must be within the deformtion region. The vlue of ech prmeter is defined s the displcement of this point reltive to the originl irfoil shpe, in the direction norml to the surfce. A positive prmeter vlue corresponds to displcement in the outwrd norml direction, nd negtive vlue corresponds to the inwrd norml direction. A spline connects ll the deformtion points to the triling-edge point nd the left side of the deformtion region to give continuous irfoil surfce. Both ends of the spline re fixed. While the surfce must be continuous nd smooth on the left side, the triling-edge ngle is free to chnge. For given set of prmeter vlues, there is unique corresponding irfoil shpe. To clculte the cost-function vlue for given shpe, mesh is generted nd the flow simultion is performed until the solution is sttisticlly converged. Becuse the flow hs unstedy vortex shedding, the cost function is oscilltory. In the optimiztion procedure, the men cost function J (cf. (2.5)) is used, nd is obtined by integrting in time until convergence. An exmple of the oscilltory cost function, nd the time-verged vlue is shown in figure 2. The cse shown corresponds to the originl irfoil shpe.

5 Optiml erocoustic shpe design J t Figure 2. Cost function (gry) nd men cost function (solid blck) vs. time. Oscilltory cost function is time-verged until the men converges. The cse shown is for the originl irfoil shpe. 4. One-prmeter results Results using single prmeter,, re presented for severl surrogte function choices. With polynomil surrogte function, t lest three initil dt points re needed. In ll one prmeter cses presented here, the llowble rnge of is.5.2. The thickness of the irfoil is.1, nd the chord length is unity. Figure 3 shows the evolution of the response surfce using third-order polynomil s the surrogte function. The upper left plot shows the three initil points, nd following plots show three itertions on the vlue of corresponding to the minimum. With ech itertion, the surrogte function evolves to include ll known cumultive dt. Convergence is reched when the function minimum does not chnge from one itertion to the next. A lest-squres fit of the polynomil is used, so the polynomil does not go exctly through the dt points. A totl of six function evlutions is required to rech convergence, nd 19% reduction in cost function is chieved. As expected, n improvement in the function fit is obtined by using fourth order polynomil, s shown in figure 4. The optimiztion procedure is the sme s for the third order cse. A more significnt reduction in cost function, 26%, is chieved. However, there is trde-off in computtionl cost, since the higher-order polynomil picks up more detil in the function but requires eight function evlutions. Figure 5 shows results for single prmeter cse using cubic spline s the surrogte function. The optimum irfoil shpe corresponding to the minimum cost-function vlue chieved with the spline fit is shown in figure 6. It is qulittively similr to the shpes obtined using the polynomil surrogtes nd these re not shown. Becuse the surrogte spline is piecewise nd fits exctly through the dt points, it cptures more detil in the function thn either polynomil cse for the present problem. Similr to the fourth-order polynomil cse, the cost function reduction is 27% with 12 function evlutions. Using only one prmeter, we hve demonstrted tht the pproximtion modeling

6 26 A. L. Mrsden, M. Wng & P. Koumoutskos J Figure 3. Third-order polynomil response surfce for one-prmeter cse. Ech plot shows men cost function J vs. shpe prmeter where = corresponds to the originl shpe. Line is surrogte function fit, dots re known function vlues J Figure 4. Fourth-order polynomil response surfce for one prmeter cse. Ech plot shows men cost function J vs. shpe prmeter where = corresponds to the originl shpe. Line is surrogte function fit, dots re known function vlues. method is robust nd converges to minimum with modest number of function evlutions. A significnt reduction in cost function hs been chieved nd the results for ll surrogte functions were qulittively similr. Comprtively, the spline surrogte function resulted in the gretest cost-function reduction. The cses using polynomils emphsize tht it is undesirble to use low-order polynomils s globl models, since they re unble to cpture detils of the function such s multiple locl minim. However, there is lso dnger in incresing the order of the polynomil, due to oscilltions between the dt points known s the Runge phenomenon. To void these problems, one my wish to use trust region method, in which the polynomil model is restricted to region ner the minimum, where the function is pproximtely qudrtic.

7 Optiml erocoustic shpe design J Figure 5. One-prmeter cse with cubic spline s surrogte function. Ech plot shows men cost function J vs. shpe prmeter, where = corresponds to the originl shpe. Line is surrogte function fit, dots re known function vlues. Figure 6. Initil (blck) nd finl (gry) irfoil shpes using one prmeter with spline s surrogte function. 5. Two prmeters Although the results using one prmeter re very promising, the true test is whether the computtionl cost remins resonble when the method is extended to more prmeters. In this section, results re presented using two prmeters, nd b, for which bihrmonic spline ws the surrogte function. The deformtion points for prmeters nd b re evenly spced in the deformtion region of the irfoil surfce. A spline is chosen s the surrogte function, bsed on the results for the one-prmeter test cse, nd the optimiztion procedure is the sme s in the one-prmeter cse. We lso study the effect of choice of initil dt on the finl solution. Three sets of initil dt were used, which we cll A, B nd C. For ll cses, the prmeters re limited by.5.2 nd.35 b.2. The lower limit corresponds to stright line connecting the left edge of the deformtion region nd the triling edge. The left side of figure 7 shows the initil dt points used for the two-prmeter cse with dt set A. Contours of the men cost function vlue, J, re shown with prmeters nd b plotted on the xes. In this cse, the initil dt points re not chosen to lie in prticulr pttern. The finl surrogte-function fit is shown on the right side of figure 7. The cluster of points ner the minimum shows the convergence of the solution, nd the surfce hs one minimum vlley. The cost-function reduction for this cse ws

8 28 A. L. Mrsden, M. Wng & P. Koumoutskos b.1 5 b Figure 7. Cse A: Contours of men cost function, J, vs. prmeters nd b for two-prmeter cse with bihrmonic spline s surrogte. Dt points mrked with. Initil dt points re shown on left, finl converged solution is shown on right. 5 5 Figure 8. Initil (blck) nd finl (gry) irfoil shpes using two prmeters with bihrmonic spline s surrogte function, dt set A. 29% which is slight improvement over the best one-prmeter cse. As expected, the two-prmeter cse requires more function evlutions; 17 evlutions were required for cse A. Figure 8 shows the initil nd finl irfoil shpes for this cse. Contours of the initil surrogte-function fit using dt set B re shown on the left of figure 9. In this cse, the initil dt were chosen in smll str pttern centered round the origin. Like cse A, the finl surrogte fit, shown in the right of figure 9, hs one minimum vlley. However, it gives solution qulittively very different from cse A, suggesting tht the ctul function hs t lest two locl minim. The optimum irfoil shpe for cse B is shown in figure 1. In contrst to cse A (Fig. 8), the triling-edge ngle in cse B hs incresed insted of decresed. Although the mgnitude of the shpe deformtion is reltively smll, the reduction in cost function is significnt t 52%. This solution ws not previously expected. Cses A nd B show tht the initil dt set cn drmticlly impct the finl solution, cusing the solutions to converge to two distinct locl minim. In both cses vible irfoil shpe ws found which resulted in significnt cost-function reduction. However, idelly, we desire the solution to converge to the globl minimum, nd to give the sme result independent of the initil dt choice. It is, of course, impossible to gurntee convergence to the globl minimum, nd the cost-function reduction is lwys limited by the prmeter spce. However, there re wys to increse our chnces of converging to globl minimum nd improve robustness. For instnce, by choosing n initil dt set which covers the entire llowble rnge of nd b, the solution is not bised towrd minimum in prticulr re. To demonstrte this, initil dt set C is chosen s

9 1 Optiml erocoustic shpe design b.1 b Figure 9. Cse B: Contours of men cost function, J, vs. prmeters nd b for two-prmeter cse with bihrmonic spline s surrogte. Dt points mrked with. Initil dt points re shown on left, finl converged solution is shown on right. Figure 1. Initil (blck) nd finl (gry) irfoil shpes using two prmeters with bihrmonic spline s surrogte function, dt set B. lrge str pttern centered round the origin nd shown by the surrogte fit in the left of figure 11. The finl surrogte fit is shown on the right of figure 11 nd this solution gives cost function reduction of 45%. We notice tht the finl fit of cse C cptures two locl minim. The finl irfoil shpe is qulittively similr to tht of cse B but the minimum is locted in slightly different position, suggesting tht solutions B nd C my not be well converged. The totl number of function evlutions hs been reduced drmticlly, from 23 evlutions required for cse B to 9 evlutions required in cse C. By introducing second prmeter to the problem, it hs been demonstrted tht results improve drmticlly nd the cost of the optimiztion problem remins mngeble. A second prmeter lso gve the flexibility to find new solutions, s in cses B nd C, which re not dmissible in the one-prmeter spce. The importnce of choosing n initil dt set which spns the prmeter spce hs lso been confirmed. It cn increse the chnce of finding the globl minimum nd reduce the number of itertions. The physicl resons for the reduction in cost function cn be explined by the vortexshedding chrcteristics nd ssocited unstedy forcing on the irfoil. Figure 12 compres vortex-shedding chrcteristics for the originl shpe nd the finl shpes of cses A nd C in terms of the instntneous stremwise velocity. We see tht the vortex shedding strength hs decresed for both cses A nd C compred to the originl. As result, the mplitude of lift fluctutions, which dominte the coustic dipole source, hs been reduced by 12% for cse A nd 24% for cse C. The lrger decrese in lift mplitude

10 21 A. L. Mrsden, M. Wng & P. Koumoutskos b b vs. prmeters nd b for two-prmeter Figure 11. Cse C: Contours of men cost function, J, cse with bihrmonic spline s surrogte. Dt points mrked with. Initil dt points re shown on left, finl converged solution is shown on right. Figure 12. Instntneous stremwise velocity contours for originl (top), cse C (middle) nd cse A (lower). Contour levels re the sme for ll cses, between < u < 1.53 for cse C explins why the reduction in cost function is greter for cses B nd C thn for cse A. The shedding frequencies for ll cses re similr. The optiml shpe found in cse A lso shows n increse in men lift of 4% over the originl vlue, wheres in cse C slight 3% decrese in men lift is observed. In prcticl ppliction it is often necessry to ensure tht the men lift is not reduced by the shpe optimiztion. To this end, erodynmic properties will need to be included using multi-objective optimiztion methods, which my slightly compromise the lrge reduction in cost function found in cse C. To extend the method to severl prmeters, it will be desirble to use surrogte function which is esy to implement nd hs good behvior in high dimensions. Mny model functions exhibit undesirble behvior such s excessive wiggles between dt

11 Optiml erocoustic shpe design 211 points when extended to high order. An lterntive is the use of kriging functions, type of interpolting model first introduced in the geosttistics community by South Africn geologist D. G. Krige. Kriging is bsed on sttistics nd rndom-function theory, nd is esy to implement in n rbitrry number of dimensions. The underlying ide is to use weighted liner combintion of vlues t the smpled dt loctions to interpolte the function. The best liner estimtor is then found by minimizing the error of the estimtion. A detiled derivtion of the method cn be found in Isks & Srivstv (1989) nd Guint (22). Kriging hs since been dopted by the optimiztion community nd is now used in mny engineering problems. 6. Discussion nd future work A summry of results using severl surrogte functions with one nd two prmeters is presented in tble 1. The tble clerly shows n improved reduction in cost function with the ddition of second shpe prmeter. Generlly, the cses with greter cost function reduction lso require more function evlutions, lthough judicious selection of the initil dt points cn speed up convergence drmticlly. It is dvntgeous to chose initil dt which spn the entire prmeter spce. Results using pproximtion modeling re nerly two-fold improvement over the results of previous methods presented in Mrsden et l. (21). In future work, the use of kriging functions with severl optimiztion prmeters will be explored. Extension to multiple prmeters will determine the sclbility of this method in terms of the number of function evlutions. It will lso determine whether further reductions in cost function re possible, nd if the trde-off in computtionl cost is significnt. Additionlly, there re severl vritions on the method used here which could improve convergence nd robustness. As shown by Booker et l. (1999) nd Serfini (1998), the use of polling step in the lgorithm rigorously gurntees convergence to locl minimum. The use of merit functions, s discussed in Torczon & Trosset (1998), to ensure good dt distribution will be explored. With this method, weighting function is employed to determine whether the dt points re well distributed throughout the domin. If the dt points re clustered together, dditionl points my be evluted to improve the surrogte function fit in res lcking dt. To speed up the optimiztion process, it my be desirble to evlute the minimum point in prllel with severl other points which improve the function fit. These vritions of the method will explore the possible cost trde-off between fst convergence to locl minimum nd incresed chnce of reching globl minimum. The use of multi-objective optimiztion to include erodynmic properties (lift nd drg) nd thickness will lso be required in the design of prcticl triling-edge shpes. Once confidence hs been gined in the specifics of the pproximtion-modeling method, optimiztion of the turbulent triling-edge flow will be performed. In the turbulent cse, the irfoil is not cousticlly compct for ll the frequencies of interest, nd the cost function my need to be reconsidered. Alterntively, n pproximtion of the cost function cn be used so long s it is well correlted with the true coustic source function. The choice of cost function will be influenced to some degree by whether the objective is to reduce noise in bnd of frequencies of primry interest, or to reduce the totl rdited power.

12 212 A. L. Mrsden, M. Wng & P. Koumoutskos surrogte function prmeters Jorig Joptimum % reduction function evlutions 3 rd order polynomil % 4 4 th order polynomil % 8 cubic spline % 12 bihrmonic spline, set A % 17 bihrmonic spline, set B % 23 bihrmonic spline, set C % 9 Tble 1. Summry of results for severl surrogte functions choices with one nd two prmeters. Cost function reduction nd number of function evlutions needed for convergence re compred for ll cses. Acknowledgments This work ws supported by the Office of Nvl Reserch under grnt No. N Computer time ws provided by NAS t NASA Ames Reserch Center nd the DoD s HPCMP through ARL/MSRC nd NRL-DC. REFERENCES Blke, W. K A sttisticl description of pressure nd velocity fields t the triling edge of flt strut.report 4241, Dvid Tylor NSRDC, Bethesd, MM. Booker, A. J., Dennis, J. E., Frnk, P. D., Serfini, D. B., Torczon, V. nd Trosset, M. W A rigorous frmework for optimiztion of expensive functions by surrogtes. Structurl Optimiztion, 17, Chung, H. nd Alonso, J. 22 Design of low-boom supersonic business jet using cokriging pproximtion models. AIAA Pper Curle, N The influence of solid boundry upon erodynmic sound. Proc. Roy. Soc. Lond. A 231, guint, A. 22 Use of dt smpling, surrogte models, nd numericl optimiztion in engineering design. AIAA Pper Isks, E. H. & Srivstv, M An Introduction to Applied Geosttistics, Oxford University Press. Jmeson, A., Mrtinelli, L. & Pierce, N. A Optimum erodynmic design using the Nvier-Stokes equtions. Theoret. Comp. Fluid Dynmics 1, Mrsden, A. L., Wng, M. & Mohmmdi, B. 21 Shpe optimiztion for erodynmic noise control. Annul Reserch Briefs, Center for Turbulence Reserch, NASA Ames/Stnford Univ., Mrsden, A. L., Wng, M., Mohmmdi, B. & Moin, P. 22 Shpe optimiztion for triling edge noise control. Workshop on Geometry, Dynmics nd Mechnics in Honour of the 6th Birthdy of J.E. Mrsden, Toronto, Cnd, August 7, Mohmmdi, B. & Pironneu, O. 21 Applied Shpe Optimiztion for Fluids, Oxford University Press.

13 Optiml erocoustic shpe design 213 Pironneu, O Optiml Shpe Design for Elliptic Systems, Springer-Verlg, New York. Serfini, D A Frmework for Mnging Models in Nonliner Optimiztion of Computtionlly Expensive Functions. Ph.D. Thesis, Rice University. citeseer.nj.nec.com/serfini99frmework.html Torczon, V. & Trosset, M. W Using pproximtions to ccelerte engineering design optimiztion. ICASE TR NASA Lngley Reserch Center. Wng, M. & Moin, P. 2 Computtion of triling-edge flow nd noise using lrgeeddy simultion. AIAA J. 38, Wng, M. & Moin, P. 22 Dynmic wll modeling for lrge-eddy simultion of complex turbulent flows. Phys. Fluids 14,