Abstract
We present a new, general design method, called designbymorphing for an object whose performance is determined by its shape due to hydrodynamic, aerodynamic, structural, or thermal requirements. To illustrate the method, we design a new leadingandtrailing car of a train by morphing existing, baseline leadingandtrailing cars to minimize the drag. In designbymorphing, the morphing is done by representing the shapes with polygonal meshes and spectrally with a truncated series of spherical harmonics. The optimal design is found by computing the optimal weights of each of the baseline shapes so that the morphed shape has minimum drag. As a result of optimization, we found that with only two baseline trains that mimic current highspeed trains with low drag that the drag of the optimal train is reduced by \(8.04\%\) with respect to the baseline train with the smaller drag. When we repeat the optimization by adding a third baseline train that underperforms compared to the other baseline train, the drag of the new optimal train is reduced by \(13.46\%\). This finding shows that bad examples of design are as useful as good examples in determining an optimal design. We show that designbymorphing can be extended to many engineering problems in which the performance of an object depends on its shape.
This is a preview of subscription content, log in to check access.
Notes
 1.
The training cost function of the ANN and the cost function of the optimized train, which in this case is the drag of the train, are two distinct functions and should not be confused with each other.
 2.
If shapes are not represented on grids in this manner, then the morphed shapes based on their grid representations will not only differ from the morphed shapes based on their spectral representations, but also will often produce unexpected shapes that look very different from the objects from which they were morphed.
 3.
The ANN provided by NUMECA is not opensource code and is written in a way that does not give the user the ability to find the ANNpredicted drag at an arbitrary design point. (The NUMECA ANN only predicts drag values of the design points in a training set.) Determining the ANN’s predicted drag at points outside the training set is important for reconstructing a response surface with a fine mesh (and not just interpolating the drag from the design points of the training set) and for predicting drag at points outside the design space as shown in Fig. 12. The latter is useful to determine whether the design space is too small; if the location of the ANNpredicted minimum drag is near the boundary of the design space or outside the design space, then the design space is too small.
References
 1.
Abadi M, Agarwal A, Barham P, Brevdo E, Chen Z, Citro C, Corrado GS, Davis A, Dean J, Devin M et al (2016) Tensorflow: largescale machine learning on heterogeneous distributed systems. arXiv preprint arXiv:1603.04467
 2.
Aider JL, Beaudoin JF, Wesfreid JE (2010) Drag and lift reduction of a 3d bluffbody using active vortex generators. Experiments in fluids 48(5):771–789
 3.
Anjum MF, Tasadduq I, AlSultan K (1997) Response surface methodology: a neural network approach. Eur J Op Res 101(1):65–73
 4.
Baker C, Cheli F, Orellano A, Paradot N, Proppe C, Rocchi D (2009) Crosswind effects on road and rail vehicles. Veh Syst Dyn 47(8):983–1022
 5.
Inc Bambardier (2010) Aeroefficient optimised train shaping. Bambardier Inc., Montreal
 6.
Baş D, Boyacı İH (2007) Modeling and optimization i: usability of response surface methodology. J Food Eng 78(3):836–845
 7.
Boyd JP (2001) Chebyshev and fourier spectral methods. Courier Corporation, North Chelmsford
 8.
Brechbühler C, Gerig G, Kübler O (1995) Parametrization of closed surfaces for 3D shape description. Comput Vis Image Underst 61(2):154–170
 9.
Canuto C, Hussaini M, Quarteroni A, Zang T (2007) Spectral methods: evolution to complex geometries and applications to fluid dynamics. Scientific computation. Springer, Berlin
 10.
Chen J, Shapiro V, Suresh K, Tsukanov I (2007) Shape optimization with topological changes and parametric control. Int J Numer Methods Eng 71(3):313–346
 11.
Chung MK, Worsley KJ, Nacewicz BM, Dalton KM, Davidson RJ (2010) General multivariate linear modeling of surface shapes using surfstat. NeuroImage 53(2):491–505
 12.
Cooper R (1981) The effect of crosswinds on trains. Journal of Fluids Engineering 103(1):170–178
 13.
Cybenko G (1989) Approximation by superpositions of a sigmoidal function. Math Control Signals Syst (MCSS) 2(4):303–314
 14.
De Berg M, Van Kreveld M, Overmars M, Schwarzkopf OC (2000) Computational geometry. In: Computational geometry, Springer, Berlin pp 1–17
 15.
Demeulenaere A, Ligout A, Hirsch C (2004) Application of multipoint optimization to the design of turbomachinery blades. In: ASME Turbo Expo 2004: power for land, sea, and air, american society of mechanical engineers, pp 1481–1489
 16.
Demeulenaere A, Bonaccorsi JC, Gutzwiller D, Hu L, Sun H (2015) Multidisciplinary multipoint optimization of a turbocharger compressor wheel. In: ASME Turbo Expo 2015: turbine technical conference and exposition, American society of mechanical engineers, pp V02CT45A020–V02CT45A020
 17.
Desbrun M, Meyer M, Alliez P (2002) Intrinsic parameterizations of surface meshes. Computer Graphics Forum. Wiley, Hoboken, pp 209–218
 18.
do Carmo MP (1976) Differential geometry of curves and surfaces. PrenticeHall
 19.
DuMouchel W, Jones B (1994) A simple bayesian modification of doptimal designs to reduce dependence on an assumed model. Technometrics 36(1):37–47
 20.
Duriez T, Aider JL, Masson E, Wesfreid JE (2009) Qualitative investigation of the main flow features over a TGV. In: EUROMECH COLLOQUIUM 50, vol 509
 21.
Elef A, Mousa M, Nassar H (2014) An efficient technique for morphing zerogenus 3D objects. Int J Phys Sci 9(13):302–308
 22.
Feng J, Ma L, Peng Q (1996) A new freeform deformation through the control of parametric surfaces. Comput Gr 20(4):531–539
 23.
Gottlieb D, Orszag SA (1977) Numerical analysis of spectral methods: theory and applications, vol 26. Siam, Philadelphia
 24.
Hemida HN (2006) Largeeddy simulation of the flow around simplified highspeed trains under side wind conditions. PhD thesis, Chalmers University of Technology Goteborg, Sweden
 25.
Hornik K, Stinchcombe M, White H (1989) Multilayer feedforward networks are universal approximators. Neural Netw 2(5):359–366
 26.
Hughes TJ, Cottrell JA, Bazilevs Y (2005) Isogeometric analysis: cad, finite elements, nurbs, exact geometry and mesh refinement. Comput Methods Appl Mech Eng 194(39):4135–4195
 27.
Jakubek D, Wagner C (2016) Adjointbased, cadfree aerodynamic shape optimization of highspeed trains. In: Dillmann A, Heller G, Krämer E, Wagner C, Breitsamter C (eds) New results in numerical and experimental fluid mechanics X. Springer, Berlin, pp 409–419
 28.
Jameson A (1989) Aerodynamic design via control theory. In: Chao CC, Orszag SA, Shyy W (eds) Recent advances in computational fluid dynamics. Springer, Berlin, pp 377–401
 29.
Jameson A, Pierce N, Martinelli L (1998) Optimum aerodynamic design using the Navierstokes equations. In: 35th aerospace sciences meeting and exhibit, p 101
 30.
Jiaqi L, Feng L (2013) Multiobjective design optimization of a transonic compressor rotor using an adjoint equation method. AIAA Paper 2732:2013
 31.
Kang J (2014) Design of marine structures through morphing method and its supporting techniques. Marine Technol Soc J 48(2):81–89. doi:10.4031/MTSJ.48.2.7 cited By 0
 32.
Khuri AI, Mukhopadhyay S (2010) Response surface methodology. Wiley Interdiscip Rev Comput Stat 2(2):128–149
 33.
Kleijnen JP (2008) Response surface methodology for constrained simulation optimization: an overview. Simul Modell Pract Theory 16(1):50–64
 34.
Ku YC, Kwak MH, Park HI, Lee DH (2010) Multiobjective optimization of highspeed train nose shape using the vehicle modeling function. In: 48th AIAA aerospace sciences meeting. Orlando, USA
 35.
Li R, Xu P, Peng Y, Ji P (2016) Multiobjective optimization of a highspeed train head based on the FFD method. J Wind Eng Ind Aerodyn 152:41–49
 36.
Long C, Marsden A, Bazilevs Y (2014) Shape optimization of pulsatile ventricular assist devices using FSI to minimize thrombotic risk. Comput Mech 54(4):921–932
 37.
Lyu Z, Kenway GK, Martins JR (2014) Aerodynamic shape optimization investigations of the common research model wing benchmark. AIAA J 53(4):968–985
 38.
Marcus PS (1986) Description and philosophy of spectral methods. In: Winkler KHA, NormanML (eds) Astrophysical Radiation Hydrodynamics. Springer, Berlin, pp 359–386
 39.
Mocanu BC (2012) 3d mesh morphing. PhD thesis, Institut National des Télécommunications
 40.
MuñozPaniagua J, García J, Crespo A (2014) Genetically aerodynamic optimization of the nose shape of a highspeed train entering a tunnel. J Wind Eng Ind Aerodyn 130:48–61
 41.
MunozPaniagua J, García J, Crespo A, Laspougeas F (2015) Aerodynamic optimization of the nose shape of a train using the adjoint method. J Appl Fluid Mech 8(3):601–612
 42.
Nemec M, Zingg DW, Pulliam TH (2004) Multipoint and multiobjective aerodynamic shape optimization. AIAA journal 42(6):1057–1065
 43.
Oh S (2016) Finding the optimal shape of an object using designbymorphing. PhD dissertation, University of California, Berkeley
 44.
Peters J (1982) Optimising aerodynamics to raise IC performance. Railw Gaz Int 10:78–91
 45.
Pironneau O (1974) On optimum design in fluid mechanics. J Fluid Mech 64(1):97–110
 46.
Poole J, Allen C, Rendall T (2014) Application of control pointbased aerodynamic shape optimization to twodimensional drag minimization. In: 52nd AIAA aerospace sciences meeting, National Harbor, Maryland, pp 2014–0413
 47.
Praun E, Sweldens W, Schröder P (2001) Consistent mesh parameterizations. In: Proceedings of the 28th annual conference on computer graphics and interactive techniques, ACM, pp 179–184
 48.
Press WH, Flannery BP, Teukolsky SA, Vetterling WT et al (1989) Numerical recipes, vol 3. Cambridge University Press, cambridge
 49.
Samareh J (2004) Aerodynamic shape optimization based on freeform deformation. In: 10th AIAA/ISSMO multidisciplinary analysis and optimization conference, p 4630
 50.
Schaeffer N (2013) Efficient spherical harmonic transforms aimed at pseudospectral numerical simulations. Geochem Geophys Geosyst 14(3):751–758
 51.
Shen L, Ford J, Makedon F, Saykin A (2004) A surfacebased approach for classification of 3D neuroanatomic structures. Intell Data Anal 8(6):519–542
 52.
Shen L, Farid H, McPeek MA (2009) Modeling threedimensional morphological structures using spherical harmonics. Evolution 63(4):1003–1016
 53.
Shojaeefard MH, Mirzaei A, Babaei A (2014) Shape optimization of draft tubes for agnew microhydro turbines. Energy Convers Manag 79:681–689
 54.
Shyy W, Papila N, Vaidyanathan R, Tucker K (2001) Global design optimization for aerodynamics and rocket propulsion components. Prog Aerosp Sci 37(1):59–118
 55.
Sorkine O, Alexa M (2007) Asrigidaspossible surface modeling. In: Symposium on geometry processing, vol 4
 56.
Styner M, Oguz I, Xu S, Brechbühler C, Pantazis D, Levitt JJ, Shenton ME, Gerig G (2006) Framework for the statistical shape analysis of brain structures using spharmpdm. Insight J 1071:242
 57.
Sun Z, Song J, An Y (2010) Optimization of the head shape of the CRH3 high speed train. Sci China Technol Sci 53(12):3356–3364
 58.
Hq Tian (2009) Formation mechanism of aerodynamic drag of highspeed train and some reduction measures. J Cent South Univ Technol 16:166–171
 59.
Vanaja K, Shobha Rani R (2007) Design of experiments: concept and applications of plackett burman design. Clin Res Regul Aff 24(1):1–23
 60.
Vassberg J, Jameson A (2014) Influence of shape parameterization on aerodynamic shape optimization. In: Verstraete T, Periaux J (eds) Introduction to optimization and multidisciplinary design in aeronautics and turbomachinery. Von Karman Institute SintGenesiusRode, pp 1–19
 61.
Viana FA, Venter G, Balabanov V (2010) An algorithm for fast optimal latin hypercube design of experiments. Int J Numer Methods Eng 82(2):135–156
 62.
Wang X, Hirsch C, Kang S, Lacor C (2011) Multiobjective optimization of turbomachinery using improved NSGAII and approximation model. Comput Methods Appl Mech Eng 200(9):883–895
 63.
Yao S, Guo D, Sun Z, Yang G (2015) A modified multiobjective sorting particle swarm optimization and its application to the design of the nose shape of a highspeed train. Eng Appl Comput Fluid Mech 9(1):513–527
 64.
Zhang WH, Beckers P, Fleury C (1995) A unified parametric design approach to structural shape optimization. Int J Numer Methods Eng 38(13):2283–2292
Acknowledgements
We thank Alain Demeulenaere and David Gutzwiller for valuable contributions to the science and engineering results and to NUMECA, USA, Inc. for software and computational support. Partial support was supplied by NSF Grants AST1009907 and AST1510703 and by NASA PATM Grants NNX10AB93G and NNX13AG56G. Partial support for computational work was provided by NSF XSEDE (NSF OCI1053575) and NASAHEC.
Author information
Affiliations
Corresponding author
Appendix
Appendix
We require that all of the designs of the leadingandtrailing cars that we create by morphing baseline leadingandtrailing cars have continuous surfaces. In particular, we require that at the interface labeled \(I_p\) in Fig. 3 where the nose joins the passenger compartment that the surfaces are sufficiently smooth that (I) the locations of the surfaces of the passenger compartment and the nose at \(I_p\) are continuous, and (II) that the slopes of the tangent planes of the surfaces at \(I_p\) also be continuous. Here we show that sufficient conditions for this smoothness are

1.
The passenger compartments of all of the baseline leadingandtrailing cars have the same shape.

2.
At the interface \(I_p\), all the baseline shapes satisfy conditions (I) and (II) listed above.

3.
The sum of the weights \(\omega _i\) used in the morph in Eq. 10 sum to unity.
Because the train’s is symmetric we need only concern ourselves with \(\pi /2 \ge \phi \ge 0\) and \(\pi /2 \ge \theta \ge 0\) (see Fig. 3). Let \(R_k(\theta , \phi )\) be the surface of the kth baseline nose. \(I_p\) is located on the \(x=0\) plane at \(\phi = \pi /2\), \(\pi /2 \ge \theta \ge 0\). The location, \(\widetilde{R}(\theta )\) of the interface \(I_p\) of each of the N baseline leadingandtrailing cars is the the same because all of the passenger compartments are identical:
The radius of a morphed train nose \(R(\theta , \phi )\) is obtained from Eqs. 9 and 10, so it can be rewritten as
Therefore, from Eqs. 15 and 16, if \(1 = \sum _{k=0}^{N1} \omega _k\), the location \(R(\theta , \pi /2)\) of the morphed nose at \(I_p\) is \(\widetilde{R}(\theta )\), so condition (I) is satisfied.
The slopes of the surfaces of two adjacent objects are continuous at their interface if the tangent planes of the two objects are the same at each each location of their interface. The latter will be true if the unit normal vectors of the tangent planes are the same at each location of the interface. Let \(\mathbf{X}_k(\theta , \phi )\) be the vector from the origin (in Fig. 3) to \(R_k(\theta , \phi )\) and \(\mathbf{X}(\theta , \phi )\) be the vector from the origin to \(R(\theta , \phi )\). So,
A standard geometrical result [14, 18] shows that at \(I_p\):
where \(\mathbf{n}_k(\theta )\) and \(\mathbf{n}(\theta )\) are the unnormalized normal vectors to the tangent planes of the noses of the kth baseline nose and of the morphed nose, respectively, at \(\theta \) at \(I_p\). These normal vectors can be normalized (indicated with ”hats”) to provide unit vectors:
Substituting Eqs. 17 into 19 gives at \(I_p\):
Assumptions (1) and (2) in the first paragraph of the Appendix require that at \(I_p\)
and
With the definition of \({{\varvec{\chi }}}\) in Eq. 24, Eq. 22 at \(I_p\) can be rewritten:
where the last equality comes from the fact that the quantity in parenthesis in Eq. 26 is \(\mathbf{n}_k\). From Eqs. 20 and 23, \(\mathbf{n}_k\) can be written as
where \(a_k\) is constant and \(a_0 \equiv 1\). Substituting Eq. 28 into 27, at \(I_p\) we obtain
Substituting Eq. 29 into 21 shows that the unit normal vector of the tangent plane of the morphed train nose at \(I_p\) is equal to that of baseline \(\mathbf{0}\) nose at \(I_p\), which itself is equal to that of all of the other baseline noses at \(I_p\) due to Eq. 23. Thus, condition (II) is satisfied.
Rights and permissions
About this article
Cite this article
Oh, S., Jiang, C., Jiang, C. et al. Finding the optimal shape of the leadingandtrailing car of a highspeed train using designbymorphing. Comput Mech 62, 23–45 (2018). https://doi.org/10.1007/s0046601714824
Received:
Accepted:
Published:
Issue Date:
Keywords
 Designbymorphing
 Train head
 Optimization
 Genetic algorithm
 Artificial neural network
 Drag reduction