Finding the optimal shape of the leading-and-trailing car of a high-speed train using design-by-morphing


We present a new, general design method, called design-by-morphing 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 leading-and-trailing car of a train by morphing existing, baseline leading-and-trailing cars to minimize the drag. In design-by-morphing, 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 high-speed 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 under-performs 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 design-by-morphing can be extended to many engineering problems in which the performance of an object depends on its shape.

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  1. 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. 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. 3.

    The ANN provided by NUMECA is not open-source code and is written in a way that does not give the user the ability to find the ANN-predicted 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 ANN-predicted minimum drag is near the boundary of the design space or outside the design space, then the design space is too small.


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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 AST-1009907 and AST-1510703 and by NASA PATM Grants NNX10AB93G and NNX13AG56G. Partial support for computational work was provided by NSF XSEDE (NSF OCI-1053575) and NASA-HEC.

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Correspondence to Philip S. Marcus.



We require that all of the designs of the leading-and-trailing cars that we create by morphing baseline leading-and-trailing 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. 1.

    The passenger compartments of all of the baseline leading-and-trailing cars have the same shape.

  2. 2.

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

  3. 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 leading-and-trailing cars is the the same because all of the passenger compartments are identical:

$$\begin{aligned} \widetilde{R}(\theta ) =R_k(\theta , \pi /2), \,\, k=0,1,\ldots ,N-1, \end{aligned}$$

The radius of a morphed train nose \(R(\theta , \phi )\) is obtained from Eqs. 9 and 10, so it can be rewritten as

$$\begin{aligned} R(\theta ,\phi ) = \sum _{k=0}^{N-1} \omega _k R_k(\theta ,\phi ). \end{aligned}$$

Therefore, from Eqs. 15 and 16, if \(1 = \sum _{k=0}^{N-1} \omega _k\), 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,

$$\begin{aligned} \mathbf{X}=\sum _{k=0}^{N-1} \omega _k \mathbf{X}_k. \end{aligned}$$

A standard geometrical result [14, 18] shows that at \(I_p\):

$$\begin{aligned} \mathbf{n}_k(\theta )= & {} \frac{\partial \mathbf{X}_k}{\partial \theta } \Big |_{\phi =\pi /2} \times \frac{\partial \mathbf{X}_k}{\partial \phi } \Big |_{\phi =\pi /2}, \end{aligned}$$
$$\begin{aligned} \mathbf{n}(\theta )= & {} \frac{\partial \mathbf{X}}{\partial \theta } \Big |_{\phi =\pi /2} \times \frac{\partial \mathbf{X}}{\partial \phi } \Big |_{\phi =\pi /2}, \end{aligned}$$

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:

$$\begin{aligned} \hat{\mathbf{n}}_k(\theta )= & {} \mathbf{n}_k(\theta )/|| \mathbf{n}_k(\theta )||, \end{aligned}$$
$$\begin{aligned} \hat{\mathbf{n}}(\theta )= & {} \mathbf{n}(\theta )/ ||\mathbf{n}(\theta )||. \end{aligned}$$

Substituting Eqs. 17 into 19 gives at \(I_p\):

$$\begin{aligned} \mathbf{n}(\theta ) = \sum _{j=0}^{N-1} \omega _j \frac{\partial \mathbf{X}_j}{\partial \theta } \Big |_{\phi =\pi /2} \times \sum _{k=0}^{N-1} \omega _k \frac{\partial \mathbf{X_k}}{\partial \phi } \Big |_{\phi =\pi /2} \end{aligned}$$

Assumptions (1) and (2) in the first paragraph of the Appendix require that at \(I_p\)

$$\begin{aligned} {\hat{\mathbf{n}}}_0 = {\hat{\mathbf{n}}}_1 = {\hat{\mathbf{n}}}_2 = \cdots = {\hat{\mathbf{n}}}_{N-1} \end{aligned}$$


$$\begin{aligned} {{\varvec{\chi }}}(\theta ) \equiv \frac{\partial \mathbf{X}_0}{\partial \theta } \Big |_{\phi =\pi /2} = \frac{\partial \mathbf{X}_1}{\partial \theta } \Big |_{\phi =\pi /2} = \cdots = \frac{\partial \mathbf{X}_{N-1}}{\partial \theta } \Big |_{\phi =\pi /2}. \end{aligned}$$

With the definition of \({{\varvec{\chi }}}\) in Eq. 24, Eq. 22 at \(I_p\) can be rewritten:

$$\begin{aligned} \mathbf{n}(\theta )= & {} \sum _{j=0}^{N-1} \omega _j {\varvec{\chi }}\times \sum _{k=0}^{N-1} \omega _k \frac{\partial \mathbf{X}_k}{\partial \phi } \Big |_{\phi =\pi /2} \end{aligned}$$
$$\begin{aligned} \mathbf{n}(\theta )= & {} \sum _{j=0}^{N-1} \omega _j \sum _{k=0}^{N-1} \omega _k \Bigl ({\varvec{\chi }}\times \frac{\partial \mathbf{X}_k}{\partial \phi } \Big |_{\phi =\pi /2} \Bigr ) \end{aligned}$$
$$\begin{aligned} \mathbf{n}(\theta )= & {} \sum _{j=0}^{N-1} \omega _j \sum _{k=0}^{N-1} \omega _k \mathbf{n}_k, \end{aligned}$$

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

$$\begin{aligned} \mathbf{n}_k(\theta ) = a_k \mathbf{n}_0(\theta ), \end{aligned}$$

where \(a_k\) is constant and \(a_0 \equiv 1\). Substituting Eq. 28 into 27, at \(I_p\) we obtain

$$\begin{aligned} \mathbf{n}(\theta ) = \mathbf{n}_0(\theta ) \sum _{j=0}^{N-1} \omega _j \sum _{k=0}^{N-1} \omega _k a_k . \end{aligned}$$

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.

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Oh, S., Jiang, C., Jiang, C. et al. Finding the optimal shape of the leading-and-trailing car of a high-speed train using design-by-morphing. Comput Mech 62, 23–45 (2018).

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  • Design-by-morphing
  • Train head
  • Optimization
  • Genetic algorithm
  • Artificial neural network
  • Drag reduction