The head shape of high-speed trains has become a critical factor in boosting the speed further. Aerodynamic simulation-based optimization is a dominant method to obtain the optimal head shape which relies on detailed train head models defined by a lot of design variables. Since aerodynamic simulation-based optimization involves heavy calculations, too many design variables not only causes high computational costs, but also makes the optimal solution difficult to obtain. Therefore, how to use few design variables to define detailed train head model is the key to success. Partial differential equation (PDE)-based geometric modelling which creates a complicated PDE patch with few design variables provides an effective solution to this problem. In addition, it also has the advantage of naturally maintaining any high-order continuities between two adjacent surfaces which is very important in designing highly smooth train heads to achieve excellent aerodynamic performance. At the present time, PDE-based geometric modelling cannot be directly applied in computer-aided design (CAD), computer-aided manufacturing (CAM), and computer-aided engineering (CAE) since it has not become an industrial standard. In contrast, non-uniform rational B-splines (NURBS) are commonly used in CAD, CAM, CAE, and many other engineering fields. They have already become part of industry wide standards. In order to apply PDE-based geometric modelling in shape design of high-speed train heads for CAD etc., how to optimally convert PDE surfaces into NURBS surfaces must be addressed. In this paper, a new method of achieving optimal conversion of PDE surfaces representing high-speed train heads into NURBS surfaces is developed. It takes control points and weight deformations of NURBS surfaces to be design variables, and the error between NURBS surfaces and PDE surfaces as the objective function. The least squares fitting and the genetic algorithm are combined to obtain the optimal conversion between PDE surfaces and NURBS surfaces. The application examples demonstrate the effectiveness of the developed method.

We present a novel but simple physics-based method to interactively manipulate surface shapes of 3D models with $$ C^1 $$ C 1 continuity in real time. A fourth-order partial differential equation involving a sculpting force originating from elastic bending of thin plates is proposed to define physics-based deformations and achieve $$ C^1 $$ C 1 continuity at the boundary of deformation regions. In order to obtain real-time physics-based surface manipulation, we construct a mapping relationship between a deformation region in a 3D coordinate space and a unit circle on a 2D parametric plane, formulate corresponding $$ C^1 $$ C 1 continuous boundary conditions for the unit circle, and obtain a simple analytical solution to describe the physics-based deformation in the unit circle caused by a sculpting force. After that, the obtained physics-based deformation is mapped back to the 3D coordinate space, and added to the original surface to create a new surface shape with $$ C^1 $$ C 1 continuity at the boundary of the deformation region. We also develop an interactive user interface as a plug-in of the 3D modelling software package Maya to achieve real-time surface manipulation. The effectiveness, easiness, real-time performance, and better realism of our proposed method is demonstrated by testing surface deformations on several 3D models and comparing with other methods and ground-truth deformations.

With the increasing running speed, the aerodynamic issues of high-speed trains are being raised and impact the running stability and energy efficiency. The optimization design of the head shape is significantly important in improving the aerodynamic performance of high-speed trains. Existing aerodynamic optimization methods are limited by the parametric modeling methods of train heads which are unable to accurately and completely parameterize both global shapes and local details. Due to this reason, they cannot optimize both global and local shapes of train heads. In order to tackle this problem, we propose a novel multi-objective aerodynamic optimization method of high-speed train heads based on the partial differential equation (PDE) parametric modeling. With this method, the half of a train head is parameterized with 17 PDE surface patches which describe global shapes and local details and keep the surface smooth. We take the aerodynamic drag and lift as optimization objectives; divide the optimization design process into two stages: global optimization and local optimization; and develop global and local optimization methods, respectively. In the first stage, the non-dominated sorting genetic algorithm (NSGA-II) is adopted to obtain the framework of the train head with an optimized global shape. In the second stage, Latin hypercube sampling (LHS) is applied in the local shape optimization of the PDE surface patches determined by the optimized framework to improve local details. The effectiveness of our proposed method is demonstrated by better aerodynamic performance achieved from the optimization solutions in global and local optimization stages in comparison with the original high-speed train head.

Surface reconstruction is an important part in automotive styling design. Existing reconstruction methods mainly rely on the proficiency of digital modelers who manually modify the surface shape to approximate the scanned data. Apart from manual modifications, the reconstructed surfaces cannot well reflect the design intent of designers since the feature curves of clay models have not been preserved accurately. In this paper, we propose a partial differential equation (PDE) based surface reconstruction method to analytically generate optimal surfaces with Cn continuity under the constraint of the feature curves. The proposed method accurately preserves automotive feature curves and achieves automatic reconstruction of Class-A surfaces without time-consuming manual work. The effectiveness of the proposed method is demonstrated by a number of experiments that reconstruct main parts of automotive exteriors.

Partial differential equation (PDE) based geometric modelling has a number of advantages such as fewer design variables, avoidance of stitching adjacent patches together to achieve required continuities, and physics-based nature. Although a lot of papers have investigated PDE-based shape creation, shape manipulation, surface blending and volume blending as well as surface reconstruction using implicit PDE surfaces, there is little work of investigating PDEbased shape reconstruction using explicit PDE surfaces, specially satisfying the constraints on four boundaries of a PDE surface patch. In this paper, we propose a new method of using an accurate closed form solution to a fourth-order partial differential equation to reconstruct 3D surfaces from point clouds. It includes selecting a fourth-order partial differential equation, obtaining the closed form solutions of the equation, investigating the errors of using one of the obtained closed form solutions to reconstruct PDE surfaces from different number of 3D points.