Non-uniform Rational B-splines Method Research Articles

Characterizing and predicting bluff-body solid fuel ramjet performances via shape design and multi-objective optimization model

In this work, we propose a rapid optimization approach to examine its application potential for the design and performance prediction and optimization of a solid fuel ramjet (SFRJ) with a bluff body. For this, the shape of the bluff body is parameterized first using the non-uniform rational B-spline method. We then develop a model for predicting SFRJ performances by incorporating both levy motion-gradient descent and support vector regression methods. It is found that a faster prediction is achievable, while the average error is maintained to be less than 5%. We then develop a multi-objective optimization model by considering the full thrust and minimum total pressure loss (TPL). The optimization model is examined using the non-dominated sorting genetic algorithm. A cost parameter is also created to facilitate the tradeoffs between the thrust and TPL in the Pareto front, when different bluff-body design configurations are considered. The present results reveal that an increase in the cost parameter will elevate the turbulence intensity within the SFRJs while drawing the incoming air closer to the fuel surface, resulting in an increase in thrust and regression rate, but the TPL will also increase. When prioritizing the TPL reduction in the design stage, the optimized solution reduces TPL by 50%. Meanwhile, the net thrust is shown to be decreased by less than 3.5%. Furthermore, flow-field investigation reveals that the improved performance of the optimized SFRJ is due to more uniform flow velocity gradients around the bluff body and a reduced rear vortex, resulting in reduced momentum loss. Our proposed optimization approach's robustness has been further confirmed with consistent performances, as the ramjet inlet speed varies over a broad range. It shows that our approach has great potential to be applied for the SFRJ performance prediction and optimization, being operated under various conditions.

A data-driven modelling and optimization framework for variable-thickness integrally stiffened shells

The integrally stiffened shells possess the advantages of high specific strength, high specific rigidity and excellent sealing performance, which have been widely used in the load-carrying structure of next-generation manned spacecraft and cargo spacecraft. However, the automatic modelling and optimization of the variable-thickness (VT) integrally stiffened shell are challenging due to the diverse stiffener configurations and the VT skin at multiple corners. To address this issue, a novel data-driven modelling and optimization framework is proposed for the VT integrally stiffened shell in this paper. Firstly, a novel mesh deformation method is proposed for modelling the VT integrally stiffened shell combining the radial basis function (RBF) surrogate model and nonuniform rational B-splines (NURBS). By means of the RBF surrogate model, the mapping relationship between the background mesh domain of curved shell and the target mesh domain of flat plate is trained, and then the equal-thickness (ET) stiffened flat plate is transformed into the VT stiffened flat plate by moving coordinates of nodes. In order to realize the accurate description of the VT stiffened flat plate, NURBS method is used to describe the coordinates of nodes. Finally, the VT stiffened flat plate is transformed into the VT integrally stiffened shell by the mapping relationship. In order to verify the accuracy of the proposed method, a skin-stiffener verticality detection method is proposed for stiffened curved shells. Moreover, based on the deep neural network (DNN) method, a data-driven stiffener layout optimization method is established to minimize the structural weight of the VT integrally stiffened shell. To illustrate the effectiveness of the proposed method, an example of the integrally stiffened shell under internal pressure is carried out. The average value of the detection results of skin-stiffener verticality for the integrally stiffened shell under internal pressure is less than 0.29°, indicating the high accuracy of the proposed modelling method. Compared with the initial result and the optimal result of the equal-thickness integrally stiffened shell, the optimal result of the VT integrally stiffened shell achieves a structural weight reduction of 14.77% and 10.31%, respectively. Optimization results indicate the effectiveness of the proposed optimization framework and the advantage of VT integrally stiffened shells in lightweight design.

An incremental form interpolation model together with the Smolyak method for multi-material topology optimization

An incremental form interpolation model integrated with the anisotropic Smolyak method and the Non-Uniform Rational B-Spline (NURBS) method is proposed for multiple material topology optimization. Only a single category of design variables is exploited to characterize the material properties for any number of material types and the Smolyak method embedded within the NURBS method is devised to represent the topology density field while keeping the number of polynomials to a minimum, leading to a significant reduction in the number of design variables. An arbitrary number of volume and/or mass constraints can be implemented with the introduction of the characterization function to indicate the existence information for each material. Finally, the effectiveness of the proposed methodology is illustrated by three numerical examples.

Bionic shape design of electric locomotive and aerodynamic drag reduction

Bionics has been widely used in many fields. Previous studies on the application of bionics in locomotives and vehicles mainly focused on shape optimisation of high-speed trains, but the research on bionic shape design in the electric locomotive field is rare. This study investigated a design method for streamlined electric locomotives according to the principles of bionics. The crocodiles were chosen as the bionic object because of their powerful and streamlined head shape. Firstly, geometric characteristic lines were extracted from the head of a crocodile by analysing the head features. Secondly, according to the actual size requirements of the electric locomotive head, a free-hand sketch of the bionic electric locomotive head was completed by adjusting the position and scale of the geometric characteristic lines. Finally, the non-uniform rational B-splines method was used to establish a 3D digital model of the crocodile bionic electric locomotive, and the main and auxiliary control lines were created. To verify the drag reduction effect of the crocodile bionic electric locomotive, numerical simulations of aerodynamic drag were performed for the crocodile bionic and bluff body electric locomotives at different speeds in open air by using the CFD software, ANSYS FLUENT16.0. The geometric models of crocodile bionic and bluff body electric locomotives were both marshalled with three cars, namely, locomotive + middle car + locomotive, and the size of the two geometric models was uniform. Dimensions and grids of the flow field were defined. And then, according to the principle of motion relativity, boundary conditions of flow field were defined. The results indicated that the crocodile bionic electric locomotive demonstrated a good aerodynamic performance. At the six sampling speeds in the range of 40–240 km/h, the aerodynamic drag coefficient of the crocodile bionic electric locomotive decreased by 7.7% on the average compared with that of the bluff body electric locomotive.

A general dynamics model under timing-sequence geometry principle

This paper presents a principle of timing-sequence geometry that is used to express any dynamic system with highly precise and dynamic properties. It provides unified theoretical support for dynamic systems of different characteristics, and it is subversive compared with the traditional and related theories. The principle uses time series that implicate dynamic characteristics as the study object , skilfully embeds critical time elements in non-uniform rational B-splines (NURBSs) through a kind of thought “timing-geometry,” and obtains a “functional” relationship between geometry and time, called S-NURBS, which quantizes the nonlinear system and its dynamical properties. Two methods were provided for different research purposes. The first is the direct time NURBS method, which is appropriate for the study of the analytic properties of the dynamic system state space, and the other is the tangent vector NURBS method for the study of differential properties a system. In terms of the process of studying chaos characteristics, we proposed a new algorithm for the maximum Lyapunov exponent of chaotic time series based on the foundation of our principle. Using a few nonlinear systems, we verified the correctness and effectiveness of these methods within acceptable error limits according to the experimental results.

Determining of the hydrodynamic forces on the multi-hull tunnel vessel in steady motion

The research described in this paper was carried out to determine the hydrodynamic forces on the multi-hull tunnel vessel in steady motion. The hull form of vessel is fairly generated by the tunnel hull form generator code using the non-uniform rational B-Spline method. Then, the hydrodynamics simulation is carried out based on finite volume discretization method using volume of fluid model to consider free surface between water and air phases around the vessel. A dynamic mesh restructuring method is applied for grid generation regarding to the heave and pitch motions of vessel in each time step. The calculations of the center of gravity arising, trim angle, pressure, resistance and effective power are studied at various vessel’s speeds. The resistance plot versus velocity has an increasing trend having a hump velocity while the power curve shows a linear-like changes respect to speed increasing. Pressure calculations show that the ratio of hydrostatic pressure to total pressure is decreased at the end point of keel from 100 to 1 % as velocity increases from 5 to 20 m/s. The proposed numerical algorithm is a promising method for hydrodynamic analyses of wide-ranging high speed vessel types, particularly tunnel vessels.

Research on Spiral Bevel Gear Tooth Surface Reconstruction Based on NURBS

The tooth surface shape of spiral bevel gear is complex. In the detection process of errors tooth surface model is constructed based on coordinate data of limited the actual tooth surface sampling points. The model is used to analysis and manufacture of gear to meet the processing requirements. NURBS (non-uniform rational B-Spline) method is the most effective reconstruction method for geometry there are both freedom and analytic curves and surfaces. NURBS method is used to calculate non-uniform knot vector, NURBS control points and B-Spline inverse and reconstruct tooth surface. Reconstructionof spiral bevel gears in using NURBS is achieved and the tooth surface reconstruction is drawn out in Open GL in VC environment. And then reconstruction error on the tooth surface is analyzed. It proves that using the method of NURBS to reconstruct spiral bevel gear tooth surface is correct and feasible.

5-axis Interpolation and Synchronous Servo Process with NURBS Method

Abstract Nowadays, NURBS (Non-Uniform Rational B-Spline) method is widely used in CAD/CAM (Computer-Aided Design/Computer-Aided Manufacturing) to represent sculptured curves or surfaces. In this paper, we develop a 5-axis NURBS real-time interpolator and realize it in our developing CNC(Computer Numerical Control) system. At first, we use two NURBS curves to represent tool-tip and tool-axis path respectively. According to feedrate and Taylor series extension, servo-controlling signals of 5 axes are obtained for each interpolating cycle. Then, generation procedure of NC(Numerical Control) code with the presented method is introduced and the method how to integrate the interpolator into our developing CNC system is given. And also, the servo-controlling structure of the CNC system is introduced. Through the illustration, it has been indicated that the proposed method can enhance the machining accuracy and the spline interpolator is feasible for 5-axis CNC system.

Representation Method and Degree Selection of Class A Surfaces for Auto-Body

To represent curves and surfaces unified, many commercial soft-wares adopt non-uniform rational B-spline (NURBS) as the method. The NURBS method includes Bézier, B-spline and so forth. In surfaces smoothing Class A surfaces of auto-body are a kind of surfaces that have high desire. While different representing methods can fit curves and surfaces with prodigiously different qualities and efficiency. So the representing methods and degree selection of curves and surfaces for auto-body are fundamental issues that auto-body engineers must be solved. Based on analyzing the character of Bézier and B-spline representing methods, Bézier method is adopted to represent Class A surfaces of auto-body according to the desire of auto-body surfaces. At the same time, the degree of curves and surfaces for auto-body is limited between 2nd-7th according to the oscillation of high degree curves and surfaces and continuity desire. The example verified that the Bézier method can improve the quality and efficiency.

Multi-objective robust airfoil optimization based on non-uniform rational B-spline (NURBS) representation

In order to improve airfoil performance under different flight conditions and to make the performance insensitive to off-design condition at the same time, a multi-objective optimization approach considering robust design has been developed and applied to airfoil design. Non-uniform rational B-spline (NURBS) representation is adopted in airfoil design process, control points and related weights around airfoil are used as design variables. Two airfoil representation cases show that the NURBS method can get airfoil geometry with max geometry error less than 0.0019. By using six-sigma robust approach in multi-objective airfoil design, each sub-objective function of the problem has robustness property. By adopting multi-objective genetic algorithm that is based on non-dominated sorting, a set of non-dominated airfoil solutions with robustness can be obtained in the design. The optimum robust airfoil can be traded off and selected in these non-dominated solutions by design tendency. By using the above methods, a multi-objective robust optimization was conducted for NASA SC0712 airfoil. After performing robust airfoil optimization, the mean value of drag coefficient at Ma0.7–0.8 and the mean value of lift coefficient at post stall regime (Ma0.3) have been improved by 12.2% and 25.4%. By comparing the aerodynamic force coefficients of optimization result, it shows that: different from single robust airfoil design which just improves the property of drag divergence at Ma0.7–0.8, multi-objective robust design can improve both the drag divergence property at Ma0.7–0.8 and stall property at low speed. The design cases show that the multi-objective robust design method makes the airfoil performance robust under different off-design conditions.