Non-uniform Rational Basis Spline Curve Research Articles

Numerical and Experimental Analysis of Torsion Springs Using NURBS Curves

Torsion springs, which transfer power through the twisting of their coil, provide advantages such as module simplification and efficient use of space. The design of a torsion spring has been formulated, but it is difficult to determine the local behaviors of torsion springs according to actual load conditions. This study proposes a torsion-spring design method through finite element analysis (FEA) using nonuniform-rational-basis-spline (NURBS) curves. Through experimentation, the angle and displacement values for the actual spring load were converted into useable data. Torsion-spring displacement values were obtained via experimentation and converted into coordinates that may be expressed using NURBS curves. The results of these experiments were then compared to those obtained via FEA, and the validity of this method was thereby verified.

Research on monitoring method for thinning spinning process of the ultra-thin-walled cylinder based on the drum shape.

In order to perceive the state of the process of reverse thinning spinning of the ultra-thin wall tube, a monitoring method based on drum shape is proposed. The method uses the non-uniform rational basis spline curve fitting method based on the moving least squares method to reconstruct the outer contour data of the spinning zone collected by the line laser displacement sensor and extracts the outer contour curve of the drum zone according to the spinning characteristics. Then, the dynamic time warping algorithm is used to analyze the similarity between the current and previous drum curves in order to judge whether the current spinning state is abrupt or not. At the same time, the current spinning state is judged by combining the curvature comb of the drum curve, the number of convex areas of the drum curve, the drum ratio, and the drum change trend. Finally, the drum shape detection and the spinning condition monitoring experiments are carried out. The experimental results show that this method can obtain the outer contour shape of the drum section at one time, and the accuracy is 0.05 mm with respect to the measured value of the three coordinates. The method can realize the visual monitoring of the abrupt change of the spinning process by judging the similarity of the drum curve at the adjacent time under different spinning states. In the state of instability, the curvature comb of the drum curve is discontinuous, the curve is G0 continuous, the drum ratio exceeds the critical drum ratio 2.0, and there is more than one convex region, which realizes the monitoring of spinning state more accurately. The experimental results verify the correctness of the proposed method and prove that the method can be used as a new method to judge the stability of the ultra-thin-walled cylinder spinning process.

A Single-Image Super-Resolution Method Based on Progressive-Iterative Approximation

In this paper, a novel single image super-resolution (SR) method based on progressive-iterative approximation is proposed. To preserve textures and clear edges, the image SR reconstruction is treated as an image progressive-iterative fitting procedure and achieved by iterative interpolation. Due to different features in different regions, we first employ the nonsubsampled contourlet transform (NSCT) to divide the image into smooth regions, texture regions, and edges. Then, a hybrid interpolation scheme based on curves and surfaces is proposed, which differs from the traditional surface interpolation methods. Specifically, smooth regions are interpolated by the non-uniform rational basis spline (NURBS) surface geometric iteration. To retain textures, control points are increased, and the progressive-iterative approximation of the NURBS surface is employed to interpolate the texture regions. By considering edges in an image as curve segments that are connected by pixels with dramatic changes, we use NURBS curve progressive-iterative approximation to interpolate the edges, which sharpens the edges and can maintain the image edge structure without jaggy and block artifacts. The experimental results demonstrate that the proposed method significantly outperforms the state-of-the-art methods in terms of both subjective and objective measures.

Bilevel Programming Weight Minimization of Composite Flying-Wing Aircraft with Curvilinear Spars and Ribs

This paper studies weight minimization of a composite flying-wing aircraft using bioinspired arbitrarily shaped spars and ribs, known as SpaRibs, for the internal structural layout design. A generalized mathematical model is developed to parameterize the shape of SpaRibs using a four-node nonuniform rational basis spline curve. Additionally, for reducing weight, composite structures and lift redistribution using multiple control surfaces are considered. A stacking sequence table is used to parameterize the wing laminate configuration with ply drops for flexible wing design. For the optimization problem including topology as well as discrete and size design variables, a bilevel programming optimization is employed by using a previously developed parallel particle swarm optimization and a gradient-based optimization, respectively, for the upper- and lower-level optimization problems. The upper-level optimization problem is employed to satisfy such design requirements as the flutter constraints for achieving a specific body freedom flutter mode. The lower-level optimization is used to optimize the control surface rotations to minimize the wing root bending moment, which is considered to be a surrogate for wing weight. Different internal structural layouts and flutter constraints are studied for comparison. Optimization studies show that using SpaRibs for aircraft wing box internal structural layout design can reduce the wing weight by 21.8% as compared to the base model and by 7.3% as compared to the wing box with traditional straight spars and ribs. The additional specific flutter result constraints lead to a 4.2% increase in the wing weight for aircraft design with SpaRibs as compared to the one obtained only with the lower bound on the flutter speed.

On optimal and near-optimal shapes of external shading of windows in apartment buildings.

We studied previously optimal shape of external shading of windows in a cellular office with an outer edge modeled by a non-uniform rational basis spline (NURBS) curve whose control points were placed uniformly around western fin, overhang and eastern fin of the window, and whose depths were allowed to vary independently. We observed there that for each climate considered in the study there exists a shading shape close to the optimal one, but with a substantially simpler structure of control points for the NURBS curve. This simpler structure was reflected in partitioning control points into six groups such that all control points in the same group have equal depths, with groups corresponding to lower part of the western fin, upper part of the western fin, joint of the western fin and the overhang, internal part of the overhang, joint of the overhang and the eastern fin and the remaining part of the eastern fin. Here we confirm that shadings with control point structure restricted in such way can perform as well as shadings with unrestricted control points by optimising shape of external shading of windows in an apartment room for both restricted and unrestricted control point structure for the same range of climates, and showing that differences in heating and cooling demands between Pareto optimal shadings in both cases are negligibly small. This grouping of control points thus gives a simple and natural division of shading into a small number of basic constituents that have most impact on its heating and cooling demands. We further consider the convex hull of the Pareto front for shadings with restricted control points, as it contains shadings that minimise equivalent source energy in terms of the ratio of efficiencies and source energy conversion factors for district heating and cooling. We show that, in cases when depths of control point groups in convex hull shadings do not experience sudden changes between their extremal values, these depths can be fitted reasonably well by a sigmoid function that results in functional shadings that satisfactorily approximate heating and cooling demands of shadings in the Pareto front.

Selective Laser Sintering of Ag Conductive Inks with a Real-Time Non-uniform Rational Basis Spline Curve Interpolator

A method is proposed for the fabrication of Ag patterns on plastic substrates by printing Ag conductive inks, followed by selective laser sintering (SLS). The printing and sintering processes were realized by a motion control system with a real-time non-uniform rational basis spline (NURBS) curve interpolator. The proposed approach achieves a constant laser scanning speed along the curve path. Ag patterns fabricated by a NURBS curve interpolator and compared to those produced using a linear interpolator. This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License .

Aerothermodynamic Design Optimization of Hypersonic Vehicles

The objective of this study is to develop a reliable and efficient design optimization method for hypersonic vehicles focused on aerothermodynamic environments. Considering the nature of hypersonic flight, a high-fidelity aerothermodynamic analysis code is used for the simulation of weakly ionized hypersonic flows in thermochemical nonequilibrium. A gradient-based method is implemented for optimization. Bezier or nonuniform rational basis spline curves are used to parametrize the geometry or the geometry change. Linear elasticity theory is implemented for mesh deformation. Penalty functions are utilized to prevent undesired geometrical changes. The design objective is to minimize drag without increasing the total heat transfer rate and the maximum values of the surface heat flux, temperature, and pressure. Design optimizations are performed at different trajectory points of the IRV-2 vehicle. The effects of parametrizations, the number of design variables, and freestream conditions on design performance are studied.

Shape computations with NURB curves

AbstractFreeform curves are commonly used in contemporary design practices, especially with digital modeling tools. We investigate facilitating shape subtraction and addition with two-dimensional (planar) non-uniform rational basis-spline (NURB) curves with the codes and conventions of modeling while preserving the visual continuity of curved shapes. Our proposed tool, developed in a common digital modeling environment, automates the adjustment of parameters for tangential continuity of curves in shape rule applications. When the user designates a curve range to subtract from an initial shape and provides a new curved shape to add to it, the tool splits the initial shape, scales and aligns the curve to be added to fit into this range, introduces additional control points at the joining ends of the new curve to preserve continuity and redraws the new curve. We present a sample set of design variations produced using this practical approach which can be utilized as a method or become part of an automated NURB curve manipulation tool for designers.

A New Multistep Technique Based on the Nonuniform Rational Basis Spline Curves for Nonlinear Transient Heat Transfer Analysis of Functionally Graded Truncated Cone

ABSTRACTA novel multistep method based on the nonuniform rational basis spline curves is developed for the solution of a system of nonlinear differential equations. Efficiency of the presented method in terms of convergence and accuracy is demonstrated through the two-dimensional nonlinear transient heat transfer analysis of multilayered functionally graded truncated cone subjected to an internal thermal load with radiative–convective boundary condition at outer surface. In order to discretize the governing equations in the spatial domain, a layerwise-differential quadrature method is implemented. This transforms the partial differential equations into a system of nonlinear ordinary differential equations in the temporal domain, which is then solved using the presented technique. Comparative studies between predictions of the presented multistep technique and various existing numerical methods demonstrate superior computational efficiency of the method with low computational cost. High efficiency of this method indicated that the presented multistep approach can be successfully employed in various nonlinear problems. Finally, the effects of different parameters on the transient temperature of the truncated cone are studied. It is expected that the presented multistep method will be applied on various practical engineering problems in future studies.

A General Hybrid Optimization Strategy for Curve Fitting in the Non-uniform Rational Basis Spline Framework

In this paper, a general methodology to approximate sets of data points through Non-uniform Rational Basis Spline (NURBS) curves is provided. The proposed approach aims at integrating and optimizing the full set of design variables (both integer and continuous) defining the shape of the NURBS curve. To this purpose, a new formulation of the curve fitting problem is required: it is stated in the form of a constrained nonlinear programming problem by introducing a suitable constraint on the curvature of the curve. In addition, the resulting optimization problem is defined over a domain having variable dimension, wherein both the number and the value of the design variables are optimized. To deal with this class of constrained nonlinear programming problems, a global optimization hybrid tool has been employed. The optimization procedure is split in two steps: firstly, an improved genetic algorithm optimizes both the value and the number of design variables by means of a two-level Darwinian strategy allowing the simultaneous evolution of individuals and species; secondly, the optimum solution provided by the genetic algorithm constitutes the initial guess for the subsequent gradient-based optimization, which aims at improving the accuracy of the fitting curve. The effectiveness of the proposed methodology is proven through some mathematical benchmarks as well as a real-world engineering problem.

Space cutter compensation method for five-axis nonuniform rational basis spline machining

In view of the good machining performance of traditional three-axis nonuniform rational basis spline interpolation and the space cutter compensation issue in multi-axis machining, this article presents a triple nonuniform rational basis spline five-axis interpolation method, which uses three nonuniform rational basis spline curves to describe cutter center location, cutter axis vector, and cutter contact point trajectory, respectively. The relative position of the cutter and workpiece is calculated under the workpiece coordinate system, and the cutter machining trajectory can be described precisely and smoothly using this method. The three nonuniform rational basis spline curves are transformed into a 12-dimentional Bézier curve to carry out discretization during the discrete process. With the cutter contact point trajectory as the precision control condition, the discretization is fast. As for different cutters and corners, the complete description method of space cutter compensation vector is presented in this article. Finally, the five-axis nonuniform rational basis spline machining method is further verified in a two-turntable five-axis machine.

A Novel Trajectory Planning Scheme for Parallel Machining Robots Enhanced with NURBS Curves

In this paper, a CAD-based trajectory planning scheme for parallel machining robots is introduced using the parametric Non-uniform rational basis spline (NURBS) curves. First, a trajectory is designed via a NURBS curve then, a motion scheduling architecture consisting of time-dependent and constant feedrate profiles is advised to generate the position commands on the represented NURBS curve as the tool path. Using the generated commands, the inverse kinematics is elaborated to obtain the joints motions of the parallel machining robot. This paper investigates the NURBS trajectory generation for a parallel robot with 4(UPS)-PU mechanism as the case study. In order to evaluate the effectiveness of the proposed method, the inverse kinematic results for the parallel machining robot of 4(UPS)-PU is compared with the simulation results obtained from the CATIA software. The results confirmed that the proposed trajectory planning scheme along with the advised motion planning architecture is not only feasible for the parallel machining robots but also yields a smooth trajectory with a satisfactory performance for all the joints.

Generation of spherical non-uniform rational basis spline curves and its application in five-axis machining

A method of generating spherical non-uniform rational basis spline curves based on De Boor’s algorithm is presented in this article. The spherical curve preserves many good properties from non-uniform rational basis spline curves in Euclidean space, such as local modification property, convex hull property, rotation invariant property, knot insertion property, and so on. Construction of closed spherical non-uniform rational basis spline curve will be discussed too. Furthermore, a progressive iterative approximation scheme is proposed to approximate the given points on unit sphere using spherical non-uniform rational basis spline curves. The convergence and curve continuity will be discussed. Since the analytical expression of the spherical non-uniform rational basis spline curve is messy, numerical differentiation is used to test its continuity at interior knots. Finally, several examples are presented to verify the effectiveness of the proposed method. The proposed method is applied on tool path generation of five-axis machining to avoid interference by adjusting fewer directions and obtain smooth tool path simultaneously.

Development of a novel adaptive nonuniform rational basis spline interpolator with limited kinematic error for hexapod machine tools

The kinematic nonlinearity of the Stewart platform–based machine tools produces a geometrical error which is shown to have significant value compared to the machining accuracy during the interpolation of non-uniform rational basis spline curves and surfaces, thus needs to be limited during the interpolation. In this study, a fast and sufficiently accurate algorithm is presented to estimate the maximum kinematic error based on the concept of the median osculating circle. Moreover, a weighted search interpolator and an adaptive non-uniform rational basis spline interpolator have been introduced, which are capable of controlling and limiting the feedrate fluctuations and limiting the maximum kinematic error within an interpolation segment. The experimental verification has shown the effectiveness of the proposed algorithm.

A look-ahead and adaptive speed control algorithm for parametric interpolation

A novel interpolation algorithm for parametric curves is proposed. The algorithm improves the machining efficiency and precision greatly with full consideration of the geometrical characteristic of the machining curves and the kinematical characteristic of the machine tools. Firstly, a look-ahead process, including a pre-interpolation and a speed adjustment, is sought to achieve an optimal machining speed profile for the whole parametric curve. The pre-interpolation stage with a constant feedrate is performed on the parametric curve. Critical zones are selected according to the chord errors and the changing values of the speed. The criterion for selection is that the chord error or the changing value of the speed exceeds the given threshold values. Then, a speed adjustment is applied to the sampling points of the critical zones. Meanwhile, the speed profile between two adjacent critical zones is developed with an acceleration/deceleration strategy to satisfy the dynamical requirements. After that, some relevant information about critical zones is stored into a dynamic memory buffer for the following fine interpolation process. Secondly, the fine interpolation is adaptively fulfilled with the data stored in the dynamic memory buffer based on the lengths of adjacent critical zones. Finally, a non-uniform rational basis spline curve is selected for simulation and experiment to demonstrate the feasibility of the proposing algorithm. The experimental results show that the algorithm can achieve not only smooth and high-speed transitions but also high accuracy with the minimization of chord errors in critical zones.

Efficient algorithm for time-optimal feedrate planning and smoothing with confined chord error and acceleration

In this paper, an efficient algorithm is proposed to generate a smooth near-time-optimal feedrate function along a parametric tool path for three-axis CNC machining under a feedrate bound, an acceleration bound for each axis, and a chord error bound. The algorithm first gives a discrete and computationally efficient algorithm to find a sequence of globally optimal velocity points under the feedrate, acceleration, and chord error bounds. A linear programming strategy is proposed to smooth the velocity points sequence. Finally, the velocity points sequence is fitted into a cubic spline to obtain a near-time-optimal and smooth feedrate function. Simulation and experiment results for Non-uniform rational basis spline curves are presented to illustrate the feasibility of the algorithm.