Non-uniform B-spline Research Articles

Research on the properties of aluminum foam material based on the establishment of engineering model

Aluminum foam material has many excellent characteristics with low density and high-ratio energy absorption. It is suitable for lightweight designs in aviation, automotive and other fields. However, the preparation process and conditions of the foamed aluminum material have an important effect on the material performance parameters. The performance parameters of different batches of products are quite different. Therefore, it brings inconvenience to material application and research. Based on a large number of material experiments, this paper studies the establishment of a finite element simulation model for foam aluminum materials. Cubic non-uniform B-Spline curve method is used to construct free-form surfaces of series materials. Unknown quantity is obtained through a large number of known parameters. At the same time, the parameter curve of the required known quantity is established. The paper provides a method for achieve the purpose of reducing the number of experiments and improving the efficiency of structural designs.

NSGA-II approach for proper choice of nodes and knots in B-spline curve interpolation

Describing data, obtained by various instruments, with an analytic function is one of the tasks that people often face in a wide variety of applications such as virtual reality, CAD design, reverse engineering, data visualization, medical imaging, and cultural relic restoration and so on. Moreover, non-uniform B-spline is an extensively-used tool for interpolation which is an effective means of describing data. In this paper, according to the geometric features reflected in the data points, a method for calculating the tangent vectors at the corresponding data points is proposed for reference. And based on the constraints on tangent vectors (calculated by the proposed reference method, can also be given beforehand) and control points, non-dominated sorting genetic algorithms-II, namely NSGA-II, is adopted for adaptive B-spline curve interpolation without knowing nodes and knots in advance. The resulting interpolation curve approximates the given tangent vectors and the data polyline, and it is more natural-looking, in general than those obtained by other methods. In addition, the new method works well in a higher degree. Testing results on the feasibility and universal applicability of the new method are also included.

A fourth-order non-uniform mesh optimal B-spline collocation method for solving a strongly nonlinear singular boundary value problem describing electrohydrodynamic flow of a fluid

In this paper, a non-uniform mesh optimal B-spline collocation method is presented for the numerical solution of a singular two-point boundary value problem describing electrohydrodynamic flow (EF) of a fluid in a circular cylindrical conduit. The EF problem is highly nonlinear and has a singularity at the point r=0. Further, this problem has a boundary layer near right end of the solution domain. We consider a grading function to construct a non-uniform mesh over the problem domain. The non-uniform mesh is constructed in such a way that the mesh is finer near the right end boundary. Convergence of the proposed method is analyzed. The EF problem is solved for small and large values of the two relevant parameters: (i) strength of non-linearity β, and (ii) Hartmann electric number H. The effects of β and H on the velocity field are investigated. The computed results have been compared with those obtained by two other B-spline collocation methods over uniform mesh to show the advantage of the present method. It is shown that the present method provides O(h4) convergent approximation.

Geometric Modeling of Compound NURBS Surfaces

The design of sculptured surfaces occupies an essential area in the field of modern industrial, aerospace, and medical applications. The challenge is to design products that have complex features efficiently with great flexibility of editing in certain regions without affecting other regions, which the designer has no intent to modify. In this paper, we propose a surface design method based on compound NURBS surface to model automotive parts with 400 control points. First, a Non-Uniform B-Spline basis function is derived with a cubic degree and 20 control points. This method is utilized to design car posterior door, car hood, and rear car door as case studies.

Reconstruction of Velocity Distribution in Partially-Filled Pipe Based on Non-Uniform Under-Sampling

In the process of research on the flow velocity distribution in a partially filled pipe, the under-sampling of measurement data often occurs. For the first time, this problem is solved by the improved non-uniform B-spline curve fitting approximation (NBSC) method. The main innovation of this method is to reconstruct the flow velocity distribution fitting curve with a small amount of non-uniform feature points containing flow velocity information. First, the curvature of a whole discrete sampled data is analyzed, then the weighted threshold is set, and the sampled points that satisfy the threshold are extracted as the initial velocity distribution feature points. Next the node vectors were constructed according to the initial feature points, and the initial interpolation fitting curves are generated. Secondly, by using the relative deviation between the initial approximation curve and each sampled point, new feature points were added where the curve allowable deviation exceeded the specified tolerance, and then a new interpolation fitting curve was obtained. The above procedure was repeated until the fitting curve reached expected accuracy, thus the appropriate feature points were determined. Experimental results showed that, in the case of the same approximation deviation, the proposed NBSC method can solve the problem of under-sampling of measurement data better.

A Heuristic Rapidly-Exploring Random Trees Method for Manipulator Motion Planning

In order to plan the robot path in 3D space efficiently, a modified Rapidly-exploring Random Trees based on heuristic probability bias-goal (PBG-RRT) is proposed. The algorithm combines heuristic probabilistic and bias-goal factor, which can get convergence quickly and avoid falling into a local minimum. Firstly, PBG-RRT is used to plan a path. After obtaining path points, path points are rarefied by the Douglas-Peucker algorithm while maintaining the original path characteristics. Then, a smooth trajectory suitable for the manipulator end effector is generated by Non-uniform B-spline interpolation. Finally, the effector is moving along the trajectory by inverse kinematics solving angle of joint. The above is a set of motion planning for the manipulator. Generally, 3D space obstacle avoidance simulation experiments show that the search efficiency of PBG-RRT is increased by 217%, while search time is dropped by 168% compared with P-RRT (Heuristic Probability RRT). After rarefying, the situation where the path oscillated around the obstacle is corrected effectively. And a smooth trajectory is fitted by spline interpolation. Ultimately, PBG-RRT is verified on the ROS (Robot Operating System) with the Robot-Anno manipulator. The results reveal that the validity and reliability of PBG-RRT are proofed in obstacle avoidance planning.

Volute profile design of the squirrel cage fan based on a B-spline curve under space limitation

The cut volute profile is far and widely used in the squirrel cage fan to meet the space limitation of range hood systems. The cut volute profile often causes unreasonable impeller–volute interference and the aerodynamic performance of the fan to drop. A numerical model combined with the neural network and the genetic algorithm of a squirrel cage fan volute for a range hood is presented in this paper. The secondary non-uniform B-spline curve represents the volute profile variation law, and its control points are used as design variables to meet space constraints. The goal of global optimization is to maximize the efficiency and volume flow rate. As a result of the optimization, the internal flow loss of the fan is reduced compared with the prototype. The volume flow rate and efficiency are increased by over 4.4% in case of optimized volute than the original configuration of the volute.

Robust and Efficient Quadrotor Trajectory Generation for Fast Autonomous Flight

In this letter, we propose a robust and efficient quadrotor motion planning system for fast flight in three-dimensional complex environments. We adopt a kinodynamic path searching method to find a safe, kinodynamic feasible, and minimum-time initial trajectory in the discretized control space. We improve the smoothness and clearance of the trajectory by a B-spline optimization, which incorporates gradient information from a Euclidean distance field and dynamic constraints efficiently utilizing the convex hull property of B-spline. Finally, by representing the final trajectory as a non-uniform B-spline, an iterative time adjustment method is adopted to guarantee dynamically feasible and non-conservative trajectories. We validate our proposed method in various complex simulational environments. The competence of the method is also validated in challenging real-world tasks. We release our code as an open-source package.

Moment Preserving Local Spline Projection Operators

This article describes an elementary construction of a dual basis for nonuniform B-splines that is local, $$L^\infty $$-stable, and reproduces polynomials of any prescribed degree. This allows one to define local projection operators with near-optimal approximation properties in any $$L^q$$, $$1 \le q \le \infty $$, and high order moment preserving properties. As the dual basis functions share the same piecewise polynomial structure as the underlying splines, simple quadrature formulas can be used to compute the projected spline coefficients.

A new high order numerical approach for a class of nonlinear derivative dependent singular boundary value problems

This paper is concerned with the design and analysis of a high order numerical method based on B-spline collocation approach to approximate the solution of a class of nonlinear derivative dependent singular two-point boundary value problems subjected to Neumann and Robin boundary conditions. Convergence result for the proposed method is established through Green's function approach. It is proved that the new method provides h6−order convergent approximation to the solution of the underlying problem. To validate this theoretical result, numerical experiments are carried out. It is shown that the rate of convergence of present method is two orders and four orders of magnitude higher when compared respectively to the uniform mesh B-spline collocation method and non-uniform mesh B-spline collocation method [41]. Moreover, the new method has shown its advantage (in terms of accuracy) over other existing methods [10,23,31,36,49] applied to the special cases of the problem.

External characteristic analysis and stator parameter optimization for a torque converter

The performance of a flattened hydraulic torque converter is optimized with orthogonal experiment and response surface method, considering parameters of its stator blade, defined with nonuniform rotational B-splines. The optimization model, with maximum of the stalling torque ratio as an objective, is determined through an external characteristic statistical analysis under the new European Driving Cycle condition. The optimization results show that the stall torque ratio is increased by 10.83%, while the highest efficiency is above 84%.

Minimum-time B-spline trajectories with corridor constraints. Application to cinematographic quadrotor flight plans

This paper proposes a novel strategy for completing a flight plan with a quadrotor UAV, in the context of aerial video making. The flight plan includes different types of waypoints to join, while respecting flight corridors and bounds on the derivatives of the position of the quadrotor. To this aim, non-uniform clamped B-splines are used to parameterize the trajectory. The latter is computed in order to minimize its overall duration, while ensuring the validation of the waypoints, satisfying the flight corridors and respecting the maximum magnitude on its derivatives. A receding waypoint horizon is used in order to split the optimization problem into smaller ones, which reduces the computation load when generating pieces of trajectories. The effectiveness of the proposed trajectory generation technique is demonstrated by simulation and through an outdoor flight experiment on a quadrotor.

تصـميم وتنفيـذ الاسطح الحـرة بأسـتخدام طريقة التصميم للمقاطـع المستعرضة لمنحنيـات الـ B-spline

هذا البحث يقدم استراتيجية لتصميم الاسطح بشكل عملي، تتم بوساطة استخدام طريقة المنحنيات المستعرضة. الاسطح التي تم دراستها تم تكوينها بوساطة سلسلة من الشرائح المتعاقبة المكونة للشكل. فضلا عن توصيف طريقة تعريف ونمذجة النقاط باستخدام تطبيقات الهندسة العكسية باستخدام مكائن قياس الاحداثيات. تم تنفيذ نموذجين من مادة (Ureol) لتمثيل الحالات المختلفة لمنحنيات الـ B-spline لإيضاح مجال التطبيق الملائم لها. البيانات المطلوبة تم تصميمها بوساطة برنامج الـ MATLAB ثم تم نقلها الى برنامج الـ UG-NX9 لغرض ربط المنحنيات وتكوين السطوح والحصول على برامج الـ G-code الخاصة بالنماذج. كذلك تم عمل محاكاة افتراضية باستخدام برنامج الـ VERICUT لا ظهار الاخطاء المزامنة لعملية التشغيل. تم تنفيذ النماذج على ماكنة تفريز لثلاثة محاور نوع (Isel). في نهاية هذه الاجراءات تم قياس النماذج باستخدام ماكنة قياس الاحداثيات (Faro Arm CMM)، وجد ان قيمة معدل الخطأ (0.0589) لنموذج المنحنيات المستعرضة باستخدام نوع الـ B-spline المنتظم، و(0.1337) لنموذج المنحنيات المستعرضة باستخدام الـ B-spline غير المنتظم. يمكن الاستنتاج ان خطوات البرنامج المقترح التي تم بناؤها يمكن برمجتها في برنامج تصميمي واحد من هذا يمكننا تنفيذ اي سطح في أقصر وقت تصميمي.

HYDRODYNAMIC ANALYSIS OF SHARK BODY HYDROFOIL USING CFD METHODS

The most efficient designs are found in the nature. Many natural events and processes have been successfully transferred to the science and technology in order to solve the problems or to increase the efficiency of the systems. The hydrodynamic principles were efficiently employed by swimming animals for many years. Excellent body shapes of fish species have enabled them to continue their generation until today. The aim of this study is to understand the 2D hydrodynamics of the tope shark (Galeorhinus galeus) and to analyze the fluid flow around its body hydrofoil. The hydrofoil of the tope shark (TSH) has been generated using a real-scale image and digitalized using NURB (Non-uniform B-spline) curves. ANSYS CFX software has been employed for CFD simulations after suitable meshing around the TSH. The pressure and velocity area around the shark hydrofoil was illustrated for different angles of attack and Re numbers. The hydrodynamic performance variables such as lift, drag and pressure coefficients for the hydrofoil were analyzed.

ALGORITHMIC APPROACH IN THE DESIGN OF REPETITIVE PATTERNS ON ARCHITECTURAL SURFACES

Nowadays, smooth, curved surfaces are very common in architectural solutions, and proper design of repetitive patterns on such surfaces is a real challenge for architects and designers. This paper presents various algorithmic approaches to pattern design and distribution, based on examples of existing solutions and on the authors’ designed algorithms and didactic experiences. Algorithmic studies are based on Rhinoceros for NURBS (Non-Uniform B-Spline) modelling and on the Grasshopper plugin for graphic algorithm editing. This approach to the topic combines geometry, aesthetics, flexibility and IT techniques, that are very useful in case of pattern distribution. The examples present algorithms that help in clear definitions and the realization of the designer’s intentions to achieve accurate results. KEYWORDS: pattern, surface, mapping, tessellation.

An efficient and automated method to generate complex blade geometries for numerical analysis

This paper presents an efficient and automated algorithm for the generation of straight and curved blades, as used in horizontal- and vertical-axis wind turbines. A single B-spline surface with both structural and aerodynamic zones is constructed, based on Non-Uniform B-splines Functions. The overall process of the approximation of both 2D blade airfoils and 3D blade surfaces is presented. The main algorithm is designed to construct a detailed B-spline surface, minimizing the segmentation data of the blade and the number of control points needed. A comparison between uniform and non-uniform B-splines blades is conducted using strain energy analysis. Additionally, the Double Reflection Method is implemented as a Rotation Minimizing Frame for a robust transformation of a 3D straight blade into a curved blade, following an arbitrary 3D pole curve. The final output of the main algorithm is a low-resolution control mesh that can regenerate a high-resolution 3D curved NURBS blade. The novel method is designed as a pre-step for numerical analysis, such as classical Finite Element Method and Isogeometric Analysis.

Optimization of freeform surfaces using intelligent deformation techniques for LED applications

Abstract For many years, optical designers have great interests in designing efficient optimization algorithms to bring significant improvement to their initial design. However, the optimization is limited due to a large number of parameters present in the Non-uniform Rationaly b-Spline Surfaces. This limitation was overcome by an indirect technique known as optimization using freeform deformation (FFD). In this approach, the optical surface is placed inside a cubical grid. The vertices of this grid are modified, which deforms the underlying optical surface during the optimization. One of the challenges in this technique is the selection of appropriate vertices of the cubical grid. This is because these vertices share no relationship with the optical performance. When irrelevant vertices are selected, the computational complexity increases. Moreover, the surfaces created by them are not always feasible to manufacture, which is the same problem faced in any optimization technique while creating freeform surfaces. Therefore, this research addresses these two important issues and provides feasible design techniques to solve them. Finally, the proposed techniques are validated using two different illumination examples: street lighting lens and stop lamp for automobiles.

Enhanced Iterative Back-Projection Based Super-Resolution Reconstruction of Digital Images

Iterative back-projection (IBP) is a popular and straightforward approach applied successfully in the field of image super-resolution reconstruction (SRR). SRR using IBP (SRR–IBP) methods efficiently satisfies the basic reconstruction constraints. In spatial domain applications, it allows easy inclusion of data and is a computationally efficient method. However, inferior convergence rate, sensitivity to the initial choice of image, the presence of different degrees of ringing artifacts are some of the major disadvantages that limit the performance of SRR–IBP. To relieve these inherent limitations, an evolutionary edge preserving IBP (EEIBP) is proposed in this paper. The proposed work introduces an improved initial choice of the digital image by interpolating the low-resolution digital image via hybridizing the notion of uniform and non-uniform B-spline interpolation. Secondly, it incorporates a spatially adaptive back-projecting kernel (SABPK) and regularization constraints in the iterative process. The SABPK utilizes covariance-based adaptation to restore the lost high-frequency details and is regulated by a control parameter to make the reconstruction process robust. The regularization constraints use different low-level feature descriptors to track the information related to shape and salient visual properties of the digital image. Finally, the overall reconstruction error is minimized via GA, PSO and cuckoo search (CS) algorithms. Experimental results demonstrate the robustness and the effectiveness of the proposed EEIBP method to provide a high-resolution solution with improved visual perception and reduced artifacts. Moreover, EEIBP method optimized via CS algorithm enables a better quality of reconstruction as compared the other search algorithms (gradient, GA and PSO).

Non-uniform interpolatory subdivision surface

This paper presents an interpolatory subdivision scheme with non-uniform parametrization for arbitrary polygon meshes with arbitrary manifold topology. This is the first attempt to generalize the non-uniform four point interpolatory curve subdivision to surface with extraordinary points. The scheme is constructed from the inspiration of the relation between the non-uniform four-point interpolatory subdivision scheme and the non-uniform B-spline refinement rule. Numerical examples and comparisons with the uniform interpolatory subdivision schemes indicate that the quality of the limit surface can be improved by using non-uniform parameter values for non-uniform initial data.

Contour propagation for lung tumor delineation in 4D-CT using tensor-product surface of uniform and non-uniform closed cubic B-splines

A robust contour propagation method is proposed to help physicians delineate lung tumors on all phase images of four-dimensional computed tomography (4D-CT) by only manually delineating the contours on a reference phase. The proposed method models the trajectory surface swept by a contour in a respiratory cycle as a tensor-product surface of two closed cubic B-spline curves: a non-uniform B-spline curve which models the contour and a uniform B-spline curve which models the trajectory of a point on the contour. The surface is treated as a deformable entity, and is optimized from an initial surface by moving its control vertices such that the sum of the intensity similarities between the sampling points on the manually delineated contour and their corresponding ones on different phases is maximized. The initial surface is constructed by fitting the manually delineated contour on the reference phase with a closed B-spline curve. In this way, the proposed method can focus the registration on the contour instead of the entire image to prevent the deformation of the contour from being smoothed by its surrounding tissues, and greatly reduce the time consumption while keeping the accuracy of the contour propagation as well as the temporal consistency of the estimated respiratory motions across all phases in 4D-CT. Eighteen 4D-CT cases with 235 gross tumor volume (GTV) contours on the maximal inhale phase and 209 GTV contours on the maximal exhale phase are manually delineated slice by slice. The maximal inhale phase is used as the reference phase, which provides the initial contours. On the maximal exhale phase, the Jaccard similarity coefficient between the propagated GTV and the manually delineated GTV is 0.881 0.026, and the Hausdorff distance is 3.07 1.08 mm. The time for propagating the GTV to all phases is 5.55 6.21 min. The results are better than those of the fast adaptive stochastic gradient descent B-spline method, the 3D + t B-spline method and the diffeomorphic demons method. The proposed method is useful for helping physicians delineate target volumes efficiently and accurately.