Non-uniform Rational B-spline Surfaces Research Articles

Spiral tool path generation method in a NURBS parameter space for the ultra-precision diamond turning of freeform surfaces

Abstract Spiral tool path generation methods for the ultra-precision diamond turning of freeform surfaces can significantly affect machining efficiency and surface quality. However, effective methods for freeform surfaces with a high slope, especially nonrotation symmetric surfaces and those determined by the recursive formula, are unavailable. Therefore, a new spiral tool path generation method based on a nonuniform rational B-spline (NURBS) parameter space is proposed for the ultra-precision diamond turning of freeform surfaces. The discrete points are used for NURBS global interpolation by discretizing the surface in a radial Cartesian coordinate system, thereby forming a NURBS surface. A more uniform spiral tool path of freeform surfaces with a higher calculation efficiency can be obtained by directly utilizing the parameters on the NURBS surface than by applying existing methods. In addition, a solution is provided for freeform surfaces determined by the recursive formula. Simulation and experiments are conducted to show the effectiveness and adaptability of our method.

T-Spline Surface Toolpath Generation Using Watershed-Based Feature Recognition

The freeform surface is treated as a single machining region for most traditional toolpath generation algorithms. However, due to the complexity of a freeform surface, it is impossible to produce a high-quality surface using one unique machining process. Hence, region-based methods are widely investigated for freeform surface machining to achieve an optimized toolpath. The Non-Uniform Rational B-spline Surface (NURBS) represented freeform surface is not suitable for region-based toolpath generation because of the surface gaps caused by NURBS trimming and merging operations. To solve the limitation of the NURBS, T-spline is proposed with the advantages of being gap-free, having less control points, and local refinement, which is an ideal tool for region-based toolpath generation. Thus, T-spline is introduced to represent a freeform surface for its toolpath generation in the paper. A region-based toolpath generation method for the T-spline surface is proposed based on watershed technology. Firstly, watershed-based feature recognition is presented to divide the T-spline surface into a set of sub-regions. Secondly, the concept of a PolyBoundingBox that consists of a set of minimum bounding boxes is proposed to describe the sub-regions, and Manufacturing-Suitable Regions are constructed with the help of T-spline local refinement and the PolyBoundingBox. In the end, an optimized multi-rectangles toolpath generation algorithm is applied for sub-regions. The proposed method is tested using three synthetic T-spline surfaces, and the comparison results show the advantage in toolpath length and toolpath reversing number.

A smooth tool path planning method on NURBS surface based on the shortest boundary geodesic map

Abstract To fill the development gap between computer-aided design (CAD) and computer-aided manufacturing (CAM) on non-uniform rational b-spline (NURBS), a general tool path planning method on NURBS surface is proposed in this study. The generated tool path is composed of a series of closed paths which are derived by mapping the contours of a shortest boundary geodesic map (SBGM) from parameter space to 3D Euclidean space. SBGM that is created in parameter space depicts the shortest boundary geodesic distance (SBG) from internal points on NURBS surface to surface boundaries. It is calculated in discrete form firstly and then reconstructed by using a bicubic smoothing spline algorithm to improve the smoothness of the generated tool path. The tool path interval is deduced according to the requirement of residual scallop height constraint and determined conveniently based on the SBG difference of the two adjacent paths. It is confirmed that the proposed tool path planning method can be applied on all kinds of NURBS surface without the requirements of any other post-processing procedures. Typical case studies and experiments are presented to verify the effectiveness of the proposed method. Comparing with other traditional approaches, the proposed tool path planning method has excellent performance in improving the smoothness and reducing the total length of tool path.

Influence of hexapod robot foot shape on sinking considering multibody dynamics

Hexapod robots have attracted attention for their excellent terrain adaptabilities. When a robot walks on soft soil, dynamic subsidence and slippage greatly reduce its walking performance. The influence of foot’s shape is usually ignored or simply studied without considering the multibody dynamics of the robot. This study is focused on the influence of the foot shape on the walking performance of a robot by coupling the sinkage with multibody dynamics. A composite contact model based on the Bekker, spring-damping, and Janosi-Hanamoto models was used to model the interaction of the robot and soft soil. Non-uniform rational B-spline (NURBS) surface and mesh were used to describe the geometries of foot and soft soil, respectively. The influences of three foot shapes on the sinkage and walking stability of the robot were analyzed by comparsion. The improved X-shaped foot reduced the robot’s sinkage and improved its walking stability.

An Improvement on the Upper Bounds of the Partial Derivatives of NURBS Surfaces

The Non-Uniform Rational B-spline (NURBS) surface not only has the characteristics of the rational Bézier surface, but also has changeable knot vectors and weights, which can express the quadric surface accurately. In this paper, we investigated new bounds of the first- and second-order partial derivatives of NURBS surfaces. A pilot study was performed using inequality theorems and degree reduction of B-spline basis functions. Theoretical analysis provides simple forms of the new bounds. Numerical examples are performed to illustrate that our method has sharper bounds than the existing ones.

Age-dependent dose calculations for common PET radionuclides and brain radiotracers in nonhuman primate computational models.

The combination of nonhuman primates (NHPs) with the state-of-the-art molecular imaging technologies allows for within-subject longitudinal research aiming at gaining new insights into human normal and disease conditions and provides an ideal foundation for future translational studies of new diagnostic tools, medical interventions, and therapies. However, radiation dose estimations for nonhuman primates from molecular imaging probes are lacking and are difficult to perform experimentally. The aim of this work is to construct age-dependent NHP computational model series to estimate the absorbed dose to NHP specimens in common molecular imaging procedures. A series of NHP models from baby to adult were constructed based on nonuniform rational B-spline surface (NURBS) representations. Particle transport was simulated using Monte Carlo calculations to estimate S-values from nine positron-emitting radionuclides and absorbed doses from PET radiotracers. Realistic age-dependent NHP computational model series were developed. For most source-target pairs in computational NHP models, differences between C-11 S-values were between -13.4% and -8.8%/kg difference in body weight while differences between F-18 S-values were between -12.9% and -8.0%/kg difference in body weight. The absorbed doses of 11 C-labeled brain receptor substances, 18 F-labeled brain receptor substances, and 18 F-FDG in the brain ranged within 0.047-0.32mGy/MBq, 0.25-1.63mGy/MBq, and 0.32-2.12mGy/MBq, respectively. The absorbed doses to organs are significantly higher in the baby NHP model than in the adult model. These results can be used in translational longitudinal studies to estimate the cumulated absorbed organ doses in NHPs at various ages.

Uncertainty RCS Computation for Multiple and Multilayer Thin Medium-Coated Conductors by an Improved TDS Approximation

An efficient uncertainty RCS computation technology is proposed to analyze the EM scattering from 3-D medium-coated targets with varying geometrical shapes. First, the scattering target is described by a nonuniform rational B-spline surface, where the uncertain units are described by several independent random variables. Then, an improved thin dielectric sheet-based method of moments is presented to model the scattering from the multiple and multilayer thin medium-coated objects. As a result, only unknowns exist on the target’s outer surface, and the computational resources can significantly be saved in comparison with solvers that associate unknowns with all the surfaces and interfaces. It should also be noted that the half RWG basis functions are introduced at the interface of different media for multiple medium-coated conductors. Finally, both the mean value and variance of the scattering electric current are derived by applying a perturbation approach to account for the target’s uncertain variable geometry. Moreover, the validity of the proposed method is verified in comparison with the traditional Monte Carlo (MC) method, which is used to solve several certain problems, and it is found that the proposed method can provide higher efficiency compared with the MC method.

Topologically robust CAD model generation for structural optimisation

Computer-aided design (CAD) models play a crucial role in the design, manufacturing and maintenance of products. Therefore, the mesh-based finite element descriptions common in structural optimisation must be first translated into CAD models. Currently, this can at best be performed semi-manually. We propose a fully automated and topologically accurate approach to synthesise a structurally-sound parametric CAD model from topology optimised finite element models. Our solution is to first convert the topology optimised structure into a spatial frame structure and then to regenerate it in a CAD system using standard constructive solid geometry (CSG) operations. The obtained parametric CAD models are compact, that is, have as few as possible geometric parameters, which makes them ideal for editing and further processing within a CAD system. The critical task of converting the topology optimised structure into an optimal spatial frame structure is accomplished in several steps. We first generate from the topology optimised voxel model a one-voxel-wide voxel chain model using a topology-preserving skeletonisation algorithm from digital topology. The weighted undirected graph defined by the voxel chain model yields a spatial frame structure after processing it with standard graph algorithms. Subsequently, we optimise the cross-sections and layout of the frame members to recover its optimality, which may have been compromised during the conversion process. At last, we generate the obtained frame structure in a CAD system by repeatedly combining primitive solids, like cylinders and spheres, using boolean operations. The resulting solid model is a boundary representation (B-Rep) consisting of trimmed non-uniform rational B-spline (NURBS) curves and surfaces.

OpenMP parallelism in computations of three-dimensional potential numerical wave tank for fully nonlinear simulation of wave-body interaction using NURBS

The open multi-processing parallelism was implemented into a serial code to compute a high-order boundary integral equation based on the second Green's identity. on symmetric multiprocessing computer platforms. The parallel code is augmented to a three-dimensional potential numerical wave tank for fully nonlinear wave-body interaction problems in the time domain. The non-uniform rational B-Spline surface was manipulated to define the geometry of the computational boundaries and the distribution of boundary values either as a high-order iso-geometric formulation. The material node approach and fourth-order Runge-Kutta time integration method were compounded for time marching the fully nonlinear free surface boundary. The propagation of a fifth-order Stokes wave was simulated to examine the scalability of the parallel constructs implementation. Afterwards, the second-order Stokes wave interaction with a cylindrical pile was modeled as a practical case study to show the benefits of the parallel code. The results were compared with former experiments and numerical studies.

An Enhanced Topology Optimization Approach Based on the Combined MMC and NURBS-Curve Boundaries

An efficient topology optimization method is developed newly in this study by combining the Non-uniform rational B-spline (NURBS) curves and moving morphable components (MMC). The MMC-based topology optimization is an explicit and geometrical method that utilizes a set of morphable components to create basic blocks for optimization. Optimum topologies may be obtained by optimizing shapes, lengths, thicknesses, orientations and layout of these components. The combined method adopts a different way in the creation of morphable components that consist of NURBS curves. Various kinds of complicated curved components can be built with NURBS curves or surfaces. Here, the NURBS curve is applied for shaping the geometries of structural basic components, and the coordinates of control points become design variables for topology optimization. A MATLAB optimization code has been developed. Four numerical examples of a short cantilever, a MBB beam, a simply supported beam with two point loadings, and a vehicle lower chassis structure subjected to crash loadings are provided to prove that the combined topology optimization approach coupled with NURBS curves and basic morphable components can get optimum topologies with clear topological boundaries successfully. As results of comparison study with other approaches, we can obtain the same topologies and faster convergence rates for the three separate cases. The combined approach can improve the smoothness of the topological boundaries that are similar to the shape optimization results obtained by post-optimization after the density-based topology optimization.

CONSTRUCTION AND APPLICATION OF BREP PHANTOM FOR CHINESE WOMEN OF CHILDBEARING AGE IN RADIATION PROTECTION.

The purpose of this study is to construct boundary representation (BREP) phantom for Chinese women of childbearing age, to estimate the external radiation dose and to analyze radiation protection scheme. The BREP phantom for Chinese women of childbearing age was constructed by image segmentation, 3D reconstruction, non-uniform rational B-spline surface construction and voxelization. The photon-irradiated organ absorbed dose-conversion coefficients (DCCK) of the three female specific organs and the photon effective dose-conversion coefficient (ECCK) were calculated by Monte-Carlo method. The results showed that age, body fat-tissue thickness, direction and area of irradiation, organ location and volume all affected the dose of women specific organs when receiving medical exposure. In the case of ensuring the quality of the diagnosis, radiation protection for female specific organs can be achieved by organ dose modulation techniques and reducing exposure area or volume.

Modeling technology of curved surface development for puffer fish

The drag reduction mechanism of puffer epidermis was closely related to its real geometry. In order to solve the modeling problem of epidermal spines on the puffer surface, a modeling method for the expansion of puffer shape was proposed. The three-dimensional scanning and non-uniform rational B-spline surface modeling technology was used to reconstruct the puffer model. According to the curvature characteristics, the surface mathematical equations including exponential, logarithmic, and sinusoidal functions were established based on the multinomial function. The surface was generated by a mathematical equation, and the surface was divided into several non-uniform rational B-spline patches according to curvature. After discretization, the point cloud Gaussian curvature and average value were calculated based on the implicit equation of moving least square surface, and whether the surface is approximately extensible or not was judged. Finally, the puffer surface was divided into 46 curved patches. In this article, the surface expansion algorithm gave priority to ensure the area unchanged, and four feature surfaces were selected according to the epidermal spines arrangement of the puffer surface. The results showed that the technique can simply and efficiently unfold the curved surface of the puffer fish, thus the mapping relationship between the epidermal spines on the surface and the plane was determined, which established a foundation for the accurate arrangement and modeling of the epidermal spines on the surface.

Algebraic Point Projection for Immersed Boundary Analysis on Low Degree NURBS Curves and Surfaces

Point projection is an important geometric need when boundaries described by parametric curves and surfaces are immersed in domains. In problems where an immersed parametric boundary evolves with time as in solidification or fracture analysis, the projection from a point in the domain to the boundary is necessary to determine the interaction of the moving boundary with the underlying domain approximation. Furthermore, during analysis, since the driving force behind interface evolution depends on locally computed curvatures and normals, it is ideal if the parametric entity is not approximated as piecewise-linear. To address this challenge, we present in this paper an algebraic procedure to project a point on to Non-uniform rational B-spline (NURBS) curves and surfaces. The developed technique utilizes the resultant theory to construct implicit forms of parametric Bézier patches, level sets of which are termed algebraic level sets (ALS). Boolean compositions of the algebraic level sets are carried out using the theory of R-functions. The algebraic level sets and their gradients at a given point on the domain are then used to project the point onto the immersed boundary. Beginning with a first-order algorithm, sequentially refined procedures culminating in a second-order projection algorithm are described for NURBS curves and surfaces. Examples are presented to illustrate the efficiency and robustness of the developed method. More importantly, the method is shown to be robust and able to generate valid solutions even for curves and surfaces with high local curvature or G 0 continuity—problems where the Newton–Raphson method fails due to discontinuity in the projected points or because the numerical iterations fail to converge to a solution, respectively.

Properties of Curved Parts Laser Cladding Based on Controlling Spot Size

In this study, a method based on controlling the laser spot size was proposed in the process of curved parts laser cladding, and the coatings obtained by this method were analysed through investigation of the microstructure, microhardness, adhesion property and wear resistance properties. The nonuniform rational B-spline surface (NURBS) reconstruction method was used to obtain the workpiece geometrical characteristics of laser cladding, and through the establishment of a mathematical model, the process of the laser beam working on the curved surface was simplified as the intersection of the cylinder and curvature sphere. Then, the spot size was transformed into the area of a cylinder intersecting with a sphere, and by adjusting the laser head, the size of the laser spot was controlled in the threshold and interpolation points were obtained. The laser cladding trajectory was ensured by these interpolation points, and the experiment was carried out to study the properties of the coating. The results showed that the average coating thickness was about 1.07 mm, and the fluctuation of coating thickness did not exceed 0.05 mm; also, there were no cracks or pores in the layer after penetrant flaw detection. The SEM showed that the grains passed through the transition of plane crystal, cellular crystal, dendrite and equiaxed crystal from the bottom to the top of the layer. After 30 cycles of thermal shock tests, the cladding layer was still well bonded with the substrate and the microhardness and wear resistance were 2 times and 1.4 times higher than that of substrate, respectively.

Thermal buckling of curvilinearly stiffened laminated composite plates with cutouts using isogeometric analysis

This research presents the Non-Uniform Rational B-Splines (NURBS) based isogeometric finite element analysis of stiffened laminated composite plates. A first-order shear deformation theory is used to derive the governing equations by employing a variational formulation. Isotropic, orthotropic and laminated composite plates stiffened with multiple curvilinear stiffeners of different profiles are investigated. A novel way to achieve displacement compatibility between the panel and stiffeners interfaces is introduced. An easy way of modeling plates with complicated cutouts by using edge curves and generating a ruled NURBS surface between them is described. Influence on the critical thermal buckling load due to the presence of circular and elliptical cutouts is also investigated. Results of parametric studies are presented which show the influence of ply orientation, size and orientation of the cutout, and the position and profile of the curvilinear stiffener. The numerical examples show high reliability and efficiency of the proposed formulation when compared to other published solutions and those obtained using ABAQUS, a commercial software.

Surface Fitting for non-Grid Data from in-Situ Measuring Systems of Aero-engine Blade

High machining accuracy of aero-engine blade largely determines the carrying capacity, endurance, acceleration and the dynamic performance of the aero-engine, so a reliable machining error inspection and evaluation technique is imperative. In order to give a reliable error evaluation, the non- uniform rational B-spline (NURBS) technique is adopted to reconstruct the surface within a specified accuracy. Usually, data points measured from aero-engine blade are non-grid data in situ measuring systems. To overcome the difficulty of NURBS surface fitting from non-grid data, a new method based on data conversion is proposed, in which chord length parameterization and uniform parameter sampling are combined together to realize the data convertation, and subsequently hierarchical fitting strategy is applied to finish the NURBS surface reconstruction. The way proposed for data conversion is easy to realize, and by which gemetrical features of original measured data are also reserved well, which make the whole method outstanding in low time cost. Experimental results show that the method is fast, effective. The source code has been implemented in VC++, while the resulting pictures are constructed in Matlab with the obtained control points, knot vectors, and the orders.

Surface reconstruction based on CAD model driven priori templates.

In this paper, a method of reconstructing 3D (three dimensional) models from the original scanned point cloud using priori templates is proposed. Different from previous reconstruction methods that triangulate and fit the original scanning point cloud directly, we construct a priori template based on the CAD (computer aided design) model and guide the reconstruction of the original scanning point cloud with the priori template. Given a CAD model, the basic geometric elements are used as the basic units to extract the set elements of 3D shapes. Then the geometric elements are meshed, and the normal vectors at the mesh nodes are extracted. The corresponding point cloud data of each basic element are extracted from the original point cloud. The point cloud data near the normal of the guide point are searched, and the Gaussian weighted average value of the searched point represents the actual geometric parameters of the part at the guide point. Finally, the geometric elements of the basic unit are reconstructed locally by Non-Uniform Rational B-Splines surface fitting, and the complete reconstruction model is obtained by integrating the local reconstruction. Experiments show that our method can solve the problems of high quality reconstruction, sharp feature preservation, and detail recovery in surface reconstruction.

Modeling of a Microscale Surface Using NURBS Technique

This article describes microscale surface modeling using the Non-Uniform Rational B-Spline (NURBS) surface interpolation technique. A three-dimensional surface model was generated on the basis of measured surface profile data. To validate this model, three brass specimens having different roughness values were used. Direct comparison between measured profiles and the curves modeled with NURBS was employed. It was identified that the proposed method allows the generation of microscale models similar to actual surfaces. Finally, a method to extract the Bearing Area Curve (BAC) from a 3D model was detailed. The proposed modeling will be useful for the characterization of bearing capacity of the surface and for contact analysis.

Biomechanical simulations with dynamic muscle paths on NURBS surfaces

AbstractIn order to model the behaviour of muscles that work around the joints in musculoskeletal simulations, information about the muscle paths are needed. Typically, musculotendon paths, their lengths, and their force directions are modeled as the shortest connection wrapping around obstacles representing bones and adjacent tissue. In this work, the shortest path problem is described via constrained variational dynamics and is applied to a biomechanical example with muscle paths on non‐uniform rational B‐spline (NURBS) surfaces representing the wrapping obstacles. The results are compared with muscle paths from a G1‐continuous combination of geodesics [2,4,5].

On the application of isogeometric finite volume method in numerical analysis of wet-steam flow through turbine cascades

The isogeometric finite volume analysis is utilized in this research to numerically simulate the two-dimensional viscous wet-steam flow between stationary cascades of a steam turbine for the first time. In this approach, the analysis-suitable computational mesh with “curved” boundaries is generated for the fluid flow by employing a non-uniform rational B-spline (NURBS) surface that describes the cascade geometry, and the governing equations are then discretized by the NURBS representation. Thanks to smooth and accurate geometry representation of the NURBS formulation, the employed isogeometric framework not only resolves issues concerning the conventional mesh generation techniques of the finite volume method in steam turbine problems, but also, as validated against well-established experiments, significantly improves the accuracy of the numerical solution. In addition, the shock location in the cascade is predicted and tracked with a sufficient accuracy.