Non-uniform Rational B-spline Surfaces Research Articles

An isogeometric multimesh design approach for size and shape optimization of multidirectional functionally graded plates

This article firstly presents a novel numerical methodology to concurrently optimize material distribution (size) and thickness variation (shape) of multidirectional functionally graded (MFG) plates under free vibration within the isogeometric analysis (IGA) framework. An isogeometric multimesh design (IMD) approach is proposed to generate two distinct non-uniform rational B-spline (NURBS) surfaces via the k-refinement strategy. A finer analysis one relied upon a combination of the IGA and a generalized shear deformation theory (GSDT) is utilized for the unknown solution approximation in finite element analyses (FEAs). Whilst the other coarser design one is employed for the exact geometry representation as well as the optimal material and thickness depiction. Size and shape design variables are in turn the ceramic volume fraction and z-axis coordinate of the top side of the MFG plate coincidentally assigned to each of control points on this surface. Flexibly utilizing such two surfaces helps diminish a large number of design variables and considerably save the computational cost in optimization problems, yet still appropriately manifesting optimal material and thickness profiles. Additionally, this definition accurately simulates mechanical behavior of MFG plates in analysis ones as well. A recently developed derivative-free adaptive hybrid evolutionary firefly algorithm (AHEFA) is used to solve constrained frequency maximization problems. Several numerical examples are executed to verify the effectiveness and robustness of the present paradigm.

Isogeometric analysis in option pricing

ABSTRACTIsogeometric analysis is a recently developed computational approach that integrates finite element analysis directly into design described by non-uniform rational B-splines (NURBS). In this paper, we show that price surfaces that occur in option pricing can be easily described by NURBS surfaces. For a class of stochastic volatility models, we develop a methodology for solving corresponding pricing partial integro-differential equations numerically by isogeometric analysis tools and show that a very small number of space discretization steps can be used to obtain sufficiently accurate results. Presented solution by finite element method is especially useful for practitioners dealing with derivatives where closed-form solution is not available.

Transient fully nonlinear ship waves using a three-dimensional NURBS numerical towing tank

A fully nonlinear Numerical Towing Tank (NTT) is developed to simulate ship waves based on non-uniform rational B-spline (NURBS) surface and potential theory in the time domain. Fully nonlinear free surface boundary conditions and the exact body boundary condition are applied to solve the boundary value problem. The free surface and the ship hull surfaces are described precisely using NURBS formulation. Mixed Eulerian–Lagrangian (MEL) method is used to update the exact free surface based on material node approach. Isoparametric direct boundary integral equation (BIE) is used to compute the flow field. The intersection of the free surface and the rigid walls are found using an adaptive methodology based on NURBS and double nodes approach at each time step. To retain the open sea condition, a sponge layer is adopted on the exact free surface adjacent the tank's walls. To examine the accuracy of NURBS surface, the NURBS description of the ship hull for different orders of the basis function and different numbers of nodal points are compared with the exact surface. To verify the present computational algorithm, the solutions are compared with the experimental records and the prior numerical study for a Wigley hull. Propagation of the ship wave within the computational domain is studied due to ship advance in different water depths.

Efficient and Anti-Aliased Trimming for Rendering Large NURBS Models.

In Computer-Aided Design (CAD), Non-Uniform Rational B-Splines (NURBS) are a common model representation for export, simulation and visualization. In this paper, we present a direct rendering method for trimmed NURBS models based on their parametric description. Our approach builds on a novel trimming method and a three-pass pipeline which both allow for a sub-pixel precise visualization. The rendering pipeline bypasses tessellation limitations of current hardware using a feedback mechanism. In contrast to existing work, our trimming method scales well with a large number of trim curves and estimates the trimmed surface's footprint in screen-space which allows for an anti-aliasing with minimal performance overhead. Fragments with trimmed edges are routed into a designated off-screen buffer for subsequent blending with background faces. The evaluation of the presented algorithms shows that our rendering system can handle CAD models with ten thousands of trimmed NURBS surfaces. The suggested two-level data structure used for trimming outperforms state-of-the-art methods while being more precise and memory efficient. Our curve coverage estimation used for anti-aliasing provides an efficient trade-off between quality and performance compared to multisampling or screen-space anti-aliasing approaches.

Bending and free vibration analyses of in-plane bi-directional functionally graded plates with variable thickness using isogeometric analysis

This article is firstly concerned with bending and free vibration analyses of in-plane bi-directional functionally graded (IBFG) plates with variable thickness in the framework of isogeometric analysis (IGA). The plate thickness is smoothly altered in both x- and y-axes by a predetermined power law. Two types of power-law material models with the symmetrical and asymmetrical volume fraction distribution are suggested to characterize the in-plane material inhomogeneity. A non-uniform rational B-spline (NURBS) surface for simultaneously representing both variable thickness and volume fraction distribution of each constituent is employed. By using the k-refinement strategy, the C0-continuous requirement at symmetrical material interfaces can be achieved, yet still ensuring material gradations elsewhere owing to the prominent advantage of NURBS basis functions in easily controlling continuity. Effective material properties are then evaluated by either the rule of mixture or the Mori-Tanaka scheme. An analysis NURBS surface separately created with the foregoing NURBS surface is utilized to exactly describe geometry and approximately solve unknown solutions in finite element analysis (FEA) based on the IGA associated with a generalized shear deformation theory (GSDT). The Galerkin C1-continuous isogeometric finite element model is therefore simply achieved due to the possibility of flexibly meeting high-order derivatives and continuity of analysis NURBS functions. In addition, no shear correction factors exist in the present formulation, although shear deformation effects are still considered. The influences of variable thickness, material property, length-to-thickness ratio, boundary condition on bending and free vibration responses are investigated and discussed in detail through several numerical examples.

Research on trajectory planning of complex curved surface parts by laser cladding remanufacturing

A method of path planning based on laser cladding process characteristics was proposed, which can improve the machining efficiency and the quality of surface parts. The effective defocusing amount range of the laser cladding process is determined by the coaxial model of laser and powder interaction and the coaxial powder feeding experiments. The points cloud data of the cladding surface was obtained by the 3D scanner and the point set on the cladding path was obtained by the point clouds slicing method, utilizing equal bow height sparse trajectory point set. And then the NURBS (non-uniform rational B-splines) surface was fitted to the cladding surface and its differential geometrical characteristics were analyzed to determine the vector processing point and biased it to a certain distance to obtain the trajectory point of the gun head. The complex movement of laser gun head with continuous variable altitude is determined by the robot flexible processing characteristics, meanwhile, keeping the laser gun head and the cladding surface processing point normal vector on the same line. From the cladding experiments and the quantitative analysis of the experimental samples, good quality of the cladding layer on the curved surface was achieved. Also, the reliability and practicability of the method were verified, thus providing a good reference for remanufacturing of other surface parts.

Collision detection of virtual plant based on bounding volume hierarchy: A case study on virtual wheat

Visualization of simulated crop growth and development is of significant interest to crop research and production. This study aims to address the phenomenon of organs cross-drawing by developing a method of collision detection for improving vivid 3D visualizations of virtual wheat crops. First, the triangular data of leaves are generated with the tessellation of non-uniform rational B-splines surfaces. Second, the bounding volumes (BVs) and bounding volume hierarchies (BVHs) of leaves are constructed based on the leaf morphological characteristics and the collision detection of two leaves are performed using the Separating Axis Theorem. Third, the detecting effect of the above method is compared with the methods of traditional BVHs, Axis-Aligned Bounding Box (AABB) tree, and Oriented Bounding Box (OBB) tree. Finally, the BVs of other organs (ear, stem, and leaf sheath) in virtual wheat plant are constructed based on their geometric morphology, and the collision detections are conducted at the organ, individual and population scales. The results indicate that the collision detection method developed in this study can accurately detect collisions between organs, especially at the plant canopy level with high collision frequency. This collision detection-based virtual crop visualization method could reduce the phenomenon of organs cross-drawing effectively and enhance the reality of visualizations.

Parameter-based spiral tool path generation for free-form surface machining

In computer-aided manufacturing systems, generation strategies are classified into two main types according to the generation domain of the tool path: real space and the parametric domain. Although generation methods in the parametric domain defined using a parametric surface, such as a non-uniform rational B-spline surface, have the advantage of rapid calculation, tool path variations are limited because few methods using this type of tool path generation recognize features of the surface in the parametric domain. This study proposes a new algorithm for tool path generation that can generate machining points in the parametric domain without losing information about the distances among the corresponding points in real space. Using this proposed method, a tool path with a constant pitch in real space and an efficient tool path, such as a spiral path, can be rapidly calculated. This paper explains the theory behind this method and describes its implementation using free-form surfaces composed of a single patch. The proposed method was then expanded to surfaces composed of multiple patches. In this case, because correspondence between each surface patch is required, parametric boundary representation using a half-edge structure is also proposed in this paper. Parametric boundary representation can be used to represent connections among the outer and inner trimming boundaries of each surface patch on the parametric domain. Finally, the proposed method was implemented for two surfaces composed of multiple patches, and its effectiveness was confirmed by simulating and machining these surfaces.

Gravitational Search Algorithm for NURBS Curve Fitting

By providing great flexibility non-uniform rational B-spline (NURBS) curves and surfaces are reason of preferability on areas like computer aided design, medical imaging and computer graphics. Knots, control points and weights provide this flexibility. Computation of these parameters makes the problem as a non-linear combinational optimization problem on a process of reverse engineering. The ability of solving these problems using meta-heuristics instead of conventional methods attracts researchers. In this paper, NURBS curve estimation is carried out by a novel optimization method namely gravitational search algorithm. Both knots and knots together weights simultaneous optimization process is implemented by using research agents. The high performance of the proposed method on NURBS curve fitting is showed by obtained results.Keywords: Non-uniform rational B-spline, gravitational search algorithm, meta-heuristic

Surface Fitting for Quasi Scattered Data from Coordinate Measuring Systems.

Non-uniform rational B-spline (NURBS) surface fitting from data points is wildly used in the fields of computer aided design (CAD), medical imaging, cultural relic representation and object-shape detection. Usually, the measured data acquired from coordinate measuring systems is neither gridded nor completely scattered. The distribution of this kind of data is scattered in physical space, but the data points are stored in a way consistent with the order of measurement, so it is named quasi scattered data in this paper. Therefore they can be organized into rows easily but the number of points in each row is random. In order to overcome the difficulty of surface fitting from this kind of data, a new method based on resampling is proposed. It consists of three major steps: (1) NURBS curve fitting for each row, (2) resampling on the fitted curve and (3) surface fitting from the resampled data. Iterative projection optimization scheme is applied in the first and third step to yield advisable parameterization and reduce the time cost of projection. A resampling approach based on parameters, local peaks and contour curvature is proposed to overcome the problems of nodes redundancy and high time consumption in the fitting of this kind of scattered data. Numerical experiments are conducted with both simulation and practical data, and the results show that the proposed method is fast, effective and robust. What’s more, by analyzing the fitting results acquired form data with different degrees of scatterness it can be demonstrated that the error introduced by resampling is negligible and therefore it is feasible.

An efficient methodology for slicing NURBS surfaces using multi-step methods

Slicing the geometric models has been a crucial computational task for various manufacturing processes in which significant academic research has been dedicated for its efficiency, accuracy, and generalizability. Meanwhile, the feasibility of direct slicing of the modeled geometric surfaces needs more investigation due to the demands for precise geometrical information in applications such as multi-axis machining, layer-based manufacturing and 3D printing, coordinate metrology, and robotics. The purpose of this study is to introduce a method to find the best patches of a non-uniform rational B-spline (NURBS) surface for a direct slicing procedure. Formulating the relation between the spatial coordinates on the surface and the surface parameters is created by a parameterization process. Adams–Bashforth multi-step method is used in this methodology to calculate the most accurate series of the slicing parameters. This paper provides the conception, implementation, and verification of the developed methodology. The resulting solution is very practical and beneficial for direct slicing of CAD models in additive manufacturing systems.

Exact evaluation of hydrodynamic loads on ships using NURBS surfaces and acceleration potential

The development of an accurate surface intersecting methodology and adaptive acceleration potential scheme, in order to precisely compute pressure on the wet surface of ships hulls, is presented. Non-Uniform Rational B-Spline (NURBS) surfaces are used to describe the ships hulls and the free surface evolutions. The intersection of free surface with ship hull is found by solving the resultant system of equations yielded from surface intersecting scheme. The pressure distribution over the exact wet surface of ship hull surface is obtained based on acceleration potential in a time domain simulation. Hence, implicit body boundary condition is applied to evaluate the time derivative of velocity potential on the hull surface in a fully nonlinear numerical wave tank. On the other hand, tangential derivatives of the velocity potential on the wet surface are approximated by B-Spline surface. Wave propagation is considered to examine the accuracy and convergence of the present solution. Motions of a Wigley hulls in linear wave are compared with experimental measurements and prior numerical solutions to verify the present numerical procedure. Also, fully nonlinear motions of a Wigley hull in nonlinear wave is simulated.

Bounds on partial derivatives of NURBS surfaces

In this paper, we estimate the partial derivative bounds for Non-Uniform Rational B-spline(NURBS) surfaces. Firstly, based on the formula of translating the product into sum of B-spline functions, discrete B-spline theory and Dir function, some derivative bounds on NURBS curves are provided. Then, the derivative bounds on the magnitudes of NURBS surfaces are proposed by regarding a rational surface as the locus of a rational curve. Finally, some numerical examples are provided to elucidate how tight the bounds are.

A novel modelling method of geometric errors for precision assembly

Geometric errors are inevitably observable on machining parts and have huge influences on the assembly quality and precision of mechanical system. The modelling of geometric errors is the key issue for the prediction of their effects on the assembly quality. Although various approaches for the representation and analysis of geometric errors have been proposed, most of them only consider orientation and position deviations but do not consider form errors. However, in mechanical system, since the non-uniform distribution of the altitudes of form errors on parts surface inevitably leads to non-uniform contact states, further causing the local surface deformations and non-uniform stresses on contact surfaces, even very small form errors propagate and accumulate through assembly parts and lead to malfunction as well as decreased assembly accuracy stability. The paper proposes a method for the representation of geometric errors based on Non-Uniform Rational B-Splines (NURBS) surface, which can consider form errors and characterize the altitude distribution of geometric errors on the machined surface. Machining Characteristic Model (MCM), representing the distribution characteristics of geometric errors on the surfaces machined under the same machining process, is established by averaging a number of surface models of geometric errors. The developed mathematical models of geometric errors can be implemented in COMPUTER-AIDED DESIGN (CAD) system so that a method to integrate geometric errors in CAD model is presented. The resulting solid model (i.e. REALISTIC GEOMETRIC SOLID MODEL) is the representation of parts geometry considering geometric errors. Realistic assembly model is established through the relative positioning of REALISTIC GEOMETRIC SOLID MODELS, which realizes the consideration of parts geometric errors in the virtual modelling of mechanical assembly. The efficiency and feasibility of the proposed methods are verified in the experimental study.

The effect of characteristics of free-form surface on the machined surface topography in milling of panel mold

In the milling process of automobile panel mold of hardened steel, the characteristic of free-form surface is one of the dominant factors for surface topography. In this paper, the trajectory of cutting edge is firstly modeled to analyze the residual height of the free-form surface in ball-end milling of hardened steel. Furthermore, the non-uniform rational B-splines (NURBS) surface reconstruction is utilized to generate the surface topography. Subsequently, the influences of surface curvature, lead angle, milling vibrations on the machined surface topography, and residual height are investigated, respectively. Finally, the accuracy of the surface topography and the roughness prediction model are validated by the milling experiments of free-form surface, where two-dimensional contour maps could be obtained. The simulation and experimental results demonstrate that the machined surface topography of hardened steel is fitted by means of NURBS surface reconstruction. In that manner, the effects of surface characteristics on the machined surface topography can be accurately predicted.

The Design of Case Products’ Shape Form Information Database Based on NURBS Surface

In order to improve the computer design of product shape design,applying the Non-uniform Rational B-splines(NURBS) of curves and surfaces surface to the representation of the product shape helps designers to design the product effectively.On the basis of the typical product image contour extraction and using Pro/Engineer(Pro/E) to extract the geometric feature of scanning mold,in order to structure the information data base system of value point,control point and node vector parameter information,this paper put forward a unified expression method of using NURBS curves and surfaces to describe products’ geometric shape and using matrix laboratory(MATLAB) to simulate when products have the same or similar function.A case study of electric vehicle’s front cover illustrates the access process of geometric shape information of case product in this paper.This method can not only greatly reduce the capacity of information debate,but also improve the effectiveness of computer aided geometric innovation modeling.

Analysis of terahertz scattering from electrically large scatterer with NURBS modeling

A novel expression based on conventional Non-Uniform Rational B-Splines surfaces is proposed to model a scatterer, which can be used to overcome the deficiencies of conventional modeling method and can also be combined perfectly with the stationary phase method (SPM). Furthermore, we propose a fast and accurate algorithm to locate inner critical points and boundary critical points of physical optics (PO) integrals, as well as to locate stationary points of edge-diffracted integrals. The terahertz (THz) scattering field from an electrically large scatterer is evaluated by the physical theory of diffraction and SPM (PTD-SPM), which consists of four parts, including the PO scattering field from inner critical points, boundary critical points, corner points of PO integrals, and edge-diffracted field from the stationary points of edge-diffracted integral. Moreover, the order of magnitude of each part contribution is derived. Numerical results demonstrate that PTD-SPM is a valid and efficient algorithm for analyzing THz scattering from an electrically large scatterer.

On the limits of non-uniform rational B-spline surfaces with varying weights

The non-uniform rational B-spline is a mathematical model commonly used in computer-aided design and manufacturing. For a non-uniform rational B-spline surface, when a single weight approaches infi...

A GEOMETRIC PROCESSING WORKFLOW FOR TRANSFORMING REALITY-BASED 3D MODELS IN VOLUMETRIC MESHES SUITABLE FOR FEA

Abstract. Conservation of Cultural Heritage is a key issue and structural changes and damages can influence the mechanical behaviour of artefacts and buildings. The use of Finite Elements Methods (FEM) for mechanical analysis is largely used in modelling stress behaviour. The typical workflow involves the use of CAD 3D models made by Non-Uniform Rational B-splines (NURBS) surfaces, representing the ideal shape of the object to be simulated. Nowadays, 3D documentation of CH has been widely developed through reality-based approaches, but the models are not suitable for a direct use in FEA: the mesh has in fact to be converted to volumetric, and the density has to be reduced since the computational complexity of a FEA grows exponentially with the number of nodes. The focus of this paper is to present a new method aiming at generate the most accurate 3D representation of a real artefact from highly accurate 3D digital models derived from reality-based techniques, maintaining the accuracy of the high-resolution polygonal models in the solid ones. The approach proposed is based on a wise use of retopology procedures and a transformation of this model to a mathematical one made by NURBS surfaces suitable for being processed by volumetric meshers typically embedded in standard FEM packages. The strong simplification with little loss of consistency possible with the retopology step is used for maintaining as much coherence as possible between the original acquired mesh and the simplified model, creating in the meantime a topology that is more favourable for the automatic NURBS conversion.

A Genetic Algorithm NURBS-based new approach for fast kinematic limit analysis of masonry vaults

The present paper proposes a new Genetic Algorithm NURBS-based approach for the limit analysis of masonry vaults based on an upper bound formulation. A given masonry vault geometry can be represented by a NURBS (Non-Uniform Rational B-Spline) parametric surface and a NURBS mesh of the given surface can be generated. Each element of the mesh is a NURBS surface itself and can be idealized as a rigid body. An upper bound limit analysis formulation, which takes into account the main characteristics of masonry material is deduced, with internal dissipation allowed exclusively along element edges. The approach is capable of well predicting the load bearing capacity of any masonry vault of generic shape. It is proved that, even by using a mesh constituted by very few elements, a good estimate of the collapse load multiplier is obtained provided that the initial mesh is adjusted by means of a meta-heuristic approach (i.e. a Genetic Algorithm, GA) in order to enforce that element edges accurately represent the actual failure mechanism. The proposed method turns out to be both accurate and much less computationally expensive than existing methods for the limit analysis of masonry vaults.