Non-uniform Rational B-spline Representation Research Articles

Discretization of Non-uniform Rational B-Spline (NURBS) Models for Meshless Isogeometric Analysis

We present an algorithm for fast generation of quasi-uniform and variable-spacing nodes on domains whose boundaries are represented as computer-aided design (CAD) models, more specifically non-uniform rational B-splines (NURBS). This new algorithm enables the solution of partial differential equations within the volumes enclosed by these CAD models using (collocation-based) meshless numerical discretizations. Our hierarchical algorithm first generates quasi-uniform node sets directly on the NURBS surfaces representing the domain boundary, then uses the NURBS representation in conjunction with the surface nodes to generate nodes within the volume enclosed by the NURBS surface. We provide evidence for the quality of these node sets by analyzing them in terms of local regularity and separation distances. Finally, we demonstrate that these node sets are well-suited (both in terms of accuracy and numerical stability) for meshless radial basis function generated finite differences discretizations of the Poisson, Navier-Cauchy, and heat equations. Our algorithm constitutes an important step in bridging the field of node generation for meshless discretizations with isogeometric analysis.

Automatic modelling of heterotypic tunnel structures via NURBS-based meshfree method

• A NURBS-based meshfree method is proposed for the automatic modelling of heterotypic tunnel structures. • The proposed method unifies continuous-discontinuous analysis in a NURBS-meshfree framework. • A premier integral cell generation approach free of boundary integral points discarding is developed, which improves the robustness of meshfree interpolation. • The proposed method eliminates the barrier between numerical theories and applied modelling in tunnel engineering. For heterotypic tunnel structures with complex geometries and diverse components, a meshfree (MF) method outperforms the finite element method in terms of modelling interfaces, cracks, and discontinuities. This paper proposes a nonuniform rational B-spline (NURBS)-based MF method (NURBS-MF) to analyze such structures. The proposed NURBS-MF combines the merits of both the NURBS representation and MF approach. By mapping the NURBS surface onto a canonical parametric domain, MF nodes in the NURBS-MF method can be generated, regardless of the geometries and discontinuities. An MF interpolation is implemented directly based on the MF nodes within this mapping domain, which creates a set of integral cells completely conforming with the geometry boundary. The integral cells along the boundaries do not split, and thus the discarding of invalid integral points is avoided. Under this framework, by simply inputting material parameters and a discretized grid size, a numerical tunnel model is automatically established based on the tunnel design model represented by NURBS in computer-aided design. Further, the modelling cost and difficulty of complex heterotypic tunnel structures are significantly reduced. Therefore, automatic modelling and calculation are possible with one click, which also eliminates the barrier between numerical theories and applied modelling in tunnel engineering. The proposed method is proven to be novel and efficient for practical engineering problems through several numerical tests.

NON-UNIFORM RATIONAL B-SPLINE BASED ISO-GEOMETRIC ANALYSIS FOR A CLASS OF HYDRODYNAMIC PROBLEMS

Herein, we present the design and development of a ‘Non-uniform Rational B-spline (NURBS)’ based iso-geometric approach for the analysis of a number of ‘Boundary Value Problems (BVPs)’ relevant in hydrodynamics. We propose a ‘Potential Function’ based ‘Boundary Element Method (BEM)’ and show that it holds the advantage of being computationally efficient over the other known numerical methods for a wide range of external flow problems. The use of NURBS is consistent, as inspired by the ‘iso-geometric analysis’, from geometric formulation for the body surface to the potential function representation to interpolation. The control parameters of NURBS are utilised and they have been explored to arrive at some preferable values and parameters for parameterization and the knot vector selection. Also, the present paper investigates the variational strength panel method, and its computational performance is analyzed in comparison with the constant strength panel method. The two variations have been considered, e.g. linear and quadratic. Finally, to illustrate the effectiveness and efficiency of the proposed NURBS based iso-geometric approach for the analysis of boundary value problems, five different problems (i.e. flow over a sphere, effect of the knot vector selection on analysis, flow over a rectangular wing section of NACA 0012 aerofoil section, performance of DTMB 4119 propeller (un-skewed), performance of DTNSDRC 4382 propeller (skewed)) are considered. The results show that in the absence of predominant viscous effects, a ‘Potential Function’ based BEM with NURBS representation performs well with very good computational efficiency and with less complexity as compared to the results available from the existing approaches and commercial software programs, i.e. low maximum errors close to 110−3 , faster convergence with even up to 75 % reduction in the number of panels and improvements in the computational efficiency up to 32.5 % even with low number of panels.

Non-Uniform Rational B-Spline Based Iso-Geometric Analysis for a Class of Hydrodynamic Problems

Herein, we present the design and development of a ‘Non-uniform Rational B-spline (NURBS)’ based iso-geometric approach for the analysis of a number of ‘Boundary Value Problems (BVPs)’ relevant in hydrodynamics. We propose a ‘Potential Function’ based ‘Boundary Element Method (BEM)’ and show that it holds the advantage of being computationally efficient over the other known numerical methods for a wide range of external flow problems. The use of NURBS is consistent, as inspired by the ‘iso-geometric analysis’, from geometric formulation for the body surface to the potential function representation to interpolation. The control parameters of NURBS are utilised and they have been explored to arrive at some preferable values and parameters for parameterization and the knot vector selection. Also, the present paper investigates the variational strength panel method, and its computational performance is analyzed in comparison with the constant strength panel method. The two variations have been considered, e.g. linear and quadratic. Finally, to illustrate the effectiveness and efficiency of the proposed NURBS based iso-geometric approach for the analysis of boundary value problems, five different problems (i.e. flow over a sphere, effect of the knot vector selection on analysis, flow over a rectangular wing section of NACA 0012 aerofoil section, performance of DTMB 4119 propeller (un-skewed), performance of DTNSDRC 4382 propeller (skewed)) are considered. The results show that in the absence of predominant viscous effects, a ‘Potential Function’ based BEM with NURBS representation performs well with very good computational efficiency and with less complexity as compared to the results available from the existing approaches and commercial software programs, i.e. low maximum errors close to 110−3 , faster convergence with even up to 75 % reduction in the number of panels and improvements in the computational efficiency up to 32.5 % even with low number of panels.

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.

NURBS-based microstructure design for organic photovoltaics

The microstructure – spatial distribution of electron donor and acceptor domains – plays an important role in determining the photo current in thin film organic solar cells (OSCs). Optimizing the microstructure can lead to higher photo current generation, and is an active area of experimental research. There has been recent progress in framing OSC microstructure design as a computational design problem. However, most current approaches to microstructure optimization are based on volumetric distribution of material, which makes the design space very large. In contrast, we frame the microstructure design optimization problem in terms of designing the interface between the donor and acceptor regions, and thus pose it as a surface representation and optimization problem. This results in substantially reduced number of design variables, thus enabling use of standard optimization tools. In this work, we address the efficient design of OSC microstructure by using surface and curve modeling techniques to model the donor–acceptor interface, and use meta-heuristic, gradient-free optimization techniques to optimize the microstructure for maximum short circuit current generation. Our modeling framework consists of three major components: (1) geometric modeling of OSC microstructure that uses Non-Uniform Rational B-spline (NURBS) curves and surfaces to construct the free-form donor–acceptor interface, (2) photo-current generation modeling that uses a parallel, finite-element based exciton–drift–diffusion (XDD) model, and (3) optimization that utilizes genetic algorithms (GA) to optimize the OSCs microstructure via exploration of the NURBS representation. We apply these methods for the optimization of both 2D and 3D microstructures. Results show substantial improvement in current density compared to the bulk-heterojunction microstructures. These results provide promising microstructures for experimental groups to fabricate. The proposed surface representation approach seems to be a promising approach for interface design in engineered systems.

New algorithms for virtual reconstruction of heterogeneous microstructures

A new set of algorithms is introduced for the virtual reconstruction of heterogeneous material microstructures, in which morphologies of embedded particles/fibers are explicitly represented. As a pre-processing phase, a shape library containing morphologies of heterogeneities, parameterized in terms of Non-Uniform Rational B-Splines (NURBS), is extracted from digital data such as micro-computed tomography images. Two packing algorithms are then introduced to reconstruct an initial (raw) periodic microstructure: In the first approach, a set of hierarchical bounding boxes approximating particle shapes are employed to check for overlap. The second approach, which is specialized for fibrous microstructures, uses the NURBS representation of fiber centerlines during the packing process. An optimization phase is applied to build the final microstructural model, relying either on the Genetic Algorithm to selectively eliminate some of the inclusions or their sequential relocation within the raw microstructure. The objective functions of this optimization phase are designed to replicate the target statistical microstructural descriptors such as the volume fraction, size distribution, and spatial arrangement of inclusions. Several example problems are presented to show the application of these algorithms for synthesizing various heterogeneous microstructures, as well as their finite element modeling and simulation.

Signed algebraic level sets on NURBS surfaces and implicit Boolean compositions for isogeometric CAD–CAE integration

Boundary-representation (B-rep) geometrical models, often mathematically represented using Non-Uniform Rational B-Spline (NURBS) surfaces, are the starting point for complex downstream product life-cycle evaluations including Computer-Aided Engineering (CAE). Boolean operations during B-rep model generation require surface intersection computations to describe the composed entity. However, for parametric NURBS surfaces, intersection operations are non-trivial and typically carried out numerically. The numerical intersection computations introduce challenges relating to the accuracy of the resulting representation, efficiency with which the computation is carried out, and robustness of the result to small variations in geometry. Often, for downstream CAE evaluations, an implicit, procedural knowledge of the Boolean operations between the composed objects that can resolve point containment queries (exact to the original NURBS bounding surfaces) maybe sufficient during quadrature. However, common point containment queries on B-rep models are numerical, iterative and relatively expensive. Thus, the first goal of the present paper is to describe a purely algebraic, and therefore non-iterative, approach to carrying out point containment queries on complex B-rep models built using low-degree NURBS surfaces. For CAE operations, the boundary representation of B-rep solids is, in general, not convenient and as a result, the B-rep model is converted to a meshed volumetric approximation. The major challenges to such a conversion include capturing the geometric features accurately when constructing the secondary (meshed) representation, apart from the efficiency of carrying out such a mesh generation step repeatedly as the geometric shape evolves. Thus, an ideal analysis procedure would operate directly on B-rep CAD models, without needing a secondary mesh, and would procedurally unify the geometric operations during CAD as well as CAE stages. Therefore, the second and broader goal of the present paper is to demonstrate CAD–CAE integration using signed algebraic level set operations directly on B-rep models by embedding or immersing the bounding surfaces within a discretized domain while preserving the geometric accuracy of the surfaces exact to the original NURBS representation during analysis.

A novel path generation method of onsite 5-axis surface inspection using the dual-cubic NURBS representation

Onsite surface inspection with a touch probe or a laser scanner is a promising technique for efficiently evaluating surface profile error. The existing work of 5-axis inspection path generation bears a serious drawback, however, as there is a drastic orientation change of the inspection axis. Such a sudden change may exceed the stringent physical limit on the speed and acceleration of the rotary motions of the machine tool. In this paper, we propose a novel path generation method for onsite 5-axis surface inspection. The accessibility cones are defined and used to generate alternative interference-free inspection directions. Then, the control points are optimally calculated to obtain the dual-cubic non-Uniform rational B-splines (NURBS) curves, which respectively determine the path points and the axis vectors in an inspection path. The generated inspection path is smooth and non-interference, which deals with the ‘mutation and shake’ problems and guarantees a stable speed and acceleration of machine tool rotary motions. Its feasibility and validity is verified by the onsite inspection experiments of impeller blade.

Global adaptive slicing of NURBS based sculptured surface for minimum texture error in rapid prototyping

Purpose – This paper aims to propose a global adaptive direct slicing technique of Non-Uniform Rational B-Spline (NURBS)-based sculptured surface for rapid prototyping where the NURBS representation is directly extracted from the computer-aided design (CAD) model. The imported NURBS surface is directly sliced to avoid inaccuracies due to tessellation methods used in common practice. The major objective is to globally optimize texture error function based on the available range of layer thicknesses of the utilized rapid prototyping machine. The total texture error is computed with the defined error function to verify slicing efficiency of this global adaptive slicing algorithm and to find the optimum number of slices. A variety of experiments are conducted to study the accuracy of the developed procedure, and the results are compared with previously developed algorithms. Design/methodology/approach – This paper proposes a new adaptive algorithm which globally optimizes a texture error function produced by staircase effect for a user-defined number of layers. The adaptive slicing algorithm dynamically calculates optimized slicing thicknesses based on the rapid prototyping machine’s specifications to minimize the texture error function. This paper also compares the results of implementing the developed methodology with the results of previously developed algorithms and presents cost-effective optimum slicing layer thicknesses. Findings – A new methodology for global adaptive direct slicing algorithm of CAD models, based on a texture error function for the final product and the possible layer thicknesses in rapid prototyping, has been developed and implemented. Comparing the results of implementation with the common practice for several case studies shows that the proposed approach has greater slicing efficiency. Typically, by utilizing this approach, the number of prototyping layers can be reduced by 20-50 per cent compared to the slicing with other algorithms, while maintaining or improving the accuracy of the final manufactured surfaces. Therefore, the developed slicing method provides a better solution to trade-off between the rapid prototyping time and the rapid prototyping accuracy. For the many advantages of global direct slicing, it can be seen as the future solution to the slicing process in rapid prototyping systems. Originality/value – This paper presents an innovative approach in direct global adaptive slicing of the additive manufacturing parts. The novel definition of an error function which comprehensively addresses the resulting manufactured surface quality of the entire product allows presenting an objective function to solve and to find the optimum selection of all the layer thicknesses during the slicing process.

A unified approach for nonlinear vibration analysis of curved structures using non-uniform rational B-spline representation

A novel procedure for the nonlinear vibration analysis of curved beam is presented. The Non-Uniform Rational B-Spline (NURBS) is combined with the Euler−Bernoulli beam theory to define the curvature of the structure. The governing equation of motion and the general frequency formula, using the NURBS variables, is applicable for any type of curvatures, is developed. The Galerkin procedure is implemented to obtain the nonlinear ordinary differential equation of curved system and the multiple time scales method is utilized to find the corresponding frequency responses. As a case study, the nonlinear vibration of carbon nanotubes with different shapes of curvature is investigated. The effect of oscillation amplitude and the waviness on the natural frequency of the curved nanotube is evaluated and the primary resonance case of system with respect to the variations of different parameters is discussed. For the sake of comparison of the results obtained with those from the molecular dynamic simulation, the natural frequencies evaluated from the proposed approach are compared with those reported in literature for few types of carbon nanotube simulation.

A NURBS interpolation method with minimal feedrate fluctuation for CNC machine tools

To reduce the feedrate fluctuation in non-uniform rational B-spline (NURBS) interpolation caused by approximation errors of interpolation methods, a new NURBS interpolation method with minimal feedrate fluctuation for CNC machine tools is proposed to obtain the sampling curve parameter by solving a quartic equation with respect to the curve parameter increment according to the scheduled feedrate using Ferrari’s formulas and Shengjin’s formulas. Firstly, the second-order Taylor expansion is applied on the numerator and denominator in the NURBS representation separately with unknown curve parameter increment. Then, a quartic equation with respect to the curve parameter increment is formed and solved by Ferrari’s formulas and Shengjin’s formulas efficiently, which can achieve minimal feedrate fluctuation in real-time interpolation. In principle, the proposed method can totally eliminate the feedrate fluctuation for second-degree NURBS curves and can reduce it to a satisfied value for other high-degree NURBS curves. Finally, numerical simulations and experiments are conducted to verify the feasibility and applicability of the proposed method.

Molecular dynamics-based triangulation algorithm of free-form parametric surfaces for computer-aided engineering

A new computational approach for automated triangulation of Computer-Aided Design (CAD) surface models, applicable to various CFD (Computational Fluid Dynamics) problems of practical interest is proposed. The complex shaped product configurations are represented by a set of Non-Uniform Rational B-Splines (NURBS) surface patches. The suggested technique is based on the molecular dynamics method. The main idea of the approach is that the mesh nodes are considered as similarly charged interacting particles which move within the region to be meshed under the influence of internal (such as particle–particle interaction forces) and external forces as well as optional additional forces. Moreover, the particles experience a medium resistance due to which the system comes to equilibrium within a relatively short period of time. The proposed 3D surface mesh generation algorithm uses a parametric NURBS representation as initial definition of the domain boundary. This method first distributes the interacting nodes into optimal locations in the parametric domain of the NURBS surface patch using molecular dynamics simulation. Then, the well-shaped triangles can be created after connecting the nodes by Delaunay triangulation. Finally, the mapping from parametric space to 3D physical space is performed. Since the presented interactive algorithm allows to control the distance between a pair of nodes depending on the curvature of the NURBS surface, the method generates high quality triangular mesh. The algorithm enables to produce uniform mesh, as well as anisotropic adaptive mesh with refinement in the large gradient regions. The mesh generation approach has the abilities to preserve the representation accuracy of the input geometry model, create a close relationship between geometry modeling and grid generation process, be automated to a large degree. Some examples are considered in order to illustrate the method’s ability to generate a surface mesh for a complicated CAD model.

Isogeometric analysis suitable trivariate NURBS representation of composite panels with a new offset algorithm

Trivariate NURBS (non-uniform rational B-splines) representation of composite panels which is suitable for three-dimensional isogeometric analysis (IGA) is constructed with a new curve/surface offset algorithm. The proposed offset algorithm, which is required by IGA, is non-existent in the CAD literature. Using the presented approach, finite element analysis of composite panels can be performed with the only input being the geometry representation of the composite surface. The method proposed provides a bi-directional system in which one can go forward from CAD to analysis and backwards from analysis to CAD. This is believed to facilitate the design of composite structures. Different parts (patches) can be parametrized independently of each other and glued together, in the finite element solver, by a discontinuous Galerkin method. A stress analysis of curved composite panel with stiffeners is provided to demonstrate the proposed framework.

Optimized stereo reconstruction of free-form space curves based on a nonuniform rational B-spline model

Analytical reconstruction of 3D curves from their stereo images is an important issue in computer vision. We present an optimization framework for such a problem based on a nonuniform rational B-spline (NURBS) curve model that converts reconstruction of a 3D curve into reconstruction of control points and weights of a NURBS representation of the curve, accordingly bypassing the error-prone point-to-point correspondence matching. Perspective invariance of NURBS curves and constraints deduced on stereo NURBS curves are employed to formulate the 3D curve reconstruction problem into a constrained nonlinear optimization. A parallel rectification technique is then adopted to simplify the constraints, and the Levenberg-Marquardt algorithm is applied to search for the optimal solution of the simplified problem. The results from our experiments show that the proposed framework works stably in the presence of different data samplings, randomly posed noise, and partial loss of data and is potentially suitable for real scenes.

A dynamic model for predicting the motion solder droplets during assembly

The dynamic motion of a solder droplet during assembly is a complex, unsteady, free surface problem involving surface tension and viscous effects. The motion of the droplet is coupled with the motion of the component or chip to be assembled and involves dynamic contact lines. A methodology based on a non-uniform rational b-spline (NURBS) discretization has been developed for the dynamic analysis of the droplet motion. A surface energy based formulation has been developed to incorporate the surface tension effects. The developed methodology leads to an updated Lagrangian scheme with a Galerkin in space and Least square in time formulation. The NURBS representation used for the spatial discretization enables the method to handle problems involving complex droplet geometries. The ability of the NURBS representation to provide both global and local control, along with the least square method used in this methodology, enables us to develop an unconditionally stable time integration scheme which can be optimized to achieve desired accuracy and numerical dissipation efficiently. A sample problem of droplet shape evolution has been solved to demonstrate the path prediction capability of the proposed methodology. In future, the method can be applied to solve various real world dynamic motion problems associated with droplets.

Optimized Nonuniform Rational B-Spline Geometrical Representation for Aerodynamic Design of Wings

The geometric representation and parameterization used in an aerodynamic wing design process determines the number of design variables and influences the smoothness of the wing representation. In an attempt to reduce the number of design variables while preserving good smoothness properties, the present research investigates the performance of an optimized nonuniform rational B-spline (NURBS) geometrical representation for the aerodynamic design of wings. As a first step, an approach is described whereby optimal spatial positions and weights of a fixed number of NURBS control points is determined using a quasi-Newton optimization algorithm to approximate a general airfoil section. The resulting optimized NURBS representation significantly reduces the number of design variables needed to define accurately a wing section while ensuring good smoothness properties. In a second step, the NURBS control point positions and weights are used as design variables in an aerodynamic optimization problem. This methodology results in a rapid and robust design process, as illustrated by examples of aerodynamic optimization for two- and three-dimensional cases

A Parametric Approach for Initial Hull Form Modeling Using NURBS Representation

An approach to initial hull form modeling using the concept of parametric generation and manipulation is introduced. The parametric approach is to directly control geometric features or to describe hydrostatic performance of a hull by utilizing naval architecture parameters that uniquely and unambiguously define a hull form. The Non Uniform Rational B-spline (NURBS) representation, which is the most common type of a hull form modeling in use, is combined with the parametric approach to represent a hull form. From a planar net of control points of a NURBS surface, a hull form satisfying the given requirements is automatically generated. The generation of the hull form is accomplished by subdividing the hull into five zones: stem, entrance, midbody, run, and stern, maintaining the continuity between adjacent zones. A library of pre-defined stems and sterns is provided to help a rapid and efficient design. Modification of the initial hull form can be accomplished by manipulating the set of NURBS control points that influences the designated part of the hull form. Refinement for precise hull form and variation to similar hull forms are easily carried out within any hull form distortion system. This proposed modeling approach is useful for an initial hull form design from scratch with minimal user requirements; therefore, it can be an effective tool for the initial design of a hull form when a suitable parent model is unavailable.

Accurate parametrization of conics by NURBS

One argument often given to explain the popularity of NURBS (nonuniform rational B-spline) is that it permits the definition of free-form curves and surfaces (as do most spline models). It also provides an exact representation of conic sections and thus of a large set of curves and surfaces used intensively in CAD: circular arcs, circles, cylinders, cones, spheres, surfaces of revolution and so forth. Nevertheless, few published works discuss the mathematical properties behind the representation of conics by NURBS except for two monographs by Piegl and Tiller (1995) and Farin (1995). The article does not pretend to fill this theoretical lack but rather deals with the following problems: all known NURBS representations of curves and surfaces based on conics have only a C/sup l/ continuity. Moreover, no technique exists that would eventually allow us to find a parametrization with a higher level of continuity. The parametrization resulting from the NURBS representation of conics can deviate significantly from the ideal are length (that is, uniform) parametrization. The only known solution to reduce this deviation is to increase the number of control points of the spline by using refinement algorithms, for instance, but such a process converges only slowly to the uniform parametrization. The solution proposed uses an original reparametrization process called zigzag reparametrization, based on a specific family of rational polynomials.

Algorithms for Computing Conic Splines

Conic splines play an important role in engineering geometry and computer graphics, in particular, in geometric and solids modeling where the majority of objects are bounded by simple surfaces such as natural quadrics (plane, cone, cylinder, and sphere). The integration of analytic and free‐form curves into solids modeling systems is one of the most important tasks in today's solids modelers. This paper presents algorithms that represent conic splines in nonuniform rational B‐spline (NURBS) form, thus incorporating them into a system based on free‐form geometry. The algorithms take as input the end points and end tangents of the segment, and one arbitrary point somewhere along the arc, and return the precise NURBS representation of the curve. It was originally developed for supporting Boolean operations on trimmed surfaces, i.e., to compute conic arc surface‐surface intersections in true 3‐D space. Since during Boolean operations all kinds of segments can pop up, special attention is paid to such cases as parallel end tangents and arcs contained outside the convex hull. Knot insertion is applied to obtain a representation that is numerically as stable as possible. The input required to generate a curve is very general, therefore the spline conversion has different kinds of applications apart from surface‐surface intersection. Some examples are font design, shape generation, geometry conversion and processing, and curve approximation.