Non-uniform Rational B-spline Surfaces Research Articles

Three-dimensional reconstruction of industrial parts from a single image

This study proposes an image-based three-dimensional (3D) vector reconstruction of industrial parts that can generate non-uniform rational B-splines (NURBS) surfaces with high fidelity and flexibility. The contributions of this study include three parts: first, a dataset of two-dimensional images is constructed for typical industrial parts, including hexagonal head bolts, cylindrical gears, shoulder rings, hexagonal nuts, and cylindrical roller bearings; second, a deep learning algorithm is developed for parameter extraction of 3D industrial parts, which can determine the final 3D parameters and pose information of the reconstructed model using two new nets, CAD-ClassNet and CAD-ReconNet; and finally, a 3D vector shape reconstruction of mechanical parts is presented to generate NURBS from the obtained shape parameters. The final reconstructed models show that the proposed approach is highly accurate, efficient, and practical.

NURBS-based path planning for aerosol jet printing of conformal electronics

Conformal printing offers a promising capability to integrate complex electronic functionality with intricate objects consisting of curved surfaces. However, motion planning to support these printing configurations presents a challenge, in part due to conventional reliance on tessellated surface representations that limit the accuracy and scalability of the printing process. In this paper, we introduce a direct approach to conformal aerosol jet printing on topographically complex surfaces, represented using non-uniform rational B-splines (NURBS). Using the NURBS surface definition directly offers a scalable and continuous approach for toolpath generation, which is well-aligned with standard computer-aided design practices. We select a patch that directly corresponds to the area of interest on the NURBS surface and extract the associated surface points. These surface points are then appropriately connected to generate the machine code (G-code) for a 3-axis printing system. We demonstrate this capability by printing a strain gauge on a curved model of a wind turbine blade. Our approach offers a promising avenue for manufacturing printed electronics with diverse applications that require conformal printing. This approach is particularly valuable for incorporating printed circuits on intricate underlying structures, including structural health monitoring for aerospace and civil infrastructure applications.

Effect of crown retention systems and loading direction on the stress magnitude of posterior implant-supported restorations: A 3D-FEA

This study aimed to investigate the effect of four retention systems for implant-supported posterior crowns under compressive loading using three-dimensional finite element analysis. A morse-taper dental implant (4.1 × 10 mm) was designed with Computer Aided Design software based on non-uniform rational B-spline surfaces. According to International Organization for Standardization 14,801:2016, the implant was positioned at 3 mm above the crestal level. Then four models were designed with different crown retention systems: screw-retained (A), cement-retained (B), lateral-screw-retained (C), and modified lateral-screw-retained (D). The models were imported to the analysis software and mesh was generated based on the coincident nodes between the juxtaposed lines. For the boundary conditions, two loads (600 N) were applied (axial to the implant fixture and oblique at 30°) totaling 8 conditions according to retention design and loading. The von-Mises stress analysis showed that different retention systems modify the stress magnitude in the implant-supported posterior crown. There is a similar stress pattern in the implant threads. However, models C and D presented higher stress concentrations in the crown margin in comparison with A and B. The oblique loading highly increased the stress magnitude for all models. In the simulated conditions, part of the stress was concentrated at the lateral screw under axial loading for model C and oblique loading for model D. The results indicate a possible new failure origin for crown retained using lateral screws in comparison to conventional cement-retained or screw-retained systems.

Full-LSPIA: A Least-Squares Progressive-Iterative Approximation Method with Optimization of Weights and Knots for NURBS Curves and Surfaces

The Least-Squares Progressive-Iterative Approximation (LSPIA) method offers a powerful and intuitive approach for data fitting. Non-Uniform Rational B-splines (NURBS) are a popular choice for approximation functions in data fitting, due to their robust capabilities in shape representation. However, a restriction of the traditional LSPIA application to NURBS is that it only iteratively adjusts control points to approximate the provided data, with weights and knots remaining static. To enhance fitting precision and overcome this constraint, we present Full-LSPIA, an innovative LSPIA method that jointly optimizes weights and knots alongside control points adjustments for superior NURBS curves and surfaces creation. We achieve this by constructing an objective function that incorporates control points, weights, and knots as variables, and solving the resultant optimization problem. Specifically, control points are adjusted using LSPIA, while weights and knots are optimized through the LBFGS method based on the analytical gradients of the objective function with respect to weights and knots. Additionally, we present a knot removal strategy known as Decremental Full-LSPIA. This strategy reduces the number of knots within a specified error tolerance, and determines optimal knot locations. The proposed Full-LSPIA and Decremental Full-LSPIA maximize the strengths of LSPIA, with numerical examples further highlighting the superior performance and effectiveness of these methods. Compared to the classical LSPIA, Full-LSPIA offers greater fitting accuracy for NURBS curves and surfaces while maintaining the same number of control points, and automatically determines suitable weights and knots. Moreover, Decremental Full-LSPIA yields fitting results with fewer knots while maintaining the same error tolerance.

Form-finding for Free-form Reticulated Shell Structure Considering Structural Performance and Regularization of Node Connection Patterns

In this study, we propose a form-finding method for the free-formed reticulated shell structure, considering both structural performance and regularization of node connection patterns. The coordinates of the nodes of the reticulated shells are represented on the non-uniform rational B-spline (NURBS) surface. According to the principal curvature, the clustering classifies the node connection patterns into specified groups. Multi-objective optimization is performed to minimize the strain energy and maximum deviation in clusters. A multi-objective genetic algorithm is used to solve the optimization problem. The mechanical properties of the optimal solution obtained in a numerical example are discussed to confirm the effectiveness of the proposed method.

Contact Analysis for Cycloid Pinwheel Mechanism by Isogeometric Finite Element

Cycloid drives are generally used in precision machinery requiring high-reduction ratios, such as robot joint (RV) reducers. The contact stress of cycloidal gears greatly affects lifetime and transmission performance. Traditional finite element method (FEM) has less computational efficiency for contact analysis of complex surface. Therefore, in this paper, isogeometric analysis (IGA) was employed to explore the multi-tooth contact problem of the cycloid pinwheel drive. Based on the nonuniform rational B spline (NURBS) curved surface generation method, the NURBS tooth profile of the cycloid gear was reconstructed. In addition, the NURBS surface of the cycloid gear–pin tooth–output pin was generated via the element splicing method. A geometrical analysis model of cycloid pinwheel drive was established to solve the contact force of the meshing pair under different input angles and compared with the finite element method in terms of convergence, resultant accuracy, and solving timeliness. The results show that isogeometric analysis has higher accuracy and efficiency than the finite element method in calculating the contact stress and contact force. The error of the IGA is only 8.8% for 10 × 10 elements in contact, while the error of the finite element method reaches about 40%. The method can improve the contact simulation accuracy of the cycloid drive and provides a reference for the design evaluation of RV reducer.

Fast Reconstruction Model of the Ship Hull NURBS Surface with Uniform Continuity for Calculating the Hydrostatic Elements

The fast reconstruction of the ship hull nonuniform rational B-spline (NURBS) surface with uniform continuity is essential for calculating hydrostatic elements such as waterplane area and molded volume in real time. Thus, this study proposes a fast reconstruction model with uniform continuity to solve the problem of uniform continuity and splicing in the separate model of hull bow and stern surfaces. The proposed model includes the NURBS curve generation (UCG) algorithm with uniform continuity and the hybrid NURBS surface generation (HSG) algorithm. The UCG algorithm initially fits the feature points using the global interpolation algorithm and then precisely constructs straight-line segments in the curve using the improved flattening algorithm. In comparison, the HSG algorithm adaptively selects the surface knot vectors according to the parameters of the section curves. In this study, the profile of discontinuous compartments is uniformly expressed, effectively avoiding various articulation problems in separation modeling. The results of comparative experiments show that the NURBS surface generated using the HSG algorithm can accurately express the characteristics of various parts of the hull with uniform continuity, and the calculation speed of the proposed model can be increased by up to 8.314% compared with the existing best-performing algorithms. Thus, the proposed model is effective and can improve computational efficiency to a certain extent. The NURBS surfaces generated by the proposed model can be further applied to calculating the hydrostatic elements of hulls and compartments.

An integrated design approach for simultaneous shape and topology optimization of shell structures

In this work, a novel design approach is developed to carry out shape and topology optimization of shell structures simultaneously. This approach effectively integrates the IsoGeometric Analysis (IGA) method, the Adaptive Bubble Method (ABM) and the Finite Cell Method (FCM) to make full use of their respective advantages in shell structure optimization. To be specific, the IGA method using Non-Uniform Rational B-Splines (NURBS) elements is adopted to analyze the shell structure and realize shape optimization with merits of exact shell surface representation geometrically and rotation-free shell formulation physically; the ABM is utilized to insert deformable holes adaptively in the parametric domain of the NURBS surface for topology optimization of the shell structure; the FCM is employed to facilitate the optimization process by simplifying significantly the discrimination and the numerical integration processes of trimmed elements involved in fixed-mesh analysis of the holed shell structure. Based on this integrated design approach of IGA/ABM/FCM, shape optimization can be implemented by directly optimizing the control-point coordinates of the NURBS surface in the physical space, while topology optimization could be carried out simultaneously in the parametric domain as easily as for 2D planar structures. Finally, representative examples are investigated to demonstrate the effectiveness and advantages of the proposed design approach.

Isogeometric degenerated shell formulation for post-buckling analysis of composite variable-stiffness shells

Variable-stiffness (VS) laminated shells with curved fibers have gradually become a promising lightweight structural concept in the aerospace field. However, as the complexity of the shell geometry and the fiber angle increases, the traditional finite element method (FEM) requires mesh refinement to guarantee accurate discretization, which significantly increases the computational cost, especially in the post-buckling analysis with multiple solving requirements. In this study, a non-uniform rational B-Splines (NURBS)-based degenerated solid shell formulation in the total Lagrangian framework is proposed for the post-buckling analysis of VS laminated shells. Since the nonlinear function related to the rotational degrees of freedom (DOFs) is additionally introduced into the displacement field expression, the restriction on the magnitude of the nodal rotations is effectively eliminated. In addition, the paper adopts the NURBS surface reconstruction technology based on the point-line-surface features to build B-spline CAD models from the point cloud with geometric imperfect features. Then, an integrated modeling-analysis framework for predicting the post-buckling behavior of VS structures with geometric imperfections is proposed. A series of numerical examples systematically demonstrate that the framework can effectively conduct the post-buckling analysis of shell structures with geometric imperfections. Meanwhile, the accuracy and robustness of the solutions by IGA are proved to be better than FEM.

Effect of crown stiffness and prosthetic screw absence on the stress distribution in implant-supported restoration: A 3D finite element analysis.

This in-silico investigation evaluated the mechanical impact of Morse tape implant-abutment interface and retention system (with and without screw) and restorative materials (composite block and monolithic zirconia) by means of a three-dimensional finite element analysis (3D-FEA). Four 3D models were designed for the lower first molar. A dental implant (4.5 × 10 mm B&B Dental Implant Company) was digitized (micro CT) and exported to computer-aided design (CAD) software. Non-uniform rational B-spline surfaces were reconstructed, generating a 3D volumetric model. Four different models were generated with the same Morse-type connection, but with a different locking system (with and without active screw) and a different crown material made of composite block and zirconia. The D2 bone type, which contains cortical and trabecular tissues, was designed using data from the database. The implants were juxtaposed inside the model after Boolean subtraction. Implant placement depth was simulated for the implant model precisely at crestal bone level. Each acquired model was then imported into the finite element analysis (FEA) software as STEP files. The Von Mises equivalent strains were calculated for the peri-implant bone and the Von Mises stress for the prosthetic structures. The highest strain values in bone tissue occurred in the peri-implant bone interface and were comparable in the four implant models (8.2918e-004-8.6622e-004 mm/mm). The stress peak in the zirconia crown (64.4 MPa) was higher than in the composite crown (52.2 MPa) regardless of the presence of the prosthetic screw. The abutment showed the lowest stress peaks (99.71-92.28 MPa) when the screw was present (126.63-114.25 MPa). Based on this linear analysis, it is suggested that the absence of prosthetic screw increases the stress inside the abutment and implant, without effect on the crown and around the bone tissue. Stiffer crowns concentrate more stress on its structure, reducing the amount of stress on the abutment.

Isogeometric-based mapping modeling and buckling analysis of stiffened panels

Modeling and analysis of stiffened panels are two key technologies in the design of aerospace thin-walled structures. For the stiffened panels with complex geometry, classical finite element analysis (FEA) and conventional isogeometric analysis (IGA) based on explicit geometry usually require time-consuming and labor-intensive geometric processing, and additional coupling matrices to be ready for analysis. In this study, a new method for modeling and buckling analysis of stiffened panels is proposed, which provides a more efficient and simpler way. During the modeling process, the stiffeners are treated as curves on surfaces, which is not explicitly defined using the control-point-based representation of curves, but implicitly defined using parameter curves in the parametric space of the surface. Mapping modeling provides more accurate geometric description and transfer the complex modeling problems (three-dimensional space) of stiffeners on free-form surface into simple modeling problems in the regular parametric space (two-dimensional space). During the buckling analysis process, a new mapped stiffener element based on mapping modeling is proposed, which can model the section of the eccentric stiffener without changing the geometry. The precise normal information of the Non-Uniform Rational B-Splines (NURBS) surface can ensure that the stiffeners are perpendicular to the skin. In addition, the coupling of the stiffener and the skin is automatic, without any additional coupling matrix. This buckling analysis framework realizes the complete integration of modeling and analysis. Furthermore, for the stiffened panels with cutouts, the trimmed surface analysis (TSA) method is extended to be used for the numerical integration of the trimmed stiffeners, which means that no additional geometric process is required. Finally, four numerical examples of different types of stiffened panels are constructed, involving metal, trimmed surface, classical grid-stiffener, free-form surface, variable-stiffness composites, and curvilinear grid-stiffener. Several numerical examples of static and buckling analysis of stiffened panels with high fidelity demonstrate the effectiveness of the proposed framework.

Ship hull surface reconstruction from scattered points cloud using an RBF neural network mapping technology

The ship hull surface reconstruction based on three-dimensional scattered points cloud is vital to Computer-Aided Design modeling of ship reverse engineering. There are several problems with traditional methods: the preprocessing of a large-scale scattered points cloud is complicated, and the result is susceptible to noise. Because of the “black box” characteristic, the surface reconstruction based on neural networks cannot apply to the practical engineering. To address these issues, combining the Radial Basis Function neural network and Non-Uniform Rational B-Spline interpolation algorithm, a new ship hull surface reconstruction method was proposed, which can satisfy the standard geometric model description in Computer-Aided Design system. Firstly, the Radial Basis Function neural network was used to pre-fit the three-dimensional scattered points cloud. Then the mathematical model of the surface was mapped. Finally, based on the bilinear interpolation algorithm, the mathematical model was transformed to a Non-Uniform Rational B-Spline surface to apply to the ship practical engineering. In addition, by comparing our method with the traditional method, the advantages of our method in surface reconstruction quality and surface repair ability were verified, which provided a new way for the application of Radial Basis Function neural network in ship reverse engineering.

A model for gastrointestinal tract motility in a 4D imaging phantom of human anatomy.

Gastrointestinal (GI) tract motility is one of the main sources for intra/inter-fraction variability and uncertainty in radiation therapy for abdominal targets. Models for GI motility can improve the assessment of delivered dose and contribute to the development, testing, and validation of deformable image registration (DIR) and dose-accumulation algorithms. To implement GI tract motion in the 4D extended cardiac-torso (XCAT) digital phantom of human anatomy. Motility modes that exhibit large amplitude changes in the diameter of the GI tract and may persist over timescales comparable to online adaptive planning and radiotherapy delivery were identified based on literature research. Search criteria included amplitude changes larger than planning risk volume expansions and durations of the order of tens of minutes. The following modes were identified: peristalsis, rhythmic segmentation, high amplitude propagating contractions (HAPCs), and tonic contractions. Peristalsis and rhythmic segmentations were modeled by traveling and standing sinusoidal waves. HAPCs and tonic contractions were modeled by traveling and stationary Gaussian waves. Wave dispersion in the temporal and spatial domain was implemented by linear, exponential, and inverse power law functions. Modeling functions were applied to the control points of the nonuniform rational B-spline surfaces defined in the reference XCAT library. GI motility was combined with the cardiac and respiratory motions available in the standard 4D-XCAT phantom. Default model parameters were estimated based on the analysis of cine MRI acquisitions in 10 patients treated in a 1.5T MR-linac. We demonstrate the ability to generate realistic 4D multimodal images that simulate GI motility combined with respiratory and cardiac motion. All modes of motility, except tonic contractions, were observed in the analysis of our cine MRI acquisitions. Peristalsis was the most common. Default parameters estimated from cine MRI were used as initial values for simulation experiments. It is shown that in patients undergoing stereotactic body radiotherapy for abdominal targets, the effects of GI motility can be comparable or larger than the effects of respiratory motion. The digital phantom provides realistic models to aid in medical imaging and radiation therapy research. The addition of GI motility will further contribute to the development, testing, and validation of DIR and dose accumulation algorithms for MR-guided radiotherapy.

Calibration method of laser scanning measurement system for large-scale component using a free-form surface reconstruction model

For large dimensional scanning measurement, the calibration method is limited by the standard part with the scale in millimeters or centermeters. In this paper, a novel calibration method suitable for large-scale components in meters has been developed based on the free-form surface reconstruction model. The Non-Uniform Rational B-Splines (NURBS) surface is approximated from the discrete measurement points obtained by high-precision detection using the least-squares method. The deviation between the measurement points and reference model is defined by statistical results and represented by the standard deviation of the distances from the measurement points to the corresponding closest points on the surface. Furthermore, the uncertainty components introduced in the procedure of coordinate registration and surface approximation are considered to estimate the combined uncertainty of a reconstructed surface. Finally, a scanning measurement device can be calibrated by the reconstruction model consisting of a NURBS surface expression and its uncertainty prediction. To validate the proposed calibration method, the surface of a standard part with a 4-meter diameter is detected to acquire a set of measurement points and subsequently reconstructed to a parametric surface, the uncertainty of which is analyzed considering errors introduced by both device accuracy and data processing. For the reconstruction model with degree of 4, the result shows that the calibration accuracy is superior to 30 μm with a tolerance error of less than 12 μm.

Parallel ray tracing through freeform lenses with NURBS surfaces

We implement Monte Carlo-based parallel ray tracing to achieve quick irradiance evaluation for freeform lenses with non-uniform rational B-splines (NURBS) surfaces. We employ the inverse transform sampling method to sample rays uniformly from the Lambertian light source and adopt the analytical form of the B-spline basis function to achieve fast surface interpolation. When performing parallel calculations for the intersections between the rays and the NURBS surfaces, we propose a parameter transformation method to avoid the parameters escaping from the defined range in the iteration process. Simulation results of two complex picture-generating freeform lenses show that our method is fast and effective.

Uncertainty EM Scattering Prediction for Inhomogeneous Dielectric Bodies of Revolution

A perturbed volumetric integral equation is fabricated to predict the uncertain scattering characteristics for the in-homogeneous dielectric bodies of revolution (BoRs) targets. The transverse surface of the body of revolution is discretized in terms of the rectangular and triangular basis functions. In this way, all the unknowns are assigned to the electric flux density and the governing equation can be solved mode by mode. Both the geometrical and dielectric uncertainties are taken into consideration for EM scattering characteristics. On one hand, the geometrical shape uncertainty is described with the nonuniform rational B-spline surface by using several independent random variables. On the other hand, the dielectric uncertainty is represented by the perturbed permittivity. Thus, the volume integral equation is reconstructed by the perturbed electric flux density. Both the mean value and variance of the radar cross Section can be derived to account for the geometrical and dielectric uncertainties. Numerical results are given to show that the calculation efficiency of the proposed method is higher than that of the traditional Monte Carlo method.

3D Reality-Based Survey and Retopology for Structural Analysis of Cultural Heritage

Cultural heritage's structural changes and damages can influence the mechanical behaviour of artefacts and buildings. The use of finite element methods (FEM) for mechanical analysis is largely used in modelling stress behaviour. The workflow involves the use of CAD 3D models and the use of non-uniform rational B-spline (NURBS) surfaces. For cultural heritage objects, altered by the time elapsed since their creation, the representation created with the CAD model may introduce an extreme level of approximation, leading to wrong simulation results. The focus of this work is to present an alternative method intending to generate the most accurate 3D representation of a real artefact from highly accurate 3D reality-based models, simplifying the original models to make them suitable for finite element analysis (FEA) software. The approach proposed, and tested on three different case studies, was based on the intelligent use of retopology procedures to create a simplified model to be converted to a mathematical one made by NURBS surfaces, which is also suitable for being processed by volumetric meshes typically embedded in standard FEM packages. This allowed us to obtain FEA results that were closer to the actual mechanical behaviour of the analysed heritage asset.

Research on the Lightweight Design of an Aircraft Support Based on Lattice-Filled Structures

This work studied the lightweight design of an aircraft support based on lattice-filled structures. Different from the traditional design process of lattice-filled structures, this work combined several approaches, including topology optimization, homogenization analysis, and Non-Uniform Rational B-splines (NURBS) surface modeling, to reduce the structural weight more effectively. The theories and implementations involved in the design process are introduced in this work. The new lattice-filled design of the aircraft support component reduced the weight by 40% compared with the original value, and its additive manufacturability was verified. Finally, the structural responses of the lattice-filled design from both a detailed model and homogenization model were determined and compared, considering both the static responses and dynamic characteristics. The results revealed that the homogenization method efficiently and accurately obtained the structural displacements and natural frequencies of the complex lattice-filled design. This indicates that the homogenization method can effectively reduce the calculation burden of the design process of lattice-filled structures, which opens a new channel for the structural optimizations of lattice-filled structures.

Fast NURBS-based parametric modeling of human calves with high-accuracy for personalized design of graduated compression stockings

Background and objectivesAccurate human body models are increasingly demanded by high-quality human-centered ergonomic applications, especially the design and manufacturing of compressive functional apparels. However, existing parametric models in related works are not capable to accurately describe detailed local shape features of human. MethodsIn this work, a high-accuracy parametric modeling approach for human limb was proposed. 3D Scans of human calves were studied. Key data points of the scanned human calves were identified according to human anatomy, forming a quasi-triangular mesh of feature points. Then, non-uniform rational B-splines (NURBS) method was implemented. Control points were calculated from the key data points, with which the human calf shapes can be reconstructed by the smooth NURBS surface, giving rise to a new parametric model of human calves. Error between the scanned and reconstructed calf shapes were analyzed to verify the effectiveness of this model. ResultsError analysis showed that, this proposed method delivers a high-efficiency and high-accuracy parametric shape modeling approach with averaged error observed as only 0.37% for all the 260 subjects, much less compared to previous relative works (around 5%). For tentative application, customized medical compression stockings were designed based on this model and proved as valid to exert desired gradient compression on the according calf mannequin. ConclusionsBy introducing the non-uniform rational B-splines method, a parametric model capable of characterizing human limbs with high-accuracy was proposed. Using very small amount of data, this model is expected to highly facilitate remote customized design and provide 3D shape references for design of compressive garments. Moreover, the proposed methods can inspire developments of other mixed modeling methods for high-accuracy applications.

Geodesic Path Computation on Connected Trimmed Surfaces

Creeping waves considerably contribute to the total high-frequency electromagnetic field in the shadow region. These waves propagate along surface geodesic paths. To trace them, geodesic differential equations have to be solved, which require special care in the case of connected trimmed surfaces. In this article, an algorithm is proposed to compute geodesic path transition across connected trimmed surfaces. The formulation includes efficient boundary recognition as well as accurately initializing geodesic equations on the next trimmed surface. For boundary recognition, inscribed or peripheral polygons are constructed to estimate the boundary and to effectively initialize numerical methods of quadratic convergence to determine the exact exit point. To get the initial point and initial direction on the parametric domain of the next trimmed surface, two optimization problems are efficiently solved. The proposed algorithm is applicable even if first/second-order surface discontinuity is encountered during the transition. The proposed method is validated on canonical surfaces and nonuniform rational B-spline (NURBS) surfaces for various trimming curves. Accurate computation of geodesic paths is observed. Furthermore, the far-field radiation pattern of a monopole antenna mounted on a sphere, a cylinder, and an ellipsoid is obtained and confirmed by numerical electromagnetic software.