NURBS Research Articles

Isogeometric collocation method to simulate phase-field crystal model

PurposeThe purpose of this study is to develop a new numerical algorithm to simulate the phase-field model.Design/methodology/approachFirst, the derivative of the temporal direction is discretized by a second-order linearized finite difference scheme where it conserves the energy stability of the mathematical model. Then, the isogeometric collocation (IGC) method is used to approximate the derivative of spacial direction. The IGC procedure can be applied on irregular physical domains. The IGC method is constructed based upon the nonuniform rational B-splines (NURBS). Each curve and surface can be approximated by the NURBS. Also, a map will be defined to project the physical domain to a simple computational domain. In this procedure, the partial derivatives will be transformed to the new domain by the Jacobian and Hessian matrices. According to the mentioned procedure, the first- and second-order differential matrices are built. Furthermore, the pseudo-spectral algorithm is used to derive the first- and second-order nodal differential matrices. In the end, the Greville Abscissae points are used to the collocation method.FindingsIn the numerical experiments, the efficiency and accuracy of the proposed method are assessed through two examples, demonstrating its performance on both rectangular and nonrectangular domains.Originality/valueThis research work introduces the IGC method as a simulation technique for the phase-field crystal model.

Two-dimensional electromagnetic scattering analysis based on the boundary element method

An effective formula for the shape-sensitivity analysis of electromagnetic scattering is presented in this paper. First, based on the boundary element method, a new electromagnetic scattering formula is derived by combining the traditional electromagnetic scattering formula with the non-uniform rational B-spline (NURBS) curve, and the geometric model is represented by NURBS, which ensures the geometric accuracy, avoids the heavy grid division in the optimization process, and realizes the fast calculation of high-fidelity numerical solutions. Second, by deducing the sensitivity variables, the electromagnetic scattering equation of shape optimization is obtained, which can provide reliable data references for shape optimization. Finally, the effectiveness and accuracy of the algorithm are demonstrated by an example, and the sensitivity data of some examples are given.

Geometry-independent spline finite element method to analyze two-dimensional heat conduction and elasticity problems

Traditional isogeometric analysis involves redundant geometric calculations and is more complex than the finite element method when imposing boundary conditions. To address this, a geometry-independent spline finite element method is proposed, which integrates the geometric construction techniques of isogeometric analysis and the element construction techniques of finite element method. Firstly, the geometrical precision element is constructed by using non-uniform rational B-spline and B-splines to describe the geometry and the physical field, where a transformation matrix is introduced to construct shape functions with interpolation properties. Secondly, calculation formats for two-dimensional heat conduction and elasticity problems are derived by parameter mapping. Finally, the feasibility and accuracy of the method are verified through several numerical examples. Unlike in isogeometric analysis, in the proposed method, geometry and physics are separated, and the shape functions have interpolation properties, which reduce redundant geometry calculations and allow the boundary conditions to be applied directly to the nodes. The numerical results indicate that the method has a higher convergence rate compared to the original isogeometric analysis.

Density-Based Isogeometric Topology Optimization of Shell Structures

Shell structures with high stiffness-to-weight ratios are desirable in various engineering applications. Topology optimization serves as a popular and effective tool for generating optimal shell structures. The solid isotropic material with penalization (SIMP) method is often chosen because of its simplicity and convenience. However, SIMP method is typically integrated with conventional Finite Element Analysis (FEA) which has limitations in computational accuracy. Achieving high accuracy with FEA necessitates a substantial number of elements, leading to computational burdens. In addition, the discrete representation of the material distribution function may result in rough boundaries. Owing to these limitations, this paper proposes an Isogeometric Analysis (IGA) based SIMP method for optimizing the topology of shell structures based on Reissner–Mindlin theory. This method uses Non-Uniform Rational B-Splines (NURBS) to represent both the shell structure and the material distribution function with the same basis functions, allowing for higher accuracy and smoother boundaries. The optimization model takes compliance as the objective function with a volume fraction constraint and the coefficients of the density function as design variables, resulting in an optimized shell structure defined by the material distribution function. To obtain fairing boundaries of the holes in the optimized shell structure, further process is conducted by fitting the boundaries with fair B-spline curves automatically. Furthermore, the proposed IGA-SIMP framework is applied to generate porous shell structures by imposing different local volume fraction constraints. Numerical examples are provided to demonstrate the feasibility and efficiency of the IGA-SIMP method, showing that it outperforms the FEA-SIMP method and produces smoother boundaries.

Comparative post-therapeutic dosimetry between 2D planar-based and hybrid-based methods for personalized Lu-177 treatment.

This study aims to compare the calculated absorbed dose in target organs and tumors obtained using the different imaging protocols and the calculation methodologies implemented by HERMES HybridViewer dosimetry software for 177Lu-PSMA I&T and 177Lu-DOTATATE therapy. Multiple time-point whole-body planar images and one SPECT/CT image were acquired from 18 patients including 177Lu-PSMA I&T (13 patients) and 177Lu-DOTATATE treatment (5 patients) after administration of 3.80-8.58GBq injected activity. The regions of interest were drawn in the whole body, kidneys, liver, urinary bladder, salivary glands, and tumors to determine the time-integrated activity (TIA) in source organs. Absorbed doses in target organs were calculated according to the Medical Internal Radiation Dose (MIRD) scheme using the HERMES HybridViewer dosimetry integrated with OLINDA/EXM V.2.1 that utilizes the non-uniform rational B-splines (NURBS) for computational digital phantom. The planar-based dosimetry showed a higher dose per injected activity compared to the hybrid-based dosimetry, primarily due to organ overlap. The highest difference in absorbed dose between the imaging scenarios was observed in the spleen with a variation of up to 51.6%, while the difference for other target organs and tumors was less than 40%. The dosimetry calculation derived from the 2D planar-based method consistently demonstrates a significantly higher absorbed dose in organs and tumors compared with the hybrid-based method. However, the hybrid method outperforms the planar method in terms of tumor visualization and overlap-free organ delineation.

A novel tool path planning method for 5-axis single-point diamond turning

5-Axis single-point diamond turning adds two linked axes based on 3-axis single-point diamond turning, which can smoothly and continuously process complex surfaces. Compared to 3-axis turning, tool path planning for 5-axis single-point diamond turning is a complex decision-making process, which is a novel problem in the existing studies and a difficulty in multi-axis turning. In this study, a novel path planning method for 5-axis single-point diamond turning is proposed. Firstly, the mapping relation from scallop height to the distribution of cutter contact (CC) points is established in the non-uniform rational B-spline (NURBS) parameter space, thereby the planning of CC points is quickly achieved under fewer constraints. Furthermore, the local interference and global interference generation mechanism of 5-axis single-point diamond turning are revealed by geometrical analysis, combining diamond tool parameters and tool shank parameters. By transforming the motion of the cutter axis posture in 3D space into swings in two planes, the posture accessibility diagram of the cutter axis is efficiently constructed. Finally, the feasibility and effectiveness of the tool path planning method are verified by 5-axis single-point diamond turning experiment. And the contour accuracy is improved through the quadratic planning of the CC points (the contour accuracy after compensation is 3.2 μm compared to normal one 9.6 μm). This study extends the traditional 3-axis turning to 5-axis turning, providing a new perspective for the machining of complex surfaces.

On the isogeometric analysis of vibration and buckling of bioinspired composite plates using inverse hyperbolic shear deformation theory

Inspired by the remarkable energy absorption and damage tolerance capabilities observed in mantis shrimp crustaceans, this study delves into the intricate realm of biological composites with helicoidal structures. However, the effect of the relative orientation of fibers within the helicoidal structure matrix on the buckling and vibration characteristics still warrants comprehensive investigation for the better design of the bioinspired structures. For the first time, a Non-Uniform Rational B-Splines (NURBS)-based isogeometric analysis (IGA) framework termed NURBio is proposed to investigate the vibration and buckling characteristics of bioinspired laminated composite structures. The framework employs NURBS basis functions for exact complex geometric representation and field approximation and offers superior computational accuracy and efficiency as compared to conventional finite element method (FEM). The present analysis is underpinned by an inverse hyperbolic shear deformation theory (IHSDT) capable of accurately capturing the interlaminar stress distribution. This research investigates the performance of various bioinspired layup configurations encompassing recursive, exponential, semicircular, and Fibonacci helicoidal designs and compares them with those of unidirectional, and cross-ply laminates. A series of numerical examples on the buckling and vibration response of bioinspired composite plates are presented to demonstrate the versatility and efficacy of the proposed isogeometric framework. The influence of different layups, loading and boundary conditions, and geometric properties on the response of different bioinspired plate structures are meticulously examined. The study offers valuable insights into the design and optimization of high-performance bioinspired structures.

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.

Inverse Geometric Reconstruction Based on MW-NURBS Curves

Currently, rational curves such as the Non-Uniform Rational B-Spline (NURBS) play a significant role in both shape representation and shape reconstruction. NURBS weights are often real in nature and are referred to as challenging to assign, with the exception of conics. ‘Matrix Weighted Rational Curves’ are the expanded form of rational curves that result from replacing these real weights with matrices, or matrix weights. The only difference between these curves and conventional curves is the geometric definition of the matrix weights. In this paper, MW-NURBS curves are used to reconstruct space curves from their stereo perspectives. In particular, MW-NURBS fitting is carried out in stereo views, and the weight matrices for the MW-NURBS curves are produced using the normal vectors provided at the control points. Instead of needing to solve a complicated system, the MW-NURBS model can reconstruct curves by choosing control points and control normals from the input data. The efficacy of the proposed strategy is verified by using many examples based on both synthetic and real images. The various error types are compared to those of conventional methods like point-based and NURBS-based approaches. The results demonstrate that the errors acquired from the proposed approach are much fewer than those obtained from the point-based method and the NURBS-based method.

DE3D-NURBS: A differential evolution-based 3D path-planner integrating kinematic constraints and obstacle avoidance

This work presents a novel path planner, DE3D-NURBS, for 3D path planning considering the maximum and minimum climb/dive angle and the maximum curvature imposed by a robot, such as an autonomous fixed-wing aircraft. To deal with this constrained problem, we propose a Differential Evolution (DE) based algorithm, assuming the partitioning of feasible and infeasible individuals and a novel mutation operator. This solution can provide a short differentiable smooth path, directly parameterized by a kth-order Non-Uniform Rational B-Spline (NURBS) curve, a generalization of B-spline curves. This representation provides more flexibility to satisfy essential kinematic constraints and to take into account obstacles and topography. This path can guide an Unmanned Aerial Vehicle (UAV) through a set of desired robot configurations (position and orientation) in 3D space. The simulation results provide a more comprehensive insight into the capabilities of the proposed planner to generate feasible paths, considering simple scenarios in a free obstacle environment and also complex scenarios. Such scenarios include topographic environments, narrow passages such as a tunnel, and a solution for closed-loop missions in a modular approach. We also provide a comparative benchmark with other nature-inspired algorithms. These results serve as a baseline for evaluating the effectiveness of the proposed DE algorithm in complex and challenging case studies.

Modelowanie cyfrowe w celu przewidywania zagrożenia osuwania się gruntu na wybranym obszarze; Przypadek testowy Ponzano (Włochy)

Creating a digital model is one of the aims of the geotechnical engineers, to predict the land sliding hazard which occur in different regions in the world. In February 2017, an extensive landslide occurred in the hamlet of Ponzano as a sloped area in the Abruzzo region in Italy. In this regard, predicting the land sliding hazard is one of the important issues to prevent hazard to the civilizations. In this project we created a model in the 3D dimension through the Plaxis 3D numerical solution software from the data based of the region Abruzzo in 2007 to evaluate the land sliding hazard before happened and then compare the results with the data from the drone data surveyed recently in the 2022. In this regard, the data from the Abruzzo resources from 2007 imported into the QGIS as the open-source cross-platform software to analysis the geospatial data and then imported into the Recap software to work on the point cloud data and then imported into the Civil 3D software to create a solid surface from the TIN surface and finally since the solid surface contains a large number of irrelative details, they were imported into the Rhinoceros software to create a NURBS surface to be smoothed for better performance in the analysis. The NURBS surface imported into the Plaxis and all of the geometry and geotechnical engineering parameters by considering the investigated geotechnical survey that was conducted in parallel in the area, defined for the model. The “Hardening Soil” model considered for the 1st layer as the “clay and lime” and the “Hoek-Brown” model defined for the 2nd layer as the “marl-flysch”. A fine mesh elements distribution assessed also for the model. The phases defined as the “gravity” to define the unit weight of the soil layers, the “plastic” phase to calculate the instant deformations and the “consolidation” to analyze the plastic deformations in the sloped area of the model. In parallel, the drone data achieved in 2022, were imported into the CloudCompare as the 3D point cloud processing software and different methods such as the “segment”, “statistical outlier Filter”, “CSF filter”, “noise filter”, “cross section” etc. were performed to clean the data and then imported into the Recap software to work on data and then imported into the Civil 3D software to create solid surface of the current data after the land sliding. In this regard, to evaluate the displacement occurred from the year 2007 toward the 2022, a TIN volume surface as the colored map created through the Civil 3D software to show the displacements in the z direction and all of the results were compared with the Plaxis 3D numerical solution software. The results showed that the colored map with the displacement in the positive and negative direction of the z is the same of the analyzed model and the values match each other’s and we created a digital model of the selected area to predict the land sliding hazard in the region in the following.

Preform design to reduce forging load and grain size simultaneously in disk forging of IN718

In this study, we propose a new methodology for designing preform shapes for turbofan blisk and disk forgings of IN718 superalloy using a CNN (Convolutional Neural Network). The objective function of the preform design model is set to minimize grain size and reduce forging load. Various preform shape candidates were generated using NURBS (Non-Uniform Rational B-Splines) curves, defined by points on the target shape outline, and the grain size was calculated through FE (Finite Element) analysis based on the JMAK (Johnson-Mehl-Avrami-Kolmogorov) model to train the CNN model. The preform shapes derived from the proposed load and grain size-aware CNN model were compared and validated against those from previous studies regarding grain size and load reduction. An objective function f, which is capable of quantitatively expressing the load and grain size improvement, was developed for comparison and validation. The results showed an improvement of 19.64% for the blisk and 18.96% for the disk compared to previous studies. This research is expected to significantly contribute to the forging of aerospace components.

Sensitivity analysis of non-uniform rational B-splines–based finite element/boundary element coupling in structural-acoustic design

For the purpose of modeling the acoustic fluid-structure interaction using direct differentiation method and conducting a structural-acoustic sensitivity analysis, a coupling approach based on the finite element method and the fast multipole boundary element method is suggested. Non-uniform rational B-splines isogeometric analysis bypasses the difficult volume parameterization procedure in the isogeometric finite element method and the time-consuming meshing process in classical finite element/boundary element method, allowing numerical analysis on computer-aided design models to be completed directly. The finite element/fast multipole boundary element method based on non-uniform rational B-splines isogeometric analysis enables the numerical prediction of the effects of arbitrarily formed vibrating structures on the sound field. Several numerical examples are shown to demonstrate the usefulness and efficiency of the proposed method.

DIFFERENCE IN SURFACE FITTING WITH STANDARD AND MATRIX TOPOLOGY

This paper presents the impact of the surface topology of the scanned 3D object on parametric fitting. Whether it is a simple NURBS (Non-uniform rational B-spline) or a more complex hierarchical spline version, it is important to apply the fitting procedure. Here we describe the differences between fitting a surface with a given topology as a result of a 3D scanning system and a matrix topology of the surface, where the original surface is replaced by the result of a preset number of sections of the original geometry. We use the matrix and the free-form distribution. The former is more stable with respect to the distribution of the control point, the latter is numerically more suitable. In the future, we plan to adopt the free-form distribution to utilize the advantages of both distributions.

Shape optimization of autonomous underwater helicopters based on different parameter curves and various optimization algorithms

This study aimed to compare the non-uniform rational B-spline (NURBS) and Bezier curves to design the shape of an autonomous underwater helicopter (AUH). Currently, there is no established method for AUH shape design. The optimal solutions obtained from both curves were compared to evaluate their performance. A parametric model was constructed to determine the design variables of the two curves. Computational fluid dynamics was used to solve the resistance values of different shapes. Sample points were generated using the design of the experimental approach, and an approximate model was established using the radial basis function neural network method. Based on the combinatorial optimization strategy, three intelligent optimization algorithms were used for the global search. Then, the resulting solution was used as the starting point for a local search to obtain the global optimal solution. The resistance optimal solutions achieved by employing the NURBS and Bezier curves were 25.2% and 24.1% smaller than the initial values, respectively. These findings indicate that, within the design parameters of this study, the shape design approach using the NURBS curve yields superior performance for the AUH.

Optimization of machining path for integral impeller side milling based on SA-PSO fusion algorithm in CNC machine tools

The five axis linkage Computer Numerical Control machine tool for integral impeller can achieve blade machining through side milling, which is of great significance for improving the machining accuracy, production efficiency, and long-term stability of integral impeller blades. This study is based on non-uniform rational B-spline curves and aims to reduce the surface over cutting or under cutting of integral turbine blades. The path planning of non deployable ruled surfaces was analyzed in depth through side milling, and the path planning model of the side milling cutter axis was solved through a fusion algorithm of simulated annealing algorithm and particle swarm optimization algorithm, in order to find the optimal path through iterative process. As the number of iterations increased, the error values of particle swarm optimization algorithm and simulated annealing particle swarm optimization fusion algorithm gradually decreased, with convergence times of about 7 and 6, respectively. The stable error value of the fusion algorithm was 0.253, which is 30.45% lower than that of the particle swarm optimization algorithm. The optimal number of iterations for solving the model using particle swarm optimization algorithm and fusion algorithm was the 7th, with range values of 0.0213 and 0.0165 mm, respectively. The tool axis trajectory surface optimized by the fusion algorithm was closer to the tool axis motion state compared to the initial tool axis trajectory surface. The range of the sum of mean squared deviations for single and global cutting was 0.0011–0.0198 and 0.046–0.0341, but the overall error value was relatively small. This study effectively reduces the envelope error of machining tools and improves machining accuracy, thereby solving the principle error of non expandable ruled surfaces in the motion trajectory of the blade axis of the integral turbine. This provides new research ideas for the intelligent development of Computer Numerical Control machining technology.

Generalized particle domain method: An extension of material point method generates particles from the CAD files

SummaryIn this paper, a generalized particle domain method (GPDM) is proposed and developed within the framework of the convected particle domain interpolation method. This new method generates particles directly from non‐uniform rational B‐spline (NURBS)‐based CAD file of a continuum body. The particle domain corresponds to a NURBS element even for trimmed elements of solids with complex geometries. The shape functions and the gradient of shape functions are evaluated using NURBS basis functions to map material properties between particles and grid nodes. It approves that this proposed GPDM can track the domain of particles accurately and avoid the issue of cell‐crossing instability. Several numerical examples are presented to demonstrate the high performance of this proposed new particle domain method. It is shown that the results obtained using the proposed GPDM are consistent with the experimental data reported in the literature. Further development of the generalized particle domain method can provide a link to the material point method and isogeometric analysis.

Isogeometric material optimization for shape control of bi-directional functionally graded plates with piezoelectric layers

This paper proposes an effective numerical method for shape control of bi-directional functionally graded plates (2D-FGPs) with piezoelectric layers. Isogeometric analysis (IGA) based on non-uniform rational B-splines (NURBS) related to third-order shear deformation theory (TSDT) is employed for the static analysis of the 2D-FGPs with piezoelectric layers. The B-spline basis functions are utilized to represent the distribution of the ceramic volume fractions, where the control points placed along the plane corresponding to the ceramic volume fraction and the applied voltages are taken as the design variables. In addition, an improved moth flame optimization algorithm is utilized to solve the optimization problem of minimizing the static shape error, which effectively balances the exploratory and exploitative capabilities of the algorithm. Various numerical examples of square, skew, and dart-shaped 2D-FGPs are analyzed to validate the proposed method and demonstrated the superior mechanical performance of 2D-FGPs over 1D-FGPs.

A Multi-UAV cooperative mission planning method based on SA-WOA algorithm for three-dimensional space atmospheric environment detection

AbstractIn the application of rotorcraft atmospheric environment detection, to reflect the distribution of atmospheric pollutants more realistically and completely, the sampling points must be spread throughout the entire three-dimensional space, and the cooperation of multiple unmanned aerial vehicles (multi-UAVs) can ensure real-time performance and increase operational efficiency. In view of the problem of coordinated detection by multi-UAVs, the region division and global coverage path planning of the stereo space to be detected are studied. A whale optimization algorithm based on the simulated annealing-whale optimization algorithm (SA-WOA) is proposed, which introduces adaptive weights with the Levy flight mechanism, improves the metropolis criterion, and introduces an adaptive tempering mechanism in the SA stage. Path smoothing is subsequently performed with the help of nonuniform rational B-spline (NURBS) curves. The comparison of algorithms using the eil76 dataset shows that the path length planned by the SA-WOA algorithm in this paper is 10.15% shorter than that of the WOA algorithm, 13.25% shorter than the SA planning result, and only 0.95% difference from the optimal path length in the dataset. From the perspective of planning time, its speed is similar to WOA, with a relative speed increase of 27.15% compared to SA, proving that the algorithm proposed in this paper has good planning performance. A hardware system platform is designed and built, and environmental gas measurement experiments were conducted. The experimental results indicate that the multi-UAV collaborative environment detection task planning method proposed in this paper has certain practical value in the field of atmospheric environment detection.

Small-sample linear profile error uncertainty assessment based on grey system

The uncertainty assessment of the profile error of the cam profile, as defined in the national standard method, is difficult to carry out under conditions of small sample size and absence of probability distribution assumptions. This paper proposes a small-sample assessment model for the uncertainty of the profile error based on grey system. Firstly, the coordinate transformation is conducted using Vector Alignment Method to reduce systematic errors, and the non-uniform rational B-splines curve interpolation is utilized to fit the cam profile curve and perform error assessment. Subsequently, based on the error assessment results, Grey Information Measurement Model (GIMM) for the uncertainty of the profile error in small samples is established. This model employs Grey Relational Analysis to eliminate outliers and evaluates the uncertainty of the profile error by solving grey correlation coefficients. Maximum-Minimum Information Measure Method is used to assess the optimal sample size. Finally, numerical experiments and experimental tests were conducted on the uncertainty of camshaft profile error in automobiles. A total of 15 sets of profile data were compared with Guide to the Representation of Uncertainty in Measurement (GUM) and Monte Carlo Method (MCM) under different sample sizes. The results showed that GIMM achieved evaluation with only 8 sets of data samples under small sample and poor information conditions, with an uncertainty of 0.6338 μm, compared to 0.6346 μm for GUM and 0.6391 μm for MCM. The acceptance rate of GIMM reached 95.2%. This model outperforms other methods, providing a simplified and reliable assessment of cam profile error uncertainty.