Non-uniform Rational B-spline Surfaces Research Articles

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.

3D scene reconstruction of landslide topography based on data fusion between laser point cloud and UAV image

The main topic of this research was to propose an optimizing modeling method of landslide topography. This method fuses high-resolution imagery from an unmanned aerial vehicle (UAV) with laser point cloud. Due to the relatively high efficiency and large coverage area, the UAV system is suitable for emergency investigation of landslide. Based on the UAV images and laser point cloud, the use of partial differential equation (PDE) combines the hole repair technology with non-uniform rational B-splines (NURBS) method to rapidly reconstruct a three-dimensional (3D) scene. The mean square error (MSE) determined by comparison with measured data from the laser scanner technology is 16.1274 mm. And the error of plane calculated by the NURBS surface fitting is smallest by comparing with the least-squares plane fitting, Delaunay triangulation and polygon mesh method. In addition, the realistic 3D landslide model is obtained by texture mapping. The XiaPa landslide in Liangshan District, Xichang City, Sichuan Province China, was selected as a case study, in which the accuracy is evaluated to prove that the model has higher precision than traditional methods. The experiment results show that the integrated approach in this paper could be implemented in qualitative and quantitative risk assessment procedures pertaining to landslides.

Nonuniform sampling pitch acquisition method in myocardial single-photon emission computed tomography.

Single-photon emission computed tomography is a tomographic imaging method that acquires a projection image by rotating a gamma camera around by 380° or 180°. For myocardial single-photon emission computed tomography, 180° acquisition is common, but it has limitations including an incomplete reconstruction, which can distort the resulting image. It is possible to produce a complete reconstruction using 360° acquisition, but the testing time is long and is burdensome to patients. The nonuniform sampling pitch acquisition (NUSPA) method devised in this study involves reducing the total sampling count using NUSPA that reduces the sampling pitch in the range in which the gamma cameras are closer to the myocardium (RAO45-LPO45) and increases it elsewhere. The NUSPA-1 method based on a 6° sampling pitch had 20 views fewer than 360° acquisition. In addition, the NUSPA-2 method based on a 3.75° sampling pitch had 60 views fewer than 360° acquisition, considerably reducing the testing time. The acquired sinograms from the NUSPA methods were subjected to nonuniform rational B-spline surface interpolation processing, producing data with a uniform sampling pitch, after which image reconstruction was performed. The images after nonuniform rational B-spline interpolation for both the line sources and heart-liver phantom investigated in this study were not found to have the distortion observed from 180° acquisition or a count decrease at the center, resulting in image quality nearly equivalent to 360° acquisition. This method enabled a reduction in testing time without impacting image quality.

An improved computational framework based on the dual boundary element method for three-dimensional mixed-mode crack propagation analyses

This study presents an improved computational framework based on the Dual Boundary Element Method (DBEM) for modelling three-dimensional mixed-mode fatigue crack propagation in geometrically complex structural components. The improvements are: (i) an input data scheme based on an alternative collocation strategy, which enables the discretization of each Non-Uniform Rational B-splines (NURBS) surfaces of geometrical models without any concerning about conform meshes at the surface´s intersections. (ii) a variational formulation defined over the crack front, which accounts for both kinking and twisting angles of the during the crack propagation. (iii) a linear approximation for the Stress Intensity Factors (SIF) range, which was applied in the three-dimensional fatigue analysis and enables the discrete solution of the Paris law in accurate form. The SIF are assessed by the displacement correlation or extrapolation techniques. The maximum energy release rate or the Schöllmann's criteria are adopted for driving the propagation process. Four numerical examples are presented in order to validate the proposed computation framework and investigate its accuracy and robustness.

Development of a feature-recognition and measurement path generation system based on NURBS surfaces for 5-axis on-machine measurement

On-machine measurement (OMM), which is used to measure the machined surfaces of a workpiece during or after the machining of the workpiece, enables direct measurement within the workspace without moving the workpiece. However, although the three-dimensional geometric shape created by computer-aided design systems include various features, because such features are converted into non-uniform rational B-spline (NURBS) surfaces during OMM, measurement deviation occurs between a feature and its corresponding NURBS surface. In this paper, we suggest a method to generate the measurement path for 5-axis OMM in a way that recognizes the features necessary for measurement from NURBS surfaces whose feature data set was removed during the conversion processes. To verify the reliability of the measurement path generation system named OMV+, which was developed using the above method, we carried out an experiment to compare the measurement path that defines the measuring point on the feature and the path that defines the measuring point on the NURBS surface with the measured result using the touch probe on the machined workpiece.

Representation of NURBS surfaces by Controlled Iterated Functions System automata

Iterated Function Systems (IFS) are a standard tool to generate fractal shapes. In a more general way, they can represent most of standard surfaces like Bézier or B-Spline surfaces known as self-similar surfaces. Controlled Iterated Function Systems (CIFS) are an extension of IFS based on automata. CIFS are basically multi-states IFS, they can handle all IFS shapes but can also manage multi self-similar shapes. For example CIFS can describe subdivision surfaces around extraordinary vertices whereas IFS cannot. Having a common CIFS formalism facilitates the development of generic methods to manage interactions (junctions, differences...) between objects of different natures.This work focuses on a CIFS approach of Non-Uniform Rational B-Splines (NURBS) which are the main used representation of surfaces in CAGD systems. By analyzing the recursive generating process of basis functions, we prove the stationarity of NURBS computation. This implies that NURBS can be represented as a finite automaton: a CIFS. Subdivision transformations implied in the generating process are directly deduced from blossoming formulation and are expressed as a function of the initial nodal vector. We provide a method to construct the CIFS automata for NURBS of any-degree. Then NURBS-surfaces automata are deduced using a “tensor-product” of NURBS automata. This new representation of NURBS allows us to build a bridge between them and other surfaces already represented in CIFS formalism: fractals and subdivision surfaces.

Design of self-supporting surfaces with isogeometric analysis

Self-supporting surfaces are widely used in contemporary architecture, but their design remains a challenging problem. This paper aims to provide a heuristic strategy for the design of complex self-supporting surfaces. In our method, non-uniform rational B-spline (NURBS) surfaces are used to describe the smooth geometry of the self-supporting surface. The equilibrium state of the surface is derived with membrane shell theory and Airy stresses within the surfaces are used as tunable variables for the proposed heuristic design strategy. The corresponding self-supporting shapes to the given stress states are calculated by the nonlinear isogeometric analysis (IGA) method. Our validation using analytic catenary surfaces shows that the proposed method finds the correct self-supporting shape with a convergence rate one order higher than the degree of the applied NURBS basis function. Tests on boundary conditions show that the boundary’s influence propagates along the main stress directions in the surface. Various self-supporting masonry structures, including models with complex topology, are constructed using the presented method. Compared with existing methods such as thrust network analysis and dynamic relaxation, the proposed method benefits from the advantages of NURBS-based IGA, featuring smooth geometric description, good adaption to complex shapes and increased efficiency of computation.

Entropy generation analysis of wet-steam flow with variation of expansion rate using NURBS-based meshing technique

In this paper, the entropy generation is introduced as a criterion to analyze the impact of variation of expansion rate in a two-dimensional viscous wet-steam flow through stationary cascades of a steam turbine. In addition, for the first time the NURBS-based mesh generation method is introduced and assessed in the field of wet-steam flow. In the mentioned method, the computational grid with “curved” boundaries is generated by non-uniform rational B-spline (NURBS) surface that describes the geometry of the cascade. The results, which are validated against well-established experiments, reveal that compared to the typical H-type grid, employing the generated NURBS-based mesh significantly reduce the computational error of numerical simulation. According to the obtained results, the prediction errors of static pressure distribution and droplet average radius are reduced by 31% and 78.44%, respectively. Furthermore, it is observed that decreasing the expansion results in moving the nucleation zone toward downstream of the blade and increasing the rate of droplets nucleation. Additionally, the liquid mass fraction as well as the amount of entropy and total entropy generation in the fluid flow increases.

Multi-UAV Trajectory Optimization Utilizing a NURBS-Based Terrain Model for an Aerial Imaging Mission

Trajectory optimization precisely scanning an irregular terrain is a challenging problem since the trajectory optimizer needs to handle complex geometry topology, vehicle performance, and a sensor specification. To address these problems, this paper introduces a novel framework of a multi-UAV trajectory optimization method for an aerial imaging mission in an irregular terrain environment. The proposed framework consists of terrain modeling and multi-UAV trajectory optimization. The terrain modeling process employs a Non-Uniform Rational B-Spline (NURBS) surface fitting method based on point cloud information resulting from an airborne LiDAR sensor or other sensor systems. The NURBS-based surface model represents a computationally efficient terrain topology. In the trajectory optimization method, the framework introduces a multi-UAV vehicle routing problem enabling UAV to scan an entire area of interest, and obtains feasible trajectories based on given vehicle performance characteristics, and sensor specifications, and the approximated terrain model. The proposed multi-UAV trajectory optimization algorithm is tested by representative numerical simulations in a realistic aerial imaging environment, namely, San Diego and Death Valley, California.

Effects of surface splicing characteristics of the hardened steel mold on the machined surface topography

The car cover mold has characteristics of large structure size, complicated surface shape, dimensional precision, and high surface precision. The milling process of the splicing transition zone having different hardness can easily produce load mutation, which has an impact on the tool in the form of vibration. This vibration affects the surface topography of the splicing transition zone. In this article, we propose the surface motion model for ball-end milling cutter with impact vibration and the surface topography prediction model for splicing transition zone which is established based on the Non-Uniform Rational B-Splines (NURBS) surface control point reconstruction. Then, the effects of surface splicing characteristics (different hardness, over-seam scale, etc.) and processing parameters on the surface topography and on the residual height have been analyzed. Finally, based on the hardened steel surface splicing milling experiment, the accuracy of the surface topography model in which vibration impact has been incorporated is verified. At the same time, the corresponding roughness value and fractal dimension are calculated on the basis of two-dimensional contour graph under different splicing characteristics. The results show that the non-linear characteristics of the surface topography of the hardened steel surface are significant, and the best processing parameters can be obtained under different splicing characteristics.

Uncertainty Scattering Analysis of 3-D Objects With Varying Shape Based on Method of Moments

An effective uncertainty analytical method based on the Method of Moments is proposed for the scattering analysis of 3-D objects with varying shape. The uncertainties of the geometric shape are described by several random variables in the integral equations through the nonuniform rational B-spline surface modeling approach. All random variables are independent. Therefore, the scattering properties of the 3-D object with uncertain geometric shape can be quantified by the perturbation approach with less cost of computational resource than the standard Monte Carlo method. The numerical results demonstrate the effectiveness of the proposed method.

Fully nonlinear and linear ship waves modelling using the potential numerical towing tank and NURBS

Waves induced by the ship advance in the calm water are simulated in a potential Numerical Towing Tank based on both the fully nonlinear approach and the linear theory. In the linear theory, a modified Green function, which was derived based on linearized free surface boundary conditions and the image method is used as a fundamental solution in the boundary integral equation. High-order boundary integral equation is used to solve the boundary value problem with Non-Uniform Rational B-Spline (NURBS) basis function. On the other hand, the time domain simulation of the fully nonlinear free surface flow due to the steady flow past over a ship model is conducted based on the Mixed Eulerian-Lagrangian approach. NURBS surface is used to describe the ship hull and the fully nonlinear free surface evolution. The numerical solutions are compared with a prior experimental study to verify both numerical models. In addition, the effect of nonlinear terms in fully nonlinear free surface boundary conditions is investigated. The wave making resistance as an essential criterion is compared with the experimental data and some numerical efforts to investigate the accuracy of both models. The capability of the modified Green function is also examined to simulate the ship waves in rectangular channels.

Triangular mesh generation on free-form surfaces based on bubble dynamics simulation

Purpose Modern CAD systems facilitate the creation of any surface geometry imaginable, and complex surfaces for free-form grid shells are often represented by a set of Non-Uniform Rational B-Splines surface patches. But it remains an intractable issue how to generate high-quality grids on complex surfaces efficiently. To solve this issue, an automatic triangular mesh generation method is presented, based on bubble dynamics simulation and a modified Delaunay method. Design/methodology/approach A moderate amount of points are first distributed on a given surface. Next, by regarding the points as elastic bubbles with the same size and introducing the forces acting on bubbles, the motion control equations of bubbles are established. The equilibrium state of the bubble system is found by Verlet algorithm. Then, the Voronoi diagram on the surface is obtained by calculating the intersection between the surface and the three-dimensional (3D) Voronoi diagram of the centers of bubbles. Finally, a triangular mesh, Delaunay triangulation on the surface, is determined based on the dual change of the Voronoi diagram. Findings This method generates meshes on the surface directly, unlike mapping-based methods, avoiding the mapping distortion. Examples are given to demonstrate the successful execution of this method. The result also illustrates that this method is applicable to various surfaces in high automation level and resultant meshes are highly uniform and well-shaped. Originality/value Thus, this method provides the convenience for the geometry design of complex free-form grid structure.

Optimal NURBS conversion of PDE surface-represented high-speed train heads

The head shape of high-speed trains has become a critical factor in boosting the speed further. Aerodynamic simulation-based optimization is a dominant method to obtain the optimal head shape which relies on detailed train head models defined by a lot of design variables. Since aerodynamic simulation-based optimization involves heavy calculations, too many design variables not only causes high computational costs, but also makes the optimal solution difficult to obtain. Therefore, how to use few design variables to define detailed train head model is the key to success. Partial differential equation (PDE)-based geometric modelling which creates a complicated PDE patch with few design variables provides an effective solution to this problem. In addition, it also has the advantage of naturally maintaining any high-order continuities between two adjacent surfaces which is very important in designing highly smooth train heads to achieve excellent aerodynamic performance. At the present time, PDE-based geometric modelling cannot be directly applied in computer-aided design (CAD), computer-aided manufacturing (CAM), and computer-aided engineering (CAE) since it has not become an industrial standard. In contrast, non-uniform rational B-splines (NURBS) are commonly used in CAD, CAM, CAE, and many other engineering fields. They have already become part of industry wide standards. In order to apply PDE-based geometric modelling in shape design of high-speed train heads for CAD etc., how to optimally convert PDE surfaces into NURBS surfaces must be addressed. In this paper, a new method of achieving optimal conversion of PDE surfaces representing high-speed train heads into NURBS surfaces is developed. It takes control points and weight deformations of NURBS surfaces to be design variables, and the error between NURBS surfaces and PDE surfaces as the objective function. The least squares fitting and the genetic algorithm are combined to obtain the optimal conversion between PDE surfaces and NURBS surfaces. The application examples demonstrate the effectiveness of the developed method.

A reliability-based optimization approach for material and thickness composition of multidirectional functionally graded plates

A reliability-based design optimization (RBDO) methodology is first proposed to simultaneously optimize material and thickness distribution of multidirectional functionally graded (MFG) plates for compliance minimization under uncertainties of design variables and system parameters. The modified sequential optimization and reliability assessment (MSORA) method is integrated into the recently suggested adaptive hybrid evolutionary firefly algorithm (AHEFA) to release a novel RBDO approach which is the so-called MSORA-AHEFA. The MSORA is refined from its original for computational cost savings by eliminating the iterative process of finding most probable points (MPPs). An isogeometric multimesh design (IMD) approach is presented to form two separate non-uniform rational B-spline (NURBS) surfaces via the k−refinement strategy. In which, a coarser design NURBS one is employed to define design variables of the ceramic volume fraction and z−axis coordinate of the top side coincidentally assigned at each of control points. A finer analysis NURBS one constructed by the isogeometric analysis (IGA) and a generalized shear deformation theory (GSDT) is used for bending analyses of MFG plates with arbitrary thickness. Applying such two separately defined surfaces dramatically lessens the number of design variables and the computational cost in optimization problems, yet still manifesting optimal profiles properly. Meanwhile, mechanical behavior of the plate in analysis ones is precisely simulated as well. For the validation of the suggested MSORA-AHEFA, its results for the benchmark welded beam are compared with those of formerly published work. Several numerical examples regarding RBDO of MFG plates are then executed to attest the effectiveness of the current paradigm.

Multi-Field-Coupled Model and Solution of Active Electronically Scanned Array Antenna Based on Model Reconstruction

Active electronically scanned array antenna (AESA antenna) is capable of controlling the radiation pattern by controlling the feeding phase of the radiating elements. It has good performance and plays an important role in radar systems. With the development of AESA antenna towards high-frequency bands and high-density arrays, the structural-electromagnetic-thermal (SET) coupling becomes increasingly significant. It seriously restricts the realization of high performances of AESA antennas. However, the previously reported theoretical multi-field-coupled model for the coupling problem ignores the effect of the deformations of the feed system and array elements on the electrical performance. It only considers the positional deviations of the array elements in the coupling field. As a result, the accuracy of the numerical solution by the theoretical model is reduced. To overcome the above problems, this paper first establishes the field-circuit coupling model by introducing the deformation errors of the feed system into the existing theoretical model. Secondly, this paper proposes a new numerical solution for the multi-field-coupled problem of AESA antennas based on model reconstruction. And the model reconstruction includes the following: the NURBS (nonuniform rational B-spline) surface fitting algorithm that completes the mapping from finite element models to geometric models by the surface equations established by the node information and the local model reconstruction algorithm that determines the local geometric models by the positions and the directions. The NURBS surface fitting algorithm guarantees the accuracy of both the positions and shapes of array elements. The local model reconstruction algorithm ensures the accuracy of the amplitudes and phases of feed connectors. Finally, the numerical solution was applied to the 32-element AESA antenna and the simulations are close to the measurements.

Numerical study on hydrodynamic interaction between two tankers in shallow water based on high-order panel method

Abstract A three-dimensional high-order panel method based on Non-Uniform Rational B-Spline (NURBS) for predicting the ship–ship hydrodynamic interaction during meeting and overtaking in shallow water is developed. The NURBS surface is used to precisely represent the hull geometry. The velocity potential on the body surface is described by B-spline after the source density distribution on the boundary surface is determined. A collocation approach is applied to the boundary integral equation discretization, and the velocity potential is being solved at each time step. Under the low-speed assumption, the effect of free surface elevation is neglected in the numerical calculation, and the infinite image method is used to deal with the finite water depth effect. Two tankers in model tests are investigated, and the predicted hydrodynamic interaction forces and moments are compared with experimental measurements to clarify the validity of the proposed numerical method. Calculations are then conducted for different water depths, lateral ship–ship distances and ship speeds, and the detailed results are discussed to demonstrate the effects of these factors.

Application of endwall contouring in a high-subsonic tandem cascade with endwall boundary layer suction

A proper combination of active and passive flow control methods would be a promising way to further enhance the compressor performance. This numerical investigation presents a new attempt of application of endwall contouring in a high subsonic tandem cascade with the endwall boundary layer suction implemented in the forward blade. The study aims to further improve the corner flow in the rear blade at design and lower incidences. The planar endwall is redesigned using the optimization method. Nonuniform rational B-spline surface is employed to parameterize the endwall surface. A multi-points optimization strategy is selected to minimize the aerodynamic loss generated in the cascade. In the optimization, the suction strategy remains unchanged. To clarify the impacts of endwall contouring on the cascade performance, flow details in both mainflow passage and suction flow path at design and off-design incidences are analyzed.As a result, the total loss decreases by 8.4% at the design incidence via the endwall contouring. Furthermore, a more prominent loss reduction can be achieved at lower incidences. However, the control effects of endwall contouring is weakened towards higher positive incidences. Worse still, even a deterioration in the cascade performance can be detected at I=5° due to the presence of a small-scale corner stall in the forward blade. The results indicate that, as the gap flow is not strong enough to counteract the endwall secondary flow, the endwall contouring could be employed to further enhance the flow control ability of the previous proposed compound flow control method, while its negative impact on the cascade performance at higher positive incidences still remains to be addressed.

About Revisiting Domain Decomposition Methods for Poroelasticity

In this paper, we revisit well-established domain decomposition (DD) schemes to perform realistic simulations of coupled flow and poroelasticity problems on parallel computers. We define distinct solution schemes to take into account different transmission conditions among subdomain boundaries. Indeed, we examine two different approaches, i.e., Dirichlet-Neumann (DN) and the mortar finite element method (MFEM), and we recognize their advantages and disadvantages. The MFEM significantly lessens the computational cost of reservoir compaction and subsidence calculations by dodging the conforming Cartesian grids that arise from the pay-zone onto its vicinity. There is a manifest necessity of producing non-matching interfaces between the reservoir and its neighborhood. We thus employ MFEM over nonuniform rational B-splines (NURBS) surfaces to stick these non-conforming subdomain parts. We then decouple the mortar saddle-point problem (SPP) using the Dirichlet-Neumann domain decomposition (DNDD) scheme. We confirm that this procedure is proper for calculations at the field level. We also carry comprehensive comparisons between the conventional and non-matching solutions to prove the method’s accuracy. Examples encompass linking finite element codes for slightly compressible single-phase and poroelasticity. We have used this program to a category of problems ranking from near-borehole applications to whole field subsidence estimations.

Surface-Based Geological Reservoir Modelling Using Grid-Free NURBS Curves and Surfaces

Building geometrically realistic representations of geological heterogeneity in reservoir models is a challenging task that is limited by the inflexibility of pre-defined pillar or cornerpoint grids. The surface-based modelling workflow uses grid-free surfaces that allows efficient creation of geological models without the limitations of pre-defined grids. Surface-based reservoir modelling uses a boundary representation approach in which all heterogeneity of interest (structural, stratigraphic, sedimentological, diagenetic) is modelled by its bounding surfaces, independent of any grid. Volumes bounded by these surfaces are internally homogeneous and, thus, no additional facies or petrophysical modelling is performed within these geological domains and no grid or mesh discretisation is needed during modelling. Any heterogeneity to be modelled within such volumes is incorporated by adding surfaces. Surfaces and curves are modelled using a parametric non-uniform rational B-splines (NURBS) description. These surfaces are efficient to generate and manipulate, and allow fast creation of multiple realisations of geometrically realistic reservoir models. Multiple levels of surface hierarchy are introduced to allow modelling of all features of interest at the required level of detail; surfaces at one hierarchical level are constructed so as to truncate or conform to surfaces of a higher hierarchical level. This procedure requires the joining, terminating and stacking of surfaces to ensure that models contain “watertight” surface-bounded volumes. NURBS curves are used to represent well trajectories accurately, including multi-laterals or side-tracks. Once all surfaces and wells have been generated, they are combined into a reservoir model that takes into account geological relationships between surfaces and preserves realistic geometries.