Spline Surfaces Research Articles

On the G-Similarities of two open B-spline curves in R3

Let G be a transformation group in R3. Any two vectors x and y in R3 are called G-equivalence vectors if there exist a transformation g G such that y = gx satisfies. In this paper the transformation group G will be considered as similarity transformations group or its any subgroup. So if given two vectors x and y in R3 are G-equivalence vectors then these vectors x and y are called G-similar. i.e. rotational, reflectional, translational or scaling similarity. B-spline curves are used basically in Computer Aided Design (CAD), Computer Aided Geometric Design (CAGD), Computer Aided Modeling (CAM). In determining the invariants of spline curves and surfaces at any point, it is necessary to find the analytical equation of each curve and surface and calculate its invariants such as curvature, torsion, principal curvatures, mean and Gaussian curvatures at the desired point. However, it can be very difficult to find the curve or surface to be designed analytically. For example, when a car is designed, the aerodynamic curves in the car will be different from the known surface equation of the car. It is very difficult to write this equation exactly. For these curves and surfaces we designed, the way to overcome this difficulty is to design them with spline curves and surfaces. In this paper the G- equivalence conditions of given two open B-spline curves are studied in case G is similarity transformations group or its any subgroup.

Modeling of CHAMP satellite data according to the 3D surface spline model of geomagnetic fields

Surface observations and CHAMP measurement data are employed to develop a three-dimensional surface spline (3DSS) model of mainland China. The magnetic field distribution at the satellite level is then demonstrated using the model obtained. The results of this model are compared and verified by deriving the corresponding two (2DTY) and three-dimensional (3DTY) Taylor polynomial models. Issues such as the removal of disruptive geomagnetic fields, the data gap between the surface and satellite levels, and boundary effects are carefully considered during modeling. We then focus on evaluating the modeling effect of the satellite data. Ten satellite points not involved in the modeling procedure are selected, and the residuals, absolute change rates, and RMSEs of these points are calculated. Results show that the distribution of the magnetic field determined by the 3DSS model is highly consistent with that obtained from the IGRF12 model. Expect for component Y, the absolute change rates of other components are less than 0.5%. Specifically, the RMSE of Y of 3DSS is nearly 60% lower than those of 3DTY and 2DTY; the RMSE of other components of the former are also over 90% lower than those of the latter. This finding implies that the 3DSS model has good performance for modeling satellite data and its results are reliable. Moreover, the modeling effect of 3DTY is better than that of 2DTY.

Single-image super-resolution based on local biquadratic spline with edge constraints and adaptive optimization in transform domain

This paper proposes a novel single-image super-resolution method based on local biquadratic spline with edge constraints and adaptive optimization in transform domain. The complex internal structure of the image makes the values of adjacent pixels often differ greatly. Using surface patches to interpolate image blocks can avoid large surface oscillation. Because the quadratic spline has better shape-preserving property, we construct the biquadratic spline surface on each image block to make the interpolation more flexible. The boundary conditions have great influence on the shape of local biquadratic spline surfaces and are the keys to constructing surfaces. Using edge information as a constraint to calculate them can reduce jagged and mosaic effects. To decrease the errors caused by surface fitting, we propose a new adaptive optimization model in transform domain. Compared with the traditional iterative back-projection, this model further improves the magnification accuracy by introducing SVD-based adaptive optimization. In the optimization, we convert similar block matrices to the transform domain by SVD. Then the contraction coefficients are calculated according to the non-local self-similarity, and the singular values are contracted. Experimental comparison with the other state-of-the-art methods shows that the proposed method has better performance in both visual effect and quantitative measurement.

Characterizing flexural behaviour of panel-to-panel connections in cross-laminated timber floor systems

Cross-laminated timber floor systems are currently designed by treating them as one-way systems. The one-way assumption is conservative but two-way model would overestimate the system stiffness if the presence of semi-rigid panel-to-panel connections is not properly considered in the analysis. These connections allow for partial transfer of bending moment from one panel to the adjacent panel. The primary objective of this research was to characterize the flexural behaviour of typical panel-to-panel connections used in cross-laminated timber floors. Two types of connections employing self-tapping screws and surface splines were examined through testing. The resulting bending moment versus rotation curves were used to determine the rotational stiffness and moment capacity of connection assemblies. The influences of screw quantity, screw diameter and length, and spline thickness on connection stiffness and capacity were studied. Test results indicated that the flexural properties were significantly dependent on the number of screws used in the connection. Screw diameter and spline thickness also influenced flexural performance but to a less extent. However, the longer screws resulted in smaller stiffness and capacity with high variability for butt joints. It was found that the capability of moment transfer is very limited for these tested connections and less than 6% rotational stiffness was achieved by comparing with those of intact beams. A secondary objective of the research was to evaluate the experimental procedure developed for this research to measure flexural properties of panel-to-panel connection. Overall, the experience and results indicate that the experimental procedure used was practical.

Lung Lobe Segmentation Based on Lung Fissure Surface Classification Using a Point Cloud Region Growing Approach

In anatomy, the lung can be divided by lung fissures into several pulmonary lobe units with specific functions. Identifying the lung lobes and the distribution of various diseases among different lung lobes from CT images is important for disease diagnosis and tracking after recovery. In order to solve the problems of low tubular structure segmentation accuracy and long algorithm time in segmenting lung lobes based on lung anatomical structure information, we propose a segmentation algorithm based on lung fissure surface classification using a point cloud region growing approach. We cluster the pulmonary fissures, transformed into point cloud data, according to the differences in the pulmonary fissure surface normal vector and curvature estimated by principal component analysis. Then, a multistage spline surface fitting method is used to fill and expand the lung fissure surface to realize the lung lobe segmentation. The proposed approach was qualitatively and quantitatively evaluated on a public dataset from Lobe and Lung Analysis 2011 (LOLA11), and obtained an overall score of 0.84. Although our approach achieved a slightly lower overall score compared to the deep learning based methods (LobeNet_V2 and V-net), the inter-lobe boundaries from our approach were more accurate for the CT images with visible lung fissures.

Design of spline surface vacuum gripper for pick and place robotic arms

The gripper is the most important part in an industrial robot. It is related with the environment around the robot. Today, the industrial robot grippers have to be tuned and custom made for each application by engineers, by searching to get the desired repeatability and behaviour. Vacuum suction is one of the grippers in Watch Case Press Production (WCPP) and a mechanism to improve the efficiency of the manufacturing procedure. Pick and place are the important process for the annealing process. Thus, by implementing vacuum suction gripper, the process of pick and place can be improved. The purpose of vacuum gripper other than design vacuum suction mechanism is to compare the effectiveness of vacuum suction gripper with the conventional pick and place gripper. Vacuum suction gripper is a mechanism to transport part and which later sequencing, eliminating and reducing the activities required to complete the process. Throughout this study, the process pick and place became more effective, the impact on the production of annealing process is faster. The vacuum suction gripper can pick all part at the production which will lower the loss of the productivity. In conclusion, vacuum suction gripper reduces the cycle time about 20%. Vacuum suction gripper can help lower the cycle time of a machine and allow more frequent process in order to increase the production flexibility.

Neutronics analysis of the stellarator-type fusion-fission hybrid reactor based on the CAD optimization method

The stellarator plasmas devices have some advantages in steady state operation with less magneto hydrodynamic activities, and the development of stellarator-type FFHR (Fusion-Fission Hybird Reactor) can provide viable energy in the foreseeable future with less requirements to the plasmas technology compared with the pure fusion system. However, the neutronics design of the stellarator-type FFHR has some challenges and difficulties, since the construction of helical structures with large number of high order spline surfaces are the extremely complicated process, and the physical parameters of the complex model have to be changed frequently in the primary design stage, which make the corresponding manual input simulation work into the time consuming and error prone tasks. In this context, the CAD optimization method has been used in converting the dense mesh oriented CAD structure to the low poly style, and the CAD-PSFO code has been employed for the modeling transformation process in Monte Carlo calculation. The thorium resource has been selected as the breeding material for the stellarator-type FFHR, and the polonium and natural abundance of uranium have been used as the seed fuel for the initial loading in order to get the high energy multiplication factor. The OMCB (OpenMC basedBurnupcode) has been introduced to study the burnup features, and stellarator-type FFHR has been designed as the multi-purposes facility that can keep tritium self-sufficiency and get the high breeding ratio with steady energy output.

A general isogeometric polar approach for the optimisation of variable stiffness composites: Application to eigenvalue buckling problems

This study presents a general approach for the multi-scale design of variable stiffness composites (VSCs). The first-level problem of the multi-scale two-level optimisation strategy (MS2LOS) is solved to determine the optimal distribution of the VSC stiffness properties at the macroscopic scale satisfying the requirements of the problem at hand. In this phase, the VSC laminate is modelled as an equivalent homogeneous anisotropic plate whose behaviour is described in terms of polar parameters (PPs), which vary locally over the structure. The First-order Shear Deformation Theory is used to take into account the influence of the transverse shear stiffness on the mechanical response of the VSC and Basis Spline (B-Spline) surfaces are employed to represent the PPs fields. In this background, the expression of the gradient of the buckling factor is determined analytically by exploiting the properties of the polar formalism and of the B-Spline surfaces. Moreover, the effect of the discrete variables, involved in the definition of the B-Spline surfaces, on the performances of the optimised solution is investigated. The effectiveness of the approach is proven on two benchmark problems dealing with the maximisation of the first buckling load of a VSC laminate, subject to feasibility and geometric requirements, taken from the literature. The results obtained by means of the MS2LOS based on the polar formalism outperform those reported in the literature, which are obtained through an optimisation strategy based on lamination parameters.

Least Degree [formula omitted]-Refinable Multi-Sided Surfaces Suitable For Inclusion Into [formula omitted] Bi-2 Splines

Geometrically smooth spline surfaces, generalized to include n-sided facets or configurations of n≠4 quads, can exhibit a curious lack of additional degrees of freedom for modeling or engineering analysis when refined.This paper establishes a minimal polynomial degree for smooth constructions of multi-sided surfaces that guarantees more flexibility in all directions under refinement. Degree bi-4 is both necessary and sufficient for flexibility-increasing G1-refinability within a bi-quadratic C1 spline complex. Sufficiency is proven by two alternative flexibly G1-refinable constructions exhibiting good highlight line distributions.

Experimental investigation on the forming and evolution process of cell structure in gas counter pressure assisted chemical foaming injection molded parts

By using a standard stretch spline as the research object, the influence of gas counter pressure (GCP) technology on melt foaming behavior in chemical foaming injection molding (CFIM) process was investigated. Related experimental line for GCP assisted CFIM foam was designed, and the effect of GCP technology on melt flow front, spline surface quality and internal cell was studied. According to the results obtained from the experiment, two critical GCP pressures and one critical GCP holding time were innovation proposed. Two critical GCP pressures are the critical GCP pressure of melt flow front cell not cracking and the critical GCP pressure of melt not foaming, respectively. The critical GCP holding time is the secondary foaming behavior time. Based on the proposed critical GCP pressures and critical GCP holding time, the influence mechanism of GCP technology on melt foaming action during CFIM process was revealed.

Quad-mesh based isometric mappings and developable surfaces

We discretize isometric mappings between surfaces as correspondences between checkerboard patterns derived from quad meshes. This method captures the degrees of freedom inherent in smooth isometries and enables a natural definition of discrete developable surfaces. This definition, which is remarkably simple, leads to a class of discrete developables which is much more flexible in applications than previous concepts of discrete developables. In this paper, we employ optimization to efficiently compute isometric mappings, conformal mappings and isometric bending of surfaces. We perform geometric modeling of developables, including cutting, gluing and folding. The discrete mappings presented here have applications in both theory and practice: We propose a theory of curvatures derived from a discrete Gauss map as well as a construction of watertight CAD models consisting of developable spline surfaces.

Static Explosion Field Reconstruction Based on the Improved Biharmonic Spline Interpolation

To obtain the peak pressure distribution of explosion field near the ground, an improved reconstruction method for interpolation of blast overpressure data from the finite measuring points was studied. According to the peak attenuation law of shock wave on the propagation path and the attenuation characteristic of the wave front after passing through the measured points, the polynomial fitting function of peak attenuation of shock wave near the ground is constructed by using the least square method. Using the peak attenuation polynomial of shock wave constrain the radial interpolation of biharmonic spline surface interpolation algorithm (BSSIA) to realize the interpolation conforming the propagation law of shock wave and ensure the interpolation convergence. The root-mean-squared error (RMSE) and the maximum relative error (MRE) before and after the improvement were computed by comparing the peaks with empirical formula data. The improvement is evident as the RMSE decreased from 14.12kPa to 0.22kPa and the MRE decreased from 23.91% to 0.37%. The peak reconstructed by the improved biharmonic spline surface interpolation algorithm (IBSSIA) and BSSIA with test data, the overpressure peak distribution obtained by IBSSIA is in good agreement with empirical result and consistent with the explosion image.

Spline-based surfaces in architecture and civil engineering

Spline is the most convenient approximation tool widely used for designing curvilinear technical surfaces in various branches of technology, architecture and civil engineering. To solve such problems, bicubic splines are used. The spline function as a polynomial of the third degree contains differential coefficients. Most often, this is the first derivative in the spline nodes. These coefficients entirely determine its shape. We propose using a fourth order integro-differential spline with additional integral coefficients to construct the surface. These coefficients characterize the shape of the spline curve and the surface between the nodes. As far as computing goes, this spline is not more complicated than traditional one, since it is constructed by solving tridiagonal linear equation system. However, additional coefficients are convenient for local modifications of the curve and surface shapes. This reduces the number of pieces of a composite spline surface. In the practice of constructing geometric shapes in architecture, there are problems when the surface is more complex in one direction than in another. For modeling such surfaces, instead of bicubic spline, we propose using a heterogeneous integro-differential spline. The latter is cubic in one direction of the grid of nodes, and the fourth order polynomial in another direction.

Cyclic performance of in-plane shear cross-laminated timber panel-to-panel surface spline connections

Cross-laminated timber (CLT) panels have been used as structural diaphragm elements that form the primary lateral force-resisting system in mass-timber buildings. To better understand the overall performance of CLT diaphragms, an experimental program including monotonic and cyclic in-plane shear panel-to-panel spline connections tests was conducted. The spline connections were constructed using nails and screws with different spacings. Experimental results from the testing program allow for characterization of the performance of the connection systems in terms of elastic stiffness, strength, and ductility. Based on the results of this specific testing program, nailed spline connections displayed similar performance in terms of stiffness and strength and were more 58.7% more ductile than the screw connections, which is partly explained by the larger overstrength factors observed for the screw connections. A simplified performance model is presented to aid designers in determining the elastic, inelastic, and post-peak force parameters of the various spline connections tested.

FreeCAD based Monte Carlo modeling approach for fusion reactor facilities

Abstract MCNPX is a general purpose multi-particle Monte Carlo transport code with board ranges of energies, which has already been widely employed in the fields of radiotherapy, nuclear engineering, and nuclear fusion research. The neutronics design and analysis of fusion reactor facilities using MCNPX have been considered as the time consuming and error prone tasks. For example, the ITER-type models include a lot of nested structures and complicated high-order curve surfaces, which can not be easily described in the mathematics expressions, and the spline surfaces are also not supported by the native syntax of MCNPX. In this context, a code system called CAD-PSMC (FreeCAD based parsing script for MCNPX) has been developed in IMPCAS (Institute of Modern Physics, Chinese academy of Sciences) for the study of modeling conversion problem in ITER-type neutron calculations. Two kinds of methods have been provided and discussed in the framework of this code, including the establishing of the mapping relationship between the specification tree and the build-in boolean operations in consideration of the complicated hierarchical structures, and the demonstration of fast ray-casting technology based iteration method coupling with the mesh-oriented optimized CAD geometries in solving the high-order curve surfaces related problems. Finally, the stability and feasibility of the CAD-PSMC have been verified and validated through the comparison with the selected reference models and the neutronic simulation results of the ITER-type benchmark model.

A novel NURBS surface approach to statistically monitor manufacturing processes with point cloud data

As sensor and measurement technologies advance, there is a continual need to adapt and develop new Statistical Process Control (SPC) techniques to effectively and efficiently take advantage of these new datasets. Currently high-density noncontact measurement technologies, such as 3D laser scanners, are being implemented in industry to rapidly collect point clouds consisting of millions of data points to represent a manufactured parts' surface. For their potential to be realized, SPC methods capable of handling these datasets need to be developed. This paper presents an approach for performing SPC using high-density point clouds. The proposed approach is based on transforming the high-dimensional point clouds into Non-Uniform Rational Basis Spline (NURBS) surfaces. The control parameters for these NURBS surfaces are then monitored using a surface monitoring technique. In this paper point clouds are simulated to determine the performance of the proposed approach under varying fault scenarios.

Spline surfaces with C1 quintic PH isoparametric curves

Given two spatial C1 PH spline curves, aim of this paper is to study the construction of a tensor–product spline surface which has the two curves as assigned boundaries and which in addition incorporates a single family of isoparametric PH spline curves. Such a construction is carried over in two steps. In the first step a bi–patch is determined in a ‘Coons–like’ way having as boundaries two quintic PH curves forming a single section of given spline curves, and two polynomial quartic curves. In the second step the bi–patches are put together to form a globally C1 continuous surface. In order to determine the final shape of the resulting surface, some free parameters are set by minimizing suitable shape functionals. The method can be extended to general boundary curves by preliminary approximating them with quintic PH splines.

Higher frequency hedonic property price indices: a state-space approach

The hedonic imputation method allows characteristic shadow prices to evolve over time. These shadow prices are used to construct matched samples of predicted prices, which are inserted into standard price index formulas. We use a spatio-temporal model to improve the method’s effectiveness on housing data at higher frequencies. The problem is that at higher frequencies, there may not be enough observations per period to reliably estimate the characteristic shadow prices. In such cases, the reliability of the hedonic imputation method is improved by using a state-space formulation which yields estimates of the shadow prices that are weighted sums of previous periods’ information. In addition, the state-space representation of the model includes a geospatial spline surface which significantly reduces the number of parameters to be estimated when compared to the standard practice of including postcode dummies in the model. Empirically, using a novel criterion, we show that in higher frequency comparisons, our hedonic method outperforms competing alternatives.

Design considerations for variable stiffness, doubly curved composite plates

It is widely acknowledged that laying up composite tapes and fabrics on doubly curved surfaces can introduce wrinkling and overlap defects. It is also not a straightforward task to define fibre angle directions. The latter problem amplifies when considering curvilinear fibre paths on doubly curved surfaces. Herein, we define a new fibre orientation datum curve and use this as a reference for optimisation purposes. A novel implementation of lamination parameters is introduced whereby we constrain the values of lamination parameters based on minimum turning radii of automated fibre placement machines and maximum ply angle change for stiffness optimisation. As an example, this approach is applied to a pre-twisted plate with 68 layers subjected to combined centrifugal and pressure loading. Lamination parameters are used to reduce the complexity and multiplicity of composite laminate design with many layers to a convex and continuous design space using a reduced number of variables. Lamination parameters are varied over the geometry using spline surface interpolation over a grid of control points and the optimal lamination parameter values are found using a gradient based optimiser. The fibre paths for each ply are then designed from the optimal lamination parameter variation using a two-stage optimisation process.

On spline-based approaches to spatial linear regression for geostatistical data

For spatial linear regression, the traditional approach to capture spatial dependence is to use a parametric linear mixed-effects model. Spline surfaces can be used as an alternative to capture spatial variability, giving rise to a semiparametric method that does not require the specification of a parametric covariance structure. The spline component in such a semiparametric method, however, impacts the estimation of the regression coefficients. In this paper, we investigate such an impact in spatial linear regression with spline-based spatial effects. Statistical properties of the regression coefficient estimators are established under the model assumptions of the traditional spatial linear regression. Further, we examine the empirical properties of the regression coefficient estimators under spatial confounding via a simulation study. A data example in precision agriculture research regarding soybean yield in relation to field conditions is presented for illustration.