Spline Surfaces Research Articles

Optimizing algorithm for high-precision imaging stitching systems based on spline surfaces

As optical systems continue to advance, non-uniform rational B-spline (NURBS) surfaces increasingly being considered in asymmetric optical systems due to their localized control characteristics. However, the representation of NURBS surfaces has complicated the analysis of these systems, leading to a significant computational burden. To address this challenge, we propose an optimizing algorithm for imaging optical systems based on high-precision ray tracing of NURBS surfaces. This method initiates with getting a knot grid as prior information, in conjunction with the Newton-Raphson algorithm, to obtain high-precision numerical solutions for the intersection of rays with a NURBS surface. Building upon this methodology, we introduce an optimization technique that includes shape evaluation to generate an evaluation function specific to NURBS surfaces. This approach is then applied within a rapid optimization process that accounted for the region of ray influence. Under consistent control point grids and sampling ray conditions, we present an off-axis four-mirror system to showcase that our algorithm has achieved a computational efficiency improvement of approximately 14 times compared to the previous method. This high-precision imaging design based on spline surfaces fulfills the need for efficient and accurate algorithms for NURBS surface applications in various imaging systems, providing guidance for practical applications.

Effect of leading-edge and trailing-edge camber morphing on gust load for an elastic wing

This paper investigates the influence of the spanwise-distributed trailing-edge camber morphing on the dynamic stall characteristics of a finite-span wing at Re = 2 × 105. The mathematical model of the spanwise-distributed trailing-edge camber morphing is established based on Chebyshev polynomials, and the deformed wing surface is modeled by a spline surface according to the rib’s morphing in the chordwise direction. The Computational Fluid Dynamics (CFD) method is adopted to obtain flow-field results and aerodynamic forces. The SST-γ model is introduced and the overset mesh technique is adopted. The numerical results show that the spanwise-distributed trailing-edge morphing obviously changes the aerodynamic and energy transfer characteristics of the dynamic stall. Especially when the phase difference between the trailing-edge motion and the wing pitch is −π/2, the interaction between the three-dimensional (3-D) Leading-Edge Vortex (LEV) and Trailing-Edge Vortex (TEV) is strengthened, and the work done by the aerodynamic force turns negative. This indicates that the trailing-edge deformation has the potential to suppress the oscillation amplitude of stall flutter. We also found that as the trailing-edge camber morphing varies more complexly along the spanwise direction, the suppression effect decreases accordingly.

Bicubic Splines for Fast-Contracting Control Nets

Merging parallel quad strips facilitates narrowing surface passages, and allows a design to transition to a simpler shape. While a number of spline surface constructions exist for the isotropic case where n pieces join, few existing spline constructions deliver a good shape for control nets that merge parameter lines. Additionally, untilrecently, none provided a good shape for fast-contracting polyhedral control nets. This work improves the state-of-the-art of piecewise polynomial spline surfaces accommodating fast-contracting control nets. The new fast-contracting (FC) surface algorithm yields the industry-preferred uniform degree bi-3 (bi-cubic). The surfaces are by default differentiable, have an improved shape, measured empirically as to highlight the line distribution, and require fewer pieces compared to existing methods.

Splines for Fast-Contracting Polyhedral Control Nets

Rapid reduction in the number of quad-strips, to accommodate narrower surface passages or reduced shape fluctuation, leads to configurations that challenge existing spline surface constructions. A new spline surface construction for fast contracting polyhedral control-nets delivers good shape. A nestedly refinable construction of piecewise degree (2,4) is compared with a uniform degree (3,3) spline construction.

A general formulation of reweighted least squares fitting

We present a generalized formulation for reweighted least squares approximations. The goal of this article is twofold: firstly, to prove that the solution of such problem can be expressed as a convex combination of certain interpolants when the solution is sought in any finite-dimensional vector space; secondly, to provide a general strategy to iteratively update the weights according to the approximation error and apply it to the spline fitting problem. In the experiments, we provide numerical examples for the case of polynomials and splines spaces. Subsequently, we evaluate the performance of our fitting scheme for spline curve and surface approximation, including adaptive spline constructions.

An assessment of numerical and geometrical quality of bases on surface fitting on Powell–Sabin triangulations

It is well known that the problem of fitting a dataset by using a spline surface minimizing an energy functional can be carried out by solving a linear system. Such a linear system strongly depends on the underlying functional space and, particularly, on the basis considered. Some papers in the literature study the numerical behavior and processing of the above-mentioned linear systems in specific cases. The bases that have local support and constitute a partition of unity have been shown to be interesting in the frame of geometric problems. In this work, we investigate the numerical effects of considering these bases in the quadratic Powell–Sabin spline space. Specifically, we present a direct approach to explore different preconditioning strategies and assess whether the already known ‘good’ bases also possess favorable numerical properties. Additionally, we introduce an inverse optimization approach based on a nonlinear optimization model to identify new bases that exhibit both good geometric and numerical properties.

Estimation of the Porosity and Permeability Fields of a Three-Phase Layered Reservoir Using the Spline Surfaces

Estimation of the Porosity and Permeability Fields of a Three-Phase Layered Reservoir Using the Spline Surfaces

Sensitivity analysis of multiple time-scale building energy using Bayesian adaptive spline surfaces

Sensitivity analysis has been widely used in building energy analysis to identify key variables influencing building energy and carbon emissions. However, most previous studies only focus on the single time scale energy use of buildings when conducting sensitivity analysis. The multiple time-scale sensitivity analysis would lead to increased complexity and challenges in building energy analysis. This paper proposes a new systematical procedure for global variance-based sensitivity analysis of multiple time-scale energy use of buildings. A case study of an office building is used to demonstrate the application of this approach using the Bayesian adaptive spline surface (BASS) models at three different time scales (annual, monthly, and daily). The results indicate that the procedure proposed here can provide fast and reliable sensitivity results for the multiple time-scale building energy assessment. The BASS learning models have the advantages of high predictive performance and the fast computation of sensitivity indicators in a closed form without Monte Carlo integrals. The application of principal component analysis can deal with the highly correlated multi-output energy use at a smaller time scale, which can further reduce computational costs. Moreover, the sensitivity analysis at the different time scales can provide new insights into energy characteristics due to the intrinsic variations of weather conditions. This systematical procedure can provide useful guidance for the multiple time-scale model calibration and multi-step ahead predictions of buildings. Moreover, the method proposed here can be extended to the sensitivity analysis of energy use in different time scales (sub-hourly, hourly, or daily) for other energy systems (such as PV, solar thermal, and wind).

Aeroelastic tailoring of stiffened cantilever plate using composites and structural layouts: A parametric study

AbstractThis paper studies the aeroelasticity of a stiffened cantilever plate using composite material and novel structural layouts. A comprehensive parametric study is conducted to determine the influence of different design parameters on the aeroelastic boundaries. Design parameters include plate sweep angle, ply orientation, stringer cross-section and stringer sweep angle. Nastran is used to run the aeroelastic analysis, and the process is automated using ${\rm{Matla}}{{\rm{b}}^{}}$ . The structure of the plate is modelled using laminate elements whereas the stringers are modelled using the Euler-Bernoulli beam elements. The unsteady aerodynamic loads are modelled using doublet-lattice method (DLM) and the structural and aerodynamic meshes are connected using an infinite plate surface (IPS) spline. A mesh sensitivity analysis is conducted to ensure fine meshes for the aerodynamics and structure. The study’s findings demonstrate the benefits of employing forward swept (Fw) stringers since it increases flutter speed by almost 38% compared to the unswept stringers case and prevents divergence. Moreover, the static aeroelastic analysis illustrates that the utilisation of Fw swept stringers can reduce the average tip displacement and tip twist effectively. T-shaped stringers are recommended to stiffen the plate due to their lower impact on the total mass of the plate. In some configurations, the structural layout has a much higher effect on the aeroelastic instabilities when compared to the material effect (ply orientation). However, results suggest combining both for some cases to get balanced washin and washout effects.

The Shape Parameter in the Shifted Surface Spline—A Sharp and Friendly Approach

This is a continuation of our previous study on the shape parameter contained in the shifted surface spline. We insist that the data points be purely scattered without meshes and the domain can be of any shape when conducting function interpolation by shifted surface splines. We also endeavor to make our approach easily accessible for scientists, not only mathematicians. However, the space of interpolated functions is smaller than that used before, leading to sharper function approximation. This function space has particular significance in numerical partial differential equations, especially for equations whose solutions lie in Sobolev space. Although the Fourier transform is deeply involved, scientists without a background in Fourier analysis can easily understand and use our approach.

Adaptive Spline Surface Fitting With Arbitrary Topological Control Mesh.

Reconstructing a spline surface from a given arbitrary topological triangle mesh is a fundamental and challenging problem in computer-aided design and engineering. This article introduces a novel surface fitting method utilizing G-NURBS capable of handling control meshes with arbitrary topologies. This method employs adaptive control point adjustment, guided by the geometric attributes of the input model, ensuring precise representation of sharp features such as edges and corners. Two primary strategies are employed: A parameter correspondence approach designed for sharp features and a control mesh iterative refinement technique that incorporates geometrical feature information. The proposed method has been tested and evaluated on various CAD models to demonstrate its effectiveness. This method can achieve higher fitting accuracy while faithfully preserving the geometrical features with fewer control points.

Interactive experience of sculpture design based on virtual reality technology

Abstract The application of computer technology provides new ideas for the design of sculpture works. In this paper, with the support of virtual reality technology, the pixel set of each color block is extracted from the actual image to determine the location of the center point of the color block. Based on the extracted information, a visual color marking method is used to realize the 3D registration of the sculpture scene. In the virtual scene, through the virtual modeling of the sculpture work, a spline surface model was generated by applying sparse image sequences and optimally adjusted to transform the surface piece control point problem into a quadratic function of the network vertices. The mathematical logic properties of the virtual reality sculpture work wind direction improved by 22.5 points, the mathematical logic properties of the light design improved by 25.8 points, and the mathematical logic properties of the shadow design improved by 23.1 points. The virtual reality technology provides an interactive experience for the design of sculpture works.

Restoration of spline surface of axial pump drive shaft by electric spark doping

This paper considers the possibility of restoring the splined surfaces of the drive shafts of axial pumps having specific wear by the electric spark doping (EIL) method, with the creation of coatings on their working surfaces, with corresponding mechanical characteristics. Electrode materials and processing modes are selected. Studies were carried out to determine the processing modes to obtain the optimal roughness of the applied coating. Keywords axial pump, electric spark doping, reduction, electrode materials, reduced energy, roughness

Developable Quad Meshes and Contact Element Nets

The property of a surface being developable can be expressed in different equivalent ways, by vanishing Gauss curvature, or by the existence of isometric mappings to planar domains. Computational contributions to this topic range from special parametrizations to discrete-isometric mappings. However, so far a local criterion expressing developability of general quad meshes has been lacking. In this paper, we propose a new and efficient discrete developability criterion that is applied to quad meshes equipped with vertex weights, and which is motivated by a well-known characterization in differential geometry, namely a rank-deficient second fundamental form. We assign contact elements to the faces of meshes and ruling vectors to the edges, which in combination yield a developability condition per face. Using standard optimization procedures, we are able to perform interactive design and developable lofting. The meshes we employ are combinatorially regular quad meshes with isolated singularities but are otherwise not required to follow any special curves on a developable surface. They are thus easily embedded into a design workflow involving standard operations like remeshing, trimming, and merging operations. An important feature is that we can directly derive a watertight, rational bi-quadratic spline surface from our meshes. Remarkably, it occurs as the limit of weighted Doo-Sabin subdivision, which acts in an interpolatory manner on contact elements.

Experimental and numerical flutter analysis of a folding fin with multiple asymmetric free-plays

Experimental folding fin models with an adjustable free-play are tested in a wind tunnel. The fin structure is modeled using the free-interface component mode synthesis method, and its free-play is modeled as four independent nonlinear springs with asymmetric stiffness. A nonplanar unsteady vortex-lattice method considering compressibility is employed to address nonlinear deformation and high subsonic flow. Surface spline interpolation is improved through projection and partition. The aeroelastic characteristics of folding fins with different free-play magnitudes, initial conditions and elastic-axis positions are analyzed using an established time-marching method because of its relatively small computation scale and high precision. The results show good consistency among the presented method, the wind tunnel test and the harmonic balance method. There is a negative correlation between the critical speed of divergent motion and the ratio of the initial condition to the free-play magnitude. If either the free-play magnitude or the initial condition is extreme (tiny or vast), the system nonlinearity degenerates to linearity. Generally, the flutter prevention design of a linear model can be applied to a nonlinear model, such as moving the elastic-axis position aftward. The presented fin configuration exhibits an unstable limit cycle oscillation because the orders of coupled flutter modes do not change with variations in equivalent linear stiffness.

Blending spline surfaces over polygon mesh and their application to isogeometric analysis

Blending spline surfaces over polygon mesh and their application to isogeometric analysis

Experimental and Numerical Flutter Analysis Using Local Piston Theory with Viscous Correction

Due to the maneuver and overload requirements of aircraft, it is inevitable that supersonic fins experience high angles of attack (AOAs) and viscous effects at high altitudes. The local piston theory with viscous correction (VLPT) is introduced and modified to account for the 3-dimensional effect. With the contribution of the explicit aerodynamic force expression and enhanced surface spline interpolation, a tightly coupled state-space equation of the aeroelastic system is derived, and a flutter analysis scheme of relatively small computational complexity and high precision is established with a mode tracking algorithm. A wind tunnel test conducted on a supersonic fin confirms the validity of our approach. Notably, the VLPT predicts a more accurate flutter boundary than the local piston theory (LPT), particularly regarding the decreasing trend in flutter speed as AOA increases. This is attributed to the VLPT’s ability to provide a richer and more detailed steady flow field. Specifically, as the AOA increases, the spanwise flow evolves into a gradually pronounced spanwise vortex, yielding an additional downwash and energizing the boundary layer, which is not captured by LPT. This indicates that the precision of LPT/VLPT significantly depends on the accuracy of steady flow results.

Effect of spanwise distributed camber morphing on dynamic stall characteristics of a finite-span wing

This paper investigates the influence of the spanwise-distributed trailing edge camber morphing on the dynamic stall characteristics of a finite-span wing at Re = 2 × 105. The mathematical model of the spanwise-distributed trailing-edge camber morphing is established based on Chebyshev polynomials, and the deformed wing surface is modeled by a spline surface according to rib's morphing in the chordwise direction. The computational fluid dynamics method is adopted to obtain flow-field results and aerodynamic forces. The shear-stress transportv-γ model is introduced and the overset mesh technique is adopted. The numerical results show that the spanwise distributed trailing edge morphing obviously changes the aerodynamic and energy transfer characteristics of the dynamic stall. Especially when the phase difference between the trailing edge motion and the wing pitch is −π/2, the interaction between the three-dimensional leading-edge vortex and trailing-edge vortex is strengthened, and the work done by the aerodynamic force turns negative. This indicates that the trailing edge deformation has the potential to suppress the oscillation amplitude of stall flutter. We also found that as the trailing-edge camber morphing varies more complex along the spanwise, and the suppression effect decreases accordingly.

A Cascadia Slab Model From Receiver Functions

AbstractWe map the characteristic signature of the subducting Juan de Fuca and Gorda plates along the entire Cascadia forearc from northern Vancouver Island, Canada, to Cape Mendocino in northern California, USA, using teleseismic receiver functions. The subducting oceanic crustal complex, possibly including subcreted material, is parameterized by three horizons capable of generating mode‐converted waves: a negative velocity contrast at the top of a low velocity zone underlain by two horizons representing positive contrasts. The amplitude of the conversions varies likely due to differences in composition and/or fluid content. We analyzed the slab signature for 298 long‐running land seismic stations, estimated the depth of the three interfaces through inverse modeling and fitted regularized spline surfaces through the station control points to construct a margin‐wide, double‐layered slab model. Crystalline terranes that act as the static backstop appear to form the major structural barrier that controls the slab morphology. Where the backstop recedes landward beneath the Olympic Peninsula and Cape Mendocino, the slab subducts sub‐horizontally, while the seaward‐protruding and thickened Siletz terrane beneath central Oregon causes steepening of the slab. A tight bend in slab morphology south of the Olympic Peninsula coincides with the location of recurring large intermediate depth earthquakes. The top‐to‐Moho thickness of the slab generally exceeds the thickness of the oceanic crust by 2–12 km, suggesting thickening of the slab or underplating of slab material to the overriding North American plate.

Using low-rank approximations of gridded data for spline surface fitting

The paper is devoted to the problem of finding bivariate tensor-product spline (or similar) functions that approximate gridded data in the least-squares sense. We propose to apply a low rank approximation of matrices to the data, to find the result by solving a sequence of univariate fitting problems that can be handled efficiently. This can be seen as a generalization of the method proposed by Georgieva and Hofreither (2017) that combines cross approximation (which is a particular method for low rank matrix approximation) with spline interpolation.While the algorithm yields the best least-squares approximation after r steps, where r denotes the rank of the data matrix, terminating it earlier yields a low-rank approximation, which often provides a sufficient level of accuracy. We also present a stopping criterion (based on a lower error estimate) that allows to use the method efficiently in the situation when the required number of degrees of freedom is not known in advance.