B-spline Research Articles

A numerical algorithm based on extended cubic B-spline functions for solving time-fractional convection-diffusion-reaction equation with variable coefficients

In this research, we develop an innovative and efficient numerical method for solving a nonlinear one-dimensional time-fractional convection-diffusion-reaction equation (TFCDRE) with a variable coefficient. This method integrates the Crank-Nicolson finite difference scheme with extended cubic B-spline (ExCBS) basis functions. The Caputo fractional derivative is applied for temporal discretization, while the ExCBS functions are utilized for spatial discretization. The stability of the method was discussed by the von Neumann method, which shows unconditional stability; and the convergence analysis secure an order of $O(h^2+\triangle t^{2-\alpha})$. We demonstrated the efficiency and simplicity of our method through three numerical experiments and verified its accuracy using the absolute error $(L_2)$ and maximum error $(L_\infty)$ norms in temporal and spatial dimensions. The results are also graphically represented and show substantial agreement with the approximated and exact solution. We confirmed the effectiveness and accuracy of our approach over the local discontinuous Galerkin finite element and the compact finite difference methods, respectively, from the literature.

Scraper Point Attitude Optimization in the Scraping Process of Aviation Composite Curved Surface

Aiming at the interference problem between the tool and the surface during the contact between the blade position and the composite material during the off-line programming scraping process of aviation composite parts, the contact effect of the blade during the scraping process of the composite surface is discussed, and a rotation projection attitude optimization strategy is proposed. Firstly, by constructing the mathematical model of the contact between the scraper and the surface, the geometric properties of the scraper and its contact on the surface are analyzed. Then, the non-uniform rational B-spline (NURBS) surface is used to reconstruct the surface, and the position information of the local area of the contact point is obtained to obtain the normal direction and curvature. In order to evaluate the contact effect of the scraper, an objective function is defined to minimize the contact error between the scraper linear trajectory and its projection on the surface, thereby quantifying the contact accuracy. By comparing the position and attitude of the scraper before and after optimization, the effectiveness of the proposed optimization method is verified. The optimized scraper posture significantly improves the contact accuracy with the composite surface, better adapts to the geometric characteristics of the complex surface, and ensures the efficiency and consistency of the scraping process. The research results provide a new perspective and method for the future development of automatic scraping process, and also provide an effective solution and technical means for other industries in the application scenario of curved scraping.

Isogeometric proportional topology optimization (IGA-PTO) for multi-material problems

This study addresses the multi-material optimization problem by the effective and robust isogeometric proportional topology optimization (IGA-PTO) approach. Non-uniform rational B-spline basis functions are used for descriptions of geometry, displacement field, and density fields of material phases. The alternating active-phase algorithm is employed to convert the optimization problem with multiple constraints into a series of sub-problems of single constraint. Several examples are exhibited, and intensive comparisons with the gradient-based optimality criteria (OC) algorithm are presented to highlight the superior performance of the proposed PTO algorithm. Finally, the optimized topologies are prototyped using 3D printing technology to evidence the manufacturing feasibility.

The e-MANTIS emulator: Fast and accurate predictions of the halo mass function in f(R)CDM and wCDM cosmologies

In this work, we present a novel emulator of the halo mass function (HMF), which we implemented in the framework of the e-MANTIS emulator of $f(R)$ gravity models. We also extended e-MANTIS to cover a larger cosmological parameter space and to include models of dark energy with a constant equation of state $w$CDM. We used a Latin hypercube sampling of the $w$CDM and $f(R)$CDM cosmological parameter spaces, over a wide range, and carried out a large suite of more than $10000$ $N$-body simulations with a different volume, mass resolution, and random phase for the initial conditions. For each simulation in the suite, we generated halo catalogues using the friends-of-friends (FoF) halo finder, as well as the spherical overdensity (SO) algorithm for different overdensity thresholds (200, 500, and 1000 times the critical density). We decomposed the corresponding HMFs on a B-spline basis, while adopting a minimal set of assumptions on their shape. We used this decomposition to train an emulator based on Gaussian processes. The resulting emulator is able to predict the HMF for redshifts $ 1.5$ and for halo masses $M_h M_ The typical HMF errors for SO haloes with $ c $ at $z=0$ in $w$CDM (respectively $f(R)$CDM) are of the order of $ up to a transition mass $M_t M_ ($M_t M_ For larger masses, the errors are dominated by the shot noise and scale as $ with $ up to $M_h M_ Independently of this general trend, the emulator is able to provide an estimation of its own error as a function of the cosmological parameters, halo mass, and redshift. We have performed an extensive comparison against analytical parametrizations and shown that e-MANTIS is able to better capture the cosmological dependence of the HMF, while being complementary to other existing emulators. The e-MANTIS emulator, which is publicly available, can be used to obtain fast and accurate predictions of the HMF in the $f(R)$CDM and $w$CDM non-standard cosmological models. As such, it represents a useful theoretical tool to constrain the nature of dark energy using data from galaxy cluster surveys.

Optimal Subsampling for Functional Quasi-Mode Regression with Big Data

We propose investigating optimal subsampling for functional regression with massive datasets based on the mode value, which is referred to as functional quasi-mode regression, to reduce data volume and alleviate computational burden. Using data-adaptive weights derived from regression residuals, the suggested regression offers enhanced robustness against nonnormal errors compared to traditional least squares or maximum likelihood estimation methods. To estimate the model, we employ B-spline basis functions to approximate the functional coefficient and include a penalty term in the objective function for enforcing smoothness in the resulting estimator. We adopt a computationally efficient mode-expectation-maximization algorithm, augmented by a Gaussian kernel, for numerical estimation. Under mild regularity conditions, we derive the asymptotic distributions of both full data and subsample quasi-mode estimators. The optimal subsampling probabilities by minimizing the asymptotic variance-covariance matrix under A- and L-optimality criteria are identified. These optimal probabilities rely on the full data estimate, prompting the development of a two-step algorithm to approximate the optimal subsampling procedure. The resultant algorithm is processing-efficient and can significantly reduce computational time compared to the full data approach. We also establish the asymptotic normality of the quasi-mode estimator obtained through this two-step algorithm. To assess finite sample performance, we conduct Monte Carlo simulations and analyze air quality data, showcasing the effectiveness of the developed estimator. Supplemental materials for this article are available online.

Application of B-spline basis functions as harmonic functions for the concurrent shape and mesh morphing of airfoils

The automatic deformation of the computational mesh along with the deformed geometry in design optimization cycles is a valuable procedure, as it reduces the required time for the construction of new meshes. The introduction of harmonic coordinates for the deformation of objects included within a closed mesh (cage) has been introduced in computer graphics. Harmonic coordinates result from solutions to the Laplace’s equation (harmonic functions) using a numerical solver. In this work, a modification to the classical harmonic coordinates’ concept is introduced for the deformation of 2D geometries (and the corresponding computational mesh) which are defined as B-spline curves. The B-spline basis functions are used as harmonic functions along the mesh boundary, being also the geometry to be deformed. Thus, any deformation of the B-spline boundary, through the movement of the curve’s control points, can be successfully propagated to the interior of the computational domain, resulting in the concurrent and conformable modification of the B-spline boundary and the entire computational mesh. For the computation of harmonic coordinates a node-centered Finite-Volume based Laplace solver for unstructured meshes is used, enhanced with an agglomeration multigrid scheme. The proposed method is applied and assessed for the shape and mesh morphing of airfoils.

The free vibration of the MEE sandwich microplate with the FG-CNTRC core using the MSGT

This study deals with the free vibration of the sandwich microplate with the core made of functionally graded carbon nanotube-reinforced composites (FG-CNTRC) and magneto-electro-elastic (MEE) face sheets. The governing equation of the microplates is derived by using the refined plate theory (RPT) with two variables and the modified strain gradient theory (MSGT). The Non-Uniform Rational B-Splines (NURBS) basis function of the isogeometric approach (IGA) is used for the approximation of the displacement and electric and magnetic fields of the microplates. The paper studies the effect of the length scale parameters (LSPs), CNTs distributions, CNTs volume fraction, magnetic and electric loads and geometry on the vibrational frequency of the MEE sandwich microplate.

An isogeometric approach to dynamic structures for integrating topology optimization and optimal control at macro and micro scales

In the context of active vibration suppression, an integrated topology optimization framework is proposed for the optimal control of dynamic structures. This article formulates a new composite objective function by considering the external harmonic excitations with different excitation frequencies and the complex-valued displacement field. Within the proposed optimal control performance function, the symmetric weighting matrix, the magnitudes of which are related to the importance of the control forces, is integrated into dynamic compliance to serve the state displacement. Utilizing the non-uniform rational B-splines (NURBS) basis functions, the material interpolation model defined at control points is described by the NURBS surface, which is incorporated into the optimization formulation for optimal structural and vibration control design. The sensitivity results are deduced in terms of the pseudo-densities and the corresponding weights at control points from both macro and micro perspectives of structural dynamics. To demonstrate the effectiveness of the integrated topology optimization of dynamic structures at macro and micro scales, several numerical examples, including the topology optimization of solid beams in plane stress and Mindlin plates with 3D microstructures, are provided and discussed in terms of structural shape and topology, frequency response curve, and modal displacement.

Rational [formula omitted] cubic Powell–Sabin B-splines with application to representation of ruled surfaces

This paper defines rational C1 cubic Powell–Sabin splines and analyses their basic properties. A rational B-spline basis is established and an algorithm for determining the corresponding control points and weights by using the blossoming operator is presented. The capability of the introduced splines to represent rational cubic triangular Bézier patches and quadratic NURPS is discussed and explicit conversion formulas are provided. Moreover, the application of the rational C1 cubic Powell–Sabin splines to representation of ruled surfaces is studied, showing that the cubic splines can give smoother parametrizations than the NURPS.

Adaptive mesh extended cubic B-spline method for singularly perturbed delay Sobolev problems

The purpose of this paper is to develop a robust numerical scheme for a class of singularly perturbed delay Sobolev (pseudo-parabolic) problems that have wide application in various branches of mathematical physics and fluid mechanics.For the small perturbation parameter, the standard numerical schemes for the solution of these problems fail to resolve the boundary layer(s) and the oscillations occur near the boundary layer. Thus, in this paper to resolve the boundary layer(s), im-plicit Euler scheme for the time derivatives on uniform mesh and extended B-spline basis functions consisting of free parameter λ are presented for spatial variable on Bakhvalov type mesh. The stability and uniform convergence analyisis of the proposed method are established. The error estimation of the developed method is shown to be firts order accurate in time and second order accurate in space. Numerical exprementation is carried out to validate the applicability of the developednumerical method. The numerical results reveals that the computational result is in agreement with the theoretical estimations

Robust algorithm for automatic surface-based outlier detection in MBES point clouds

Bathymetric multibeam echosounder systems (MBES) provide high-resolution mapping of underwater topography but are highly susceptible to errors due to harsh environmental conditions and the measurement process. Traditionally, manual post-processing is required to ensure data quality, a time-consuming, expensive, and subjective task. To address this issue, we propose a surface-based algorithm for pre-processing and cleaning MBES data that reduces manual intervention and improves consistency. A surface-based algorithm models the underwater topography as a surface instead of processing individual points. By assuming a continuous surface for underwater geometry, the algorithm easily identifies observations that deviate significantly from this model. The method combines a hierarchical B-spline surface with iterative robust estimation to automate data cleaning. Preliminary results on example datasets show a balanced outlier detection accuracy of 0.99, with manual processing time reduced from 2 days to just 30 min.

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.

Efficient evaluation of Bernstein-Bézier coefficients of B-spline basis functions over one knot span

New differential-recurrence relations for B-spline basis functions are given. Using these relations, a recursive method for finding the Bernstein-Bézier coefficients of B-spline basis functions over a single knot span is proposed. The algorithm works for any knot sequence and has an asymptotically optimal computational complexity. Numerical experiments show that the new method gives results which preserve a high number of digits when compared to an approach which uses the well-known de Boor-Cox formula.

NURBS-based isogeometric formulation for linear and nonlinear buckling analysis of laminated composite plates using constrained and unconstrained TSDTs

Two isogeometric plate models employing Reddy's third-order shear deformation theory (TSDT) and unconstrained third-order shear deformation theory (UTSDT) are presented and compared for linear and nonlinear buckling analysis of laminated composite plates with and without imperfection and subjected to different inplane loads. Cubic non-uniform rational B-spline (NURBS) basis functions that easily satisfy C1 continuity of the IGA-TSDT model are employed. The total Lagrangian approach in conjunction with the principle of virtual work is used to derive the governing equations. The primary and secondary solutions are traced using a tangent based arc-length method with a simple branch switching technique. The performance of the models is evaluated by validation and comparison with solutions obtained using ANSYS, Navier method (for linear analysis only), and those in the literature. The buckling response is significantly affected by pre-buckling boundary conditions and strain-displacement relations. The nonlinear buckling approach, among other approaches, is observed to be the most accurate methodology for an arbitrarily laminated composite plate. Further, IGA-UTSDT with nine DOF gives marginal improvement over IGA-TSDT with five DOF at the cost of computation. The IGA-TSDT is observed to be superior to FEM-TSDT in terms of computation demand and performance.

Free and forced vibration analysis of piezolaminated plates via an isogeometric layerwise finite element

Isogeometric layerwise finite element (L-IGA) formulation is a recent state-of-the-art approach integrating Non-Uniform Rational B-spline (NURBS) basis functions into the quasi-static solution process of piezolaminated composite plates. This study extends the application of the L-IGA framework to encompass free, forced vibration, and displacement control analyses of laminated composite plates with straight/curvilinear fibers and piezoelectric layers. To this end, the NURBS basis functions, utilized in geometry definition, are employed to solve electromechanically coupled differential equations following Hamilton’s variational principle. The adoption of high-order continuous NURBS shape functions throughout the IGA discretization span both in-plane and through-thickness laminate dimensions. This effectively facilitates precise geometry representation directly from Computer-Aided Design (CAD). Besides, such a discretization accelerates the convergence of displacement and electric potential solution fields toward exact results. Various benchmark problems have been solved to verify the robustness and high accuracy of the proposed dynamic L-IGA method. These include comparative analyses between L-IGA dynamic solutions (i.e. employing the Newmark-Beta method), analytical solutions, and ANSYS-Solid 226 finite element results. All the results are compared across various span-to-thickness ratios, mechanical-potential loading scenarios, and fiber orientation angles. Remarkably, the L-IGA method attains almost excellently accurate time response of various fields (displacement, stress, electric potential) and modal results, with considerably fewer mesh elements than Solid 226 solutions. Overall, such an outcome reveals the high potential and practical merits of the proposed L-IGA formulation as a proficient finite element approach for the dynamic analysis of piezolaminated plates.

Optimization and knowledge discovery of profiled end walls in a turbine stage at a low Reynolds number

To comprehensively explore flow control method of profiled end wall for turbine stage at low Reynolds numbers, a surrogate model optimization platform including non-uniform rational B-spline surface parameterization method, support vector regression, and improved chaos particle swarm optimization algorithm is integrated. Optimization designs have been carried out for stator profiled end walls, rotor profiled end wall, and combined end walls, respectively. The results indicate that under the constraint of the output power, the application of various profiled end wall design cases all can effectively improve the aerodynamic performance of the turbine stage. By organizing the flow field of downstream rotor, the profiled end wall of stator can significantly affect the stage efficiency. The flow control benefits of the profiled end wall of the rotor is from the obstruction of the cross migration of the pressure side leg of the horseshoe vortex. The application of profiled end wall on stator has the most practical engineering value. Self-organizing maps and Shapley methods are used to explore potential correlation information of aerodynamic parameters and summarize design experience. The sensitive design variables of profiled end walls are extracted. Based on the local controllability of NURBS surfaces, the regions that affect the stage efficiency are mainly concentrated in the middle of the stator passage, near the stator trailing edge and near the rotor leading edge. The regions with a significant impact on the output power of the turbine stage are near the trailing edge of the rotor and stator. The corresponding design rules of end walls modeling are summarized.

Enhancing SLAM efficiency: a comparative analysis of B-spline surface mapping and grid-based approaches

Enhancing SLAM efficiency: a comparative analysis of B-spline surface mapping and grid-based approaches

A B-spline material point method for deformation failure mechanism of soft–hard interbedded rock

Geological hazards related to soft–hard interbedded rock are frequent in rock engineering. The material point method (MPM) is a mesh-free numerical approach specifically designed for analyzing large deformations. Notably, significant grid-crossing errors frequently arise when material points traverse the underlying grid. To investigate the failure mechanism of soft–hard interbedded rock, an enhanced MPM incorporating B-spline basis functions and Voronoi polygon discretization is developed and subsequently validated through comparisons with uniaxial compression test data and other numerical methods. The numerical results of soft–hard interbedded rock specimens associated with different soft layer dips (SLD) and confining pressures indicate that the SLD has a great effect on compressive strength and crack extension at low confining pressure. Rocks from SLD-30° to SLD-75° correspond to the “sliding failure along discontinuities” failure mode and have lower compressive strength than rocks with other SLD angles. It is also demonstrated that the propagation of cracks leads to a significant alteration in the internal stress state of the rock, and that stress concentrations at the crack tip exacerbate the development of failure surface. Furthermore, the failure mode of soft–hard interbedded rock can be categorized into four types: (1) sliding failure across multiple discontinuities, (2) tensile fracture across multiple discontinuities, (3) sliding failure along discontinuities, (4) tensile-split along discontinuities.

Conversion from NURBS to Bézier representation

With the help of the Cox-de Boor recursion formula and the recurrence relation of the Bernstein polynomials, two categories of recursive algorithms for calculating the conversion matrix from an arbitrary non-uniform B-spline basis to a Bernstein polynomial basis of the same degree are presented. One is to calculate the elements of the matrix one by one, and the other is to calculate the elements of the matrix in two blocks. Interestingly, the weights in the two most basic recursion formulas are directly related to the weights in the recursion definition of the B-spline basis functions. The conversion matrix is exactly the Bézier extraction operator in isogeometric analysis, and we obtain the local extraction operator directly. With the aid of the conversion matrix, it is very convenient to determine the Bézier representation of NURBS curves and surfaces on any specified domain, that is, the isogeometric Bézier elements of these curves and surfaces.

A New Numerical Simulation for Modified Camassa-Holm and Degasperis-Procesi Equations via Trigonometric Quintic B-spline

In this study, the soliton solutions of the modified Camassa-Holm (mCH) and Degasperis-Procesi (mDP) equations, called modified b-equations with important physical properties, were obtained. The soliton waves' movement and positions formed by solving the mCH and mDP equations were calculated. Ordinary differential equation systems were obtained using trigonometric quintic B-spline bases for position and time direction derivatives in the equations to obtain numerical solutions. An algebraic equation system was then created by writing Crank-Nicolson type approximations for time and position-dependent terms. The stability analysis of this system was examined using the von-Neumann Fourier series method. L₂, L_{∞} and absolute error norms were used to measure the numerical results' convergence to the real solution. The numerical results calculated were compared with the exact solution and some studies in the literature.