B-spline Functions Research Articles

An objective isogeometric mixed finite element formulation for nonlinear elastodynamic beams with incompatible warping strains

AbstractWe present a stable mixed isogeometric finite element formulation for geometrically and materially nonlinear beams in transient elastodynamics, where a Cosserat beam formulation with extensible directors is used. The extensible directors yield a linear configuration space incorporating constant in-plane cross-sectional strains. Higher-order (incompatible) strains are introduced to correct stiffness, whose additional degrees of freedom are eliminated by an element-wise condensation. Further, the present discretization of the initial director field leads to the objectivity of approximated strain measures, regardless of the degree of basis functions. For physical stress resultants and strains, we employ a global patch-wise approximation using B-spline basis functions, whose higher-order continuity enables using much fewer degrees of freedom than an element-wise approximation. For time-stepping, we employ implicit energy–momentum consistent scheme, which exhibits superior numerical stability in comparison to standard trapezoidal and mid-point rules. Several numerical examples are presented to verify the present method.

Directional wave spectrum estimation through onboard measurement data utilizing B-spline basis functions

The utilization of digital twin technology, which integrates real-world measurement data with a digital representation of the vessel, is garnering increasing attention within the realm of structural integrity management. Due to limitations in sensor deployment on ships, the integration of the digital twin framework, which merges measurement data with the ship's digital design model, becomes crucial for ensuring the effectiveness of structural integrity management. This investigation introduces a methodological approach for estimating localized responses at unmeasured locations, leveraging both measurement and design datasets. To approximate the response at unobserved sites, the directional wave spectrum is initially determined utilizing the least squares method, thereby, minimizing the disparity between the estimated and measured response spectra. The methodology notably incorporates cubic B-spline interpolation specifically to smooth the directional wave spectrum deployed on the heading-frequency domain. This deliberate choice of employing cubic B-spline interpolation underscores the emphasis on achieving a refined and continuous representation of the directional wave spectrum, thus enhancing the accuracy and reliability of the estimation process. To validate the proposed methodology, synthetic pseudo-measurement data derived from the wave spectrum and RAO of a 13,000 TEU container ship are employed. Subsequently, the directional wave spectrum is estimated utilizing the pseudo-measurement data, and the estimated directional wave spectrum, in conjunction with the Response Amplitude Operator (RAO), is harnessed to approximate the response spectrum at the location where sensors are not installed.

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.

KAN-Transformer Model for UltraShort-Term Wind Power Prediction Based on EWMA Data Processing

When using the Transformer model for wind power prediction, the presence of noise in wind power data and the model’s final layer relying solely on a simple linear output reduces the model’s ability to capture nonlinear relationships, leading to a decrease in prediction accuracy. To address these issues, this paper proposes an ultrashort-term wind power prediction model based on exponential weighted moving average (EWMA) data processing and Kolmogorov–Arnold Network (KAN)-Transformer. First, multiple variable features are smoothed using EWMA, which suppresses noise while preserving the original data trends. Then, the EWMA-processed data is input into the Encoder and Decoder modules of the Transformer model to extract features. The output from the Decoder layer is then passed through the KAN layer, built using a cubic B-spline function, to enhance the model’s ability to capture nonlinear relationships, thereby improving the prediction accuracy of the Transformer model for wind power. Finally, experimental analysis is conducted, and it shows that the proposed model achieves the highest prediction accuracy, with a mean absolute error of 4.38 MW, a root mean squared error of 7.37 MW, and a coefficient of determination of 98.73%.

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.

Functional data analysis to describe and classify southern resident killer whale calls

The Southern Resident Killer Whale (SRKW) is an endangered population of whales found in the northeast Pacific. They have a vocal dialect unique from other killer whales, having a repertoire of distinct stereotyped calls. A framework for distinguishing SRKW call types using the frequency traces of the amplitude ridges from their spectrograms (termed frequency ridges) is proposed. The first step is the extraction of these ridges of SRKW calls using an Sequential Monte Carlo approach. Next, they are converted into functional data using B-spline functions. They are analyzed with a functional principal component (FPC) analysis to characterise the intrinsic variability of frequency ridges within a call type. The FPCs are able to capture the general patterns in the frequency ridges of the different SRKW call types. The FPCs are also used as the basis for call classification. Using a cross-validation procedure to assess the robustness of the classification, this framework proves to be successful for classification with some call types having an F1-score ≥80%, but other calls less well discriminated. On balance, this approach showed reasonable performance given the small sample size available, and provides a useful contribution towards the development of a universal tool for call classification.

KAN-HyperMP: An Enhanced Fault Diagnosis Model for Rolling Bearings in Noisy Environments

Rolling bearings often produce non-stationary signals that are easily obscured by noise, particularly in high-noise environments, making fault detection a challenging task. To address this challenge, a novel fault diagnosis approach based on the Kolmogorov–Arnold Network-based Hypergraph Message Passing (KAN-HyperMP) model is proposed. The KAN-HyperMP model is composed of three key components: a neighbor feature aggregation block, a feature fusion block, and a KANLinear block. Firstly, the neighbor feature aggregation block leverages hypergraph theory to integrate information from more distant neighbors, aiding in the reduction of noise impact, even when nearby neighbors are severely affected. Subsequently, the feature fusion block combines the features of these higher-order neighbors with the target node’s own features, enabling the model to capture the complete structure of the hypergraph. Finally, the smoothness properties of B-spline functions within the Kolmogorov–Arnold Network (KAN) are employed to extract critical diagnostic features from noisy signals. The proposed model is trained and evaluated on the Southeast University (SEU) and Jiangnan University (JNU) Datasets, achieving accuracy rates of 99.70% and 99.10%, respectively, demonstrating its effectiveness in fault diagnosis under both noise-free and noisy conditions.

A data-driven maximum entropy method for probability uncertainty analysis based on the B-spline theory

The probability density function (PDF) is quite important for structural reliability analysis; thus, accurate PDF modeling methods draw increasing attention. This paper proposes a novel metaheuristic data-driven paradigm of the maximum entropy method (MEM) based on the B-spline function theory. Firstly, a B-spline proxy of the MEM PDF is proposed for probability uncertainty analysis. We derive the parameter calculation formulation and calculate the undetermined parameters via the raw data of structural responses. Then, to determine the knots of the B-spline functions, we propose a novel data-driven approach with the aid of a powerful metaheuristic algorithm and the response data information. Different from the traditional MEM, the proposed method is a complete data-driven solution approach and does not involve the statistical moment calculation and the nonlinear equations composed of statistical moments. Combining the advantages of the B-spline theory and the MEM, the proposed method can reconstruct the response PDF with a complex shape, such as the PDF with multiple peaks or heavy tails. For verification, two numerical examples and one engineering example are analyzed, and compared with some classical PDF modeling methods. The results show that the proposed method is superior to the compared methods in terms of computational accuracy, when the same sample data is used.

Application of the Cubic Exponential B-spline Collocation Operator Splitting Method for Numerical solutions of Burgers Equation

In this study, the cubic exponential B-spline collocation method has been proposed for the numerical solutions of the Burgers equation with the operator splitting. To apply the operator splitting method, the Burgers' equation has decomposed into two sub-equations based on the time term: the linear part (diffusion) and the nonlinear part (convection). Subsequently, for each sub-equation, Crank-Nicolson finite difference schemes in the temporal direction and cubic exponential B-spline functions and their derivatives, have applied at the x_m nodal points in the spatial direction. The algebraic equation systems obtained have been solved numerically using the Lie-Trotter and Strang splitting schemes to get the solutions of the main equation. Some advantages of the splitting methods include preserving the physical characteristics of the solution, yielding more convergent results over long time intervals, enabling simpler algorithms, and facilitating the storage of solution vectors on computer. To assess the accuracy of the computed numerical results the L_2 and L_∞ error norms have been used. Additionally, the obtained results have been compared with some studies in the literature. The stability analysis of the applied method has been investigated using the von Neumann Fourier series method.

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

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.

A hyperelastic beam model for the photo-induced response of nematic liquid crystal elastomers

Liquid crystal elastomers (LCEs) are a novel class of materials created by combining polymeric solids with stiff, rod-like molecules known as nematic mesogens. These materials exhibit large, reversible deformations under mechanical, thermal, and optical stimuli. In this work, we develop a nonlinear beam formulation for analyzing the finite elastic deformation of beam-like structures made of LCEs under photo-actuation. This formulation applies to ideal, non-ideal, isotropic genesis, and nematic genesis LCEs. We establish the variational form of the problem based on the principle of virtual work. To solve numerical examples, we also develop a nonlinear finite element formulation based on B-spline functions. Several numerical examples are presented to demonstrate the applicability of the proposed formulation.

High-Order B-Spline Finite Difference Approach for Schrodinger Equation in Quantum Mechanics

This paper presents a new numerical method for solving the quantum mechanical complex-valued Schrodinger equation (CSE). The technique combines a second-order Crank-Nicolson scheme based on the finite element method (FEM) for temporal discretisation with nonic B-spline functions for spatial discretisation. This method is unconditionally stable with the help of Von-Neumann stability analysis. To verify our methodology, we examined an experiment utilising a range of error norms to compare experimental outcomes with analytical solutions. Our investigation verifies that the suggested approach works better than current methods, providing better accuracy and efficiency in quantum mechanical error analysis.

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.

Simulation of coupled groundwater flow and contaminant transport using quintic B-spline collocation method

PurposeThe purpose of this study is to introduce a new numerical scheme with no stability condition and high-order accuracy for the solution of two-dimensional coupled groundwater flow and transport simulation problems with regular and irregular geometries and compare the results with widely acceptable programs such as Modular Three-Dimensional Finite-Difference Ground-Water Flow Model (MODFLOW) and Modular Three-Dimensional Multispecies Transport Model (MT3DMS).Design/methodology/approachThe newly proposed numerical scheme is based on the method of lines (MOL) approach and uses high-order approximations both in space and time. Quintic B-spline (QBS) functions are used in space to transform partial differential equations, representing the relevant physical phenomena in the system of ordinary differential equations. Then this system is solved with the DOPRI5 algorithm that requires no stability condition. The obtained results are compared with the results of the MODFLOW and MT3DMS programs to verify the accuracy of the proposed scheme.FindingsThe results indicate that the proposed numerical scheme can successfully simulate the two-dimensional coupled groundwater flow and transport problems with complex geometry and parameter structures. All the results are in good agreement with the reference solutions.Originality/valueTo the best of the authors' knowledge, the QBS-DOPRI5 method is used for the first time for solving two-dimensional coupled groundwater flow and transport problems with complex geometries and can be extended to high-dimensional problems. In the future, considering the success of the proposed numerical scheme, it can be used successfully for the identification of groundwater contaminant source characteristics.

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.

Isogeometric Topology Optimization of Multi-Material Structures under Thermal-Mechanical Loadings Using Neural Networks

An isogeometric topology optimization (ITO) model for multi-material structures under thermal-mechanical loadings using neural networks is proposed. In the proposed model, a non-uniform rational B-spline (NURBS) function is employed for geometric description and analytical calculation, which realizes the unification of the geometry and computational models. Neural networks replace the optimization algorithms of traditional topology optimization to update the relative densities of multi-material structures. The weights and biases of neural networks are taken as design variables and updated by automatic differentiation without derivation of the sensitivity formula. In addition, the grid elements can be refined directly by increasing the number of refinement nodes, resulting in high-resolution optimal topology without extra computational costs. To obtain comprehensive performance from ITO for multi-material structures, a weighting coefficient is introduced to regulate the proportion between thermal compliance and compliance in the loss function. Some numerical examples are given and the validity is verified by performance analysis. The optimal topological structures obtained based on the proposed model exhibit both excellent heat dissipation and stiffness performance under thermal-mechanical loadings.

Detecting Alzheimer’s Patients using Features in Differential Waveforms of Pupil Light Reflex for Chromatic Stimuli

A procedure to detect irregular signal responses to pupil light reflex (PLR) was developed to detect Alzheimer’s Disease (AD) using a functional data analysis (FDA) technique and classification with an Elastic Net. In considering the differences in features of PLRs between AD and normal control (NC) participants, signals of summations and differentials between experimental conditions were analysed. The coefficient vectors for B-spline basis functions were introduced, and the number of basis was controlled for an optimised model. Model trained data was created using a data extension technique in order to enhance the number of participant observations. In the results, the required number of basis functions for differential signals is larger than the number for the their summation signals, and the features of differential signals contribute to classification performance.

Examination of optimized ultrashort three-color waveforms for generating short and intense isolated attosecond pulses in soft x rays

Isolated attosecond pulses (IAPs) can be readily generated via high-order harmonic generation driven by an ultrashort laser pulse. Here, it is shown that the best way to obtain the ultrashort waveform for producing a short and intense IAP in the soft x rays is to optimize the three-color (TC) laser pulse consisting of the fundamental field and its second and third harmonic fields. To calibrate it, another way of constructing the ultrashort waveform directly in time using a truncated basis set of B-spline functions is first proposed. The calibration waveform (CW) contains more frequency components up to the eighth harmonic order. It is found that the IAP by the TC waveform has a shorter duration after macroscopic propagation in a nonlinear gas medium compared to that by the CW field. It is uncovered that the CW field is additionally modified by the higher-order frequency components during propagation, dominated by the neutral atom dispersion. The effect of phase jitter in the TC waveform and the extension of the TC scheme into higher photon energies are also discussed. Currently, precise control of TC laser waveform synthesis is already achievable in the labs, thus paving an effective way for generating a useful attosecond light source in the soft x rays.

Numerical Solution to the Time-Fractional Burgers–Huxley Equation Involving the Mittag-Leffler Function

Fractional differential equations play a significant role in various scientific and engineering disciplines, offering a more sophisticated framework for modeling complex behaviors and phenomena that involve multiple independent variables and non-integer-order derivatives. In the current research, an effective cubic B-spline collocation method is used to obtain the numerical solution of the nonlinear inhomogeneous time-fractional Burgers–Huxley equation. It is implemented with the help of a θ-weighted scheme to solve the proposed problem. The spatial derivative is interpolated using cubic B-spline functions, whereas the temporal derivative is discretized by the Atangana–Baleanu operator and finite difference scheme. The proposed approach is stable across each temporal direction as well as second-order convergent. The study investigates the convergence order, error norms, and graphical visualization of the solution for various values of the non-integer parameter. The efficacy of the technique is assessed by implementing it on three test examples and we find that it is more efficient than some existing methods in the literature. To our knowledge, no prior application of this approach has been made for the numerical solution of the given problem, making it a first in this regard.