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Efficient evaluation of bernstein-bézier coefficients of b-spline basis functions over one knot span

COMPUTER-AIDED DESIGN 2025年 178卷

作者： Chudy, Filip Wozny, Pawel Univ Wroclaw Inst Comp Sci Ul Joliot Curie 15 PL-50383 Wroclaw Poland

New differential-recurrence relations for b-spline basis functions are given. Using these relations, a recursive method for finding the bernstein-b & eacute;zier 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.

关键词： b-spline basis functions bernstein-b & eacute zier form Recurrence relations b-spline curves b-spline surfaces de boor-Cox algorithm

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LDG approximation of a nonlinear fractional convection-diffusion equation using b-spline basis functions

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APPLIED NUMERICAL MATHEMATICS 2022年 171卷 45-57页

作者： Safdari, Hamid Rajabzadeh, Majid Khalighi, Moein Shahid Rajaee Teacher Training Univ Fac Math Sci Dept Appl Math Tehran Iran Univ Turku Fac Technol Dept Comp Turku Finland

This paper develops new numerical schemes for solution to nonlinear fractional convection diffusion equations of order beta is an element of [1, 2]. We propose the local discontinuous Galerkin methods by adopting linear, quadratic, and cubic b-spline basis functions and prove stability and optimal order of convergence O(h(k+1)) for the fractional diffusion problem. This method transforms the equation into a system of first-order equations and approximates the solution of the equation by selecting the appropriate basis functions. The b-spline functions significantly improve the accuracy and stability of the method. The performed numerical results demonstrate the efficiency and accuracy of the proposed scheme in different conditions and confirm the optimal order of convergence. (C) 2021 IMACS. Published by Elsevier b.V. All rights reserved.

关键词： Local discontinuous Galerkin method Convection-diffusion equation Fractional derivatives b-spline basis functions Stability

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Enhancement of the material point method using b-spline basis functions

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INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING 2018年第3期113卷 411-431页

作者： Gan, Yong Sun, Zheng Chen, Zhen Zhang, Xiong Liu, Yu Zhejiang Univ Fac Engn Hangzhou 310027 Zhejiang Peoples R China Beijing Inst Technol State Key Lab Explos Sci & Technol Beijing 100081 Peoples R China Dalian Univ Technol Fac Vehicle Engn & Mech State Key Lab Struct Anal Ind Equipment Dalian 116024 Peoples R China Univ Missouri Dept Civil & Environm Engn Columbia MO 65211 USA Tsinghua Univ Sch Aerosp Engn Beijing 100084 Peoples R China

The material point method (MPM) enhanced with b-spline basis functions, referred to as b-spline MPM (bSMPM), is developed and demonstrated using representative quasi-static and dynamic example problems. Smooth b-spline basis functions could significantly reduce the cell-crossing error as known for the original MPM. A Gauss quadrature scheme is designed and shown to be able to diminish the quadrature error in the bSMPM analysis of large-deformation problems for the improved accuracy and convergence, especially with the quadratic b-splines. Moreover, the increase in the order of the b-spline basis function is also found to be an effective way to reduce the quadrature error and to improve accuracy and convergence. For plate impact examples, it is demonstrated that the bSMPM outperforms the generalized interpolation material point (GIMP) and convected particle domain interpolation (CPDI) methods in term of the accuracy of representing stress waves. Thus, the bSMPM could become a promising alternative to the MPM, GIMP, and CPDI in solving certain types of transient problems.

关键词： b-spline basis functions Gauss quadrature large deformation material point method mesh refinement transient problems

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Higher-continuity s-version of finite element method with b-spline functions

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JOURNAL OF COMPUTATIONAL PHYSICS 2024年 497卷

作者： Magome, Nozomi Morita, Naoki Kaneko, Shigeki Mitsume, Naoto Univ Tsukuba Degree Programs Syst & Informat Engn Tennodai 1-1-1 Tsukuba Ibaraki 3058573 Japan Univ Tsukuba Inst Syst & Informat Engn Tennodai 1-1-1 Tsukuba Ibaraki 3058573 Japan Univ Tokyo Sch Engn Dept Syst Innovat Hongo 7-3-1Bunkyo Ku Tokyo 1138656 Japan

This paper proposes a strategy to solve the problems of the conventional s-version of finite element method (SFEM) fundamentally. because SFEM can reasonably model an analytical domain by superimposing meshes with different spatial resolutions, it has intrinsic advantages of local high accuracy, low computation time, and simple meshing procedure. However, it has disadvantages such as accuracy of numerical integration and matrix singularity. Although several additional techniques have been proposed to mitigate these limitations, they are computationally expensive or ad-hoc, and detract from the method's strengths. To solve these issues, we propose a novel strategy called b-spline based SFEM. To improve the accuracy of numerical integration, we employed cubic b-spline basis functions with C-2-continuity across element boundaries as the global basis functions. To avoid matrix singularity, we applied different basis functions to different meshes. Specifically, we employed the Lagrange basis functions as local basis functions. The numerical results indicate that using the proposed method, numerical integration can be calculated with sufficient accuracy without any additional techniques used in conventional SFEM. Furthermore, the proposed method avoids matrix singularity and is superior to conventional methods in terms of convergence for solving linear equations. Therefore, the proposed method has the potential to reduce computation time while maintaining a comparable accuracy to conventional SFEM.

关键词： s-version of finite element method Mesh superposition method b-spline basis functions Localized mesh refinement

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EM estimation of the b-spline copula with penalized pseudo-likelihood functions

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STATISTICAL PAPERS 2025年第1期66卷 1-34页

作者： Dou, Xiaoling Kuriki, Satoshi Lin, Gwo Dong Richards, Donald Japan Womens Univ Fac Sci 2-8-1 Bunkyo Ku Tokyo 1128681 Japan Inst Stat Math 10-3 Midoricho Tachikawa Tokyo 1908562 Japan Acad Sinica Inst Stat Sci 128Sect 2Acad Rd Taipei 11529 Taiwan Penn State Univ Dept Stat University Pk PA 16802 USA

The b-spline copula function is defined by a linear combination of elements of the normalized b-spline basis. We develop a modified EM algorithm, to maximize the penalized pseudo-likelihood function, wherein we use the smoothly clipped absolute deviation (SCAD) penalty function for the penalization term. We conduct simulation studies to demonstrate the stability of the proposed numerical procedure, show that penalization yields estimates with smaller mean-square errors when the true parameter matrix is sparse, and provide methods for determining tuning parameters and for model selection. We analyze as an example a data set consisting of birth and death rates from 237 countries, available at the website, "Our World in Data," and we estimate the marginal density and distribution functions of those rates together with all parameters of our b-spline copula model.

关键词： AIC bernstein copula b-spline basis functions b-spline copula EM algorithm Model selection SCAD penalty Tuning parameter selection

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b-spline-based material point method with dynamic load balancing technique for large-scale simulation

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ENGINEERING WITH COMPUTERS 2025年 1-18页

作者： Hidano, Soma Pan, Shaoyuan Yoshida, Keina Nomura, Reika Miki, Yohei Kawai, Masatoshi Moriguchi, Shuji Nakajima, Kengo Terada, Kenjiro Tohoku Univ Grad Sch Engn Dept Civil & Environm Engn 6-6-06 AzaaobaAramakiAoba Ku Sendai 9808572 Japan Tohoku Univ Int Res Inst Disaster Sci 468-1 AzaaobaAramakiAoba Ku Sendai 9808572 Japan Univ Tokyo Informat Technol Ctr 5-1-5 Kashiwanoha Kashiwa Chiba 2778589 Japan Nagoya Univ Informat Technol Ctr Furo ChoChikusa Ku Nagoya Aichi 4648603 Japan

In this study, a dynamic load-balancing (DLb) technique based on the sampling method is developed for MPMs using higher-order b-spline basis functions for parallel MPI calculations based on domain decomposition, in order to perform large-scale, long-duration landslide simulations in realistic computation time. Higher-order b-spline basis functions use a range of influence across cells compared to general basis functions, but this DLb technique dynamically adjusts the size of the computational domain according to the material point distribution, so that the material points are almost equally distributed across all cores. This allows the load bias between cores to be mitigated and the advantages of parallel computation to be fully exploited. Specifically, the novel contribution of this study is that the domain decomposition allows for proper communication between control points, even if the physical regions of the cores are staggered or non-adjacent, and even if the area of influence of b-spline basis functions spans multiple subdomains at this time. In numerical examples, the quasi-3D benchmark solid column collapse problem is computed for multiple core configurations to verify the effectiveness of the DLb method in terms of scalability and parallelization efficiency. The simulation of the full 3D column collapse problem also illustrates the applicability of the proposed DLb method to large-scale disaster simulations. Finally, to demonstrate the promise and capability of the DLb technique in the MPM algorithm, a full-scale size landslide disaster simulation is carried out to illustrate that it can withstand some practical size calculations.

关键词： Material point method b-spline basis functions Domain decomposition Dynamic load balancing Large-scale landslide simulation

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b-spline convected particle domain interpolation method

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ENGINEERING ANALYSIS WITH bOUNDARY ELEMENTS 2024年 160卷 106-133页

作者： Sadeghirad, Alireza Amirkabir Univ Technol Dept Civil & Environm Engn Tehran Iran

The b-spline material point method (bSMPM) has been shown to be a promising approach to effectively mitigate the cell-crossing instability by using high-order b-spline basis functions. However, the bSMPM still suffers from the quadrature error, another well-known source of inaccuracy in the material point methods. On the other hand, the convected particle domain interpolation (CPDI) method effectively reduces the quadrature error by using alternative basis functions. The b-spline basis functions can be naturally used within the CPDI formulation, here called the bSCPDI method. The bSCPDI may be prone to the cell-crossing instability since the alternative basis functions in the bSCPDI are always C0-continuous despite the high-continuity of the b-spline basis functions used as the original grid basis functions. In this paper, the reproducibility of the bSCPDI effective basis function gradients and the susceptibility of the method to the cell-crossing instability are investigated analytically and numerically. The effectiveness of the bSCPDI in mitigating the quadrature error is also investigated through numerical simulations.

关键词： Material point method convected particle domain interpolation (CPDI) method b-spline basis functions cell-crossing instability quadrature error

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b-spline signature responses in structural change detection: method development

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STRUCTURAL HEALTH MONITORING-AN INTERNATIONAL JOURNAL 2024年

作者： Gedney, brennan L. Naseri, Reza Al Kharousi, Shadhan Rizos, Dimitrios C. Univ South Carolina 300 Main Str Columbia SC 29208 USA

Damage detection is perhaps the most critical objective of structural health monitoring. Proper documentation of the progression of structural damage from onset to failure enables performance-based maintenance that mitigates risk and contains socioeconomic impact. Recently, damage detection methods that focus on change detection through analyses of the structural global response in the frequency or time domain have garnered attention due to their efficiency and accuracy. This work proposes a novel time-domain, nonparametric, data-driven change detection method. The time-domain b-spline transfer function is defined as a signature time history response of the structure, referred to in this work as the b-spline signature response (bSR). bSR may be computed directly through a physics-based simulation, or it be extracted in a discrete form from measurements of displacement, velocity, or acceleration time histories of structural response to any external excitation. The bSR is independent of loading, is a characteristic of the system, and captures the condition of the structure at the time of acquisition without the need for identifying structural properties. In this work, the fundamentals of the bSR concept are introduced first along with the process of extracting a bSR from structural responses to arbitrary excitations. Verification and validation studies are conducted based on computer simulations and laboratory testing and are discussed in detail. Subsequently, an approach is proposed to detect change through correlation of bSR signals acquired at different times during the monitoring period. Any changes in the intrinsic characteristics affecting the response cause bSR changes and correlation loss. The change detection methodology is demonstrated through computer simulations and experimental investigations. Initial parametric studies are conducted to investigate the suitability of the proposed approach to detect small structural changes. This article demonstrates the v

关键词： Change detection signature response data-driven approach direct time domain b-spline basis functions

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Conversion from NURbS to bézier representation

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COMPUTER AIDED GEOMETRIC DESIGN 2024年 113卷

作者： Yan, Lanlan East China Univ Technol Coll Sci Nanchang 330013 Jiangxi Peoples R China

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 b & eacute;zier 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 b & eacute;zier representation of NURbS curves and surfaces on any specified domain, that is, the isogeometric b & eacute;zier elements of these curves and surfaces.

关键词： NURbS b-spline basis functions bernstein basis functions Conversion matrix b & eacute zier extraction operator Recurrence relation

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Isogeometric buckling analysis of orthotropic thin-walled laminated beams using the Carrera unified formulation

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MECHANICS OF ADVANCED MATERIALS AND STRUCTURES 2023年第14期30卷 2926-2940页

作者： Sanati, Shadi Daraei, behnam Shojaee, Saeed Hamzehei-Javaran, Saleh Shahid Bahonar Univ Kerman Civil Engn Dept Kerman *** Iran

The current work presents the combination of isogeometric approach (IGA) and Carrera unified formulation (CUF) to perform the buckling analysis of isotropic thin-walled and orthotropic beams. Toward this end, the layerwise (LW) theory based CUF and b-spline basis functions in IGA have been used. The LW theory uses Lagrangian expansion (LE) to estimate the displacement field in the cross-section. However, in the present study, b-spline basis functions in both longitudinal direction and cross-section have substituted the corresponding functions. The CUF expresses the 3 D displacement fields as a set of 1 D generalized unknown displacement. Combined model in the present study obliterates the costly use of 3 D finite element analysis with its high accuracy and loyalty to 3 D problems. The principle of virtual displacement (PVD) is used to obtain the fundamental nuclei. One of the features of CUF is that it's higher-order expansion is free of the Poisson locking phenomenon. Also, there is no need to use the shear correction factor. Some examples of the isogeometric analysis of laminated composite beams with different geometries and boundary conditions are conducted and comparison of the critical buckling load results illustrate the proper validation and formulation of the present study.

关键词： Isogeometric approach Carrera unified formulation (CUF) buckling analysis layerwise theory (LW) b-spline basis functions laminated composite beams

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