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ENGINEERING REPORTS 2019年第1期1卷

作者： Meira Menandro, Fernando Cesar Univ Fed Espirito Santo Mech Engn Dept Vitoria ES Brazil Av Fernando Ferrari 514 BR-29075910 Vitoria ES Brazil

radial basis functions (RBFs), first proposed for interpolation of scattered data, have gotten the scientific community interest in the past two decades, with applications ranging from interpolation with the dual reciprocity approach of the boundary element method to mesh-free finite element applications. The use of compactly supported RBFs (CSRBFs) has spread due to their localization properties. However, the mathematical derivation of continuous polynomial functions for different continuity requirements can be quite cumbersome, and although a few classes of functions have already been proposed, there is still room for nonpolynomial trial functions. In this paper, two new classes of CSRBFs are presented for which the continuity requirements are guaranteed. To justify the novelty claim, a view of RBF literature is conducted. The first function class proposed consists of a combination of different inverse polynomial functions, similar to inverse quadric functions, and is thus called inverse class. The second class is called the rational class, in which the functions are obtained as a ratio of two polynomial functions. The proposed functions are used on an approximation software, which takes advantage of their simplicity. The results demonstrate the accuracy and convergence of the proposed functions when compared to some referenced CSRBFs.

关键词： approximation compactly supported radial basis functions least squares methods nonpolynomial radial basis functions

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A new approach based on generalised multiquadric and compactly supported radial basis functions for solving two-dimensional Volterra-Fredholm integral equations

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INTERNATIONAL JOURNAL OF COMPUTATIONAL SCIENCE AND ENGINEERING 2022年第5期25卷 532-547页

作者： Takouk, Dalila Zeghdane, Rebiha Lakehali, Belkacem Univ Bordj Bou Arreridj Fac Math & Informat Dept Math El Anceur Algeria Univ Msila Fac Math & Informat Dept Math Msila Algeria

This article describes a numerical scheme to solve two-dimensional nonlinear Volterra-Fredholm integral equations (IEs). The method estimates the solution by compactly supported radial basis functions and compared with the approximation of the solution by generalised multiquadric radial basis function with the optimal strategy for the exponent beta. Integrals appearing in the procedure of the solution are approximated using shifted Legendre-Gauss-Lobatto nodes and weights. The method is mathematically simple and truly meshless. It can be used for high-dimensional problems because it does not require any cell structures. Finally, numerical experiments are given to show and test the applicability of the presented approach and confirm the theoretical analysis.

关键词： Volterra-Fredholm integral equations two-dimensional integral equations generalised multiquadric radial basis functions compactly supported radial basis functions interpolation method shifted Legendre-Gauss-Lobatto nodes and weights

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On Computation of Bessel and Airy Oscillatory Integral Transforms

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MATHEMATICAL METHODS IN THE APPLIED SCIENCES 2025年

作者： Khan, Suliman Arshad, Muhammad Wang, Bin Qian, Zhenghua Nanjing Univ Aeronaut & Astronaut State Key Lab Mech & Control Aerosp Struct Nanjing Peoples R China Abbottabad Univ Sci & Technol Dept Math Abbottabad Pakistan

In applied sciences, the analysis of Bessel and Airy oscillatory integrals is a demanding problem, particularly for large-scale data points and large frequency parameters. The Levin method, with global radial basis functions, is an accurate tool for approximating these integrals. But as the interpolation points or frequency increase, the interpolation matrix becomes dense and highly ill-conditioned. To ensure a stable and efficient computation of Bessel and Airy integrals, we implement the Levin method with compactly supported radial basis functions. Although the accuracy of the new algorithm has not significantly improved compared to the counterpart methods. Alternatively, the method exhibits faster and well-conditioned behavior, even for large numbers of data points and large frequency parameters. The convergence analysis of the method is performed and numerically verified with several benchmark problems.

关键词： Airy oscillatory integrals Bessel integral transforms compactly supported radial basis functions Levin method stable algorithms

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A local meshless collocation method for solving certain inverse problems

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ENGINEERING ANALYSIS WITH BOUNDARY ELEMENTS 2015年 57卷 9-15页

作者： Li, Wen Liu, Xiaoyan Yao, Guangming Taiyuan Univ Technol Coll Math Taiyuan 030024 Peoples R China Clarkson Univ Dept Math Potsdam NY USA

In this paper, we propose a meshless scheme based on compactly supported radial basis functions (CSRBFs) for solving the Cauchy problem of Poisson's equation and the inverse heat conduction problems in 2D. By assuming the unknown boundary condition to be a polynomial function, the inverse problems can be solved using a procedure similar to the process for solving forward problems. We employ Tikhonov regularization technique under L-curve regularization parameter to obtain a stable numerical solution. Numerical results verify the effectiveness and stability of this method. (C) 2014 Elsevier Ltd. All rights reserved.

关键词： Inverse problem compactly supported radial basis functions Local meshless method Tikhonov regularization L-curve

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Multimodal 3D Shape Reconstruction under Calibration Uncertainty Using Parametric Level Set Methods

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SIAM JOURNAL ON IMAGING SCIENCES 2020年第1期13卷 265-290页

作者： Eliasof, Moshe Sharf, Andrei Treister, Eran Ben Gurion Univ Negev Comp Sci Dept IL-8410501 Beer Sheva Israel

We consider the problem of 3D shape reconstruction from multimodal data, given uncertain calibration parameters. Typically, 3D data modalities can come in diverse forms such as sparse point sets, volumetric slices, and 2D photos. To jointly process these data modalities, we exploit a parametric level set method that utilizes ellipsoidal radial basis functions. This method not only allows us to analytically and compactly represent the object;it also confers on us the ability to overcome calibration-related noise that originates from inaccurate acquisition parameters. This essentially implicit regularization leads to a highly robust and scalable reconstruction, surpassing other traditional methods. In our results we first demonstrate the ability of the method to compactly represent complex objects. We then show that our reconstruction method is robust both to a small number of measurements and to noise in the acquisition parameters. Finally, we demonstrate our reconstruction abilities from diverse modalities such as volume slices obtained from liquid displacement (similar to CT scans and X-rays) and visual measurements obtained from shape silhouettes as well as point clouds.

关键词： 3D shape reconstruction parametric level sets dip transform joint reconstruction shape from silhouettes point clouds compactly supported radial basis functions

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Compact RBF meshless methods for photonic crystal modelling

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JOURNAL OF COMPUTATIONAL PHYSICS 2011年第12期230卷 4910-4921页

作者： Hart, E. E. Cox, S. J. Djidjeli, K. Univ Southampton CED Grp Sch Engn Sci Southampton SO17 1BJ Hants England

Meshless methods based on compact radial basis functions (RBFs) are proposed for modelling photonic crystals (PhCs). When modelling two-dimensional PhCs two generalised eigenvalue problems are formed, one for the transverse-electric (TE) mode and the other for the transverse-magnetic (TM) mode. Conventionally, the Band Diagrams for two-dimensional PhCs are calculated by either the plane wave expansion method (PWEM) or the finite element method (FEM). Here, the eigenvalue equations for the two-dimensional PhCs are solved using RBFs based meshless methods. For the TM mode a meshless local strong form method (RBF collocation) is used, while for the tricker TE mode a meshless local weak form method (RBF Galerkin) is used (so that the discontinuity of the dielectric function epsilon(x) can naturally be modelled). The results obtained from the meshless methods are found to be in good agreement with the standard PWEM. Thus, the meshless methods are proved to be a promising scheme for predicting photonic band gaps. (C) 2011 Elsevier Inc. All rights reserved.

关键词： Photonic crystal Meshless methods compactly supported radial basis functions

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Multiscale Support Vector Approach for Solving Ill-Posed Problems

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JOURNAL OF SCIENTIFIC COMPUTING 2015年第2期64卷 317-340页

作者： Zhong, Min Hon, Yiu Chung Lu, Shuai Southeast Univ Dept Math Nanjing 210096 Jiangsu Peoples R China City Univ Hong Kong Dept Math Hong Kong Hong Kong Peoples R China Fudan Univ Sch Math Sci Shanghai 200433 Peoples R China

Based on the use of compactly supported radial basis functions, we extend in this paper the support vector approach to a multiscale support vector approach (MSVA) scheme for approximating the solution of a moderately ill-posed problem on bounded domain. The Vapnik's -intensive function is adopted to replace the standard loss function in using the regularization technique to reduce the error induced by noisy data. Convergence proof for the case of noise-free data is then derived under an appropriate choice of the Vapnik's cut-off parameter and the regularization parameter. For noisy data case, we demonstrate that a corresponding choice for the Vapnik's cut-off parameter gives the same order of error estimate as both the a posteriori strategy based on discrepancy principle and the noise-free a priori strategy. Numerical examples are constructed to verify the efficiency of the proposed MSVA approach and the effectiveness of the parameter choices.

关键词： Multiscale support vector approach compactly supported radial basis functions Ill-posed problems Regularization methods

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A level-set-based topology and shape optimization method for continuum structure under geometric constraints

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STRUCTURAL AND MULTIDISCIPLINARY OPTIMIZATION 2014年第2期50卷 253-273页

作者： Liu, Tao Wang, Shuting Li, Bin Gao, Liang Huazhong Univ Sci & Technol Sch Mech Sci & Engn State Key Lab Digital Mfg Equipment & Technol Wuhan 430074 Hubei Peoples R China Huazhong Univ Sci & Technol Sch Mech Sci & Engn Wuhan 430074 Hubei Peoples R China

Recent advances in level-set-based shape and topology optimization rely on free-form implicit representations to support boundary deformations and topological changes. In practice, a continuum structure is usually designed to meet parametric shape optimization, which is formulated directly in terms of meaningful geometric design variables, but usually does not support free-form boundary and topological changes. In order to solve the disadvantage of traditional step-type structural optimization, a unified optimization method which can fulfill the structural topology, shape, and sizing optimization at the same time is presented. The unified structural optimization model is described by a parameterized level set function that applies compactly supported radial basis functions (CS-RBFs) with favorable smoothness and accuracy for interpolation. The expansion coefficients of the interpolation function are treated as the design variables, which reflect the structural performance impacts of the topology, shape, and geometric constraints. Accordingly, the original topological shape optimization problem under geometric constraint is fully transformed into a simple parameter optimization problem;in other words, the optimization contains the expansion coefficients of the interpolation function in terms of limited design variables. This parameterization transforms the difficult shape and topology optimization problems with geometric constraints into a relatively straightforward parameterized problem to which many gradient-based optimization techniques can be applied. More specifically, the extended finite element method (XFEM) is adopted to improve the accuracy of boundary resolution. At last, combined with the optimality criteria method, several numerical examples are presented to demonstrate the applicability and potential of the presented method.

关键词： Topological shape optimization Geometric constraints Level set method compactly supported radial basis functions R-function Extended finite element method

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Multilevel compact radial functions based computational schemes for some elliptic problems

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COMPUTERS & MATHEMATICS WITH APPLICATIONS 2002年第3-5期43卷 359-378页

作者： Chen, CS Ganesh, M Golberg, MA Cheng, AHD Univ Nevada Dept Math Sci Las Vegas NV 89154 USA Univ New S Wales Sch Math Sydney NSW 2052 Australia Univ Mississippi Dept Civil Engn University MS 38677 USA

compactly supported radial basis functions (CS-RBFs) have been recently introduced in the context of the dual reciprocity method as a possible cure of dense matrices and ill-conditioning problems when using the classical radial basis functions. However, the support scaling factor and slow convergence rate of the CS-RBFs have also raised issues on the effectiveness of the CS-RBFs. In this paper, two multilevel schemes have been proposed to alleviate these problems. (C) 2002 Elsevier Science Ltd. All rights reserved.

关键词： the method of fundamental solutions particular solutions compactly supported radial basis functions multilevel scheme dual reciprocity method boundary element method

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A novel single variable based topology optimization method for band gaps of multi-material phononic crystals

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STRUCTURAL AND MULTIDISCIPLINARY OPTIMIZATION 2022年第9期65卷 1-19页

作者： Chen, Hongfang Fu, Yu Hu, Yujin Ling, Ling Huazhong Univ Sci & Technol Sch Mech Sci & Engn State Key Lab Digital Mfg Equipment & Technol Wuhan 430074 Peoples R China

Multi-material phononic crystals have the potential to not only provide wide enough band gaps to fulfill sound insulation and shock isolation, but also reduce cost, weight, or enhance rigidity. A single variable based topology optimization method for band gaps of multi-material phononic crystals is proposed in this paper. compactly supported radial basis functions (CSRBF) integrated with a variant of Heaviside functions are utilized to describe the multiple materials fields, which can make the method free from traditional filtering procedures. An adjustable stair form interpolation model is adopted to construct the mapping from the single design variable field to the multi-material physical parameter field. It reduces the dependency of the dimension of design variables on the number of material types, which can significantly decrease the number of design variables and improve the computational efficiency. A nested form avoiding overlap among each material is adopted to characterize the multiple material phases. Several numerical examples are used to demonstrate the effectiveness of the presented method.

关键词： Multi-material topology optimization Phononic crystal band gaps Single variable-based stair form compactly supported radial basis functions

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