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Article
Publication date: 25 May 2021

Miaomiao Yang, Xinkun Du and Yongbin Ge

This meshless collocation method is applicable not only to the Helmholtz equation with Dirichlet boundary condition but also mixed boundary conditions. It can calculate not only…

Abstract

Purpose

This meshless collocation method is applicable not only to the Helmholtz equation with Dirichlet boundary condition but also mixed boundary conditions. It can calculate not only the high wavenumber problems, but also the variable wave number problems.

Design/methodology/approach

In this paper, the authors developed a meshless collocation method by using barycentric Lagrange interpolation basis function based on the Chebyshev nodes to deduce the scheme for solving the three-dimensional Helmholtz equation. First, the spatial variables and their partial derivatives are treated by interpolation basis functions, and the collocation method is established for solving second order differential equations. Then the differential matrix is employed to simplify the differential equations which is on a given test node. Finally, numerical experiments show the accuracy and effectiveness of the proposed method.

Findings

The numerical experiments show the advantages of the present method, such as less number of collocation nodes needed, shorter calculation time, higher precision, smaller error and higher efficiency. What is more, the numerical solutions agree well with the exact solutions.

Research limitations/implications

Compared with finite element method, finite difference method and other traditional numerical methods based on grid solution, meshless method can reduce or eliminate the dependence on grid and make the numerical implementation more flexible.

Practical implications

The Helmholtz equation has a wide application background in many fields, such as physics, mechanics, engineering and so on.

Originality/value

This meshless method is first time applied for solving the 3D Helmholtz equation. What is more the present work not only gives the relationship of interpolation nodes but also the test nodes.

Article
Publication date: 10 July 2009

R. Dyczij‐Edlinger and O. Farle

The purpose of this paper is to enable fast finite element (FE) analysis of electromagnetic structures with multiple geometric design variables.

Abstract

Purpose

The purpose of this paper is to enable fast finite element (FE) analysis of electromagnetic structures with multiple geometric design variables.

Design/methodology/approach

The proposed methodology combines multi‐variable model‐order reduction with mesh perturbation techniques and polynomial interpolation of parameter‐dependent FE matrices.

Findings

The resulting reduced‐order models are of comparable accuracy as but much smaller size than the original FE systems and preserve important system properties such as reciprocity.

Research limitations/implications

The method is limited to mesh variations that are obtained from a nominal discretization by continuous deformation. Topological changes in the mesh are not permissible.

Practical implications

In contrast to the underlying FE models, the resulting reduced‐order systems are very cheap to analyze. Possible applications include parametric libraries, design optimization, and real‐time control.

Originality/value

The paper extends the scope of moment‐matching order‐reduction techniques to a class of non‐polynomial systems arising from FE models with geometric parameters.

Details

COMPEL - The international journal for computation and mathematics in electrical and electronic engineering, vol. 28 no. 4
Type: Research Article
ISSN: 0332-1649

Keywords

Article
Publication date: 6 February 2017

Mohammad Heydari, Ghasem Barid Loghmani and Abdul-Majid Wazwaz

The main purpose of this paper is to implement the piecewise spectral-variational iteration method (PSVIM) to solve the nonlinear Lane-Emden equations arising in mathematical…

Abstract

Purpose

The main purpose of this paper is to implement the piecewise spectral-variational iteration method (PSVIM) to solve the nonlinear Lane-Emden equations arising in mathematical physics and astrophysics.

Design/methodology/approach

This method is based on a combination of Chebyshev interpolation and standard variational iteration method (VIM) and matching it to a sequence of subintervals. Unlike the spectral method and the VIM, the proposed PSVIM does not require the solution of any linear or nonlinear system of equations and analytical integration.

Findings

Some well-known classes of Lane-Emden type equations are solved as examples to demonstrate the accuracy and easy implementation of this technique.

Originality/value

In this paper, a new and efficient technique is proposed to solve the nonlinear Lane-Emden equations. The proposed method overcomes the difficulties arising in calculating complicated and time-consuming integrals and terms that are not needed in the standard VIM.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 27 no. 2
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 1 January 1996

I. Raspo, J. Ouazzani and R. Peyret

This paper presents a spectral multidomain method for solving theNavier‐Stokes equations in the vorticity‐stream function formulation. Thealgorithm is based on an extensive use of…

Abstract

This paper presents a spectral multidomain method for solving the Navier‐Stokes equations in the vorticity‐stream function formulation. The algorithm is based on an extensive use of the influence matrix technique and so leads to a direct method without any iterative process. Numerical results concerning the Czochralski melt configuration are reported and compared with spectral monodomain solutions to show the advantage of the domain decomposition for such a problem which solution presents a singular behaviour.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 6 no. 1
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 31 July 2019

S. Saha Ray and S. Behera

A novel technique based on Bernoulli wavelets has been proposed to solve two-dimensional Fredholm integral equation of second kind. Bernoulli wavelets have been created by…

Abstract

Purpose

A novel technique based on Bernoulli wavelets has been proposed to solve two-dimensional Fredholm integral equation of second kind. Bernoulli wavelets have been created by dilation and translation of Bernoulli polynomials. This paper aims to introduce properties of Bernoulli wavelets and Bernoulli polynomials.

Design/methodology/approach

To solve the two-dimensional Fredholm integral equation of second kind, the proposed method has been used to transform the integral equation into a system of algebraic equations.

Findings

Numerical experiments shows that the proposed two-dimensional wavelets technique can give high-accurate solutions and good convergence rate.

Originality/value

The efficiency of newly developed two-dimensional wavelets technique has been validated by different illustrative numerical examples to solve two-dimensional Fredholm integral equations.

Article
Publication date: 16 April 2018

Jinglai Wu, Zhen Luo, Nong Zhang and Wei Gao

This paper aims to study the sampling methods (or design of experiments) which have a large influence on the performance of the surrogate model. To improve the adaptability of…

Abstract

Purpose

This paper aims to study the sampling methods (or design of experiments) which have a large influence on the performance of the surrogate model. To improve the adaptability of modelling, a new sequential sampling method termed as sequential Chebyshev sampling method (SCSM) is proposed in this study.

Design/methodology/approach

The high-order polynomials are used to construct the global surrogated model, which retains the advantages of the traditional low-order polynomial models while overcoming their disadvantage in accuracy. First, the zeros of Chebyshev polynomials with the highest allowable order will be used as sampling candidates to improve the stability and accuracy of the high-order polynomial model. In the second step, some initial sampling points will be selected from the candidates by using a coordinate alternation algorithm, which keeps the initial sampling set uniformly distributed. Third, a fast sequential sampling scheme based on the space-filling principle is developed to collect more samples from the candidates, and the order of polynomial model is also updated in this procedure. The final surrogate model will be determined as the polynomial that has the largest adjusted R-square after the sequential sampling is terminated.

Findings

The SCSM has better performance in efficiency, accuracy and stability compared with several popular sequential sampling methods, e.g. LOLA-Voronoi algorithm and global Monte Carlo method from the SED toolbox, and the Halton sequence.

Originality/value

The SCSM has good performance in building the high-order surrogate model, including the high stability and accuracy, which may save a large amount of cost in solving complicated engineering design or optimisation problems.

Details

Engineering Computations, vol. 35 no. 2
Type: Research Article
ISSN: 0264-4401

Keywords

Article
Publication date: 29 May 2020

Wu Qin, Hui Yin, D.J. Yu and Wen-Bin Shangguan

This paper aims to develop an efficient numerical method for mid-frequency analysis of built-up structures with large convex uncertainties.

Abstract

Purpose

This paper aims to develop an efficient numerical method for mid-frequency analysis of built-up structures with large convex uncertainties.

Design/methodology/approach

Based on the Chebyshev polynomial approximation technique, a Chebyshev convex method (CCM) combined with the hybrid finite element/statistical energy analysis (FE-SEA) framework is proposed to fulfil the purpose. In CCM, the Chebyshev polynomials for approximating the response functions of built-up structures are constructed over the uncertain domain by using the marginal intervals of convex parameters; the bounds of the response functions are calculated by applying the convex Monte–Carlo simulation to the approximate functions. A relative improvement method is introduced to evaluate the truncated order of CCM.

Findings

CCM has an advantage in accuracy over CPM when the considered order is the same. Furthermore, it is readily to consider the CCM with the higher order terms of the Chebyshev polynomials for handling the larger convex parametric uncertainty, and the truncated order can be effectively evaluated by the relative improvement method.

Originality/value

The proposed CCM combined with FE-SEA is the first endeavor to efficiently handling large convex uncertainty in mid-frequency vibro-acoustic analysis of built-up structures. It also has the potential to serve as a powerful tool for other kinds of system analysis when large convex uncertainty is involved.

Details

Engineering Computations, vol. 37 no. 9
Type: Research Article
ISSN: 0264-4401

Keywords

Article
Publication date: 12 October 2018

Umer Saeed, Mujeeb ur Rehman and Qamar Din

The purpose of this paper is to propose a method for solving nonlinear fractional partial differential equations on the semi-infinite domain and to get better and more accurate…

Abstract

Purpose

The purpose of this paper is to propose a method for solving nonlinear fractional partial differential equations on the semi-infinite domain and to get better and more accurate results.

Design/methodology/approach

The authors proposed a method by using the Chebyshev wavelets in conjunction with differential quadrature technique. The operational matrices for the method are derived, constructed and used for the solution of nonlinear fractional partial differential equations.

Findings

The operational matrices contain many zero entries, which lead to the high efficiency of the method and reasonable accuracy is achieved even with less number of grid points. The results are in good agreement with exact solutions and more accurate as compared to Haar wavelet method.

Originality/value

Many engineers can use the presented method for solving their nonlinear fractional models.

Article
Publication date: 16 August 2022

Zibo Li, Zhengxiang Yan, Shicheng Li, Guangmin Sun, Xin Wang, Dequn Zhao, Yu Li and Xiucheng Liu

The purpose of this paper is to overcome the application limitations of other multi-variable regression based on polynomials due to the huge computation room and time cost.

Abstract

Purpose

The purpose of this paper is to overcome the application limitations of other multi-variable regression based on polynomials due to the huge computation room and time cost.

Design/methodology/approach

In this paper, based on the idea of feature selection and cascaded regression, two strategies including Laguerre polynomials and manifolds optimization are proposed to enhance the accuracy of multi-variable regression. Laguerre polynomials were combined with the genetic algorithm to enhance the capacity of polynomials approximation and the manifolds optimization method was introduced to solve the co-related optimization problem.

Findings

Two multi-variable Laguerre polynomials regression methods are designed. Firstly, Laguerre polynomials are combined with feature selection method. Secondly, manifolds component analysis is adopted in cascaded Laguerre polynomials regression method. Two methods are brought to enhance the accuracy of multi-variable regression method.

Research limitations/implications

With the increasing number of variables in regression problem, the stable accuracy performance might not be kept by using manifold-based optimization method. Moreover, the methods mentioned in this paper are not suitable for the classification problem.

Originality/value

Experiments are conducted on three types of datasets to evaluate the performance of the proposed regression methods. The best accuracy was achieved by the combination of cascade, manifold optimization and Chebyshev polynomials, which implies that the manifolds optimization has stronger contribution than the genetic algorithm and Laguerre polynomials.

Details

Engineering Computations, vol. 39 no. 8
Type: Research Article
ISSN: 0264-4401

Keywords

Article
Publication date: 18 July 2019

Gilberto Gomes, Alvaro Martins Delgado Neto, Luciano Mendes Bezerra and Ramon Silva

The purpose of this paper is to describe further developments on a novel formulation of the boundary element method (BEM) for inelastic problems using the dual reciprocity method…

Abstract

Purpose

The purpose of this paper is to describe further developments on a novel formulation of the boundary element method (BEM) for inelastic problems using the dual reciprocity method (DRM) but using object-oriented programming (OOP). As the BEM formulation generates a domain integral due to the inelastic stresses, the DRM is employed in a modified form using polyharmonic spline approximating functions with polynomial augmentation. These approximating functions produced accurate results in BEM applications for a range of problems tested, and have been shown to converge linearly as the order of the function increases.

Design/methodology/approach

A programming class named DRMOOP, written in C++ language and based on OOP, was developed in this research. With such programming, general matrix equations can be easily established and applied to different inelastic problems. A vector that accounts for the influence of the inelastic strains on the displacements and boundary forces is obtained.

Findings

The C++ DRMOOP class has been implemented and tested with the BEM formulation applied to classical elastoplastic problem and the results are reported at the end of the paper.

Originality/value

An object-oriented technology and the C++ DRMOOP class applied to elastoplastic problems.

Details

Multidiscipline Modeling in Materials and Structures, vol. 15 no. 5
Type: Research Article
ISSN: 1573-6105

Keywords

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