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1 – 10 of 98This paper aims to propose an efficient and convenient numerical algorithm for two-dimensional nonlinear Volterra-Fredholm integral equations and fractional integro-differential…
Abstract
Purpose
This paper aims to propose an efficient and convenient numerical algorithm for two-dimensional nonlinear Volterra-Fredholm integral equations and fractional integro-differential equations (of Hammerstein and mixed types).
Design/methodology/approach
The main idea of the presented algorithm is to combine Bernoulli polynomials approximation with Caputo fractional derivative and numerical integral transformation to reduce the studied two-dimensional nonlinear Volterra-Fredholm integral equations and fractional integro-differential equations to easily solved algebraic equations.
Findings
Without considering the integral operational matrix, this algorithm will adopt straightforward discrete data integral transformation, which can do good work to less computation and high precision. Besides, combining the convenient fractional differential operator of Bernoulli basis polynomials with the least-squares method, numerical solutions of the studied equations can be obtained quickly. Illustrative examples are given to show that the proposed technique has better precision than other numerical methods.
Originality/value
The proposed algorithm is efficient for the considered two-dimensional nonlinear Volterra-Fredholm integral equations and fractional integro-differential equations. As its convenience, the computation of numerical solutions is time-saving and more accurate.
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Farshid Mirzaee and Nasrin Samadyar
The purpose of this paper is to develop a new method based on operational matrices of Bernoulli wavelet for solving linear stochastic Itô-Volterra integral equations, numerically.
Abstract
Purpose
The purpose of this paper is to develop a new method based on operational matrices of Bernoulli wavelet for solving linear stochastic Itô-Volterra integral equations, numerically.
Design/methodology/approach
For this aim, Bernoulli polynomials and Bernoulli wavelet are introduced, and their properties are expressed. Then, the operational matrix and the stochastic operational matrix of integration based on Bernoulli wavelet are calculated for the first time.
Findings
By applying these matrices, the main problem would be transformed into a linear system of algebraic equations which can be solved by using a suitable numerical method. Also, a few results related to error estimate and convergence analysis of the proposed scheme are investigated.
Originality/value
Two numerical examples are included to demonstrate the accuracy and efficiency of the proposed method. All of the numerical calculation is performed on a personal computer by running some codes written in MATLAB software.
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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.
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Marcos Arndt, Roberto Dalledone Machado and Adriano Scremin
The purpose of this paper is devoted to present an accurate assessment for determine natural frequencies for uniform and non-uniform Euler-Bernoulli beams and frames by an…
Abstract
Purpose
The purpose of this paper is devoted to present an accurate assessment for determine natural frequencies for uniform and non-uniform Euler-Bernoulli beams and frames by an adaptive generalized finite element method (GFEM). The present paper concentrates on developing the C1 element of the adaptive GFEM for vibration analysis of Euler-Bernoulli beams and frames.
Design/methodology/approach
The variational problem of free vibration is formulated and the main aspects of the adaptive GFEM are presented and discussed. The efficiency and convergence of the proposed method in vibration analysis of uniform and non-uniform Euler-Bernoulli beams are checked. The application of this technique in a frame is also presented.
Findings
The present paper concentrates on developing the C1 element of the adaptive GFEM for vibration analysis of Euler-Bernoulli beams and frames. The GFEM, which was conceived on the basis of the partition of unity method, allows the inclusion of enrichment functions that contain a priori knowledge about the fundamental solution of the governing differential equation. The proposed enrichment functions are dependent on the geometric and mechanical properties of the element. This approach converges very fast and is able to approximate the frequency related to any vibration mode.
Originality/value
The main contribution of the present study consisted in proposing an adaptive GFEM for vibration analysis of Euler-Bernoulli uniform and non-uniform beams and frames. The GFEM results were compared with those obtained by the h and p-versions of FEM and the c-version of the CEM. The adaptive GFEM has shown to be efficient in the vibration analysis of beams and has indicated that it can be applied even for a coarse discretization scheme in complex practical problems.
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Velinda Calvert and Mohsen Razzaghi
This paper aims to propose a new numerical method for the solution of the Blasius and magnetohydrodynamic (MHD) Falkner-Skan boundary-layer equations. The Blasius and MHD…
Abstract
Purpose
This paper aims to propose a new numerical method for the solution of the Blasius and magnetohydrodynamic (MHD) Falkner-Skan boundary-layer equations. The Blasius and MHD Falkner-Skan equations are third-order nonlinear boundary value problems on the semi-infinite domain.
Design/methodology/approach
The approach is based upon modified rational Bernoulli functions. The operational matrices of derivative and product of modified rational Bernoulli functions are presented. These matrices together with the collocation method are then utilized to reduce the solution of the Blasius and MHD Falkner-Skan boundary-layer equations to the solution of a system of algebraic equations.
Findings
The method is computationally very attractive and gives very accurate results.
Originality/value
Many problems in science and engineering are set in unbounded domains. One approach to solve these problems is based on rational functions. In this work, a new rational function is used to find solutions of the Blasius and MHD Falkner-Skan boundary-layer equations.
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Teaches by example the application of finite element templates in constructing high performance elements. The example discusses the improvement of the mass and geometric stiffness…
Abstract
Teaches by example the application of finite element templates in constructing high performance elements. The example discusses the improvement of the mass and geometric stiffness matrices of a Bernoulli‐Euler plane beam. This process interweaves classical techniques (Fourier analysis and weighted orthogonal polynomials) with newer tools (finite element templates and computer algebra systems). Templates are parameterized algebraic forms that uniquely characterize an element population by a “genetic signature” defined by the set of free parameters. Specific elements are obtained by assigning numeric values to the parameters. This freedom of choice can be used to design “custom” elements. For this example weighted orthogonal polynomials are used to construct templates for the beam material stiffness, geometric stiffness and mass matrices. Fourier analysis carried out through symbolic computation searches for template signatures of mass and geometric stiffness that deliver matrices with desirable properties when used in conjunction with the well‐known Hermitian beam material stiffness. For mass‐stiffness combinations, three objectives are noted: high accuracy for vibration analysis, wide separation of acoustic and optical branches, and low sensitivity to mesh distortion and boundary conditions. Only the first objective is examined in detail.
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Mateus Rauen, Roberto Dalledone Machado and Marcos Arndt
The purpose of this paper is to check the efficiency of isogeometric analysis (IGA) by comparing its results with classical finite element method (FEM), generalized finite element…
Abstract
Purpose
The purpose of this paper is to check the efficiency of isogeometric analysis (IGA) by comparing its results with classical finite element method (FEM), generalized finite element method (GFEM) and other enriched versions of FEM through numerical examples of free vibration problems.
Design/methodology/approach
Since its conception, IGA was widely applied in several problems. In this paper, IGA is applied for free vibration of elastic rods, beams and trusses. The results are compared with FEM, GFEM and the enriched methods, concerning frequency spectra and convergence rates.
Findings
The results show advantages of IGA over FEM and GFEM in the frequency error spectra, mostly in the higher frequencies.
Originality/value
Isogeometric analysis shows a feasible tool in structural analysis, with emphasis for problems that requires a high amount of vibration modes.
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Mário Rui Tiago Arruda and Dragos Ionut Moldovan
– The purpose of this paper is to report the implementation of an alternative time integration procedure for the dynamic non-linear analysis of structures.
Abstract
Purpose
The purpose of this paper is to report the implementation of an alternative time integration procedure for the dynamic non-linear analysis of structures.
Design/methodology/approach
The time integration algorithm discussed in this work corresponds to a spectral decomposition technique implemented in the time domain. As in the case of the modal decomposition in space, the numerical efficiency of the resulting integration scheme depends on the possibility of uncoupling the equations of motion. This is achieved by solving an eigenvalue problem in the time domain that only depends on the approximation basis being implemented. Complete sets of orthogonal Legendre polynomials are used to define the time approximation basis required by the model.
Findings
A classical example with known analytical solution is presented to validate the model, in linear and non-linear analysis. The efficiency of the numerical technique is assessed. Comparisons are made with the classical Newmark method applied to the solution of both linear and non-linear dynamics. The mixed time integration technique presents some interesting features making very attractive its application to the analysis of non-linear dynamic systems. It corresponds in essence to a modal decomposition technique implemented in the time domain. As in the case of the modal decomposition in space, the numerical efficiency of the resulting integration scheme depends on the possibility of uncoupling the equations of motion.
Originality/value
One of the main advantages of this technique is the possibility of considering relatively large time step increments which enhances the computational efficiency of the numerical procedure. Due to its characteristics, this method is well suited to parallel processing, one of the features that have to be conveniently explored in the near future.
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Dongliang Qi, Dongdong Wang, Like Deng, Xiaolan Xu and Cheng-Tang Wu
Although high-order smooth reproducing kernel mesh-free approximation enables the analysis of structural vibrations in an efficient collocation formulation, there is still a lack…
Abstract
Purpose
Although high-order smooth reproducing kernel mesh-free approximation enables the analysis of structural vibrations in an efficient collocation formulation, there is still a lack of systematic theoretical accuracy assessment for such approach. The purpose of this paper is to present a detailed accuracy analysis for the reproducing kernel mesh-free collocation method regarding structural vibrations.
Design/methodology/approach
Both second-order problems such as one-dimensional (1D) rod and two-dimensional (2D) membrane and fourth-order problems such as Euler–Bernoulli beam and Kirchhoff plate are considered. Staring from a generic equation of motion deduced from the reproducing kernel mesh-free collocation method, a frequency error measure is rationally attained through properly introducing the consistency conditions of reproducing kernel mesh-free shape functions.
Findings
This paper reveals that for the second-order structural vibration problems, the frequency accuracy orders are p and (p − 1) for even and odd degree basis functions; for the fourth-order structural vibration problems, the frequency accuracy orders are (p − 2) and (p − 3) for even and odd degree basis functions, respectively, where p denotes the degree of the basis function used in mesh-free approximation.
Originality/value
A frequency accuracy estimation is achieved for the reproducing kernel mesh-free collocation analysis of structural vibrations, which can effectively underpin the practical applications of this method.
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Shashank Vadlamani and Arun C.O.
The purpose of this paper is to discuss about evaluating the integrals involving B-spline wavelet on the interval (BSWI), in wavelet finite element formulations, using Gauss…
Abstract
Purpose
The purpose of this paper is to discuss about evaluating the integrals involving B-spline wavelet on the interval (BSWI), in wavelet finite element formulations, using Gauss Quadrature.
Design/methodology/approach
In the proposed scheme, background cells are placed over each BSWI element and Gauss quadrature rule is defined for each of these cells. The nodal discretization used for BSWI WFEM element is independent to the selection of number of background cells used for the integration process. During the analysis, background cells of various lengths are used for evaluating the integrals for various combination of order and resolution of BSWI scaling functions. Numerical examples based on one-dimensional (1D) and two-dimensional (2D) plane elasto-statics are solved. Problems on beams based on Euler Bernoulli and Timoshenko beam theory under different boundary conditions are also examined. The condition number and sparseness of the formulated stiffness matrices are analyzed.
Findings
It is found that to form a well-conditioned stiffness matrix, the support domain of every wavelet scaling function should possess sufficient number of integration points. The results are analyzed and validated against the existing analytical solutions. Numerical examples demonstrate that the accuracy of displacements and stresses is dependent on the size of the background cell and number of Gauss points considered per background cell during the analysis.
Originality/value
The current paper gives the details on implementation of Gauss Quadrature scheme, using a background cell-based approach, for evaluating the integrals involved in BSWI-based wavelet finite element method, which is missing in the existing literature.
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