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Article
Publication date: 19 June 2020

## Solving second-order nonlinear boundary value problem with nonlinear boundary conditions by an iterative method

The purpose of this paper is to solve the second-order nonlinear boundary value problem with nonlinear boundary conditions by an iterative numerical method.

### Abstract

#### Purpose

The purpose of this paper is to solve the second-order nonlinear boundary value problem with nonlinear boundary conditions by an iterative numerical method.

#### Design/methodology/approach

The authors introduce eigenfunctions as test functions, such that a weak-form integral equation is derived. By expanding the numerical solution in terms of the weighted eigenfunctions and using the orthogonality of eigenfunctions with respect to a weight function, and together with the non-separated/mixed boundary conditions, one can obtain the closed-form expansion coefficients with the aid of Drazin inversion formula.

#### Findings

When the authors develop the iterative algorithm, removing the time-varying terms as well as the nonlinear terms to the right-hand sides, to solve the nonlinear boundary value problem, it is convergent very fast and also provides very accurate numerical solutions.

#### Research limitations/implications

Basically, the authors’ strategy for the iterative numerical algorithm is putting the time-varying terms as well as the nonlinear terms on the right-hand sides.

#### Practical implications

Starting from an initial guess with zero value, the authors used the closed-form formula to quickly generate the new solution, until the convergence is satisfied.

#### Originality/value

Through the tests by six numerical experiments, the authors have demonstrated that the proposed iterative algorithm is applicable to the highly complex nonlinear boundary value problems with nonlinear boundary conditions. Because the coefficient matrix is set up outside the iterative loop, and due to the property of closed-form expansion coefficients, the presented iterative algorithm is very time saving and converges very fast.

### Details

Engineering Computations, vol. 38 no. 1
Type: Research Article
ISSN: 0264-4401

### Keywords

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Article
Publication date: 1 June 1959

## An Approximation to the Influence Function for Plate‐like Wings: A General Solution for Small Deflexions

A general solution for the small deflexions of thin plates of slowly varying thickness under lateral loading in the form of an influence function is briefly presented. It…

### Abstract

A general solution for the small deflexions of thin plates of slowly varying thickness under lateral loading in the form of an influence function is briefly presented. It is known that the influence function may be represented as an infinite series in terms of the eigenfunctions and eigenvalues associated with a homogeneous form of the plate differential equation. It is suggested that the series may give an acceptable approximation to the influence function when summed over a small number of terms when also the eigenfunctions and eigenvalues involved are deduced by an approximate procedure of the Rayleigh‐Ritz type. In order to test this assertion a numerical example is given for a uniform canti‐lever plate and the results are compared with experiment and with similar results deduced by an alternative theoretical procedure. Thus the calculation of a sufficient number of approximate normal vibration modes and frequencies for the plate as normally required for aeroelastic investigations may in this way be made to serve as the basis for a complete analysis of the plate. A simple approximate allowance for shear deflexion of the plate is presented and illustrated.

### Details

Aircraft Engineering and Aerospace Technology, vol. 31 no. 6
Type: Research Article
ISSN: 0002-2667

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Article
Publication date: 8 September 2020

## An h-version adaptive FEM for eigenproblems in system of second order ODEs: vector Sturm-Liouville problems and free vibration of curved beams

This study aims to overcome the involved challenging issues and provide high-precision eigensolutions. General eigenproblems in the system of ordinary differential…

### Abstract

#### Purpose

This study aims to overcome the involved challenging issues and provide high-precision eigensolutions. General eigenproblems in the system of ordinary differential equations (ODEs) serve as mathematical models for vector Sturm-Liouville (SL) and free vibration problems. High-precision eigenvalue and eigenfunction solutions are crucial bases for the reliable dynamic analysis of structures. However, solutions that meet the error tolerances specified are difficult to obtain for issues such as coefficients of variable matrices, coincident and adjacent approximate eigenvalues, continuous orders of eigenpairs and varying boundary conditions.

#### Design/methodology/approach

This study presents an h-version adaptive finite element method based on the superconvergent patch recovery displacement method for eigenproblems in system of second-order ODEs. The high-order shape function interpolation technique is further introduced to acquire superconvergent solution of eigenfunction, and superconvergent solution of eigenvalue is obtained by computing the Rayleigh quotient. Superconvergent solution of eigenfunction is used to estimate the error of finite element solution in the energy norm. The mesh is then, subdivided to generate an improved mesh, based on the error.

#### Findings

Representative eigenproblems examples, containing typical vector SL and free vibration of beams problems involved the aforementioned challenging issues, are selected to evaluate the accuracy and reliability of the proposed method. Non-uniform refined meshes are established to suit eigenfunctions change, and numerical solutions satisfy the pre-specified error tolerance.

#### Originality/value

The proposed combination of methodologies described in the paper, leads to a powerful h-version mesh refinement algorithm for eigenproblems in system of second-order ODEs, that can be extended to other classes of applications in damage detection of multiple cracks in structures based on the high-precision eigensolutions.

### Details

Engineering Computations, vol. ahead-of-print no. ahead-of-print
Type: Research Article
ISSN: 0264-4401

### Keywords

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Article
Publication date: 12 August 2020

## Steklov approximations of Green’s functions for Laplace equations

This paper aims to present a meshless technique to find the Green’s functions for solutions of Laplacian boundary value problems on rectangular domains. This paper also…

### Abstract

#### Purpose

This paper aims to present a meshless technique to find the Green’s functions for solutions of Laplacian boundary value problems on rectangular domains. This paper also investigates a theoretical basis for the Steklov series expansion methods to reduce and estimate the error of numerical approaches for the boundary correction kernel of the Laplace operator.

#### Design/methodology/approach

The main interest is how the Green's functions differ from the fundamental solution of the Laplace operator. Steklov expansion methods for finding the correction term are supported by the analysis that bases of the class of all finite harmonic functions can be formed using harmonic Steklov eigenfunctions. These functions construct a basis of the space of solutions of harmonic boundary value problems and their boundary traces generate an orthogonal basis of the trace space of solutions on the boundary.

#### Findings

The main conclusion is that the boundary correction term for the Green's functions is well-approximated by Steklov expansions with a few Steklov eigenfunctions. The error estimates for the Steklov approximations of the boundary correction term involved in Dirichlet or Robin boundary value problems are found. They appear to provide very good approximations in the interior of the region and become quite oscillatory close to the boundary.

#### Originality/value

This paper concentrates to document the first attempt to find the Green's function for various harmonic boundary value problems with the explicit Steklov eigenfunctions without concerns regarding discretizations when the region is a rectangle.

### Details

COMPEL - The international journal for computation and mathematics in electrical and electronic engineering , vol. 39 no. 4
Type: Research Article
ISSN: 0332-1649

### Keywords

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Article
Publication date: 6 November 2017

## Adaptive finite element method for eigensolutions of regular second and fourth order Sturm-Liouville problems via the element energy projection technique

The purpose of this paper is to present a numerically adaptive finite element (FE) method for accurate, efficient and reliable eigensolutions of regular second- and…

### Abstract

#### Purpose

The purpose of this paper is to present a numerically adaptive finite element (FE) method for accurate, efficient and reliable eigensolutions of regular second- and fourth-order Sturm–Liouville (SL) problems with variable coefficients.

#### Design/methodology/approach

After the conventional FE solution for an eigenpair (i.e. eigenvalue and eigenfunction) of a particular order has been obtained on a given mesh, a novel strategy is introduced, in which the FE solution of the eigenproblem is equivalently viewed as the FE solution of an associated linear problem. This strategy allows the element energy projection (EEP) technique for linear problems to calculate the super-convergent FE solutions for eigenfunctions anywhere on any element. These EEP super-convergent solutions are used to estimate the FE solution errors and to guide mesh refinements, until the accuracy matches user-preset error tolerance on both eigenvalues and eigenfunctions.

#### Findings

Numerical results for a number of representative and challenging SL problems are presented to demonstrate the effectiveness, efficiency, accuracy and reliability of the proposed method.

#### Research limitations/implications

The method is limited to regular SL problems, but it can also solve some singular SL problems in an indirect way.

#### Originality/value

Comprehensive utilization of the EEP technique yields a simple, efficient and reliable adaptive FE procedure that finds sufficiently fine meshes for preset error tolerances on eigenvalues and eigenfunctions to be achieved, even on problems which proved troublesome to competing methods. The method can readily be extended to vector SL problems.

### Details

Engineering Computations, vol. 34 no. 8
Type: Research Article
ISSN: 0264-4401

### Keywords

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Article
Publication date: 15 August 2019

## Vector eigenfunction expansion in the integral transform solution of transient natural convection

The purpose of this work is to revisit the integral transform solution of transient natural convection in differentially heated cavities considering a novel vector…

### Abstract

#### Purpose

The purpose of this work is to revisit the integral transform solution of transient natural convection in differentially heated cavities considering a novel vector eigenfunction expansion for handling the Navier-Stokes equations on the primitive variables formulation.

#### Design/methodology/approach

The proposed expansion base automatically satisfies the continuity equation and, upon integral transformation, eliminates the pressure field and reduces the momentum conservation equations to a single set of ordinary differential equations for the transformed time-variable potentials. The resulting eigenvalue problem for the velocity field expansion is readily solved by the integral transform method itself, while a traditional Sturm–Liouville base is chosen for expanding the temperature field. The coupled transformed initial value problem is numerically solved with a well-established solver based on a backward differentiation scheme.

#### Findings

A thorough convergence analysis is undertaken, in terms of truncation orders of the expansions for the vector eigenfunction and for the velocity and temperature fields. Finally, numerical results for selected quantities are critically compared to available benchmarks in both steady and transient states, and the overall physical behavior of the transient solution is examined for further verification.

#### Originality/value

A novel vector eigenfunction expansion is proposed for the integral transform solution of the Navier–Stokes equations in transient regime. The new physically inspired eigenvalue problem with the associated integmaral transformation fully shares the advantages of the previously obtained integral transform solutions based on the streamfunction-only formulation of the Navier–Stokes equations, while offering a direct and formal extension to three-dimensional flows.

### Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 29 no. 8
Type: Research Article
ISSN: 0961-5539

### Keywords

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Article
Publication date: 3 July 2017

## Modeling of spatial uncertainties in the magnetic reluctivity

The purpose of this paper is to present a computationally efficient approach for the stochastic modeling of an inhomogeneous reluctivity of magnetic materials. These…

### Abstract

#### Purpose

The purpose of this paper is to present a computationally efficient approach for the stochastic modeling of an inhomogeneous reluctivity of magnetic materials. These materials can be part of electrical machines such as a single-phase transformer (a benchmark example that is considered in this paper). The approach is based on the Karhunen–Loève expansion (KLE). The stochastic model is further used to study the statistics of the self-inductance of the primary coil as a quantity of interest (QoI).

#### Design/methodology/approach

The computation of the KLE requires solving a generalized eigenvalue problem with dense matrices. The eigenvalues and the eigenfunction are computed by using the Lanczos method that needs only matrix vector multiplications. The complexity of performing matrix vector multiplications with dense matrices is reduced by using hierarchical matrices.

#### Findings

The suggested approach is used to study the impact of the spatial variability in the magnetic reluctivity on the QoI. The statistics of this parameter are influenced by the correlation lengths of the random reluctivity. Both, the mean value and the standard deviation increase as the correlation length of the random reluctivity increases.

#### Originality/value

The KLE, computed by using hierarchical matrices, is used for uncertainty quantification of low frequency electrical machines as a computationally efficient approach in terms of memory requirement, as well as computation time.

### Details

COMPEL - The international journal for computation and mathematics in electrical and electronic engineering, vol. 36 no. 4
Type: Research Article
ISSN: 0332-1649

### Keywords

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Article
Publication date: 12 September 2018

## Integral transform solution of natural convection in a cylinder cavity with uniform internal heat generation

The purpose of this study is to propose the generalised integral transform technique to investigate the natural convection behaviour in a vertical cylinder under different…

### Abstract

#### Purpose

The purpose of this study is to propose the generalised integral transform technique to investigate the natural convection behaviour in a vertical cylinder under different boundary conditions, adiabatic and isothermal walls and various aspect ratios.

#### Design/methodology/approach

GITT was used to investigate the steady-state natural convection behaviour in a vertical cylinder with internal uniformed heat generation. The governing equations of natural convection were transferred to a set of ordinary differential equations by using the GITT methodology. The coefficients of the ODEs were determined by the integration of the eigenfunction of the auxiliary eigenvalue problems in the present natural convection problem. The ordinary differential equations were solved numerically by using the DBVPFD subroutine from the IMSL numerical library. The convergence was achieved reasonably by using low truncation orders.

#### Findings

GITT is a powerful computational tool to explain the convection phenomena in the cylindrical cavity. The convergence analysis shows that the hybrid analytical–numerical technique (GITT) has a good convergence performance in relatively low truncation orders in the stream-function and temperature fields. The effect of the Rayleigh number and aspect ratio on the natural convection behaviour under adiabatic and isothermal boundary conditions has been discussed in detail.

#### Originality/value

The present hybrid analytical–numerical methodology can be extended to solve various convection problems with more involved nonlinearities. It exhibits potential application to solve the convection problem in the nuclear, oil and gas industries.

### Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 28 no. 7
Type: Research Article
ISSN: 0961-5539

### Keywords

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Article
Publication date: 3 May 2016

## Nonlinear eigenvalue problem in the integral transforms solution of convection-diffusion with nonlinear boundary conditions

The purpose of this paper is to propose the generalized integral transform technique (GITT) to the solution of convection-diffusion problems with nonlinear boundary…

### Abstract

#### Purpose

The purpose of this paper is to propose the generalized integral transform technique (GITT) to the solution of convection-diffusion problems with nonlinear boundary conditions by employing the corresponding nonlinear eigenvalue problem in the construction of the expansion basis.

#### Design/methodology/approach

The original nonlinear boundary condition coefficients in the problem formulation are all incorporated into the adopted eigenvalue problem, which may be itself integral transformed through a representative linear auxiliary problem, yielding a nonlinear algebraic eigenvalue problem for the associated eigenvalues and eigenvectors, to be solved along with the transformed ordinary differential system. The nonlinear eigenvalues computation may also be accomplished by rewriting the corresponding transcendental equation as an ordinary differential system for the eigenvalues, which is then simultaneously solved with the transformed potentials.

#### Findings

An application on one-dimensional transient diffusion with nonlinear boundary condition coefficients is selected for illustrating some important computational aspects and the convergence behavior of the proposed eigenfunction expansions. For comparison purposes, an alternative solution with a linear eigenvalue problem basis is also presented and implemented.

#### Originality/value

This novel approach can be further extended to various classes of nonlinear convection-diffusion problems, either already solved by the GITT with a linear coefficients basis, or new challenging applications with more involved nonlinearities.

### Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 26 no. 3/4
Type: Research Article
ISSN: 0961-5539

### Keywords

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Article
Publication date: 22 May 2007

## Hybrid solution for transient internal convection with axial diffusion: Integral transforms and local instantaneous filtering

This paper seeks to analyze transient convection‐diffusion by employing the generalized integral transform technique (GITT) combined with an arbitrary transient filtering…

### Abstract

#### Purpose

This paper seeks to analyze transient convection‐diffusion by employing the generalized integral transform technique (GITT) combined with an arbitrary transient filtering solution, aimed at enhancing the convergence behavior of the associated eigenfunction expansions. The idea is to consider analytical approximations of the original problem as filtering solutions, defined within specific ranges of the time variable, which act diminishing the importance of the source terms in the original formulation and yielding a filtered problem for which the integral transformation procedure results in faster converging eigenfunction expansions. An analytical local instantaneous filtering is then more closely considered to offer a hybrid numerical‐analytical solution scheme for linear or nonlinear convection‐diffusion problems.

#### Design/methodology/approach

The approach is illustrated for a test‐case related to transient laminar convection within a parallel‐plates channel with axial diffusion effects.

#### Findings

The developing thermal problem is solved for the fully developed flow situation and a step change in inlet temperature. An analysis is performed on the variation of Peclet number, so as to investigate the importance of the axial heat or mass diffusion on convergence rates.

#### Originality/value

This paper succeeds in analyzing transient convection‐diffusion via GITT, combined with an arbitrary transient filtering solution, aimed at enhancing the convergence behaviour of the associated eigenfunction expansions.

### Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 17 no. 4
Type: Research Article
ISSN: 0961-5539

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