Dual Reciprocity Boundary Element Method Research Articles

Development and implementation of some BEM variants—A critical review

Due to rapid development of boundary element method (BEM), this article explores the evolution of BEM over the past half century. We here summarize the overall development and implementation of several well-known BEM variants that includes collocation BEM, galerkin BEM, dual reciprocity BEM, complex variable BEM and analog equation method. Their theoretical and mathematical backgrounds are carefully described and a generalized Laplace’s equation (and Poisson’s equation) is utilized in demonstrating the different approaches involved. An up-to-date review on characteristics and implementation for each of the five variants is presented and also highlighted their significant contributions in boundary element research. In addition, this article tries to cover whole aspect of interests including efficiency, applicability and accuracy in order to give better understanding of BEM evolution. Comparisons and techniques of improvement for these variants are also discussed.

DRBEM Solution of the Natural Convective Flow of Micropolar Fluids

The main purpose of this article is to present the use of the dual reciprocity boundary element method (DRBEM) in the analysis of the unsteady natural convective flow of micropolar fluids in a differentially heated rectangular cavity. The finite-difference method (FDM) is used for time discretization. All the convective terms and vorticity boundary condition are evaluated in terms of DRBEM coordinate matrix. Solutions are obtained for several values of microstructure parameter (k), Rayleigh number (Ra), and aspect ratio (A). Prandtl number values are taken as 0.71 and 7.0. The heat transfer rate (average Nusselt number) of micropolar fluids is found to be smaller than that of Newtonian fluid. Numerical results at steady-state are given in terms of streamlines, isotherms, vorticity contours, and velocity profiles, as well as a table containing Nusselt number values for several Ra and k.

Application of the dual reciprocity boundary integral equation technique to solve the nonlinear Klein–Gordon equation

This paper aims to obtain approximate solutions of the Nonlinear Klein–Gordon (NLKG) equation by employing the Boundary Integral Equation (BIE) method and the Dual Reciprocity Boundary Element Method (DRBEM). This method is improved by using a predictor–corrector scheme to the nonlinearity which appears in the problem. We employ the time stepping scheme to approximate the time derivative, and the Linear Radial Basis Functions (LRBFs), are used in the Dual Reciprocity (DR) technique. To confirm the accuracy of the new approach, the numerical results of a Double-Soliton and a problem with inhomogeneous terms are compared with analytical solutions and for the examples possessing single and periodic waves, two conserved quantities associated to the (NLKG) equation, the energy and the momentum are investigated.

A dual-reciprocity boundary element approach for axisymmetric nonlinear time-dependent heat conduction in a nonhomogeneous solid

A dual-reciprocity boundary element method is presented for the numerical solution of a nonsteady axisymmetric heat conduction problem involving a nonhomogeneous solid with temperature dependent properties. It is applied to solve some specific problems including one which involves the laser heating of a cylindrical solid.

Stability of Euler's method for evaluating large deformation of shear deformable plates by dual reciprocity boundary element method

In this paper the dual reciprocity boundary element method (DRBEM) is employed to evaluate large deformations of shear deformable plates. Incremental approaches utilizing the Euler method and the total incremental method are used for treating the nonlinear problems. A simple numerical algorithm for analyzing the problems is presented. Two numerical examples of square plates with different boundary conditions subjected to transverse loadings are presented to demonstrate the accuracy of the formulation. The Euler method seems to be a more stable method for treating nonlinear problems of the shear deformable plate if the dual reciprocity method is employed to evaluate the domain integrals that appear in the formulations.

DRBEM solution of natural convection flow of nanofluids with a heat source

This paper presents the dual reciprocity boundary element method (DRBEM) solution of the unsteady natural convective flow of nanofluids in enclosures with a heat source. The implicit Euler scheme is used for time integration. All the convective terms are evaluated in terms of DRBEM coordinate matrix. The vorticity boundary conditions are obtained from the Taylor series expansion of stream function equation. The results report that the average Nusselt number increases with the increase in both volume fraction and Rayleigh number. It is also observed that an increase in heater length reduces the heat transfer. The average Nusselt number of aluminum oxide-water based nanofluid is found to be smaller than that of copper-water based nanofluid. Results are given in terms of streamlines, isotherms, vorticity contours, velocity profiles and tables containing average Nusselt number for several values of Rayleigh number, heater length, volume fraction, and number of iterations together with CPU times.

DRBEM solution of exterior nonlinear wave problem using FDM and LSM time integrations

The nonlinear wave equation is solved numerically in an exterior region. For the discretization of the space derivatives dual reciprocity boundary element method (DRBEM) is applied using the fundamental solution of Laplace equation. The time derivative and the nonlinearity are treated as the nonhomogenity. The boundary integrals coming from the far boundary are eliminated using rational and exponential interpolation functions which have decay properties far away from the region of interest. The resulting system of ordinary differential equations in time are solved using finite difference method (FDM) with a relaxation parameter and least squares method (LSM). The proposed methods are examined with numerical test problems in which the behaviours of solutions are known. Although it gives almost the same accuracy with the DRBEM+FDM procedure, DRBEM+LSM solution procedure is preferred, since it is a direct method without the need of a parameter.

CV dual reciprocity BEM for transient flow in blocky systems with singular points and lines of discontinuities

The advantages of the complex variable (CV) and dual reciprocity method (DRM) are combined to effectively solve plane problems for heat (fluid, charge) flow in blocky systems with potential, flux, and/or conductivity discontinuity at contacts and with singular points such as crack tips, common apexes of structural elements and thin inclusions. We establish that in the problems considered it is reasonable to (i) distinguish the holomorphic “quasi-steady” part of the solution, (ii) use Gauss-type radial basis functions (RBF) to simplify further the employment of a found solution of a potential problem for solving thermo- (poro-) elastic problems, (iii) locate the centers of RBF outside of the contour of integration, (iv) account for the sensitivity of the solution to the distance between the centers of RBF by taking the normalized distance as d = t * / 1 . 2 8 to have the accuracy of one percent at a normalized time t *, and account for the sensitivity to the scaling parameter a sc by taking it within the range 1.6 d< a sc <8 d to avoid growing inaccuracy of the solution for small values of a sc ( a sc <1.6 d) and growing instability for large values of a sc ( a sc >8 d), and (v) use the solution of the ordinary differential equations (ODE) with respect to approximation parameters rather than with respect to the temperature at the centers of RBF. Numerical tests and examples illustrate the accuracy and abilities of the developed technique.

Simulation of Laser-Induced Thermotherapy Using a Dual-Reciprocity Boundary Element Model With Dynamic Tissue Properties

This paper presents a nonlinear dual-reciprocity boundary element method (DRBEM) for bioheat transfer in laser-induced thermotherapy. The nonlinearity stems from the dynamic changes of tissue thermophysical and optical properties and the blood perfusion rate during laser heating. The proposed DRBEM is coupled with a modified Monte Carlo method and the Arrhenius rate equation to investigate laser light propagation, bioheat transfer, and irreversible thermal damage in tumors. The computer code is justified by comparing the DRBEM results with the finite-difference results. The photothermal processes in interstitial laser thermotherapy with single or double laser fiber scattering applicators are chosen as the demonstrative examples. The dynamic nature, together with the unique advantages of "boundary-only" and excellent adaptability to complex anatomical geometries that the DRBEM method offers, makes the present nonlinear DRBEM a powerful tool for analysis and optimization of the parameters in laser surgical procedure.

Optimization of 3D cooling channels in injection molding using DRBEM and model reduction

Today, around 30% of manufactured plastic goods rely on injection moulding. The cooling time can represent more than 70% of the injection cycle. In this process, heat transfer during the cooling step has a great influence both on the quality of the final parts that are produced, and on the moulding cycle time. In the numerical solution of three-dimensional boundary value problems, the matrix size can be so large that it is beyond a computer capacity to solve it. To overcome this difficulty, we develop an iterative dual reciprocity boundary element method (DRBEM) to solve Poisson’s equation without the need of assembling a matrix. This yields a reduction of the computational space dimension from 3D to 2D, avoiding full 3D remeshing. Only the surface of the cooling channels needs to be remeshed at each evaluation required by the optimisation algorithm. For more efficiency, DRBEM computing results are extracted stored and exploited in order to construct a model with very few degrees of freedom. This approach is based on a model reduction technique known as proper orthogonal (POD) or Karhunen-Loeve decompositions. We introduce in this paper a practical methodology to optimise both the position and the shape of the cooling channels in 3D injection moulding processes. First, we propose an implementation of the model reduction in the 3D transient BEM solver. This reduction permits to reduce considerably the computing time required by each direct computation. Secondly, we present an optimisation methodology applied to different injection cooling problems. For example, we can minimize the maximal temperature on the cavity surface subject to a temperature uniformity constraint. Thirdly, we compare our results obtained by our approach with experimental results to show that our optimisation methodology is viable.

Solution for wave propagation through a circular cylinder mounted on different topography ripple-bed profile shoals using DRBEM

In this paper, the dual reciprocity boundary element method wave model that is governed by a modified mild-slope equation is developed. Instead of addressing the infinite boundary condition, the model automatically satisfies the Sommerfeld radiation condition. This work examines the effects of both the bottom curvature and the slope-squared terms. The conducted numerical experiments have a topography that is composed of a cylinder mounted on two kinds of basic shoal: types 1 and 2 cubic. In addition, numerical experiments that combine two cubic type islands with ripple-beds are performed. The results show that the curvature term dominates the calculation of the sinusoidal-varying topography; even a disturbance in the topography is insignificant. In contrast to smooth topographies, the bottom curvature and the slope-squared terms are not dominant. Thus, the extended terms can be neglected if the seabed is smooth and gentle. When the topographies are combined with ripple-beds, the difference in the dimensionless wave amplitudes along the sidewall of the types 1 and 2 cubic islands is found to be 2.9 and 3.02, respectively. According to our study of cubic islands that are combined with ripple-beds, we can conclude with certainty that even though the variation of the seabed is very slight, the curvature and the slope-squared terms cannot be neglected for an uneven bottom.

Determination of the Far Field Sound Radiation from Flames Using the Dual Reciprocity Boundary Element Method

The present work addresses the calculation of the radiated sound power spectrum in the far field of open or enclosed flames based on the application of the dual reciprocity boundary element method (DRBEM). The focus of this work is the development of a technique that allows a good representation of the sound field, assuming that the sound sources originated by the combustion are known. Following the acoustic analogy approach, the sound radiation of open flames can be evaluated by a volume integral over the equivalent acoustic source terms in the combustion zone. The DRBEM allows replacing the volume integral by a series of surface integrals. The technique is applied to the theoretical model of a spherical flame. The accuracy of the DRBEM model can be studied by comparing it with available analytical solutions for the sound radiation of this flame type. For enclosed flames, the DRBEM is used to calculate the sound propagation in the region of temperature gradients that appears at the exhaust of the combustion chamber. The sensitivity of the radiated sound field to this adjacent inhomogeneous temperature field is investigated. The non-uniqueness problem of the boundary integral equation method is tackled by the Burton and Miller approach. The DRBEM allows an effective determination of the radiated acoustic field because it can take into account inhomogeneities and satisfies automatically the radiation condition at infinity.

Study of convergence of homotopy perturbation method for systems of partial differential equations

The aim of this paper is convergence study of homotopy perturbation method for systems of nonlinear partial differential equations. The sufficient condition for convergence of the method is addressed. Since mathematical modeling of numerous scientific and engineering experiments lead to Brusselator and Burgers’ system of equations, it is worth trying new methods to solve these systems. We construct a new efficient recurrent relation to solve nonlinear Burgers’ and Brusselator systems of equations. Comparison of the results obtained by homotopy perturbation method with those of Adomian’s decomposition method and dual-reciprocity boundary element method leads to significant consequences. Two standard problems are used to validate the method.

Nonlinear heat equation for nonhomogeneous anisotropic materials: A dual‐reciprocity boundary element solution

AbstractA dual‐reciprocity boundary element method is presented for the numerical solution of initial‐boundary value problems governed by a nonlinear partial differential equation for heat conduction in nonhomogeneous anisotropic materials. To assess the validity and accuracy of the method, some specific problems are solved. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010

A coupled dual reciprocity BEM/genetic algorithm for identification of blood perfusion parameters

PurposeThe purpose of this paper is to present an inverse analysis procedure based on a coupled numerical formulation through which the coefficients describing non‐linear thermal properties of blood perfusion may be identified.Design/methodology/approachThe coupled numerical technique involves a combination of the dual reciprocity boundary element method (DRBEM) and a genetic algorithm (GA) for the solution of the Pennes bioheat equation. Both linear and quadratic temperature‐dependent variations are considered for the blood perfusion.FindingsThe proposed DRBEM formulation requires no internal discretisation and, in this case, no internal nodes either, apart from those defining the interface tissue/tumour. It is seen that the skin temperature variation changes as the blood perfusion increases, and in certain cases flat or nearly flat curves are produced. The proposed algorithm has difficulty to identify the perfusion parameters in these cases, although a more advanced genetic algorithm may provide improved results.Practical implicationsThe coupled technique allows accurate inverse solutions of the Pennes bioheat equation for quantitative diagnostics on the physiological conditions of biological bodies and for optimisation of hyperthermia for cancer therapy.Originality/valueThe proposed technique can be used to guide hyperthermia cancer treatment, which normally involves heating tissue to 42‐43°C. When heated up to this range of temperatures, the blood flow in normal tissues, e.g. skin and muscle, increases significantly, while blood flow in the tumour zone decreases. Therefore, the consideration of temperature‐dependent blood perfusion in this case is not only essential for the correct modelling of the problem, but also should provide larger skin temperature variations, making the identification problem easier.

Transient heat conduction in rolling/sliding components by a dual reciprocity boundary element method

A transfer matrix method associated with the dual reciprocity boundary element method (DRBEM) is developed for the study of transient heat conduction problems in presence of moving heat sources. In this method, the time integration is processed by an iteration transfer matrix method, the coefficient matrices are calculated only once, and no domain integration is required. It is shown that the application of DRBEM results in considerable savings in computation time and data preparation. Numerical examples are presented to demonstrate the efficiency and accuracy of the method by comparing the computed results with either published results or solutions using the finite element method. While only two-dimensional problems are presented in the paper, the method can be readily extended to three-dimensional problems to handle more complicated contact problems.

A numerical method based on the boundary integral equation and dual reciprocity methods for one-dimensional Cahn–Hilliard equation

This paper describes a numerical method based on the boundary integral equation and dual reciprocity methods for solving the one-dimensional Cahn–Hilliard (C–H) equation. The idea behind this approach comes from the dual reciprocity boundary element method that introduced for higher order dimensional problems. A time-stepping method and a predictor–corrector scheme are employed to deal with the time derivative and the nonlinearity respectively. Numerical results are presented for some examples to demonstrate the usefulness and accuracy of this approach. For these problems the energy functional dissipation and the mass conservation properties are investigated.

A time-stepping DRBEM for transient heat conduction in anisotropic solids

This paper is concerned with a dual reciprocity boundary element method (DRBEM) applied to transient heat conduction problems for time-dependent, anisotropic materials. The integral equation formulation employs the fundamental solution of the Laplace equation for linear isotropic materials. As a result, from the nonlinear and also from the anisotropic parts of the governing differential equation, domain integrals arise in the boundary integral equation. These domain integrals are transformed into boundary integrals by using a new radial basis function. Furthermore, the time derivative is approximated by a time-stepping scheme, which yields additional domain integrals in the formulation. Such a domain integral at each time-step can also be transformed into boundary integrals via the same dual reciprocity method. The details of the proposed DRBEM are presented in what follows. Because of the domain integral arising from material nonlinearity, the solution is inevitably performed in an iterative manner. The usefulness of the present iterative method is demonstrated through comparison of the results obtained by a special computer code developed by the authors with those obtained by the finite element method.

Boundary element solution of the two-dimensional sine-Gordon equation using continuous linear elements

This article studies the boundary element solution of two-dimensional sine-Gordon (SG) equation using continuous linear elements approximation. Non-linear and in-homogenous terms are converted to the boundary by the dual reciprocity method and a predictor–corrector scheme is employed to eliminate the non-linearity. The procedure developed in this paper, is applied to various problems involving line and ring solitons where considered in references [Argyris J, Haase M, Heinrich JC. Finite element approximation to two-dimensional sine-Gordon solitons. Comput Methods Appl Mech Eng 1991;86:1–26; Bratsos AG. An explicit numerical scheme for the sine-Gordon equation in 2+1 dimensions. Appl Numer Anal Comput Math 2005;2(2):189–211, Bratsos AG. A modified predictor–corrector scheme for the two-dimensional sine-Gordon equation. Numer Algorithms 2006;43:295–308; Bratsos AG. The solution of the two-dimensional sine-Gordon equation using the method of lines. J Comput Appl Math 2007;206:251–77; Bratsos AG. A third order numerical scheme for the two-dimensional sine-Gordon equation. Math Comput Simul 2007;76:271–8; Christiansen PL, Lomdahl PS. Numerical solutions of 2+1 dimensional sine-Gordon solitons. Physica D: Nonlinear Phenom 1981;2(3):482–94; Djidjeli K, Price WG, Twizell EH. Numerical solutions of a damped sine-Gordon equation in two space variables. J Eng Math 1995;29:347–69; Dehghan M, Mirzaei D. The dual reciprocity boundary element method (DRBEM) for two-dimensional sine-Gordon equation. Comput Methods Appl Mech Eng 2008;197:476–86]. Using continuous linear elements approximation produces more accurate results than constant ones. By using this approach all cases associated to SG equation, which exist in literature, are investigated.

A perturbation DRBEM model for weakly nonlinear wave run-ups around islands

In this paper, the dual reciprocity boundary element method (DRBEM) based on the perturbation method is presented for calculating run-ups of weakly nonlinear long waves scattered by islands. Under the assumption that the incident waves are harmonic, the time-dependent nonlinear Boussinesq equations are transformed into three time-independent linear equations by using the perturbation method, where, besides nonlinearity ε , the dispersion μ 2 is also included in the perturbed expansion. The first-order solution η 0 is found by using the linear long-wave equations. Then η 0 is used in two second-order governing equations related to the dispersion and nonlinearity, respectively. Since no any omission and approximation for the seabed slope ∇ h and its derivatives is made, there are the third- and fourth-order partial derivatives of η 0 appeared in the right-hand sides of two governing equations of the second-order. By employing a transformation, those third- and fourth-order partial derivatives are removed therefore large errors in approximating these derivatives are eliminated. To validate the new model, wave diffractions around a large vertical cylinder for 13 cases are first considered. It is found that the nonlinear contributions to the new model are significant for weakly nonlinear waves with a much better comparison with experimental results obtained than for the linear diffraction theory. It is also found that the dispersive effects play an important role in improving the accuracy of the new model as numerical results obtained from the Boussinesq equations (with dispersion terms) are more accurate than those from the Airy's equations (without dispersion term). Then the combined wave diffraction and refraction by a conical island is also modelled and discussed. Our model is not only accurate as the dispersive effects have been included but also computationally efficient since the domain integrals are merely evaluated by distributing collocation points over that surface.