Dual Reciprocity Boundary Element Method Research Articles

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.

Solution to transient Navier–Stokes equations by the coupling of differential quadrature time integration scheme with dual reciprocity boundary element method

AbstractThe two‐dimensional time‐dependent Navier–Stokes equations in terms of the vorticity and the stream function are solved numerically by using the coupling of the dual reciprocity boundary element method (DRBEM) in space with the differential quadrature method (DQM) in time. In DRBEM application, the convective and the time derivative terms in the vorticity transport equation are considered as the nonhomogeneity in the equation and are approximated by radial basis functions. The solution to the Poisson equation, which links stream function and vorticity with an initial vorticity guess, produces velocity components in turn for the solution to vorticity transport equation. The DRBEM formulation of the vorticity transport equation results in an initial value problem represented by a system of first‐order ordinary differential equations in time. When the DQM discretizes this system in time direction, we obtain a system of linear algebraic equations, which gives the solution vector for vorticity at any required time level. The procedure outlined here is also applied to solve the problem of two‐dimensional natural convection in a cavity by utilizing an iteration among the stream function, the vorticity transport and the energy equations as well. The test problems include two‐dimensional flow in a cavity when a force is present, the lid‐driven cavity and the natural convection in a square cavity. The numerical results are visualized in terms of stream function, vorticity and temperature contours for several values of Reynolds (Re) and Rayleigh (Ra) numbers. Copyright © 2008 John Wiley & Sons, Ltd.

Application of dual reciprocity boundary element method to predict acoustic attenuation characteristics of marine engine exhaust silencers

In marine engine exhaust silencing systems, the presence of exhaust gas flow influences the sound propagation inside the systems and the acoustic attenuation performance of silencers. In order to investigate the effects of three-dimensional gas flow and acoustic damping on the acoustic attenuation characteristics of marine engine exhaust silencers, a dual reciprocity boundary element method (DRBEM) was developed. The acoustic governing equation in three-dimensional potential flow was derived first, and then the DRBEM numerical procedure is given. Compared to the conventional boundary element method (CBEM), the DRBEM considers the second order terms of flow Mach number in the acoustic governing equation, so it is suitable for the cases with higher Mach number subsonic flow. For complex exhaust silencers, it is difficult to apply the single-domain boundary element method, so a substructure approach based on the dual reciprocity boundary element method is presented. The experiments for measuring transmission loss of silencers are conducted, and the experimental setup and measurements are explained. The transmission loss of a single expansion chamber silencer with extended inlet and outlet were predicted by DRBEM and compared with the measurements. The good agreements between predictions and measurements are observed, which demonstrated that the derived acoustic governing equation and the DRBEM numerical procedure in the present study are correct.

Dynamic analysis of 3-D structures with adaptivity in RBF of dual reciprocity BEM

A new adaptive dual reciprocity boundary element method for dynamic analysis of 3-D structures is presented in this paper. It is based on finding the best approximation function of a radial basis function (RBF) group <TEX>$f=r^n+c$</TEX> which minimize the error of displacement field expansion. Also, the effects of some parameters such as the existence of internal points, number of RBF functions and position of collocation nodes in discontinuous elements are investigated in this adaptive procedure. Three numerical examples show improvement in dynamic response of structures with adaptive RBF in dual reciprocity with respect to ordinary BEM.

Numerical solution to steady and transient problems in thermohydrodynamic lubrication using a combination of finite element, finite volume and boundary element methods

A coupled model for pressure and temperature computations in a plain journal bearing which includes shaft and bushing thermal exchange with the external environment is presented and numerically solved here, both for the steady and the time-dependent case. The numerical solution to the global problem includes a combination of ad hoc techniques for the solution to each submodel. A finite element method for the hydrodynamic Reynolds equation with a cavitation model of Elrod–Adams is applied; the energy equation in the lubricating film is solved by a second order cell-vertex volume method, and the bushing temperature distribution is computed by using a (collocation) boundary element—dual reciprocity method.

Influence of Temperature Gradients on the Sound radiated from Flames

The far field pressure of a turbulent flame can be determined using the standard boundary element method (BEM) if the sound pressure or its derivative is known at a closed surface (control surface) surrounding the flame, as long as the medium outside the control surface is homogeneous. If temperature gradients are present, the homogeneous Helmholtz equation is no more valid. In that case, the wave equation can be rewritten in form of an inhomogeneous Helmholtz equation with a source term that depends also on the unknown pressure. Using the "Dual Reciprocity BEM" the integral form of this wave equation can be solved involving only surface integrals, so that the sound field can still be computed from field values at the control surface. The cases under study consider a volume of hot gas with a temperature distribution that is prescribed or obtained from a CFD simulation. The influence of the temperature gradients on the sound field can be evaluated by comparison of characteristic quantities like sound power and radiation patterns, with and without temperature gradient.

A coupled numerical scheme of dual reciprocity BEM with DQM for the transient elastodynamic problems

AbstractThe two‐dimensional transient elastodynamic problems are solved numerically by using the coupling of the dual reciprocity boundary element method (DRBEM) in spatial domain with the differential quadrature method (DQM) in time domain. The DRBEM with the fundamental solution of the Laplace equation transforms the domain integrals into the boundary integrals that contain the first‐ and the second‐order time derivative terms. Thus, the application of DRBEM to elastodynamic problems results in a system of second‐order ordinary differential equations in time. This system is then discretized by the polynomial‐based DQM with respect to time, which gives a system of linear algebraic equations after the imposition of both the boundary and the initial conditions. Therefore, the solution is obtained at any required time level at one stroke without the use of an iterative scheme and without the need of very small step size in time direction. The numerical results are visualized in terms of graphics. Copyright © 2008 John Wiley &amp; Sons, Ltd.

A dual reciprocity boundary element method for photothermal interactions in laser-induced thermotherapy

Laser-induced thermotherapy (LITT) is a minimally invasive laser hyperthermia procedure for the treatment of localized tumors. Mathematical modeling of the photothermal processes in laser-irradiated tissues is essential for optimal treatment planning. In this study, A Monte Carlo method is introduced to simulate photon transport in the tumor tissues with complex geometries. The dual reciprocity boundary element method (DRBEM) is then formulated to solve the bioheat transfer equation in the tumors. The model is validated with the finite difference solutions. To illustrate the applications of the proposed DRBEM, several laser delivery schemes, including external laser irradiation, single or multiple laser fiber delivery applicators, are studied for tumors with regular or irregular geometric shapes. The temperature transients, laser energy distribution and coagulation patterns for different laser delivery modes are demonstrated. The unique advantages of the DRBEM, such as easy adaptability to complex tumor geometries and no need to discretize the inner domain, may make it well-suited and robust approach for predicting and controlling the temperature evolution in laser-induced thermotherapy procedure.

Three-dimensional transient half-space dynamics using the dual reciprocity boundary element method

The dual reciprocity boundary element method (DRBEM) is studied thoroughly in the present paper. To the best knowledge of the authors, the DRBEM has never been applied to 3D half-space dynamics previously. In the present paper, the mathematical derivation of the method is presented, with stress placed on peculiarities of the method when applied to transient half-space dynamics. It has been found that semi-infinite domains (half-space) are more difficult to simulate, since the truncation of the discretization of the half-space surface by boundary elements results in excessive spurious wave reflection on the border of the discretized region. Mathematical derivation of the method is followed by its numerical implementation. Wave propagation due to various kinds of loading in the time-domain is studied afterward, aimed at the validation of the method. The main advantage of the DRBEM over its counterparts, used to model infinite and semi-infinite domains (such as classical BEM formulation, integral transformations, thin layer method), is that it produces time- and frequency-independent matrices (mass and stiffness matrix), by preserving a boundary-only discretization (no internal nodes necessary). This makes the method very attractive, since this feature is very close to the common engineering understanding and the final equation of motion has a similar form like the one known from the finite element method. Moreover, the formulation allows for seamless incorporation of non-homogeneous initial conditions, i.e. non-zero initial displacements and velocities and surface tractions can be prescribed.

Numerical simulation of non‐isothermal phase change problem using ADRBEM with augmented items

AbstractIn this paper, the phase change moving interface problem along the axial direction of the cylinder in the lead alloys containing tin is simulated by the axisymmetric dual reciprocity boundary element method (ADRBEM) with augmented items. The numerical method is verified by comparing with the analytical solution under a certain condition. The calculating results show that the ADRBEM with augmented items is an effective numerical method to solve the analogous problem of non‐isothermal phase change, which occurs in the crystal growth process. © 2007 Wiley Periodicals, Inc. Heat Trans Asian Res, 36(7): 408– 416, 2007; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/htj.20173

The dual reciprocity boundary element method (DRBEM) for two-dimensional sine-Gordon equation

This paper presents the dual reciprocity boundary element method (DRBEM) for solving two-dimensional sine-Gordon (SG) equation. The integral equation formulation employs the fundamental solution of the Laplace equation, and hence a domain integral arises in the boundary integral equation. Furthermore, the time derivatives are approximated by the time-stepping method, and the domain integral also appears from these approximations. The domain integral is transformed into boundary integral by using the dual reciprocity method (DRM). The linear radial basis function (RBF) is employed for DRM. The dynamics of line solitons and ring solitons of circular and elliptic shapes are studied. Numerical results are presented for some problems involving line and ring solitons to demonstrate the usefulness and accuracy of this approach.

A comparative study of the direct boundary element method and the dual reciprocity boundary element method in solving the Helmholtz equation

In this paper, we compare the direct boundary element method (BEM) and the dual reciprocity boundary element method (DRBEM) for solving the direct interior Helmholtz problem, in terms of their numerical accuracy and efficiency, as well as their applicability and reliability in the frequency domain. For BEM formulation, there are two possible choices for fundamental solutions, which can lead to quite different conclusions in terms of their reliability in the frequency domain. For DRBEM formulation, it is shown that although the DBREM can correctly predict eigenfrequencies even for higher modes, it fails to yield a reasonably accurate numerical solution for the problem when the frequency is higher than the first eigenfrequency. 2000 Mathematics subject classification: primary 65N38; secondary 35Q35. Keywords and phrases: the dual reciprocity boundary element method (DRBEM), Helmholtz equation, irregular frequencies.

The inverse source problem for the variable coefficients convection-diffusion equation

This article is concerned with the numerical solution of an inverse source problem associated with the variable coefficients convection-diffusion equation. The Dual Reciprocity Boundary Element Method (DRBEM) is combined with an iterative Sequential Quadratic Programming (SQP) procedure in order to identify the locations and strengths of several point sources. As the number of sources needs to be estimated a priori, there are three possible cases, namely correct, over and under estimation of this number. All the three situations are investigated and a study of stability of the proposed numerical technique is performed by adding random noise into the exact input data.

Preliminary study of non-isothermal phase change phenomena in vertical Bridgman crystal growth

Axisymmetric dual reciprocity boundary element method (DRBEM) with augmented items is extended to simulate the heat and mass transfer problems in the vertical Bridgman method (VBM) crystal growth of HgCdTe and CdZnTe. Axial solute concentration redistribution of three regions numerically reap-pears, and the influence of the pulling rate of the ampoule on it is further studied. Secondly, one di-mensional transient phase change phenomena is studied, and non-isothermal phase change phe-nomena is obtained from the initial transient region through the steady growth region to the final tran-sient region. Thirdly, the two-dimensional axisymmetric phase change interface position, interfacial shape and the temperature field of the melt and the crystal are numerically captured under the condi-tion to arrive at the steady state with zero pulling rate of the ampoule. Finally, the study of transient axisymmetric non-isothermal phase change phenomena is stressed and the results are compared with those in isothermal phase change. The influence of the pulling rate on non-isothermal phase change phenomena is revealed.

Dynamic Analysis of Piezoelectric Structures by the Dual Reciprocity Boundary Element Method

Dynamic Analysis of Piezoelectric Structures by the Dual Reciprocity Boundary Element Method

Dual reciprocity BEM for dynamic piezoelectricity

AbstractThe piezoelectric phenomenon is widely utilized in many devices. An analysis of piezoelectric devices requires a solution of coupled mechanical and electrical partial differential equations. In this paper the boundary element method (BEM) is implemented to solve the eigenvalue two‐dimensional problem of linear piezoelectricity. The piezoelectric material is modeled as: homogenous, transversal isotropic, linear elastic and dielectric. The numerical solution by the BEM requires fundamental solutions,which are derived using the Stroh formalism. The mass matrix of piezoelectric is computed by the dual reciprocity BEM. A numerical example shows that the dual reciprocity BEM formulation allows to analyze accurately dynamic problems of linear piezoelectricity. (© 2006 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Parameter identification in Helmholtz-type equations with a variable coefficient using a regularized DRBEM

The identification of the space-dependent coefficient in two-dimensional Helmholtz-type equations from a complete knowledge of the field modelled by these equations and its normal derivative on the boundary and additional internal measurements of the field is investigated. This problem is approached by combining a regularized dual reciprocity boundary element method (DRBEM) with a constrained non-linear minimisation technique. The optimal regularization parameter is chosen according to Morozov's discrepancy principle. The accuracy, convergence and stability of the proposed numerical method are carefully analysed and a sensitivity analysis with respect to the initial guess for the minimisation process is also performed. The numerical results obtained show that the combined regularized DRBEM-constrained non-linear minimisation technique is accurate, convergent, stable and robust.

A boundary element solution approach for the conjugate heat transfer problem in thermally developing region of a thick walled pipe

This paper presents a sole application of boundary element method to the conjugate heat transfer problem of thermally developing laminar flow in a thick walled pipe when the fluid velocities are fully developed. Due to the coupled mechanism of heat conduction in the solid region and heat convection in the fluid region, two separate solutions in the solid and fluid regions are sought to match the solid-fluid interface continuity condition. In this method, the dual reciprocity boundary element method (DRBEM) with the axial direction marching scheme is used to solve the heat convection problem and the conventional boundary element method (BEM) of axisymmetric model is applied to solve the heat conduction problem. An iterative and numerically stable BEM solution algorithm is presented, which uses the coupled interface conditions explicitly instead of uncoupled conditions. Both the local convective heat transfer coefficient at solid-fluid interface and the local mean fluid temperature are initially guessed and updated as the unknown interface thermal conditions in the iterative solution procedure. Two examples imposing uniform temperature and heat flux boundary conditions are tested in thermally developing region and compared with analytic solutions where available. The benchmark test results are shown to be in good agreement with the analytic solutions for both examples with different boundary conditions.

An analytical solution for long-wave scattering by a submerged circular truncated shoal

An analytical solution of the long-wave (or shallow-water) equation in closed-form is obtained for simple harmonic waves scattered by a submerged circular truncated shoal. This analytical solution is firstly validated against Longuet-Higgins’s classical analytical solution for a submerged cylinder, and then validated against numerical solutions obtained by using the DRBEM (dual reciprocity boundary-element method) model for a submerged circular truncated cone. Finally, the analytical solution is used to investigate the changing trend of maximum wave amplification, the trace pattern of focal position of wave-energy versus the wave period and the influence of shoal submergence on wave-energy trapping.

DRM-MD approach for modeling laser–material interaction with axial symmetry

A dual reciprocity method multi-domain (DRM-MD) approach for modeling laser–material interaction with axial symmetry was developed. The proposed approach is based on the fundamental solution for the Laplace equation in 2D and is much simpler for implementation than the dual reciprocity boundary element method (DRBEM) based on the fundamental solution for axisymmetric problems incorporating elliptic integrals. The thermal model of laser–material interaction was applied for the cases of mono as well as multi-layer structures. Different aspects of interaction up to the melting point of considered materials are presented. The effect of temperature dependence of the absorption coefficients on the process of laser heating was considered. Numerical results for spatial as well as temporal temperature distribution inside the material bulk are presented and compared to analytical solutions.