Dual Reciprocity Method Research Articles

MHD natural convection in a semi-annulus enclosure filled with water-based nanofluid using DRBEM

Natural convection of a water-based nanofluid consisting Cu nanoparticles in a semi-annulus enclosure subjected to a magnetic field is considered. The inner wall has constant heat flux, the outer wall is maintained at a cold temperature and the other two walls are thermally insulated. The governing equations derived by considering the effect of magnetic field are solved numerically using the dual reciprocity boundary method (DRBEM) with constant elements. The boundary only nature of the dual reciprocity method results in considerably small computational cost in obtaining the solution. The simulations focus on the effects of the Rayleigh number, the Hartmann number, the nanoparticle volume fraction, and the inclination angles of the enclosure and the magnetic field on the flow and heat transfer characteristics. The results show that the convection and the average Nusselt number (Nu¯) decrease with an increase in the strength of the magnetic field whereas they increase by increasing the Rayleigh number. Furthermore, an increase in Hartmann number accelerates the rate of decrease in Nu¯ as inclination angles decrease. Thus, the inclined magnetic field and the inclination angle of the enclosure can be used as control parameters for enhancing the heat transfer convection and flow behavior in an enclosure.

A new approach to non-homogeneous hyperbolic boundary value problems using hybrid-Trefftz stress finite elements

A new approach to the solution of non-homogeneous hyperbolic boundary value problems is casted here using the hybrid-Trefftz stress/flux elements. Similarly to the Dual Reciprocity Method, the technique adopted in this paper uses a Trefftz-compliant set of functions to approximate the complementary solution of the problem and an additional trial basis to approximate its particular solution. However, the particular and complementary solutions are obtained here in a single step, and not sequentially, as typical of the Dual Reciprocity Method. The trial functions used for both particular and complementary solutions are merged together in the same basis and offered full flexibility to combine so as to recover the enforced equations in the best possible way. This option enables Trefftz-compliant functions to compensate for deficiencies of the particular solution basis, meaning that accurate total solutions can be obtained with relatively poor particular solution approximations. The formulation preserves the Hermitian, sparse and localized structure that typifies the matrix of coefficients of hybrid-Trefftz finite elements and avoids the drawbacks of the collocation procedures that arise in the Dual Reciprocity Method. Moreover, all terms of the matrix of coefficients are reduced to boundary integral expressions provided the particular solution trial functions satisfy the static problem obtained after discarding both non-homogeneous and time derivative terms from the governing equation.

DRBEM Applied to the 3D Helmholtz Equation and Its Particular Solutions with Various Radial Basis Functions

This paper presents to solve the 3D Helmholtz equation using dual reciprocity boundary element method (DRBEM) and its particular solutions with various radial basis functions (RBFs). The important function in this method is to employ the RBF. Here, we find the particular solution of the Helmholtz equation (∇2±k2)h= f(r), where f(r) is the RBF. Various RBFs are chosen and the particular solutions are obtained. The dual reciprocity method (DRM) is a method that converts the domain integral into the boundary integral. Mathematical formulations and discretization forms are described and discussed. Numerical results with three RBF with and without polynomial terms are presented and discussed. Algorithm of the method is also presented.

The dual reciprocity method and the numerical Green’s function for BEM fracture mechanic problems

The boundary element method has been applied with success to linear elastic fracture mechanic problems, involving static and dynamic cases. In order to solve body force problems (e.g., gravitational forces and transient problems with velocities and accelerations), Nardini and Brebbia presented, in 1982, the dual reciprocity formulation. Originally with the intention of solving transient problems using fundamental solutions of the static formulation, the procedure was found to be very efficient in the solution of body force problems as well. Also, a Green’s function corresponding to an embedded crack within the infinite medium can be introduced into the boundary element formulation as the fundamental solution. This yields accurate means of calculating only the external boundary unknown displacements and tractions and, in a post-processing scheme, determining the crack opening displacements. This paper introduces an approach that involves the numerical Green’s function procedure, of Telles and coworkers, and the dual reciprocity formulation. It compares beam solutions with the simulated effect of the total weight applied as a concentrated boundary force, the actual self-weight as a body force and a frequency- and time-dependent transient Heaviside load applied to a plate with a central crack.

PDE centres enhancement in the Localized Regular Dual Reciprocity Method

In this work we present an improvement of the Localized Regular Dual Reciprocity Method (LRDRM). LRDRM is an integral domain decomposition method with two distinguishing features, the boundary conditions are imposed at the local interpolation level and all the calculated integrals are regular. In this work we present an enhancement of this method where the interpolation functions themselves satisfy the partial differential equation to be solved. Results for 1D and 2D convection‐diffusion, 2D Helmholtz and 2D Poisson equations are presented, attaining accuracies two to three orders of magnitude higher than the original version of the LRDRM.

Solutions of the Acoustic Problem in the 3D Form of the Helmholtz Equation Using DRBEM

This paper presents to the solutions of the acoustic problem using dual reciprocity boundary element method (DRBEM). The acoustic propagates inside the silencer and its sound can be formulated by the three dimensional (3D) Helmholtz equation. The form of the equation is which can be solved by the integral form on the boundary. The conventional boundary element method (BEM) is not suitable for solving the acoustic problems of silencers with higher Mach number subsonic flow, due to the presence of domain integral. The dual reciprocity method (DRM) is a method that converts the domain integral into the boundary integral. Mathematical formulations discretisation form and evaluation of the integrals are described and discussed. Algorithm of the method is also presented.

Reduced order model analysis method via proper orthogonal decomposition for variable coefficient of transient heat conduction based on boundary element method

Boundary element method (BEM) is widely used in engineering analysis, especially in solving the transient heat conduction problem because of the advantage that only boundary of the problem needs to be discretized into elements. The general procedure of solving the variable-coefficient transient heat conduction problem by using the BEM is as follows. First, the governing differential equations are transformed into the boundary-domain integral equations by adopting the basic solution of the linear and homogeneous heat conduction problemGreen function. Second, domain integrals in the integral equation are converted into boundary integrals by the radial integral method or the dual reciprocity method. Finally, the time difference propulsion technology is used to solve the discrete time differential equations. A large number of practical examples verify the correctness and validity of the BEM in solving the variable coefficient of transient heat conduction problem. However, two deficiencies are encountered when the system of time differential equations is solved with the time difference method, i.e., one is the stability of the algorithm, which is closely related to the time step size, and the other is time-consuming when the freedom degree of the problem is large and all specified time steps are considered, because a system of linear equations needs to be solved in each time step. Therefore, in this paper we present a reduced order model analysis method of solving the variable-coefficient transient heat conduction problem based on BEM by using the model reduction method of proper orthogonal decomposition (POD). For variable-coefficient transient heat conduction problems, the discrete integral equations which are suitable for order reduction operation are deduced by using the BEM, the reduced order model is established by using the model reduction method of POD, and a lowdimensional approximate description of the transient heat conduction problem under time-varying boundary condition is obtained by projection of the initial discrete integral equations on some few dominant POD modes obtained from the problem under constant boundary conditions. First, for a variable coefficient transient heat conduction problem, boundary-domain integral equations are established and the domain integrals are transformed into boundary integrals by using the radial integration method. Second, the time differential equations with discrete format which is suitable for order reduction operation are obtained by reorganizing the integral equations. Third, the POD modes are developed by calculating the eigenvectors of an autocorrelation matrix composed of snapshots which are clustered by the given results obtained from experiments, BEM or other numerical methods for transient heat transfer problem with constant boundary conditions. Finally, the reduced order model is established and solved by projecting the time differential equations on reduced POD modes. Examples show that the method developed in this paper is correct and effective. It is shown that 1) the low order POD modes determined under constant boundary conditions can be used to accurately analyze the temperature field of transient heat conduction problems with the same geometric domain but a variety of smooth and time-varying boundary conditions; 2) the establishment of low order model solves the problem of heavy workload encountered in BEM where a set of large linear equations will be formed and solved in each time step when using the time difference method to solve the large time differential equations.

Trefftz‐based dual reciprocity method for hyperbolic boundary value problems

SummaryA new dual reciprocity‐type approach to approximating the solution of non‐homogeneous hyperbolic boundary value problems is presented in this paper. Typical variants of the dual reciprocity method obtain approximate particular solutions of boundary value problems in two steps. In the first step, the source function is approximated, typically using radial basis, trigonometric or polynomial functions. In the second step, the particular solution is obtained by analytically solving the non‐homogeneous equation having the approximation of the source function as the non‐homogeneous term. However, the particular solution trial functions obtained in this way typically have complicated expressions and, in the case of hyperbolic problems, points of singularity. Conversely, the method presented here uses the same trial functions for both source function and particular solution approximations. These functions have simple expressions and need not be singular, unless a singular particular solution is physically justified. The approximation is shown to be highly convergent and robust to mesh distortion.Any boundary method can be used to approximate the complementary solution of the boundary value problem, once its particular solution is known. The option here is to use hybrid‐Trefftz finite elements for this purpose. This option secures a domain integral‐free formulation and endorses the use of super‐sized finite elements as the (hierarchical) Trefftz bases contain relevant physical information on the modeled problem. Copyright © 2015 John Wiley & Sons, Ltd.

A meshless method based on the dual reciprocity method for one‐dimensional stochastic partial differential equations

This article describes a new meshless method based on the dual reciprocity method (DRM) for the numerical solution of one‐dimensional stochastic heat and advection–diffusion equations. First, the time derivative is approximated by the time–stepping method to transforming the original stochastic partial differential equations (SPDEs) into elliptic SPDEs. The resulting elliptic SPDEs have been approximated with the new method, which is a combination of radial basis functions (RBFs) method and the DRM method. We have used inverse multiquadrics (IMQ) and generalized IMQ (GIMQ) RBFs, to approximate functions in the presented method. The noise term has been approximated at the source points, at each time step. The developed formulation is verified in two test problems with investigating the convergence and accuracy of numerical results. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 292–306, 2016

A collocation and least squares p-singular boundary method without fictitious boundary

This study proposes a new version, p-SBM, of the numerical singular boundary method (SBM) to solve general classes of elliptic PDEs such as: Laplace, Helmholtz and diffusion equations. In SBM, the fundamental solution (FS) of the problem must be given but unlike the method of fundamental solutions (MFS), a fictitious boundary is not required. Instead, the inverse interpolation technique (IIT) and least squares method for the calculation of the singular diagonal elements of the interpolation matrix allows us to avoid the singularity at origin. In this study, we enrich the traditional SBM by adding a constant parameter or a linear combination to the previous MFS approximation and use various types of internal, external and boundary nodes. The p-SBM is applied to some homogeneous Laplace, Helmholtz and Diffusion problems to show its ability and solution accuracy. The non-homogeneous problems can be handled by using the dual reciprocity method (DRM).

Efficient numerical methods for boundary data and right‐hand side reconstructions in elliptic partial differential equations

In this article, we discuss the application of two important numerical methods, Ritz–Galerkin and Method of Fundamental Solutions (MFS), for solving some inverse problems, arising in the context of two‐dimensional elliptic equations. The main incentive for studying the considered problems is their wide applications in engineering fields. In the previous literature, the use of these methods, particularly MFS for right hand side reconstruction has been limited, partly due to stability concerns. We demonstrate that these diculties may be surmounted if the aforementioned methods are combined with techniques such as dual reciprocity method(DRM). Moreover, we incorporate some iterative regularization techniques. This fact is especially veried by taking into account the noisy data with boundary conditions. In addition, parts of this article are dedicated to the problem of boundary data approximation and the issue of numerical stability, ending with a general discussion on the advantages and disadvantages of various methods. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1995–2026, 2015

An efficient and accurate implementation of the Localized Regular Dual Reciprocity Method

In this work we present an efficient and accurate implementation of the LRDRM. This integral domain decomposition method exploits two advantages: first, it imposes the boundary conditions at the Local RBF interpolation. Second, the integrals to compute are always regular. The approximation of the derivative of the field variable is computed in a posteriori way, directly differentiating the Local RBF interpolation or the local integral equation. The efficient and accurate behaviour of this method are demonstrated by performing numerical examples, with special emphasis on a 1D benchmark convective-diffusion equation. Results for 2D convection–diffusion, 2D Helmholtz and 2D Poisson equations are also presented.

The modified dual reciprocity boundary elements method and its application for solving stochastic partial differential equations

This paper proposes a numerical method based on the dual reciprocity boundary elements method (DRBEM) to solve the stochastic partial differential equations (SPDEs). The concept of dual reciprocity method is used to convert the domain integral to the boundary. The conventional DRBEM starts with approximation of the source term of the original PDEs with radial basis functions (RBFs). Due to the fact that the nonhomogeneous term of SPDEs considered in this paper involves Wiener process, the traditional DRBEM cannot be applied. So a modification of it is suggested that has some advantages in comparison with the traditional DRBEM and can be developed for solving the SPDEs.The time evolution is discretized by using the finite difference method, while the modified DRBEM is proposed for spatial variations of field variables. The noise term is approximated at the collocation points at each time step. We employ the generalized inverse multiquadrics (GIMQ) RBFs to approximate functions in the presented technique. To confirm the accuracy of the new approach, several examples are employed and simulation results are reported. Also the convergence of the new technique is studied numerically.

A new extension of boundary elements method for classical elliptic partial differential equations with constant coefficients

This article is devoted to an extension of boundary elements method (BEM) for solving elliptic partial differential equations of general type with constant coefficients. As the fundamental solution of these equations was not available in the literature, BEM was not able to handle them, directly. So the dual reciprocity method (DRM) has been applied to tackle these problems. In this work, a fundamental solution for these equations is obtained and a new formulation is derived to solve them. Besides, we show that the rate of convergence of the new scheme is quadratic when singular (boundary and domain) integrals are calculated, accurately. The new scheme is applicable on complex domains, without needing internal nodes, just same as conventional BEM. So the CPU time of the new scheme is much less than that of the DRM. Numerical examples presented in the article show ability and efficiency of the new scheme in solving two‐dimensional nonhomogenous elliptic boundary value problems, clearly. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 2027–2042, 2015

Dual boundary element model coupled with the dual reciprocity method to determine wave scattering by a concentric cylindrical system mounted on a conical shoal

In this study, the dual boundary element method (DBEM) is coupled with the dual reciprocity method (DRM) to investigate wave scattering by a concentric porous cylinder system, which consists of a circular inner cylinder and semicircular porous outer cylinder mounted on a conical shoal. The complex porous-effect parameter proposed by Yu and Chwang [2] is used to describe the permeability of the porous outer cylinder. The effect of topography has been considered by applying the extended mild-slope equation (EMSE) which was proposed by Chandrasekera and Cheung [10] to the entire fluid domain. The wave force, wave amplification, and increased performance configuration are illustrated through examples.

A new particular solution strategy for hyperbolic boundary value problems using hybrid-Trefftz displacement elements

Summary A new indirect approach to the problem of approximating the particular solution of non-homogeneous hyperbolic boundary value problems is presented. Unlike the dual reciprocity method, which constructs approximate particular solutions using radial basis functions, polynomials or trigonometric functions, the method reported here uses the homogeneous solutions of the problem obtained by discarding all time-derivative terms from the governing equation. Nevertheless, what typifies the present approach from a conceptual standpoint is the option of not using these trial functions exclusively for the approximation of the particular solution but to fully integrate them with the (Trefftz-compliant) homogeneous solution basis. The particular solution trial basis is capable of significantly improving the Trefftz solution even when the original equation is genuinely homogeneous, an advantage that is lost if the basis is used exclusively for the recovery of the source terms. Similarly, a sufficiently refined Trefftz-compliant basis is able to compensate for possible weaknesses of the particular solution approximation. The method is implemented using the displacement model of the hybrid-Trefftz finite element method. The functions used in the particular solution basis reduce most terms of the matrix of coefficients to boundary integral expressions and preserve the Hermitian, sparse and localized structure of the solving system that typifies hybrid-Trefftz formulations. Even when domain integrals are present, they are generally easy to handle, because the integrand presents no singularity. Copyright © 2015 John Wiley & Sons, Ltd.

METHOD OF FUNDAMENTAL SOLUTIONS FOR NONLINEAR SKIN BIOHEAT MODEL

In this paper, the method of fundamental solution (MFS) coupling with the dual reciprocity method (DRM) is developed to solve nonlinear steady state bioheat transfer problems. A two-dimensional nonlinear skin model with temperature-dependent blood perfusion rate is studied. Firstly, the original bioheat transfer governing equation with nonlinear term induced by temperature-dependent blood perfusion rate is linearized with the Taylor's expansion technique. Then, the linearized governing equation with specified boundary conditions is solved using a meshless approach, in which the DRM and the MFS are employed to obtain particular and homogeneous solutions, respectively. Several numerical examples involving linear, quadratic and exponential relations between temperature and blood perfusion rate are tested to verify the efficiency and accuracy of the proposed meshless model in solving nonlinear steady state bioheat transfer problems, and also the sensitivity of coefficients in the expression of temperature-dependent blood perfusion rate is analyzed for investigating the influence of blood perfusion rate to temperature distribution in skin tissues.

A boundary integral equation formulation for the thermal creep gas flow at finite Peclet numbers

The integral equation formulation developed previously by the authors to study isothermal micro flows under shear slip boundary condition is extended in this work to consider the case of non-isothermal micro gas flows with thermal creep effects at finite Peclet numbers. The effect of thermal creep over the flow patterns with and without considering the effect of shear slip is investigated in detail using a boundary element method. In this work the boundary integral approaches for both fluid velocity and temperatures fields are used to solve the problem of shear-driven cavity flow at finite Peclet number. This is obtained by considering the diffusive-convective heat equation using the Dual Reciprocity Method to transform the corresponding volume integral of the convective terms into equivalent surface integrals.

Numerical modeling of cavitation characteristics and sensitivity curves for reversible hydraulic machinery

Abstract Details on the existence, extent and effects of cavitation can be helpful during the design stages of hydraulic machinery to minimize the effects of cavitation and optimize the design. Fluid flow in reversible hydraulic machinery elements is a complex three-dimensional problem. In this paper, an explicit numerical model based on finite element method (FEM) - dual reciprocity method (DRM) coupling is developed for the 2D simulation of the flow velocity and pressure distributions on the runner blade of a Francis-type reversible radial-axial hydraulic machine. The model is based on the assumptions of an ideal incompressible fluid and relative rotational motion. Instead of directly simulating the 3D flow, the proposed model is based on the analysis of specific 2D flows: fluid axial-symmetric motion and the stream function in an associated plane. This model is applied to reversible radial–axial hydraulic machinery that operates as a pump. The blade exhibits the basic profile NP205. The skeleton of this profile is defined by a quadratic equation, and its thickness function exhibits a NACA profile with a maximum relative thickness of five percent. The numerical results for different discharge values can be used to determine the cavitation characteristics and sensitivity curves for reversible hydraulic machinery.

Trefftz- type FEM for solving orthotropic potential problems

A simple Trefftz-type finite element method (TFEM) is proposed for solving certain potential problems in orthotropic media. The body force, which is induced by internal sources or sinks, may produce domain integrals in the standard Trefftz finite element formulation. This will make the advantage only-boundary integration of TFEM lose entirely. To overcome this difficulty, the dual reciprocity method (DRM) is employed to transfer the original problem into a homogeneous one. Then, a particular solution (PS) Trefftz-type finite element model is established based on the modified functional. Three benchmark examples are investigated by the proposed approach and compared with the analytical solutions.