Dual Reciprocity Boundary Element Method Research Articles

A new implementation of DRM with dual interpolation boundary face method for Poisson equation

We present a new numerical algorithm combining the dual reciprocity method with the dual interpolation boundary face method for solving the two-dimensional Poisson equations. In this new combined approach, the boundary physical variables and particular solutions are approximated by dual interpolation elements which include source and virtual points. Additionally, this algorithm is implemented by employing the augmented thin plate spline interpolation function to approximate non-homogeneous term and the Hermite-type moving-least-square approximation to approximate physical variables for virtual nodes. Compared with the traditional dual reciprocity boundary element method, this combined method possesses higher accuracy and efficiency and is better suited for structures with small feature sizes. Several numerical examples are given to illustrate the superiority of the proposed method for solving Poisson equations.

The Kriging integration method applied to the boundary element analysis of Poisson problems

A novel efficient technique is presented for the evaluation of domain integrals that appear in the boundary element method (BEM). Herein, the source term is approximated with the use of radial basis functions, as in the dual reciprocity BEM. The proposed technique, called the Kriging Integration Method (KIM), comprises the use of the Simple Kriging Method in non-overlapping patches for obtaining the weights of the integration points located inside. As it is necessary to compute the integrals of the covariance function prior to obtaining these weights, this can be efficiently realized using the Cartesian Transformation Method. The domain integrals over all the generated partitions are then computed and added to obtain the value of the whole-domain integral. Using KIM, it is possible to evaluate approximately weakly singular domain integrals over simply or multiply connected domains without applying any transformation or regularization method to the singular integrand. The numerical results obtained in several 2D potential problems demonstrate that this integration scheme is as accurate as both the dual reciprocity method and RIM and less time consuming than the RIM.

The Dual-Reciprocity Boundary Element Method Solution for Gas Recovery from Unconventional Reservoirs with Discrete Fracture Networks

SummaryIn this paper, we present a novel application of the dual-reciprocity boundary-element formulation (DRBEM) to model compressible (gas) fluid flow in tight and shale-gas reservoirs containing arbitrary distributed finite- or infinite-conductivity discrete fractures. Compared with the standard boundary-element method (BEM), the DRBEM transforms the nonlinear domain integrals at the righthand side (RHS) of BEM formulations for nonlinear partial differential equations into equivalent boundary integrals. This transformation allows retention of the domain-integral-free, boundary-integral-only character of standard BEM approaches. The proposed approach is based on coupling DRBEM with the finite-volume method (FVM) in which a multidimensional system is solved by integrating over a line with random fractures. The resulting system of equations is solved simultaneously for fracture and matrix boundary conditions by combining DRBEM and FVM without invoking any approximation for pressure-dependent nonlinear terms such as gas viscosity and compressibility. Numerical examples and field cases are presented to test the validity and showcase the capabilities of the proposed approach. The proposed model provides a general framework that can be applied to a variety of well and fracture geometries and operating schedules, and it is used to analyze production behavior for these complex systems. To the best of the authors’ knowledge, this is the first successful application of the dual-reciprocity principle to the BEM analysis of massively fractured horizontal wells (MFHWs) performance in natural-gas formations in which nonlinear, pressure-dependent gas properties are captured without approximation.

A Preconditioned Krylov Subspace Iterative Methods for Inverse Source Problem by Virtue of a Regularizing LM-DRBEM

In this paper, we investigate an effectual methodology to solve the problem of identifying the source term standing on a combined discontinuous dual reciprocity boundary element method with a regularized Levenberg–Marquardt algorithm. The mathematical model is described by optimizing the variation between the measured potential in some points within the problem domain and the estimated one by the discontinuous dual reciprocity boundary element approximation. To provide a qualitative evaluation, the derived dense and non symmetric matrix system of the forward problem is solved by virtue of the LU factorization and using some preconditioned Krylov subspace iterative algorithms, expressly the biconjugate gradient stabilized, the generalized minimum residual and the squared conjugate gradient method. Due to noise interference, a regularized Levenberg–Marquardt algorithm is adopted to retrieve the unknown source term. Extensive experiments evince greater effectiveness and stability criteria of the proposed approach.

Error estimations of the heat flux reconstruction for ITER divertor infrared thermography.

This study developed a new heat flux reconstruction code based on the dual reciprocity boundary element method for the International Thermonuclear Experimental Reactor (ITER) divertor infrared (IR) thermography system. To use divertor heat flux reconstruction in ITER, we modeled the boundary condition between the coolant pipe inner wall and the coolant based on the temperature-dependent heat transfer coefficient and also considered the temperature dependence of tungsten thermal properties. Using this reconstruction code, we evaluated the sensitivity of the input data errors, divertor coolant temperature, and surface temperature errors on the accuracy of heat flux reconstruction by using simplified heat flux profiles, which are spatially uniform on the top surface of the monoblocks. This heat flux reconstruction method demonstrated that highly accurate heat flux reconstruction can be achieved for high heat flux cases in ITER; however, further studies are needed for low heat flux reconstruction.

Analysis Infiltration Waters in Various Forms of Irrigation Channels by Using Dual Reciprocity Boundary Element Method

This research discusses the infiltration of furrow irrigation invarious forms of irrigation channels in homogeneous soils. The governing equation of the problems is a Richard’s Equation. This equation is transformed using a set of transformation including Kirchhoff and dimensionless variables into Helmholtz modified equations. Furthermore with Dual Reciprocity Boundary Element Method (DRBEM), numerical solution of modified Helmholtz equation obtained. The proposed method is tested on problem involved infiltration from periodic flat channels, non-flat channels without impermeable and non-flat channels with impermeable. The greatest value of suction potential and water content is located below the channel surface. The most water consecutively is a non-flat channel without impermeable, non-flat channel with impermeable and flat channel on Lakish Clay soils.

DRBEM Solution of MHD flow in a rectangular duct with time-varied external magnetic field

This paper investigates the flow behavior of a viscous, incompressible and electrically conducting fluid in a long channel subjected to a time-varied oblique magnetic field B0(t)=B0f(t). The time-dependent MHD equations are solved by using the dual reciprocity boundary element method (DRBEM). The transient level velocity and induced magnetic field profiles are presented for moderate Hartmann number values, several direction of applied magnetic field and for several functions f(t) as polynomial, exponential, trigonometric, impulse and step functions. It is observed that, when f(t) is a polynomial or exponential function, the velocity magnitude increases up to a certain time level where the flow shows an elliptical elongation and then starts to decrease for all values of Hartmann numbers. For the trigonometric function f(t), the flow repeats its behavior with a period. An impulse function changes flow behavior with an elongation at the level where the impulse is applied to the magnetic field. Then, it behaves as if a uniform magnetic field is applied. Having a step function f(t), the behavior of the flow shows both impulse and polynomial functions features. This study shows that time-varied magnetic field changes the flow behavior significantly at all the transient levels.

A Novel Full Boundary Element Formulation for Harmonic Analysis of Elastic Membranes Coupled to Acoustics Fluids

A novel full Boundary Element Formulation for the harmonic analysis of elastic membranes coupled to acoustics fluid is presented. The elastic membranes is modeled using the classical linear elastic pre-stretched membrane theory. The acoustic fluid is modeled using the acoustic-wave equation for homogeneous, isotropic, inviscid and irrotational fluids. Elastostatic fundamental solution is used in the boundary element formulation for the elastic membrane. The boundary element formulation for the acoustic fluid is based on the fundamental solution of three dimensional Poisson equation. Domain integrals related to inertial terms and those related with distributed pressure on membrane, were treated using the Dual Reciprocity Boundary Element Method. Fluid-structure coupling equations were established considering the continuity of the normal acceleration of the particles and dynamic pressure at fluid-structure interfaces. Results obtained shows the accuracy and efficiency of the proposed boundary element formulation.

A Novel Full Boundary Element Formulation for Transient Analysis of Elastic Membranes Coupled to Acoustics Fluids

A full time-direct Boundary Element Formulation for the dynamic analysis of elastic membranes coupled to acoustics fluid is presented. The elastic membranes is modeled using the classical linear elastic pre-stretched membrane theory. The acoustic fluid is modeled using the acoustic-wave equation for homogeneous, isotropic, inviscid and irrotational fluids. Elastostatic fundamental solution is used in the boundary element formulation for the elastic membrane. The boundary element formulation for the acoustic fluid is based on the fundamental solution of three dimensional Poisson equation. Domain integrals related to inertial terms and those related with distributed pressure on membrane, were treated using the Dual Reciprocity Boundary Element Method. Fluid-structure coupling equations were established considering the continuity of the normal acceleration of the particles and dynamic pressure at fluid-structure interfaces. The time integration is carried out using the Newmark method. Results obtained shows the accuracy and efficiency of the proposed boundary element formulation.

A Discontinuous Dual Reciprocity Method in Conjunction with a Regularized Levenberg–Marquardt Method for Source Term Recovery in Inhomogeneous Anisotropic Materials

This paper outlines a new approach to identify a source term of a [Formula: see text]D elliptic equation for anisotropic nonhomogenous media. The proposed methodology is based on the minimization of an objective function representing differences between the measured potential and those calculated by using the discontinuous dual reciprocity boundary element method, the measurements are required to render a unique solution and supposed to be pointwise in the problem domain. Since the additional data may be contaminated by measurement noises or the numerical computing errors, we adopt a regularizing Levenberg–Marquardt method to solve the nonlinear least-squares problem attained from the inverse source problem. The numerical performance of the proposed approach is studied at the end for both geometries: smooth and piecewise smooth one. The results show a very good agreement with the analytical solutions under exact and noisy data.

Fracture Dynamic Analysis of Cracked Reissner Plates Using the Boundary Element Method

This work presents a Dual Boundary Element Method/Dual Reciprocity Boundary Element Method formulation for the dynamic analysis of fractured shear deformable plates. Equations for the Dual Boundary Element Method, including the direct boundary and the traction boundary equation for Reissner plates, were used for crack modeling. The Dual Reciprocity Boundary Element Method was used to treat both domain integrals related to distributed forces and inertial terms. Proposed formulation considers the rotary inertia of the plate. A general crack modeling strategy is presented. The J-Integral is used to evaluate the Dynamic Stress Resultant Intensity Factors. Test problems, including comparisons to Finite Element Method solutions, are presented. Results demonstrate, proposed formulation is a reliable method for dynamic analysis of shear deformable cracked plate bending problems.

Simulation of heat distribution in the human eye using discontinuous dual reciprocity boundary element method and non-overlapping domain decomposition approach

Simulation of heat distribution in the human eye using discontinuous dual reciprocity boundary element method and non-overlapping domain decomposition approach

BOUNDARY ELEMENT METHOD FOR THE LID-DRIVEN PROBLEM RELATED TO THE DARCY-FORCHHEIMER-BRINKMAN SYSTEM

The Darcy-Forchheimer-Brinkman system is one mathematical model that describes the motion of a viscous fluid located in a porous domain. Recently, many papers have established the existence of a unique solution for this system depending on the boundary condition imposed. In this paper we investigate numerically a special Dirichlet-Robin boundary value problem associated with the nonlinear Darcy-Forchheimer-Brinkman system, i.e., the lid-driven cavity flow problem, by using the dual reciprocity boundary element method (DRBEM). The physical properties of such a flow are discussed by the geometry of the streamlines of the fluid flow for different Reynolds and Darcy numbers. Moreover, an additional sliding parameter is imposed on the moving wall, and its importance is highlighted by segmenting the upper lid into two opposite moving parts.

Effect of depth and width of periodic trapezoidal channels to water content in soil: a numerical study

Time-independent or steady infiltration problems from periodic furrows are studied. There are variations in the depth and width of the periodic furrows or channels. The problems are modelled using Richards equation. To solve the problems using the Dual Reciprocity Boundary Element Methods (DRBEM), the mathematical model must be changed into a modified Helmholtz equation. Solutions obtained are then compared and discussed to observe effects of the depth and width of the channels to soil’s moisture content.

Effect of an uniform magnetic field on unsteady natural convection of nanofluid

In this work, the problem of the unsteady natural convection in an -water filled nanofluids influenced by an uniform magnetic field is analysed numerically. Governing equations are given in terms of the streamlines, the vorticity and the temperature of the fluid. Dual Reciprocity Boundary Element Method (DRBEM) is performed as a solution technique. Both the fundamental solutions of Laplace equation and modified Helmholtz equation are used in the derivation of the method. The numerical analysis are done for physical parameters Rayleigh number (), Hartmann number () and particle volume fraction (). From the results, it is observed that the behaviours of all the variables are influenced by the changing values of parameters. The solution procedure of the all variables needs considerably small number of iterations and large time increments with suitable values of relaxation parameters which occur in the argument of Bessel function .

Numerical investigation of two-dimensional heat transfer of an absorbing plate of a flat-plate solar collector using dual-reciprocity method based on boundary element

In this paper, heat transfer in an absorbing plate of a flat plate solar collector has been investigated upon using dual-reciprocity method. In order to validate the boundary element method, the results of this method have been compared with the numerical results obtained by finite difference method. Due to the presence of non-homogeneous terms in the governing equation, it has been numerically solved using the dual-reciprocity boundary element method. The temperature distribution in the absorbing plate is investigated upon considering different values of mass flow rate in the tube and the results obtained by the dual-reciprocity boundary element and finite difference methods are compared. Additionally, the fluid temperature along the tube has been studied, using two numerical methods. After being compared, the results of two methods well coincide, which proves the accuracy of numerical methods. Moreover, effect of changes in various parameters such as conductivity of the absorbing plate, mass flow rate, distance between two tubes, the fluid inlet temperature, as well as the change in the diameter of the tubes on the efficiency of the absorbing plate has been thoroughly investigated. Another notable point is that the boundary element method can lead to less computational cost, compared to other numerical methods such as finite difference method. The reason for this is connected to the capability of the boundary element method, without need for points within the computational domain.

The DRBEM solution of Cauchy MHD duct flow with a slipping and variably conducting wall using the well-posed iterations

The present study focuses on the numerical investigation of the Cauchy Magnetohyrodynamic (MHD) duct flow in the presence of an externally applied oblique magnetic field, with a slipping and variably conducting wall portion of the duct walls. The underspecified and overspecified boundary informations for the velocity of the fluid and the induced magnetic field on both slipping and variably conducting duct wall and its opposite part, respectively, constitutes the Cauchy MHD duct flow problem. This study aims to recompute the velocity of the fluid and induced magnetic field with specified slip length and conductivity constant, respectively, on the underspecified wall which is both slipping and variably conducting. The governing coupled convection-diffusion type MHD equations for the direct and inverse formulations are solved in one stroke using the dual reciprocity boundary element method (DRBEM). Both the velocity and induced magnetic field and their normal derivatives to be used as overspecified boundary conditions for the construction of Cauchy problem are obtained through the direct formulation of the problem. The well-posed iterations are used in the regularization of the ill-conditioned systems of linear algebraic equations resulting from the DRBEM discretization of Cauchy problem (inverse problem). Numerical solutions for the slip velocity and induced magnetic field are obtained for Hartmann number values $M$=5, 10, 50. The main advantages of the DRBEM are its boundary only nature and the capability of providing both the unknowns and their normal derivatives on the underspecified walls so that the conductivity constant and the slip length between them can be recovered at a low computational expense.

An object-oriented approach to dual reciprocity boundary element method applied to 2D elastoplastic problems

PurposeThe purpose of this paper is to describe further developments on a novel formulation of the boundary element method (BEM) for inelastic problems using the dual reciprocity method (DRM) but using object-oriented programming (OOP). As the BEM formulation generates a domain integral due to the inelastic stresses, the DRM is employed in a modified form using polyharmonic spline approximating functions with polynomial augmentation. These approximating functions produced accurate results in BEM applications for a range of problems tested, and have been shown to converge linearly as the order of the function increases.Design/methodology/approachA programming class named DRMOOP, written in C++ language and based on OOP, was developed in this research. With such programming, general matrix equations can be easily established and applied to different inelastic problems. A vector that accounts for the influence of the inelastic strains on the displacements and boundary forces is obtained.FindingsThe C++ DRMOOP class has been implemented and tested with the BEM formulation applied to classical elastoplastic problem and the results are reported at the end of the paper.Originality/valueAn object-oriented technology and the C++ DRMOOP class applied to elastoplastic problems.

DRBEM solution to MHD flow in ducts with thin slipping side walls and separated by conducting thick Hartmann walls

In this study, the dual reciprocity boundary element method (DRBEM) solution to magnetohydrodynamic (MHD) flow is given in a single and two ducts stacked in the direction of external magnetic field. The duct walls perpendicular to the applied magnetic field (Hartmann walls) are conducting, thick and no-slip whereas the horizontal walls (side walls) are insulated, thin and allow the velocity slip. The DRBEM transforms the convection diffusion type MHD equations in the duct and Laplace equation in the thick walls to boundary integral equations which are discretized by using constant elements. The resultant matrix–vector equations are solved as a whole with the coupled boundary conditions on the common boundaries between the fluid and the thick walls. The effects of the slip length, thickness of conducting walls and the applied magnetic field strength are shown inthe flow and the induced magnetic field. It is found that, in the absence of the slip, as Hartmann number (Ha) increases, the flow is concentrated in front of the side walls in terms of two side layers, and this separation is happened for much smaller value of Ha when the thickness of the conducting walls is increased. The continuation of induced magnetic fields to the thick walls is well observed in both co-flow and counter flow cases. When the side walls admit slip, opposite to the no-slip case, an increase in the conductivity of the fluid enlarges the core region where the fluid is stagnant. The proposed numerical scheme DRBEM is capable of capturing the well known MHD flow characteristics in ducts with thick walls as well as perturbations in the behaviors of the flow and the induced magnetic field due to the thin slip walls.

Numerical Simulation of Convection-Diffusion Phenomena by Four Inverse-Quadratic-RBF Domain-Meshfree Schemes

The convection-diffusion type of PDEs is numerically solved by four numerical methods in this work. These four comparatively young numerical approaches are categorized as ‘domain-meshfree’ as they require no internal meshing but rely only on the collocation process amongst nodes via. the inverse-quadratic radial basis function (IQ-RBF). They are; the well-known Kansa Collocation Method (KCM), the Hermite Collocation Method (HCM), the Radial Point Interpolation Method (RPIM), and the Dual Reciprocity Boundary Element Method (DRBEM). The work aims to demonstrate the use of IQ-RBF as well as to compare the practical use of the methods. Moreover, engineering senses of criteria judging the quality of the methods are considered. It is found in this work that while KCM is the simplest to construct and deploys, it is highly sensitive to the number of nodes and the IQ-RBF shape parameter. The unsymmetric and populated matrix problem are alleviated when HCM or DRBEM are in use yet more CPU-time and storage seem to be the price to pay, particularly HCM. However, it is actually RPIM that has appeared to be an optimal choice under all the criteria imposed.