Dual Reciprocity Method Research Articles

Numerical solution of the boundary value problem of elliptic equation by Levi function scheme

AbstractFor boundary value problem of an elliptic equation with variable coefficients describing the physical field distribution in inhomogeneous media, the parametrix can represent the solution in terms of volume and surface potentials, with the drawback that the volume potential involving in the solution expression requires heavy computational costs as well as the solvability of the integral equations with respect to the density pair. We introduce an modified integral expression for the solution to an elliptic equation in divergence form under the parametrix framework. The well‐posedness of the linear integral system with respect to the density functions to be determined is rigorously proved. Based on the singularity decomposition for the parametrix, we propose two schemes to deal with the volume integrals so that the density functions can be solved efficiently. One method is an adaptive discretization scheme for computing the integrals with continuous integrands, leading to the uniform accuracy of the integrals in the whole domain, and consequently the efficient computations for the density functions. The other method is the dual reciprocity method which is a meshless approach converting the volume integrals into boundary integrals equivalently by expressing the volume density as the combination of the radial basis functions determined by the interior grids. The proposed schemes are justified numerically to be of satisfactory computation costs. Numerical examples in 2‐dimensional and 3‐dimensional cases are presented to show the validity of the proposed schemes.

Improved dynamic analysis for plane elasticity problems with a boundary meshfree method and distributed sources

In this paper, we present an improved dynamic analysis method to address plane elasticity problems. Distributed sources are employed in a boundary-free mesh-based approach known as Boundary Distributed Source (BDS). This method, which builds upon a modified Method of Fundamental Solutions (MFS), aims to eliminate singularities without the reliance on fictitious boundaries. Distributed sources are strategically positioned directly on the real boundary, centered over disks. The method's efficiency and simplicity are highlighted by its ability to operate without the need for precise data concerning adjacent points. The dual reciprocity method (DRM) is utilized to approximate the non-homogeneous inertial term. Furthermore, the present formulation investigates several factors that influence the convergence of the proposed method, including the number of collocation points, internal points, the radius of the circular disk, and the time step size. Based on the numerical results, it is evident that the method exhibits remarkable accuracy and efficiency when applied to plane elasticity problems including forced, harmonic, and free analyses. As compared to conventional numerical methods for dynamic analysis, the method outperforms in both accuracy and convergence rate.

Band structure calculations of three-dimensional solid-fluid coupling phononic crystals using dual reciprocity boundary element method and wavelet compression method

The algorithm for calculating the band structures of two-dimensional phononic crystals (PCs) using the boundary element method (BEM) has been proposed for many years. However, it has not yet been extended to three-dimensional (3D) PCs because the fundamental solutions of 3D dynamics are complex and are related to angular frequency. In this study, the BEM is applied to calculate the band structures of 3D solid-fluid coupling PCs by combining the dual reciprocity method. The use of the dual reciprocity method can help avoid nonlinear eigenvalue problems. To address the limitation of fully populated matrices in BEM, the wavelet compression method based on B-spline wavelet on the interval is adopted. Some small matrix entries, which are generated by the vanishing moment and local support characteristics, are set to zero using the provided truncation technique. This process results in the production of sparse matrices. The constructed generalized linear eigenvalue equations are modified to tackle the ill-conditioned matrix issues arising from the distinct fundamental solutions of solids and fluids. The results show that this method is superior to the finite element method in terms of calculation efficiency. Compared to conventional BEM, the current wavelet BEM can not only generate sparse matrices but also reduce the integration calculation time when handling large-scale problems.

Method of Fundamental Solutions without Fictitious Boundary in Elastodynamic Behavior Using Dual Reciprocity Method

Method of Fundamental Solutions without Fictitious Boundary in Elastodynamic Behavior Using Dual Reciprocity Method

On applying polyharmonic radial basis functions to solve 3D uncoupled anisotropic thermoelasticity problems using Boundary Element and Dual Reciprocity Method

On applying polyharmonic radial basis functions to solve 3D uncoupled anisotropic thermoelasticity problems using Boundary Element and Dual Reciprocity Method

Methodology for simulating the elastoplastic behavior in crack propagation problems using dual formulations of the boundary element method

Due to the application of cyclic loads, the existing crack length in a structure increases with time, resulting in greater stress concentration around the crack tip and therefore causing an increase in propagation velocity and a decrease in residual strength of the structure which, at a given time, becomes so low that the structure can no longer support the service loads. This paper presents a new approach to evaluate two-dimensional elastoplastic models in a crack propagation scenario, based on the boundary element method and its dual formulations. The methodology adopted consists of two stages: firstly, the simulation of the elastoplastic behavior, which considers the process of initial stresses and allows the treatment of several flow criteria, without taking into account the incompressibility of inelastic deformations. In this step, the domain integral due to non-homogeneous terms in the plastification region is transformed into a boundary integral by the Dual Reciprocity Method (DRM) using the polyharmonic splines functions due to its local behavior and, from the plastic calculus, the J-Integral is used to compute the SIF's. The second stage aims to simulate, in an incremental way, the crack propagation path using homemade software for crack modeling and analysis based on ElastoPlastic Fracture Mechanics (EPFM) and the Dual Boundary Element Method (DBEM). To validate the adopted methodology, as well as correctly simulate the mechanical behavior of cracks in the plastic regime of the material, three 2D models with straight and inclined cracks are used. Initially, the von Mises criterion for perfectly plastic materials is used and the numerical results compared with classic models from the literature, showing the effectiveness of the program in crack treatment and in the prediction of the propagation path, as well as the non-linear algorithm in the region plastification using the DRM.

Analysis of singularity in advection-diffusion-reaction equation with semi-analytical boundary elements

In order to solve the advection-diffusion-reaction equation with singularity by the boundary element method, the domain integral is handled by dual reciprocity method and the semi-analytical elements are implemented to approximate the unknown physical fields. By collocating internal points in the domain and approximating the primer field with radial basis function, the integral of domain can be settled and boundary-only integral equation can be derived. In this paper, we propose two new elements containing asymptotic expressions to model the unknown temperature and singular flux fields, while the known fields are simulated by linear interpolation. The unknowns in the new elements are amplitude coefficients which can be used to evaluate the intensity of flux. Compared with conventional boundary element, the proposed method allows to avoid direct evaluation of singular values. Thus, the accuracy can be guaranteed since all the unknowns have finite values. Benefiting from the semi-analytical elements, the mesh refinement for the region with singular physical field can be avoided. Moreover, the first three amplitude coefficients as well as the whole physical fields are obtained at the same time without any post process. We demonstrate the excellent numerical results through several examples in plates with singularities.

Approximation of three‐dimensional nonlinear wave equations by fundamental solutions and weighted residuals process

In this paper, the localized method of fundamental solutions (LMFS) with coupling the dual reciprocity method (DRM) is applied to simulate three‐space dimensional nonlinear wave equations. First, DRM, which is a popular meshless method based on polyharmonic splines (PhS), is applied to obtain the particular solution. After evaluating the particular solution, 3D‐LMFS method can be employed to evaluate the homogeneous solution. The proposed 3D‐LMFS algorithms construct 3D artificial surfaces where source points and then the collocation procedure propose by using the fundamental solutions in each 3D surface. Straightforwardly, a solution to the 3D wave equation is approximated by a sum of the combination of PhS and fundamental solutions using LMFS. Eventually, the scheme has been prosperously tested with selected examples.

Band structures analysis of fluid–solid phononic crystals using wavelet‐based boundary element method and frequency‐independent fundamental solutions

AbstractA novel method of combination of wavelet‐based boundary element method (WBEM) with frequency‐independent fundamental solutions is proposed to determine the band structures of fluid–solid phononic crystals (PCs) with square and triangular lattices. Integral equations established are based on the frequency‐independent fundamental solutions, which can avoid nonlinear eigenvalue problems and reduce computing time. Domain integral terms arising from the use of frequency‐independent fundamental solutions are handled with the radial integration method (RIM) and dual reciprocity method (DRM), respectively. The results show the lower precision in high frequency domain of using RIM to handle domain integral terms than that of using DRM, which can be solved by increasing Gauss point. The B‐spline wavelet on the interval and wavelet coefficients are applied to approximate the physical boundary conditions. It is proved that coupling conditions between matrix and scatterers and Bloch theorem are also applicable to wavelet coefficients. Some small matrix entries generated by wavelet vanishing moment characteristics are truncated by the provided matrix compression technique, and the influence of compressed matrices on the results is studied. Furthermore, the final Eigen equations constructed are modified to avoid numerical instability. Some examples are provided to demonstrate the accuracy and efficiency of the proposed method.

A NUMERICAL STUDY OF SUBSTANCE SPREAD IN THE POLL FROM TWO POINT SOURCES

Problems related to the purification of holding pool or reservoir become an interesting discussion in real events. In this paper, the author will modeling the distribution of the substance/purifier in a pool model with turbulent water flow using the diffusion-convection equation. The Dual Reciprocity Method is applied to the diffusion-convection equation whose derivation will be discussed in this paper. This method is chosen because the problem cannot be solved analytically, so it must be solved numerically. The Dual Reciprocity Method has good flexibility in problems of water infiltration, pollutant spread, and heat transfer. In this paper also discuss velocity profile of turbuelent flow from upcoming part of pool region. So before using DRM, will be used numerical solution of turbulent flow by k-epsilon turbulent model. In numerical calculations, two source points are selected whose positions are combined to see the most effective way to make the substance/purifier evenly distributed in the pool.

WBEM-based analysis of band structures of solid-solid and fluid-fluid phononic crystals with frequency-independent fundamental solutions

A novel method is presented to determine the band structures of two-dimensional solid-solid and fluid-fluid phononic crystals (PCs) with square and triangular lattices using the wavelet-based boundary element method (WBEM). Both sweeping frequency technique and sweeping wave vector technique are used to determine the band structures of PCs. The fundamental solutions for integral equations established in primitive cell are independent of angular frequency, resulting in an avoidance of solving nonlinear eigenvalue problems and much reduction of computing time. The radial integration method and dual reciprocity method are respectively applied to handle the domain integral terms arising from the use of frequency-independent fundamental solutions, and to make the dimension reduction. The physical fields are approximated by B-spline wavelet on the interval and wavelet coefficients in wavelet space. It is proved that the wavelet coefficients still meet the relationship of their corresponding physical quantities, such as Bloch theorem. By means of the vanishing moment characteristic of wavelets, some small entries are produced in the system matrices which are further compressed into sparse ones, and the influence of sparse matrices on the results is discussed. Finally, the high computing speed and accuracy of the proposed method are shown in examples.

A DRM FOR HOMOGENEOUS HELMHOLTZ EQUATIONS IN ANISOTROPIC MATERIALS

We consider the problems involving homogeneous Helmholtz equations in anisotropic materials. The problems considered are solved numerically using a dual reciprocity method (DRM). To test the accuracy of the method, the method is tested using problems with analytic solutions. Numerical solutions obtained by the DRM and the corresponding analytical solutions are compared and discussed.

A Novel Localized Meshless Method for Solving Transient Heat Conduction Problems in Complicated Domains

This paper first attempts to solve the transient heat conduction problem by combining the recently proposed local knot method (LKM) with the dual reciprocity method (DRM). Firstly, the temporal derivative is discretized by a finite difference scheme, and thus the governing equation of transient heat transfer is transformed into a non-homogeneous modified Helmholtz equation. Secondly, the solution of the non-homogeneous modified Helmholtz equation is decomposed into a particular solution and a homogeneous solution. And then, the DRM and LKM are used to solve the particular solution of the non-homogeneous equation and the homogeneous solution of the modified Helmholtz equation, respectively. The LKM is a recently proposed local radial basis function collocation method with the merits of being simple, accurate, and free of mesh and integration. Compared with the traditional domain-type and boundary-type schemes, the present coupling algorithm could be treated as a really good alternative for the analysis of transient heat conduction on high-dimensional and complicated domains. Numerical experiments, including two- and three-dimensional heat transfer models, demonstrated the effectiveness and accuracy of the new methodology.

Isogeometric dual reciprocity BEM for solving non-Fourier transient heat transfer problems in FGMs with uncertainty analysis

In this paper, the precise integration multi-patch isogeometric dual reciprocity boundary element method (IG-DRBEM) of the non-Fourier transient heat transfer problem in functionally graded materials (FGMs) is proposed, and the stochastic analysis of multidimensional material uncertainty by hyperbolic truncated adaptive sparse polynomial chaos expansion (PCE) is established based on the determine theory. Up to now, most research results of the isogeometric analysis boundary element method (IGABEM) in transient heat transfer problems are based on Fourier's law. The main reason is the higher-order derivative term with respect to time in the non-Fourier heat transfer control equation leads to additional domain integrals in the boundary-domain integral equation. The dual reciprocity method (DRM) can transform the domain integral to the boundary conveniently and maintain the dimension reduction advantage of the BEM. The introduction of the precise integration method (PIM) can improve the stability and accuracy of the time-domain solution. In the random analysis based on IG-DRBEM, a sparse PCE strategy with hyperbolic truncation is constructed to overcome the dimensional disaster difficulty of traditional PCE. Through the proposed adaptive iterative process, a relatively small number of PCE terms are retained compared with the full PCE on the premise of meeting the error estimation requirements. The presented method is verified to have good accuracy and high efficiency by the complex model similar to actual engineering, which further expands the application extent of IG-DRBEM.

A non‐overlapping domain decomposition dual reciprocity method for solving the forward–backward heat equation in two‐dimension

AbstractWe apply a boundary element dual reciprocity method (DRBEM) to the numerical solution of the forward–backward heat equation in a two‐dimensional case. The method is employed for the spatial variable via the fundamental solution of the Laplace equation and the Crank–Nicolson finite difference scheme is utilized to treat the time variable. The physical domain is divided into two non‐overlapping subdomains resulting in two standard forward and backward parabolic equations. The subproblems are then treated by the underlying method assuming a virtual boundary in the interface and starting with an initial approximate solution on this boundary followed by updating the solution by an iterative procedure. In addition, we show that the time discrete scheme is unconditionally stable and convergent using the energy method. Furthermore, some computational aspects will be suggested to efficiently deal with the formulation of the proposed method. Finally, two forward–backward problems, for which the exact solution is available, will be numerically solved for two different domains to demonstrate the efficiency of the proposed approach.

Indirect boundary element formulation with dual reciprocity in elastodynamics

In this paper, a new indirect boundary element formulation for two-dimensional elastodynamic problems is presented. The formulation makes use of a static fundamental solution. For accurate solution, the location of the fictitious boundary is chosen to be the same as problem boundary. The relevant fundamental solution is integrated over lines. The method is combined with the Dual Reciprocity Method (DRM), for handling the domain integral concerning to the domain inertia terms. A suitable approximation (traditional function C-R is used) for the inertia terms (or, domain accelerations), is used to transform the domain integral to the boundary. The discretization is performed on the boundary, or a boundary-like geometry. The method is easy to code and efficient as presented by the numerical examples.

THE NUMERICAL MESHLESS APPROACH FOR SOLVING THE 2D TIME NONLINEAR MULTI-TERM FRACTIONAL CABLE EQUATION IN COMPLEX GEOMETRIES

The cable equation plays a prominent role in biological neuron models, for instance, in spiking neuron models and electrophysiology. Thus, the current investigation scrutinizes the 2D time nonlinear multi-term fractional cable equation. By adopting a valid meshfree technique, the nonlinear multi-term time-fractional cable equations (NM-TTFCEs) that consist of government equations and their boundary conditions are transformed into the boundary value problems. For this purpose, the finite difference method is derived for temporal discretization such that the considered NM-TTFCEs can be transformed into a sequence of boundary value problems in inhomogeneous Helmholtz-type equations. The dual reciprocity method (DRM) is implemented to obtain a particular solution and the improved singular boundary method (ISBM) is employed to evaluate the homogeneous solution. Moreover, we apply the meshless method for solving two-dimensional NM-TTFCEs on regular and irregular distribution points with several computational domains. The numerical results vouch for the accuracy and high efficiency of the proposed method. Finally, we will conclude that the ISBM/DRM method can be considered a potential alternative to existing meshless strong form approaches in solving multi-term fractional equation problems with complex geometries.

A novel isogeometric boundary element approach for solving phase change problems with the level set method

In this study a new algorithm is proposed for numerically solving phase change problems. The algorithm (PTE-IGBEM) integrates the precise time domain expanding method (PTE) and the isogeometric boundary element method (IGBEM). The governing equations are transformed into recursive form by expanding the variables within the time period. The domain integral is transformed into boundary integral by the dual reciprocity method (DRM). An adaptive checking scheme is used to determine the number of expanding items required for a time period. In addition, the level set method (LSM) is used to capture the trajectory of the front separating the solid and liquid phases. Several classical examples are given to demonstrate the accuracy and robustness of the algorithm.

SOFTWARE FOR SIMULATION OF NON-STATIONARY HEAT TRANSFER IN ANISOTROPIC SOLIDS

The software «AnisotropicHeatTransfer3D» for the numerical solution of three-dimensional non-stationary heat conduction problems in anisotropic solids of complex domain by meshless method [1] was developed. The numerical solution of the non-stationary anisotropic heat conduction problem in the software «AnisotropicHeatTransfer3D» is based on a combination of the dual reciprocity method [2] with anisotropic radial basis functions [3] and the method of fundamental solutions [4]. The dual reciprocity method with anisotropic radial basis functions is used to obtain particular solution, and the method of fundamental solutions is used to obtain a homogeneous solution of boundary-value problem.

The localized method of fundamental solutions for 2D and 3D second-order nonlinear boundary value problems

In this paper, a new framework for the numerical solutions of general nonlinear problems is presented. By employing the analog equation method, the actual problem governed by a nonlinear differential operator is converted into an equivalent problem described by a simple linear equation with unknown fictitious body forces. The solution of the substitute problem is then obtained by using the localized method of fundamental solutions, where the fictitious nonhomogeneous term is approximated using the dual reciprocity method using the radial basis functions. The main difference between the classical and the present localized method of fundamental solutions is that the latter produces sparse and banded stiffness matrix which makes the method very suitable for large-scale nonlinear simulations, since sparse matrices are much cheaper to inverse at each iterative step of the Newton's method. The present method is simple in derivation, efficient in calculation, and may be viewed as a completive alternative for nonlinear analysis, especially for large-scale problems with complex-shape geometries. Preliminary numerical experiments involving second-order nonlinear boundary value problems in both two- and three-dimensions are presented to demonstrate the accuracy and efficiency of the present method.