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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.

In this paper, a new methodology for time domain analysis of buildings on raft foundations considering soil – structure interaction is proposed. Sub-structuring technique is used to separate the building as super-structure and the underneath soil as sub-structure. The super-structure can be modeled using any numerical method. However, in this paper the super-structure is modeled via the BEM to consider the real interaction area between column and slabs. The dynamic load is considered as an earthquake acceleration record that can be transformed to equivalent dynamic loads acting on the super-structure floors. The sub-structure is analyzed using the dual reciprocity boundary element method as closed domain. New iterative coupling technique is proposed between the super and sub-structures to reduce the computational effort and required storage. An example is presented to demonstrate the strength and the practicality of the proposed methodology.

This study delves into the investigation of steady infiltration problems within a two-layered soil, featuring periodic channels, and taking into account the presence of rootwater uptake, with four different types of root-water uptake considered. The problems are governed by a set of Richards' equations accompanied by boundary interface conditions. To tackle these problems, we transform the system of Richards' equations, along with the corresponding boundary conditions, into a set of steady diffusion-convection equations with transformed boundary conditions. This mathematical model is then addressed using a numerical approach that leverages the Iterative Dual Reciprocity Method (IDRM). Through this numerical method, we obtain solutions that characterize the distribution of hydraulic conductivity within the soil and root-water uptake in the root zone. Furthermore, we conduct comparisons and analyze the root-water uptake resulting from the four different types of root-water uptake. The findings of this study provide insights into how the type of root influences the amount of water absorbed by the roots.

A novel truly meshless method called dual reciprocity hybrid radial boundary node method (DHRBNM) is developed in present, which combines dual reciprocity method (DRM), hybrid boundary node method (HBNM) and radial point interpolation method (RPIM). Compared to the dual reciprocity hybrid boundary node method (DHBNM), RPIM is exploited to replace the moving least square in DHRBNM, unlike HBNM, the shape function obtained by present method has the delta function property, so the boundary conditions can be applied directly and easily, and computational expense is greatly reduced. In order to get the interpolation property of different basis function in DRM, different approximate functions are applied in DRM for comparison, and the accuracy and efficiency of them are discussed. Besides, RPIM is also exploited in DRM, which can greatly improve the accuracy of present method. Moreover, the accuracy of DRM is greatly influenced by the nodes number and their location, hence, some examples are investigated to show that the internal node number is equal to boundary node number and they are arranged parallel to the high gradient direction of the problem are the best choice. Finally, DHBNM is applied for comparison and some selected numerical examples are given to illustrate that the present method is efficient and less computational expense than that of DHBNM.

In this study, the dispersion of some substance in a river with turbulent flow is considered. In this case, the effect of three factors consist of length, width, and flow velocity on substance dispersion is investigated. The dispersion problem is modeled using diffusion-convection equation. Since the problem may not be solved analytically, a numerical method is chosen, namely the Dual Reciprocity Method (DRM). Numerical solutions are presented to describe the effect of the three factors on the distribution of the substance in the turbulent water flow.

It has a positive impact on the machining accuracy to predict precisely the thermal error caused by the temperature change for the high-speed spindle-bearing system. In this paper, the dual reciprocity method (DRM) based on compactly supported radial basis functions (CSRBFs) and the line integration boundary element method (LIM-BEM) are presented for the thermal-deformation coupling calculation. The essential idea of this method is building the thermal-deformation coupling model only by the boundary information and obtaining results by line integrals. In this process, the boundary element model discretized by the discontinuous iso-parametric quadratic boundary element is established. Then, the transient temperature is calculated by the CSRBFs-DRM, and the thermo-elastic deformation is done by the LIM-BEM, under the exact calculation of the heat generation and the thermal contact resistance. To validate the effectiveness, thermal-deformation coupling experiments are conducted. The proposed method is compared with experimental data and the finite element method. The result shows that the proposed model is more appropriate for the thermal-deformation coupling calculation for the satisfactory universality and accuracy.

The paper deals with two-dimensional boundary-value problems for the degenerate nonlinear parabolic equation with a source term, which describes the process of heat conduction in the case of the power-law temperature dependence of the heat conductivity coefficient. We consider a heat wave propagation problem with a specified zero front in the case of two spatial variables. The solution existence and uniqueness theorem is proved in the class of analytic functions. The solution is constructed as a power series with coefficients to be calculated by a proposed constructive recurrent procedure. An algorithm based on the boundary element method using the dual reciprocity method is developed to solve the problem numerically. The efficiency of the application of the dual reciprocity method for various systems of radial basis functions is analyzed. An approach to constructing invariant solutions of the problem in the case of central symmetry is proposed. The constructed solutions are used to verify the developed numerical algorithm. The test calculations have shown the high efficiency of the algorithm.

The authors describe a meshless method for solving three-dimensional nonstationary heat conduction problems in anisotropic materials. A combination of dual reciprocity method using anisotropic radial basis function and method of fundamental solutions is used to solve the boundary-value problem. The method of fundamental solutions is used to obtain the homogenous part of the solution; the dual reciprocity method with the use of anisotropic radial basis functions allows obtaining a partial solution. The article shows the results of numerical solutions of two benchmark problems obtained by the developed numerical method; average relative, average absolute, and maximum errors are calculated.

One of the attractive and practical techniques to transform the domain integrals to equivalent boundary integrals is the dual reciprocity method (DRM). The success of DRM relies on the proper treatment of the non-homogeneous term in the governing differential equation. For this purpose, radial basis functions (RBFs) interpolations are performed to approximate the non-homogeneous term accurately. Moreover, when the interpolation points are large, the global RBFs produced dense and ill-conditioned interpolation matrix, which poses severe stability and computational issues. Fortunately, there exist interpolation functions with local support known as compactly supported radial basis functions (CSRBFs). These functions produce a sparse and well-conditioned interpolation matrix, especially for large-scale problems. Therefore, this paper aims to apply DRM based on multiquadrics (MQ) RBFs and CSRBFs for evaluation of the Poisson equation, especially for large-scale problems. Furthermore, the convergence analysis of DRM with MQ and CSRBFs is performed, along with error estimate and stability analysis. Several experiments are performed to ensure the well-conditioned, efficient, and accurate behavior of the CSRBFs compared to the MQ-RBFs, especially for large-scale interpolation points.

The buckling of square perforated plates is analysed with the effect of shear deformation in the bending model. The relationship between the buckling load and the plate thickness is better assessed with this bending model instead of the classical model. The simply supported condition at all sides was adopted in the buckling problem such that opposite sides of the plates were uniformly compressed in one direction. The plates had a thickness to length ratio between 1/1000 and 1/5. This study adopted the dual reciprocity method (DRM) to obtain a formulation without domain integrals. The numerical results obtained were compared with those available in the literature.