Reciprocity Method Research Articles

Isogeometric dual reciprocity boundary element method for solving transient heat conduction problems with heat sources

Up to now, the isogeometric boundary element method (IGBEM) has been widely applied in different fields, and the solved problems are basically independent of time. But an excellent numerical method is more than that, so it is necessary to explore a new IGBEM which can solve time-domain problems. Based on this, the isogeometric dual reciprocity boundary element method (IG-DRBEM) is proposed to solve transient heat transfer problems with heat sources. The introduction of the dual reciprocal method enables the IGBEM to solve the transient heat transfer problem conveniently. At the same time, it does not need to divide elements within the domain, which maintains the advantage of the IGBEM. First, the boundary domain integral equation is established by the weighted residual method and the field variables are discretized by NURBS basis functions. Then, the domain integral in the integral equation is transformed into the boundary by the classical dual reciprocity method. Finally, the standard first-order ordinary differential equations are formed. In order to examine the accuracy of the proposed method, several typical numerical examples are discussed carefully. The presented method can provide a new idea for solving time-dependent problems by IGBEM.

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.

Boundary interface water infiltration into layered soils using dual reciprocity methods

In this paper, some steady infiltration problems from periodic channels into two-layered soils are considered. The problems are governed by a system of Richards’ equations with boundary interface conditions. To solve the problems, the system of Richards’ equations with respect to some boundary conditions are transformed into a system of steady diffusion-convection equations with respect to transformed boundary conditions. The mathematical model is solved numerically using a proposed Dual Reciprocity Method (DRM). Numerical solutions are obtained to describe the distribution of soil water potential and hydraulic conductivity in the soils. The results demonstrate how soil coarseness and saturated hydraulic conductivity affect the values of soil water potential and hydraulic conductivity.

Application of the radial integration method for the buckling analysis of plates with shear deformation

This work presents a novel formulation of the Boundary Element Method (BEM) with the Radial Integration Method (RIM) to calculate the critical loads of the plate buckling problem with shear deformation. An alternative formulation is adopted where the effect of the geometric non-linearity is described by using the first derivative of the function for the out-of-plane displacements. The RIM is developed for this problem and used to convert the resulting domain integrals into equivalent boundary integrals. The results are compared with other results available in the literature and with the results obtained with the Dual Reciprocity Method (DRM). The advantages of using the RIM are discussed at the end of this work.

BEM-FORM Model for the Probabilistic Response of Circular Tunnels in Elastic Media

Problems involving cavities or excavations are widely addressed in geomechanics, in both analytical and numerical approaches. The boundary element method (BEM) is well-known as an interesting choice for half plane problems, providing accurate results at a low computational cost. This work deals with the probabilistic analysis of circular tunnels embedded in elastic media, coupling a BEM formulation to a structural reliability model. The gravitational loading and material parameters are treated as random variables, whose statistical description is taken from the literature. The loadings considered include the vertical overburden stress and the lateral earth pressure. Regarding the reliability evaluation, first order reliability method (FORM) and Monte Carlo simulation technique are employed, being compared in terms of accuracy. Regarding the BEM model, the Multiple Reciprocity Method (MRM) is used in the evaluation of domain integrals, and the subregion technique is employed for the analysis of the tunnel lining. Some analyses are presented, in order to validate the coupled BEM-FORM model and apply it to the estimation of failure probability, evaluating the influence of the random variables taken into account in the probabilistic response.

On the Analytical and Numerical Study of a Two-Dimensional Nonlinear Heat Equation with a Source Term

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.

Multiple reciprocity singular boundary method for 3D inhomogeneous problems

In this study, the singular boundary method (SBM) is extended to solve 3D inhomogeneous elliptic boundary value problems. Before applying the SBM, the multiple reciprocity method (MRM) removes the inhomogeneous terms and establishes an equivalent high-order homogeneous problem, which is then solved by the SBM, a boundary-only fundamental-solution based collocation method. It is worth-noting that the conventional MRM is not applicable in most cases because polyharmonic operators cannot deal with most of the source terms. Thus, in this study a recursive multiple reciprocity method is developed via different operators. The high-order fundamental solutions are involved as kernel functions in solving a high-order homogeneous problem by the SBM. It is of great importance to remove the source singularity of the kernels when the collocation points coincide with the source points. In this study, simple formulas are analytically derived to desingularize the source singularity of kernel functions. The accuracy and efficiency of the proposed method are illustrated via several numerical examples governed by different partial differential equations.

A boundary collocation method for anomalous heat conduction analysis in functionally graded materials

This paper applies a semi-analytical boundary collocation method, the singular boundary method (SBM), in conjunction with the dual reciprocity method (DRM) and Laplace transformation technique to solve anomalous heat conduction problems under functionally graded materials (FGMs). In this study, transient heat conduction equation with Caputo time fractional derivative is considered to describe anomalous heat conduction phenomena. In the present numerical implementation, Laplace transformation and numerical inverse Laplace transformation are used to deal with the Caputo time fractional derivative, which avoid the effect of time step on the computational efficiency of the time fractional derivation approximation. The SBM in conjunction with the DRM is employed to obtain the high accurate results in the solution of Laplace-transformed time-independent nonhomogeneous problems. To demonstrate the effectiveness of the proposed method for anomalous heat conduction analysis under functionally graded materials, three numerical examples are considered and the present results are compared with known analytical solutions and COMSOL simulation.

Improved singular boundary method and dual reciprocity method for fractional derivative Rayleigh–Stokes problem

The improved singular boundary method (ISBM) and dual reciprocity method (DRM) are coupled to solve fractional derivative the Rayleigh–Stokes problem with nonhomogeneous term. This method is free of mesh and integration, mathematically simple, and easy to program. Also, origin intensity factors (OIFs) significant techniques in ISBM make the method as a strong meshless method. First, the time-fractional derivative term in mentioned equation is discretized; then, ISBM–DRM is utilized to solve consequent equation. It is proved the method is unconditionally stable and convergent with convergence order $${\mathcal {O}}(\tau ^{1+\alpha })$$. In addition, numerical results confirm the accuracy and efficiency of the presented scheme.

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.

Triple reciprocity method for unknown function's domain integral in boundary integral equation

In this paper, triple reciprocity method (TRM) is employed to transform the unknown function's domain integral appearing in boundary integral equation (BIE) to boundary integrals. The three-dimensional interior Helmholtz equation is represented as a Poisson equation with an unknown function in the right-hand side. The fundamental solution of Laplace equation is used to derive the BIE and there is a domain integral containing unknown field function in the BIE. The triple reciprocity method (TRM) is used to transform this domain integral to boundary integrals. The improved triple reciprocity approximation (TRA) is used to approximate the unknown domain function and solve the high-order derivatives of it virtually. Employing the reciprocity theory, the domain integral is transferred into the boundary integrals containing the unknown field function. Finally, the BIE can be solved without internal cells. The proposed formulas can be employed to treat the domain integral containing time derivatives of unknown variable, both known and unknown functions and other similar types. Three numerical examples are presented to demonstrate the efficiency and accuracy of the proposed formulas. Results show that TRM can solve the BIE containing unknown function's domain integral efficiently and accurately.

The use of dual reciprocity method for 2D laminar viscous flow

The paper presents the use of the dual reciprocity multidomain singular boundary method (SBMDR) for the solution of the laminar viscous flow problem described by Navier-Stokes equations. A homogeneous part of the solution is solved using a singular boundary method with the 2D Stokes fundamental solution - Stokeslet. The dual reciprocity approach has been chosen because it is ideal for the treatment of the nonhomogeneous and nonlinear terms of Navier-Stokes equations. The presented SBMDR approach to the solution of the 2D flow problem is demonstrated on a standard benchmark problem - lid-driven cavity.

Lid-driven cavity flow using dual reciprocity

The paper presents the use of the multi-domain dual reciprocity method of fundamental solutions (MD-MFSDR) for the analysis of the laminar viscous flow problem described by Navier-Stokes equations. A homogeneous part of the solution is solved using the method of fundamental solutions with the 2D Stokes fundamental solution Stokeslet. The dual reciprocity approach has been chosen because it is ideal for the treatment of the non-homogeneous and nonlinear terms of Navier-Stokes equations. The presented DR-MFS approach to the solution of the 2D flow problem is demonstrated on a standard benchmark problem - lid-driven cavity.

Spectroscopic properties of Dy3+ ions in La3Ga5.5Ta0.5O14 single crystal

Single crystal of La3Ga5.5Ta0.5O14 (LGT) doped with Dy3+ was fabricated by the Czochralski technique and investigated employing methods of optical spectroscopy. Polarized optical absorption spectra and luminescence spectra were recorded at room temperature. Unpolarized absorption and luminescence spectra were recorded at 10 K. Spectra of blue luminescence related to transitions 4I15/2, 4F9/2 → 6H15/2 were recorded at several temperatures in the 300–773 K region. Dy3+ ions have the same symmetry sites in LGT but their spectra show inhomogeneous broadening due to structural disorder of the host and are markedly anisotropic. Results of the Judd-Ofelt treatment made it possible to assess emission cross sections for the 4F9/2 → 6H15/2 blue luminescence and for the 4F9/2 → 6H13/2 yellow luminescence employing Fuchtbauer-Ladenburg formula and reciprocity method. In particular, a spectral distribution of La3Ga5.5Ta0.5O14: Dy3+ σ-polarized luminescence band with an emission cross section σem = 0.51 × 10−20 cm2 at 573.5 nm is favorable for a laser operation in the visible. Interpretation of the effect of temperature on the blue emission spectra leads to the conclusion that La3Ga5.5Ta0.5O14: Dy3+ is promising for application as luminescent thermometer.

Evaluation of singular integrals for anisotropic elastic boundary element analysis

To improve the numerical evaluation of weakly singular integrals appearing in the boundary element method, a logarithmic Gaussian quadrature formula is usually suggested in the literature. In this formula the singular function is expressed in terms of the distance between source point and field point, which is a real variable. When an anisotropic elastic solid is considered, most of the existing fundamental solutions are written in terms of complex variables. When the problems with holes, cracks, inclusions, or interfaces are considered, to suit for the shape of the boundaries usually a mapping function is introduced and then the solutions are expressed in terms of mapped complex variables. To deal with the trouble induced by the complex variables, in this study through proper change of variables we develop a simple way to improve the evaluation of weakly singular integrals, especially for the problems of anisotropic elastic solids containing holes, cracks, inclusions, or interfaces. By simple matrix expansion, the proposed method is extended to the problems with piezoelectric or magneto-electro-elastic solids. By using the dual reciprocity method, the proposed method employed for the elastostatic fundamental solution can also be applied to the elastodynamic analysis.

Experimental characterization of the coupled conductive and radiative heat transfer in ceramic foams with a flash method at high temperature

During recent decades, numerous studies on ceramic foams improved their efficiency and made their use possible in a myriad of engineering applications. Some studies showed that a good understanding of the link between structural parameters (porosity, pore diameter, or tortuosity) and thermal properties could lead to a better system performance. Ceramic foams are often used at high temperatures, where radiation plays an important role. This paper explores, both conductive and radiative heat transfers in such media, using a flash method experiment. The direct model developed is based on the resolution of the Energy Balance Equation (EBE), solved by a Finite Volume Method and the Radiative Transfer Equation (RTE), computed with an Optimized Emission Reciprocity Method (OERM). An inversion procedure made it possible to simultaneously characterize both equivalent conductivity for conduction and equivalent optical thickness for radiation of structured and random foams with different pore size distributions at 800 ∘C. Results showed that the influence of each heat transfer mode can be characterized separately with single experiment and a good agreement was found between the developed model and the experimental measurements. Radiative transfers can be described by a single equivalent, Beerian and gray extinction coefficient dependings strongly on the size of the cells and the organization of the solid matrix. The equivalent conductivity depends on the porosity, the ceramic constituting the solid matrix, the organization of the porous medium and the internal porosity. We estimated the associated equivalent total and radiative conductivities, allowing the quantification of the conduction/radiation ratio, for a wide variety of structures.

Analysis of the Field-Line Coupling of the Transmission Line Above an Orthogonal Ground Plane Through Electromagnetic Reciprocity Theorem

The far-field radiation of the transmission line (TL) operating as a transmitter and the induced voltage and current at the terminals of the TL regarded as a receiver can be correlated via the electromagnetic (EM) reciprocity theorem. For a TL above an orthogonal ground plane, the far-field radiation formulas of the TL in the testing state (where the TL is deemed as a transmitter) and the electric field expressions around the TL in the receiving state (where the TL is considered as a receiver) are derived in this study. Besides, an expression for the induced voltage at the terminal of the TL is obtained for each state and the equivalence between the two expressions is proved through the EM reciprocity theorem and method of image. A voltage source is selected as the excitation in the testing state and an effective selection method is proposed. Moreover, numerical examples are provided to verify the correctness of analyzing the field line coupling problem of the TL above an orthogonal ground plane through the EM reciprocity theorem and to certify the effectiveness of the novel expression of the excitation voltage source.

Steady state conditions predictions of time-dependent infiltration problems: an LTDRM with a predictor-corrector scheme approach

Time-dependent infiltration problems from periodic irrigation channels with root-water uptake processes, which is modelled using a Richards equation, are studied. The problems is solved numerically. To solve the problems, a set of transformations and a Laplace Transform Dual Reciprocity Method (LTDRM) with a predictor corrector scheme are employed. Finally, employing Gaver-Stehfest formula and diffusivity factor, the numerical solutions of these problems are obtained. Predictions of time needed to achieve steady state conditions for different homogeneous soils are presented and discussed.

A numerical study of Poisson equations: steady heat distribution with one-point source

In this paper, Poisson equations are considered. Especially, equations which governed problems involving steady heat distribution with a point source. The heat distribution problems studied are over thin plates. Generally, these problems may not be solved analytically. Hence, a numerical method called Dual Reciprocity Method (DRM) is employed to solve the problems. Using this method, some of the numerical results are presented and discussed to investigate the influence of source positions to heat distribution over thin plates.