Dual Reciprocity Formulation Research Articles

A novel direct interpolation boundary element method formulation for solving diffusive–advective problems

The capability of dealing with the advective transport term constitutes a challenging issue for the performance of the majority of the numerical techniques, which significantly lose their precision with the increasing in the relative magnitude of this term. Recently, a new boundary element technique called direct interpolation (DIBEM) has emerged, mainly characterized by the approximation of the entire kernel of the remaining domain integrals. The DIBEM model has been successfully applied to several scalar field problems and recently applied to certain diffusive–advective problems with superior accuracy compared to dual reciprocity formulation (DRBEM). Using DIBEM, a broader range of precise responses for flows with higher Peclet numbers can be reached beyond a lower computational cost due to the simplicity of its matrix operations. These advantages are due DIBEM concept that employs a simple interpolation procedure to approximate the kernel of the advective domain integral. However, it was noticed that in some physical scenarios with spatial variation in the velocity field, the accuracy of DIBEM did not have the same quality observed in other applications. Therefore, this work presents a new DIBEM model so that the integral equations include the explicit calculation of spatial derivatives and simultaneously solve the variation of the divergent velocity field if this is not zero. Conversely, reactive and source terms were also included to expand numerical comparisons. Thus, in the proposed examples, the results of two DIBEM models, the standard (DIBEM-S) and the new, so-called alternative model (DIBEM-A), are compared with DRBEM and also with available analytical solutions.

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.

Fast multipole accelerated dual reciprocity boundary element method for diffusion problems

A fast multipole boundary element method (FMBEM) is presented for diffusion problems based on a dual reciprocity formulation. In the dual reciprocity formulation, domain integrals that arise from solving the time-dependent boundary value problems are transformed into boundary integrals by constructing particular solutions. The time-derivatives in the governing differential equation are approximated with a first-order finite difference time-stepping scheme. Discontinuous linear elements, which are known to give more accurate results than constant or linear elements, are used in the implementation to discretize the boundary integral equations, in combination with the fast multipole method for speeding up the solution. Three numerical examples of diffusion are presented. The performance of the developed FMBEM is compared with that of a conventional BEM and a commercial finite element program. The results show that the developed FMBEM can be a reliable and efficient tool for solving diffusion problems.

Modal analysis of acoustic enclosures by dual reciprocity–boundary element method.

This work presents a modal analysis of acoustic enclosures by dual reciprocity–boundary element method (DR‐BEM). The dual reciprocity formulation allows the use of known Laplace fundamental solutions. The boundary element method possesses the main characteristic that reduces the dimension of the problem upon the discretization of the boundary to calculate the acoustic eigenfrequencies. The DR‐BEM formulation has been evaluated with the use of interior nodes and some known radial basis functions. The results obtained by this technique have been compared with the more extensively diffused competitor, the finite element method (FEM). Results shows good agreement between both methods.

Boundary element—dual reciprocity formulation for bound electron states in semiconductor quantum wires

Using the boundary element dual reciprocity method-multi-domain (DRM-MD) a bound electron states and corresponding wave functions in semiconductor quantum wires embedded in a matrix were considered. The single circular and rectangular as well as the two near circular quantum wires were analyzed. In the case of two coupled quantum wires the dependence of the resulting wave function and eigenenergies as a function of the distance between wires was calculated. The DRM-MD gave a linear electron state model and developed numerical approach accurately captured the symmetry breaking and splitting of the degenerated energy states due to presence of additional wire. According to the symmetry of the structures a suitable mesh reduction was employed and different modes were considered separately. For a case of hetero-structures domain decomposition was used.

Elastodynamic antiplane deformation of a bimaterial with an imperfect viscoelastic interface: A dual reciprocity hypersingular boundary integral solution

A boundary element method based on a hypersingular integral and dual-reciprocity formulation is proposed for the numerical solution of a dynamic antiplane problem involving an elastic bimaterial with an imperfect viscoelastic interface. The interface is modelled using linear springs and dashpots with a jump in the interfacial displacement. To reduce the problem approximately to a system of linear algebraic equations, the Laplace transform is employed to suppress the time derivatives of the unknown functions. Once the linear algebraic equations are solved, the physical solution is recovered through the use of a numerical method for inverting Laplace transform. The proposed method is applied to solve some specific problems.

Quasi‐dual reciprocity boundary‐element method for incompressible flow: Application to the diffusive–advective equation

AbstractThis work presents a new boundary‐element method formulation called quasi‐dual reciprocity formulation for heat transfer problems, considering diffusive and advective terms. The present approach has some characteristics similar to those of the so‐called dual‐reciprocity formulation; however, the mathematical developments of the quasi‐dual reciprocity approach reduces approximation errors due to global domain interpolation. Some one‐ and two‐dimensional examples are presented, the results being compared against those obtained from analytical and dual‐reciprocity formulations. The method convergence is evaluated through analyses where the mesh is successively refined for various Peclet numbers, in order to assess the effect of the advective term. Copyright © 2003 John Wiley & Sons, Ltd.

Earthquake‐induced hydrodynamic pressures on a 3D rigid dam–reservoir system using DRBEM and a radiation matrix

AbstractThe dual reciprocity method is applied to determine the hydrodynamic pressure distribution in a three‐dimensional dam–reservoir system subjected to earthquake excitation. The reservoir domain is idealized as a finite region of irregular geometry adjacent to the dam followed by an infinite domain of uniform cross‐section in the upstream direction. The reservoir hydrodynamic pressure response is governed by the Helmholtz equation subject to free surface, dam–reservoir interface, absorbing bottom/banks and radiation boundary conditions. A three‐dimensional (3D) dual reciprocity model is developed to determine the hydrodynamic pressure in the finite reservoir domain. A radiation matrix is developed and introduced to relate the hydrodynamic pressure and its normal derivative on the interface between the finite and infinite domains. The three‐dimensional radiation model used is developed by applying a two‐dimensional dual reciprocity formulation along the interface of the finite and infinite reservoirs together with a continuum solution in the upstream direction of the infinite domain. The model is compared for the hydrodynamic response of a three‐dimensional rectangular reservoir and found to be in excellent agreement with results obtained from a model based on the analytical formulation existing in the literature. Copyright © 2003 John Wiley & Sons, Ltd.

The boundary element method applied to time dependent problems in anisotropic materials

In this paper a multi-domain boundary element formulation is developed for the solution of two-dimensional dynamic anisotropic problems with or without cracks. The inertia effects are treated as body forces and the dual reciprocity formulation is used to transform the domain integral into a boundary integral. In fracture problems, traction singular quarter point elements are used in order to give a correct representation of displacement and traction fields around the crack tip. The dynamic stress intensity factors are computed directly from the tractions at the crack tip node. The results obtained demonstrate good agreement with other reported results and show strong dependence on the material anisotropy.

A dual reciprocity formulation for elasticity problems with body forces using augmented thin plate splines

This paper presents a novel dual reciprocity formulation for elasticity problems with body forces in which the approximating functions are given in terms of augmented thin plate splines (ATPS). It is shown that the ATPS approximation is capable of correctly representing gravitational and centrifugal body forces, and provides superior accuracy for general load cases.

A dual reciprocity formulation for elasticity problems with body forces using augmented thin plate splines

This paper presents a novel dual reciprocity formulation for elasticity problems with body forces in which the approximating functions are given in terms of augmented thin plate splines (ATPS). It is shown that the ATPS approximation is capable of correctly representing gravitational and centrifugal body forces, and provides superior accuracy for general load cases.

A comparative study of three boundary element approaches to transient dynamic crack problems

Using displacement integral representations for time dependent elastic problems, three different boundary element (BE) approaches can be considered. One is based on the time domain formulation, a second one on the frequency domain formulation, and the third on the dual reciprocity formulation. In this paper, the three BE approaches are applied and compared when used for the computation of stress intensity factors (SIF) of stationary cracks under dynamic loading. In the three cases a quadratic discretization in space and a singular quarter-point (SQP) element are used. Several problems are solved and the results obtained by the three approaches compared. The dual reciprocity approach for dynamic SIF computation is presented in this paper. Conclusions and recommendations on the capabilities of the three approaches in dynamic fracture mechanics are presented.

An improved boundary element formulation for wave propagation problems

In this paper the dual reciprocity formulation for scalar wave equations and elastodynamic problems developed by Nardini & Brebbia is extended to the problem of waves propagating in an infinite domain by applying the Sommerfeld's radiation condition on a suitable artificial boundary. The free surface condition of first order can also be taken into consideration. To validate the present scheme, some examples have been worked out and compared with analytical solutions.

The dual reciprocity boundary element method for spontaneous ignition

AbstractThis paper studies the non‐linear heat conduction problem of spontaneous ignition of a reactive solid by using the dual reciprocity formulation of the boundary element method. First, critical values of the Frank‐Kamenetskii parameter δ are computed for some common geometrical shapes. Then, transient analyses are carried out for values of δ below and above δc, producing the expected behaviour, with ignition occurring for δ > δc.