The radial integration method for evaluation of domain integrals with boundary-only discretization

In this paper, a simple and robust method, called the radial integration method, is presented for transforming domain integrals into equivalent boundary integrals. Any two- or three-dimensional domain integral can be evaluated in a unified way without the need to discretize the domain into internal cells. Domain integrals consisting of known functions can be directly and accurately transformed to the boundary, while for domain integrals including unknown variables, the transformation is accomplished by approximating these variables using radial basis functions. In the proposed method, weak singularities involved in the domain integrals are also explicitly transformed to the boundary integrals, so no singularities exist at internal points. Some analytical and numerical examples are presented to verify the validity of this method.     

Solution of the second-order one-dimensional hyperbolic telegraph equation by using the dual reciprocity boundary integral equation (DRBIE) method

In this paper, we use a numerical method based on the boundary integral equation (BIE) and an application of the dual reciprocity method (DRM) to solve the second-order one space-dimensional hyperbolic telegraph equation. Also the time stepping scheme is employed to deal with the time derivative. In this study, we have used three different types of radial basis functions (cubic, thin plate spline and linear RBFs), to approximate functions in the dual reciprocity method (DRM). To confirm the accuracy of the new approach and to show the performance of each of the RBFs, several examples are presented. The convergence of the DRBIE method is studied numerically by comparison with the exact solutions of the problems.     

Modal analysis of plates using the dual reciprocity boundary element method

This paper presents a new method for determining the natural frequencies and mode shapes for the free vibration of thin elastic plates using the boundary element and dual reciprocity methods. The solution to the plate's equation of motion is assumed to be of separable form. The problem is further simplified by using the fundamental solution of an infinite plate in the reciprocity theorem. Except for the inertia term, all domain integrals are transformed into boundary integrals using the reciprocity theorem. However, the inertia domain integral is evaluated in terms of the boundary nodes by using the dual reciprocity method. In this method, a set of interior points is selected and the deflection at these points is assumed to be a series of approximating functions. The reciprocity theorem is applied to reduce the domain integrals to a boundary integral. To evaluate the boundary integrals, the displacements and rotations are assumed to vary linearly along the boundary. The boundary integrals are discretized and evaluated numerically. The resulting matrix equations are significantly smaller than the finite element formulation for an equivalent problem. Mode shapes for the free vibration of circular and rectangular plates are obtained and compared with analytical and finite element results.     

The dual reciprocity boundary element method for solving Cauchy problems associated to the Poisson equation

This paper presents an application of the dual reciprocity method (DRM) to a class of inverse problems governed by the Poisson equation. Here the term inverse refers to the fact that the boundary conditions are not fully specified, i.e. they are not known for the entire boundary of the solution domain. In order to investigate the ability of the DRM to reconstruct the unknown boundary conditions using overspecified conditions on the accessible part of the boundary we consider some test problems involving circular, annular and square domains. Due to the ill-posed nature of the problem, i.e. the instabilities in the solution of these problems, the DRM is combined with the Tikhonov regularization method.     

Solving general field equations in infinite domains with dual reciprocity boundary element method

The dual reciprocity boundary element method has been successfully employed to solve general field equations posed in a closed domain, i.e. interior problems. Up to now, however, little effort has been made to extend it to exterior problems (i.e. general field equations posed in an infinite region), which are commonly encountered in engineering practice. In this paper, the interpolation functions associated with exterior problems, which were proposed by Loeffler and Mansur (in Boundary Elements X, Vol. 2, Springer, 1988), are first examined. We have found that the choice of the arbitrary constant, the inclusion of which is necessary in those interpolation functions, has clear effects on the accuracy of the numerical results. A mapping transformation, through which any exterior problem can be solved by solving an equivalent interior problem, is then proposed. Although there are certain regularity conditions attached to such a mapping, they can be easily satisfied if the unknown function satisfies certain regularity conditions at infinity in the original exterior problem. A successful application of this mapping transformation to a transient heat transfer problem demonstrates the good performance of this approach.     

The dual reciprocity method applied to free vibrations of 2D structures using compact supported radial basis functions

This paper presents a new boundary element application for free vibration analysis of 2D elastic structures. The dual reciprocity method is applied using four compact supported radial basis functions for approximating the domain inertia terms. The eigen-problem of displacement is then solved considering the traction contribution by means of static condensation. The formulation is also extended to consider additional internal nodes to improve accuracy. Three numerical problems are studied to demonstrate the validity and accuracy of the developed formulation. The results are compared to those obtained from analytical and other numerical solutions. A parametric study is set up to demonstrate the effect of the compact support radius on the final results and on the sparsity of system matrices.     

Explicit boundary element method for nonlinear solid mechanics using domain integral reduction

Applied to solid mechanics problems with geometric nonlinearity, current finite element and boundary element methods face difficulties if the domain is highly distorted. Furthermore, current boundary element method (BEM) methods for geometrically nonlinear problems are implicit: the source term depends on the unknowns within the arguments of domain integrals. In the current study, a new BEM method is formulated which is explicit and whose stiffness matrices require no domain function evaluations. It exploits a rigorous incremental equilibrium equation. The method is also based on a Domain Integral Reduction Algorithm (DIRA), exploiting the Helmholtz decomposition to obviate domain function evaluations. The current version of DIRA introduces a major improvement compared to the initial version.     

A domain decomposition method for concurrent coupling of multiscale models

Motivated by atomistic‐to‐continuum coupling, we consider a fine‐scale problem defined on a small region embedded in a much larger coarse‐scale domain and propose an efficient solution technique on the basis of the domain decomposition framework. Specifically, we develop a nonoverlapping Schwarz method with two important features: (i) the use of an efficient approximation of the Dirichlet‐to‐Neumann map for the interface conditions; and (ii) the utilization of the inherent scale separation in the solution. The paper includes a detailed formulation of the proposed interface condition, along with the illustration of its effectiveness by using simple but representative numerical experiments. Copyright © 2012 John Wiley & Sons, Ltd.     

Trefftz‐based dual reciprocity method for hyperbolic boundary value problems

A new dual reciprocity‐type approach to approximating the solution of non‐homogeneous hyperbolic boundary value problems is presented in this paper. Typical variants of the dual reciprocity method obtain approximate particular solutions of boundary value problems in two steps. In the first step, the source function is approximated, typically using radial basis, trigonometric or polynomial functions. In the second step, the particular solution is obtained by analytically solving the non‐homogeneous equation having the approximation of the source function as the non‐homogeneous term. However, the particular solution trial functions obtained in this way typically have complicated expressions and, in the case of hyperbolic problems, points of singularity. Conversely, the method presented here uses the same trial functions for both source function and particular solution approximations. These functions have simple expressions and need not be singular, unless a singular particular solution is physically justified. The approximation is shown to be highly convergent and robust to mesh distortion. Any boundary method can be used to approximate the complementary solution of the boundary value problem, once its particular solution is known. The option here is to use hybrid‐Trefftz finite elements for this purpose. This option secures a domain integral‐free formulation and endorses the use of super‐sized finite elements as the (hierarchical) Trefftz bases contain relevant physical information on the modeled problem. Copyright © 2015 John Wiley & Sons, Ltd.     

An overlapping domain decomposition method for the simulation of elastoplastic incremental forming processes

The implicit finite element (FE) simulation of incremental metal cold forming processes is still a challenging task. We introduce a dynamic, overlapping domain decomposition method to reduce the computational cost and to circumvent the need for sophisticated remeshing procedures. The two FE domains interchange information using the elastoplastic operator split and the mortar method. Copyright © 2008 John Wiley & Sons, Ltd.     

A novel domain decomposition method for highly oscillating partial differential equations

This paper is devoted to designing a novel domain decomposition method (DDM) for highly oscillating partial differential equations (PDE), especially, where the asymmetric meshless collocation method using radial basis functions (RBF), also Kansa's method is applied for a numerical solutions. It is found that the numerical error become worse when the original solution become more oscillating. To conquer this defect, we use a novel domain decomposition method which is motivated by time parallel algorithm. This DDM is based on a decomposition of computational domain by a coarse centers and a finer distribution of distinct centers. A corrector is designed to obtain better numerical solution after several iteration. Theoretical analysis and numerical examples are given to demonstrate the accuracy and efficiency of the proposed algorithm.     

Initial value problem formulation of time domain boundary element method for electromagnetic microwave simulations

To aim to obtain more stable solutions and wider area applications for the Time Domain Boundary Element Method (TDBEM), initial value problem formulation of the TDBEM is newly introduced for microwave simulations. The initial value problem formulation of the TDBEM allows us to solve transient microwave phenomena as interior region problems, which gives us well matrix property and interior resonance free solutions. This paper concentrates on applying the initial value problem formulation of the TDBEM to wake field phenomena in particle accelerator cavities.     

An inverse estimation of multi-dimensional load distributions in thermoelasticity problems via dual reciprocity BEM

Abstact The paper deals with inverse thermoelasticity problems. A general, robust, and numerically efficient technique, for retrieving the multi-dimensional, highly varying distributions of boundary conditions is presented. In the class of considered inverse problems, both input data and sought-for quantities are usually specified at the domain boundary, only. As a sequence of forward sub-solutions underlies the inverse analysis, the numerical method of choice for solving the field problem is the boundary element method (BEM). The derived inverse technique is capable of retrieving boundary condition distributions in steady state and transient problems. The accuracy and stability of the algorithm are verified by considering problems involving constant, functionally graded, and temperature dependent material properties. Strain components and temperatures, subject to uncertainties, are used as input data. Presented numerical examples show that the method is capable of reconstructing mechanical and thermal loads with reasonable accuracy. while on leave from Cracow University of Technology, 31-155 Cracow, Poland     

An exact conserving algorithm for nonlinear dynamics with rotational DOFs and general hyperelasticity. Part 1: Rods

Conventional numerical implementation of the boundary element method (BEM) for elasto-plastic analysis requires a domain discretization into cells. This requires more effort for the discretization of the problem and additional computational effort. A new technique is proposed here for the analysis of 2D and 3D elasto-plastic problems with the boundary element method. In this approach the domain does not need to be discretised into cells prior to the analysis. Plasticity is assumed to start from the boundary and the cells are generated from the boundary data automatically during the analysis. Using the cell generation process, elasto-plastic analysis with the BEM becomes much more user friendly and efficient than the standard approach with a pre-definition of cells. The accuracy and efficiency of the solution obtained by the new approach is verified by several numerical examples.     

A mesh free approach using radial basis functions and parallel domain decomposition for solving three‐dimensional diffusion equations

The analysis of transient heat conduction problems in large, complex computational domains is a problem of interest in many technological applications including electronic cooling, encapsulation using functionally graded composite materials, and cryogenics. In many of these applications, the domains may be multiply connected and contain moving boundaries making it desirable to consider meshless methods of analysis. The method of fundamental solutions along with a parallel domain decomposition method is developed for the solution of three‐dimensional parabolic differential equations. In the current approach, time is discretized using the generalized trapezoidal rule transforming the original parabolic partial differential equation into a sequence of non‐homogeneous modified Helmholtz equations. An approximate particular solution is derived using polyharmonic splines. Interfacial conditions between subdomains are satisfied using a Schwarz Neumann–Neumann iteration scheme. Outside of the first time step where zero initial flux is assumed, the initial estimates for the interfacial flux is given from the converged solution obtained during the previous time step. This significantly reduces the number of iterations required to meet the convergence criterion. The accuracy of the method of fundamental solutions approach is demonstrated through two benchmark problems. The parallel efficiency of the domain decomposition method is evaluated by considering cases with 8, 27, and 64 subdomains. Copyright 2004 © John Wiley & Sons, Ltd.     

Iterative solution of systems of equations in the dual reciprocity boundary element method for the diffusion equation

In this paper the diffusion equation is solved in two-dimensional geometry by the dual reciprocity boundary element method (DRBEM). It is structured by fully implicit discretization over time and by weighting with the fundamental solution of the Laplace equation. The resulting domain integral of the diffusive term is transformed into two boundary integrals by using Green's second identity, and the domain integral of the transience term is converted into a finite series of boundary integrals by using dual reciprocity interpolation based on scaled augmented thin plate spline global approximation functions. Straight line geometry and constant field shape functions for boundary discretization are employed. The described procedure results in systems of equations with fully populated unsymmetric matrices. In the case of solving large problems, the solution of these systems by direct methods may be very time consuming. The present study investigates the possibility of using iterative methods for solving these systems of equations. It was demonstrated that Krylov-type methods like CGS and GMRES with simple Jacobi preconditioning appeared to be efficient and robust with respect to the problem size and time step magnitude. This paper can be considered as a logical starting point for research of iterative solutions to DRBEM systems of equations. © 1998 John Wiley & Sons, Ltd.     

Coupling of a non‐overlapping domain decomposition method for a nodal finite element method with a boundary element method

Non‐overlapping domain decomposition techniques are used to solve the finite element equations and to couple them with a boundary element method. A suitable approach dealing with finite element nodes common to more than two subdomains, the so‐called cross‐points, endows the method with the following advantages. It yields a robust and efficient procedure to solve the equations resulting from the discretization process. Only small size finite element linear systems and a dense linear system related to a simple boundary integral equation are solved at each iteration and each of them can be solved in a stable way. We also show how to choose the parameter defining the augmented local matrices in order to improve the convergence. Several numerical simulations in 2D and 3D validating the treatment of the cross‐points and illustrating the strategy to accelerate the iterative procedure are presented. Copyright © 2007 John Wiley & Sons, Ltd.     

Dual reciprocity BEM: local versus global approximation functions for diffusion, convection and other problems

The use of the global approximation functions (elements of Pascal's triangle, sine expansions and others) in the dual reciprocity boundary element method is compared to the better known local radial basis functions for convection, diffusion and other problems in which the volume integrals considered contain first and second derivatives of the problem variables, time derivatives and sums and products of functions, including nonlinear terms. It will be shown that whilst it is possible to obtain accurate solutions to the problems considered using the global functions, a successful solution to a given problem depends very much on the function chosen, as well as other factors.     

On the L2 and the H1 couplings for an overlapping domain decomposition method using Lagrange multipliers

In this paper, a comparison of the L² and the H¹ couplings is made for an overlapping domain decomposition method using Lagrange multipliers. The analysis of the local equations arising from the formulation of the coupling of two mechanical models shows that continuous weight functions are required for the L² coupling term whereas both discontinuous and continuous weight functions can be used for the H¹ coupling. The choice of the Lagrange multiplier space is discussed and numerically studied. The paper ends with some numerical examples of an end‐loaded cantilever beam and a cracked plate under tension and shear. It is shown that the continuity enforced with the H¹ coupling leads to a link with a flexibility that can be beneficial for coupling a very coarse mesh with a very fine one. To limit the effect of the volume coupling on the global response, a narrow coupling zone is recommended. In this case, volume coupling tends to a surface coupling, especially with a L² coupling. Copyright © 2006 John Wiley & Sons, Ltd.