A boundary knot method for harmonic elastic and viscoelastic problems using single-domain approach Alfredo Canelas à , Berardi Sensale Instituto de Estructuras y Transporte, Facultad de Ingenierı ´a, Universidad de la Repu ´blica. J. Herrera y Reissig 565 – 11300 Montevideo, Uruguay article info Article history: Received 22 December 2009 Accepted 18 May 2010 Available online 8 June 2010 Keywords: Boundary knot method Viscoelasticity Meshfree methods Collocation technique Trefftz functions abstract The boundary knot method is a promising meshfree, integration-free, boundary-type technique for the solution of partial differential equations. It looks for an approximation of the solution in the linear span of a set of specialized radial basis functions that satisfy the governing equation of the problem. The boundary conditions are taken into account through the collocation technique. The specialized radial basis function for harmonic elastic and viscoelastic problems is derived, and a boundary knot method for the solution of these problems is proposed. The completeness issue regarding the proposed set of radial basis functions is discussed, and a formal proof of incompleteness for the circular ring problem is presented. In order to address the numerical performance of the proposed method, some numerical examples considering simple and complex domains are solved. & 2010 Elsevier Ltd. All rights reserved. 1. Introduction Meshfree methods for the solution of partial differential equations have gained large attention in recent years. These methods avoid the computational effort associated to the mesh generation in mesh-based methods like FEM or BEM. For high dimensional complex-shaped problems, the mesh generation process is usually very time consuming, especially for problems where remeshing is necessary, e.g., for accuracy reasons or when the application requires moving boundaries, like in dynamic free-boundary problems or in shape optimization. One of the first developed meshfree method for partial differential equations is the Trefftz method [1]. This method builds the solution as a linear combination of a previously known complete and linearly independent set of solutions (Trefftz basis). Another approach, that has the advantage that does not require to know a Trefftz basis for the problem, is to use radial basis functions (RBF). RBFs are a general and powerful tool in multi- variable approximation [2]. Using a particular radial solution of the governing equation (from now on called radial Trefftz function) we can define a specialized RBF method that, in conjunction with the collocation technique, constitutes a truly meshless method, since it only needs a set of distinct centers in the boundary of the domain to build the solution. The first studied of such RBF-based methods is the well-known method of fundamental solutions (MFS) introduced by Kupradze and Aleksidze [3]. It has been recognized as being highly accurate and fast convergent. However, it requires to set a controversial artificial boundary outside the physical domain [4,5]. In recent years, some techniques that avoid the use of the artificial boundary were proposed. For instance, the approach based on the indirect BEM proposed by Young et al. [6], and the boundary knot method (BKM), introduced by Kang et al. [7] and by Chen and Tanaka [8]. The BKM has been used to solve many problems in mathematical physics and engineering, like plate vibration [9], Poisson [10], convection–diffusion [11], inhomogeneous Helmholtz [12], Laplace and biharmonic problems [13], etc. In this paper we derive the radial Trefftz function for harmonic elasticity and viscoelasticity problems, allowing the implementa- tion of the BKM for the solution of these problems. To our best knowledge, this is the first time the BKM is considered for the solution of problems with a vector governing equation. The completeness issue about the set of radial Trefftz functions has usually held the attention of the researchers, but until now has remained obscure as no formal proof of completeness or incompleteness has been obtained. We investigate analytically and numerically the completeness of the proposed specialized RBF. For the circular ring problem we present a formal proof of incompleteness that can be easily adapted for all the governing equations considered previously by the researchers. This is the first time a formal proof of incompleteness is presented for the BKM, supporting the effort done by many researchers in the direction of developing a more suitable BKM technique based on domain decomposition for problems with holes or with compli- cated geometries [14–16]. We also present some numerical results and comparison of numerical performance between the proposed method and the implementation of the classic BEM described in [17]. ARTICLE IN PRESS Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/enganabound Engineering Analysis with Boundary Elements 0955-7997/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.enganabound.2010.05.010 à Corresponding author. E-mail addresses: acanelas@fing.edu.uy (A. Canelas), sensale@fing.edu.uy (B. Sensale). Engineering Analysis with Boundary Elements 34 (2010) 845–855