Arch Appl Mech (2005) 74: 780–789 DOI 10.1007/s00419-005-0390-9 ORIGINAL John T. Katsikadelis The BEM for nonhomogeneous bodies Received: 19 January 2005 / Accepted: 8 April 2005 / Published online: 18 July 2005 © Springer-Verlag 2005 Abstract The boundary element method (BEM) is developed for nonhomogeneous bodies. The static or steady- state response of such bodies leads to boundary value problems for partial differential equations (PDEs) of elliptic type with variable coefficients. The conventional BEM can be employed only if the fundamental solu- tion of the governing equation is known or can be established. This is, however, out of question for differential equations with variable coefficients. The presented method uses simple, known fundamental solutions for homogeneous isotropic bodies to establish the integral equation. An additional field function is introduced, which is determined from a supplementary domain integral boundary equation. The latter is converted to a boundary integral by employing a domain meshless technique based on global approximation by radial basis function series. Then the solution is evaluated from its integral representation based on the known fundamental solution. The presented method maintains the pure boundary character of the BEM, since the discretization into elements and the integrations are limited only to the boundary. Without restricting its generality, the method is illustrated for problems described by second-order differential equations. Therefore, the employed funda- mental solution is that of the Laplace equation. Several problems are studied. The obtained numerical results demonstrate the effectiveness and accuracy of the method. A significant advantage of the proposed method is that the same computer program is utilized to obtain numerical results regardless of the specific form of the governing differential operator. Keywords Boundary element method · Nonhomogeneous bodies · Meshless · Partial differential equations · Analog equation 1 Introduction The boundary element method (BEM) has emerged as a powerful alternative to the so-called domain methods, such as the finite difference method (FDM) and finite element method (FEM), particularly in cases where better accuracy is required or the domain methods are inefficient, for example, infinite domains. The most important feature of the BEM, however, is that it requires discretization of the boundary rather than the domain. Hence BEM computer codes are easier to use. This advantage is particularly important for design, since the pro- cess involves shape modifications and thus complete remeshing, which are difficult to carry out using FEM. To implement BEM for a given problem the integral representation of its solution is required, which can be derived if the fundamental solution of the governing differential equation is known or can be established. For many differential equations the fundamental solution is known and interesting engineering problems have been successfully solved using the BEM. Difficulties arise when the fundamental solution cannot be determined or it is too complicated and thus impractical to evaluate numerically. These difficulties become practically insurmountable when we come across problems pertaining to nonhomogeneous bodies where the coefficients J.T. Katsikadelis School of Civil Engineering, National Technical University of Athens Zografou Campus, Athens GR-15773, Greece. E-mail: jkats@central.ntua.gr