Sub-regions without subdomain partition with boundary elements Carlos Friedrich Loeffler a,n , Webe João Mansur b,c a Programa de Pós-Graduação em Engenharia Mecânica, UFES, CentroTecnológico, Av. Fernando Ferrari, 540, Bairro Goiabeiras, 29075-910 Vitoria, ES, Brazil b Programa de Pós-Graduação em Engenharia Civil, COPPE/UFRJ, Centro de Tecnologia, Bloco B, Ilha do Fundão, 21941-972 Rio de Janeiro, RJ, Brazil c LAMEMO Laboratory/COPPE/Federal University of Rio de Janeiro, Av. Pedro Calmon s/n, Anexo ao Centro de Tecnologia, Ilha do Fundão, 21941-596 Rio de Janeiro, RJ, Brazil article info Article history: Received 16 November 2015 Received in revised form 23 May 2016 Accepted 28 July 2016 Keywords: Sectorial heterogeneous problems Sub-regions Boundary Element Method abstract This paper presents a simple and effective numerical procedure to model domains with sectorial het- erogeneous properties using the boundary element method. The physical problem is divided into a complete homogenous domain and other non homogeneous sectors. Matrices similar to the standard H boundary element matrix are constructed for each sector, which are related to the energy stored in them. The assembly of the coefficients related to the sectorial matrices in the final boundary element system is done through the direct introduction of unknown variables at internal auxiliary points. Comparatively to the sub-regions technique, the proposed procedure is advantageous, since the effect of interface ap- proximations is attenuated and computer implementation of its corresponding numerical model is simpler. & 2016 Elsevier Ltd. All rights reserved. 1. Introduction One of the most unsuitable applications to the Boundary Ele- ment Method (BEM) concerns the modeling of non-homogeneous medium problems, very common cases in soil mechanics. Con- sidering that the heterogeneity is located sectorally, the use of sub- regions is still the most efficient approach [1]. As the procedure is based on a simple concept considering domain partition, mean- ingful changes were not observed concerning it along the time [2,3]. However, due the insertion of internal boundaries, in certain complex situations, the sub-regions technique is unsatisfactory because it becomes costly and inadequate for programming. Unfortunately, there are no other efficient BEM approach re- garding the solution of this important class of problems; thus domain methods such as the Finite Element Method [4,5] and the Finite Difference Method [6], are usually chosen to model it. In other important engineering branches such as fracture me- chanics, in which the BEM is more efficient than other traditional discrete methods, similar restrictions can occur. If layered-mate- rials are involved, sub-regions technique must be introduced and the size of the final matrix can be increased significantly. To pre- serve the advantages of the BEM in this case, occasionally some strategies are proposed to reduce the size of the final matrix when a large number of elements need to be used [7]. Anyway, the creation of additional internal boundaries and consequent dis- turbance in the approximation still persists. This work presents an alternative to treat this category of problems that, due to its simplicity, can accredit the BEM to fur- ther elaborate applications without implementation difficulties, highlighting interesting problems such as plasticity and time de- pendent cases that have distinct properties on the domain. With the proposed technique, once the complete domain is constituted by sectorial homogeneities, it is modeled superposing a homogeneous complete domain and other sub-domains with different properties. All sectors are linked through influence coefficients, which in the BEM procedure are generated by in- tegrations performed in non homogeneous boundaries, with the source points located in the complete or surrounding domain. Particularly, energy principles support the proposed procedure, which makes it even more distinct to the sub-region concept. In a way the idea of this work is approximately presented in models that consider localized domain actions, which consist of another kind of problem for which the BEM requires additional auxiliary techniques to become efficient. Loeffler and Mansur [8] used this approach to account for sectorial loads with the Dual Reciprocity technique [9]. Also the solution of problems of inter- action between soil and structure employing the FEM and the BEM together [10], as well as cases of integration between zoned plate bending [11,12] and plate-beam-column integrated systems [13,14] employ partially the idea shown in this article, in which common nodal points of different systems are assembled in a general matrix. It is worth mentioning that Wagdy and Rashed [15] also propose a formulation where they deduce and assemble Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/enganabound Engineering Analysis with Boundary Elements http://dx.doi.org/10.1016/j.enganabound.2016.07.018 0955-7997/& 2016 Elsevier Ltd. All rights reserved. n Corresponding author. E-mail addresses: carlosloeffler@bol.com.br (C.F. Loeffler), webe@coc.ufrj.br (W.J. Mansur). Engineering Analysis with Boundary Elements 71 (2016) 169–173