Brief note DOI 10.1007/s00158-004-0403-2 Struct Multidisc Optim 28, 221–227 (2004) Structural shape optimisation using boundary elements and the biological growth method C. Wessel, A. Cisilino and B. Sensale Abstract A numerical evolutionary procedure for the structural optimisation for stress reduction of two- dimensional structures is presented in this paper. The proposed procedure couples the biological growth method (BGM) with the boundary element method (BEM). The boundary-only intrinsic characteristic of BEM together with its accuracy in the boundary dis- placement and stress solutions make BEM especially attractive for solving shape-optimisation problems. Two formulations of BEM are used in this work: the standard for two-dimensional elastostatics for the stress analysis and the dual reciprocity method (DRM), which is used to model the swelling or shrinking of the material. Two examples are analysed to illustrate the proposed method- ology and to demonstrate its versatility and robustness. Key words biological growth method, boundary elem- ents method, dual reciprocity method, shape optimisa- tion 1 Introduction The failure of structures under service conditions fre- quently takes place at highly stressed points. Therefore, it is crucial for designers to avoid stress peaks in order Received: 26 June 2002 Revised manuscript received: 28 January 2004 Published online: 6 July 2004  Springer-Verlag 2004 C. Wessel 1, ✉ , A. Cisilino 2 and B. Sensale 3 1 Facultad de Ingenier´ ıa, Universidad Austral, Garay 125, Ciu- dad Aut´ onoma de Buenos Aires, Argentina e-mail: cwessel@fi.mdp.edu.ar 2 Facultad de Ingenier´ ıa, Universidad Nacional de Mar del Plata – CONICET, Av. Juan B. Justo 4302 (7600) Mar del Plata, Argentina 3 Instituto de Estructuras y Transporte, Facultad de Inge- nier´ ıa, Universidad de la Rep´ ublica, J. Herrera y Reissig 665 – 11300 Montevideo, Uruguay to maximise a component service life, a fact that justifies the importance given to the subject so far. The determin- istic strategies for the solution of the structural optimi- sation problems are, on the one hand, the mathematical programming methods and, on the other, the optimality criterion-based methods. In the former, well-established mathematical tools are used, like direct search (Trosset 1997), derivative-free methods (Lucidi et al. 2002; Lewis et al. 2000) and gradi- entless methods (Schnack 1988; Schnack et al. 1988; Iancu and Schnack 1989; Schnack and Iancu 1989). The latter, that is, optimality criterion-based methods, take advantage of the knowledge on the physics and me- chanics of the particular problem (Sauter et al. 1996), providing a necessary condition for a minimum of the ob- jective function. The physical and mechanical knowledge put to use in optimality criterion-based methods is also their principal drawback, as it limits their application to certain definite areas. Biological structures, such as bones and trees, provide a simple example for shape optimisation, as they change their contour to adapt to external loads while reducing stress peaks. In this line, Mattheck (1990) introduced an optimality criterion method called the biological growth method (BGM), related to Schnack’s gradientless method (Iancu 1991; Schnack et al. 1988; Sp¨ orl 1985; Schnack 1978). Based on his observations in Nature, Mattheck posits that biological structures always self-optimise their geometry to attain a state of constant stress at part of or the whole of the surface of the structure. The process of self-optimisation is carried out through the swelling or shrinking of the soft outermost layer of material, which yields the levelling of local stresses. Since the original work by Mattheck and Burkhardt (1990) was published, some papers have appeared coup- ling the BGM with the finite element method (FEM) for structural shape optimisation (Chen and Tsai 1993; Sauter 1993; Tekkaya and G¨ uneri 1998). At the same time, the boundary element method (BEM) has become a popular alternative in structural shape optimisation (Baron and Yang 1988; Kane and Saigal 1988; Mellings and Aliabadi 1995) due to its accuracy in the bound- ary displacement and stress solutions, as well as the fact that remeshing is simpler for BEM than for FEM.