On the two-dimensional analysis of reinforced materials using the BEM/1D-BEM coupling

Resumo

The design of structures composed of different materials—aiming to harness their individual advantages in the final product—has become increasingly common due to the demands for economic feasibility and high performance in new projects. Among the numerical alternatives for evaluating the mechanical quantities of such structures, the coupling of the Boundary Element Method (BEM) with its one-dimensional version, the 1D-BEM, has shown excellent results when compared to commercial software. Moreover, this coupling proves to be numerically more stable in comparison to the classical one with reinforcements represented by the Finite Element Method. The BEM/1D-BEM formulation presented herein accounts perfect adherence between the matrix domain and the reinforcements. To build the global system of equations, this coupling scheme requires the integral equation for assessing the displacements in internal points, which may be computationally costly. This study proposes a more efficient discretization methodology for the reinforcement domain, reducing the number of internal points required for evaluating the aforementioned integral. Additionally, methods are proposed to avoid accuracy loss due to near singularities resulting from the proximity of sources in a reinforcement domain to other domains. This issue is more common in two-dimensional analyses than in three-dimensional ones and is rarely avoided without proper treatment when analyzing randomly distributed reinforcements. Examples are presented using both the classical BEM formulation, with Lagrangian basis functions for boundary discretization, and the isogeometric one (IGABEM), with discretization based on Non-Uniform Rational B-Splines (NURBS) curves.