A formulation of the fast multipole boundary element method applied to the analysis of anisotropic materials under body forces

Resumo

This work presents a formulation of the boundary element method with fast multipole expansion (MECMP) applied to the analysis of anisotropic elastic materials subject to body forces. Integral equations are obtained using the Somigliana identity. Integrals are divided into near field and far field. Near field, when the source points and integration elements are close, are treated as in the standard boundary element method, that is, integrating along the element and considering the interaction between source points (nodes) and the elements. On the other hand, in the far field, when the source points and integration elements are far away, the fast multipole method is applied. In this case, the fundamental solution is expanded in a Laurent series and the node-to-node interaction is replaced by a cell-to-cell interaction. Cells are generated by a hierarchical decomposition of the domain using the quad-tree algorithm. Different fast multipole operations are used to take advantage of the hierarchical domain decomposition and expansions of the fundamental solutions. Influence matrices are never explicitly obtained and the matrix-vector product are carried out with linear complexity. The linear system is solved by an iterative method. A preconditioning matrix is used to reduce the number of iterations to obtain a result with an specified accuracy. Effectiveness and efficiency in solving large-scale problems are discussed. The treatment of problems involving body forces are taken into account by the utilization of the modified boundary condition method. This approach entails augmenting the boundary condition with a specific solution tailored to the problem. Following the solution of the linear system, the particular solution is subsequently subtracted from both displacements and tractions. Importantly, this procedural step obviates the need for generating additional vectors or matrices within the matrix equation. The formulation presented in this article is based on a representation of complex variables of the integrands, similar to the formulation previously developed for potential (scalar) problems. Validation is carried out by comparing the results obtained by the two formulations: the standard boundary element method and the boundary element method with fast multipole expansion. The influence of the number of terms of the series expansion in the calculation of fundamental solutions and influence matrices is analyzed. Numerical examples are presented to demonstrate the efficiency, accuracy, and potential of the boundary element method with fast multipole expansion to solve large-scale problems, i.e., with tens of thousands of degrees of freedom.