Implementation of Hierarchical Tree Structure in Fast Multipole Method in 2-D Seismic Elastic Domain

Document Type : Geotechnical Earthquake Engineering


International Institute of Earthquake Engineering and Seismology (IIEES)


A numerical boundary element, as an appurtenance of integral equation method, has some useful characteristics that facilitate the solutions of numerical equations, but asymmetrical and sparse structure of formed stiffness matrix in large-scale boundary element method related to high degree of freedom problems make it unpractical, especially in seismic analysis of large-scale surface topographies with irregularities. Nowadays, fast algorithms such as fast multi-pole method present new media in numerical solutions with the aim of revolutionary changes in geometric definitions. In contrary with the usual node-to-node or element-toelement interconnection implementation, the cell-to-cell relation along hierarchy tree structure is applied. In most papers, the fast algorithm uses a two-level hierarchical tree structure as a part of algorithm internally without detail illustration. Therefore, a comprehensive detail of hierarchical tree structure is requested. In this paper, a multi-level (level definition is dynamic) hierarchical tree structure is presented with graphical theme and examples. This paper presents the relation between conventional boundary element method geometric structure with hierarchical tree model, and later, explains the method along with its abilities and limitations.


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