A piecewise partitioning Scaled Boundary Finite Element algorithm to solve viscoelastic problems with cyclic symmetry

Abstract

Scaled Boundary Finite Element Method (SBFEM) and a temporally adaptive algorithm are combined to solve viscoelastic problems. By expanding variables at a discretized time interval, a spatially and temporally coupled viscoelastic problem is decoupled into a series of recursive spatial problems, which are solved by SBFEM, the computing accuracy in the time domain is controlled via a self-adaptive process. For the cyclic symmetric structures, the cyclic symmetry is exploited to reduce the computational expense of SBFEM, both the eigenvalue and system equations of SBFEM are partitioned into a number of smaller independent problems, which are solved by a partitioning algorithm. Two numerical examples are given to verify and illustrate the proposed approach.

A scaled boundary finite element partitioning based reduced order algorithm for the elastic analysis of cyclically symmetric structures

“…Furthermore, if the coefficient matrix C A is block-circulant, the Equation ( 27) will have the same form as Equation (24). The proof process of the block-circulant property of C A is as follows:…”

“…22 When SBFEM works in BEM, some efforts have been made to reduce the solution scales of eigenvalue problems and system equation. For example, Yang et al developed a partitioning algorithm in the elastic, 23 viscoelastic, 24 and heat transfer 25 analysis for the cyclically symmetric structures; Song 26,27 proposed a reduced set of base functions for the unbounded problems, only a small part of the eigenvalue problems is required to be solved. Recently, Song et al and collaborators proposed the polygonal SBFEM 11,28,29 and the quadtree SBFEM, [30][31][32] in which the advantages of SBFEM in building element and conventional FEM framework is combined, and a series of successfully applications were achieved.…”