Accuracy of the fast multipole boundary element method with quadratic elements in the analysis of 3D porous structures

Abstract

In this work, a fast multipole boundary element method for 3D elasticity problem was developed by the application of the fast multipole algorithm and isoparametric 8-node boundary elements with quadratic shape functions. The problem is described by the boundary integral equation involving the Kelvin solutions. In order to keep the numerical integration error on appropriate level, an adaptive method with subdivision of boundary elements into subelements, described in the literature, was applied. An extension of the neighbour list of boundary element clusters, corresponding to near-field computations, was proposed in order to reduce the truncation error of expansions in problems with high stress concentration. Efficiency of the method is illustrated by numerical examples including a solid with single spherical cavity, solids with two interacting spherical cavities, and numerical homogenization of solids with cubic arrangement of spherical cavities. All results agree with analytical models available in the literature. The examples show that the method can be applied to the analysis of porous structures.

Fast modelling of field evaporation in atom probe tomography using level set methods

“…Existing methods for accelerating the BEM boundary calculation step include the fast multipole method [36,65], adaptive cross-approximation [65], and pre-corrected fast fourier transform methods [65]. The fast multipole BEM could also be used to potentially accelerate ion trajectory integrations, which requires calculation of external domain field points, although an octree structure would have to be constructed over this space [66]. Iterative methods to solving the FEM-BEM system could further accelerate the computation [67] by allowing the use of specialised solvers for the separate FEM and BEM components such as the conjugate gradient method and GMRES.…”

Evaluation of the FMBEM efficiency in the analysis of porous structures

“…The structure can be modelled as infinite by using the BEM. Such analysis was presented in previous work (Ptaszny, 2015), where the structure was modelled by using the superposition method and discretizing only the boundaries of the cavities. However, the purpose of the present contribution is to compare the FMBEM efficiency with the FEM, which requires the discretization of the whole domain.…”

“…Thus, only the boundaries of the cavities are discretized. Details of the method are given in Ptaszny (2015). The boundary element mesh on the cavities is the same as in the FMBEM-3 model [Figure 12(b)], and has 1,676 elements, 5,032 nodes and 15,096 DOF.…”

“…The BEM was successfully applied in the analysis of 2D and 3D nonhomogeneous materials containing pores, by Hu et al (2000), Liu (2009), Ptaszny and Fedeliński (2007), Fedeliński et al (2014), Ptaszny et al (2014), Ptaszny and Fedeliński (2011a), Rejwer et al (2014) and Ptaszny (2015), cracks, by Yoshida et al (2001), Liu (2009), Fedeliński et al (2014), Rejwer et al (2014) and Trinh et al (2015) and composite materials, by Kamiński (1999), Liu et al (2005), Chen and Liu (2005), Lei et al (2006), Liu (2009), Ptaszny and Fedeliński (2011b), Fedeliński et al (2014) and Huang et al (2015). However, as far as we know, there is a lack of comprehensive comparison between the higher-order approximation BEM and FEM in 3D large-scale analysis in the literature.…”

Identification of elastic properties of individual material phases by cou-pling of micromechanical model and evolutionary algorithm

“…The stiffness of multi-phase material can be estimated with the help of various approaches that have already been proposed. One of the most versatile method that can deal with the finite number of phases of any morphology is numerical homogenization based on finite [8][9][10] or boundary [11,12] element analysis of representative volume element (RVE). On the other hand, this approach generally requires relatively high computational cost.…”