Three-dimensional image-based modeling by combining SBFEM and transfinite element shape functions
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ORIGINAL PAPER
Three-dimensional image-based modeling by combining SBFEM and transfinite element shape functions Hauke Gravenkamp1
· Albert A. Saputra2
· Sascha Eisenträger2
Received: 22 October 2019 / Accepted: 15 July 2020 © The Author(s) 2020
Abstract The scaled boundary finite element method (SBFEM) has recently been employed as an efficient tool to model threedimensional structures, in particular when the geometry is provided as a voxel-based image. To this end, an octree decomposition of the computational domain is deployed, and each cubic cell is treated as an SBFE subdomain. The surfaces of each subdomain are discretized in the finite element sense. We improve on this idea by combining the semi-analytical concept of the SBFEM with a particular class of transition elements on the subdomains’ surfaces. Thus, a triangulation of these surfaces as executed in previous works is avoided, and consequently, the number of surface elements and degrees of freedom is reduced. In addition, these discretizations allow coupling elements of arbitrary order such that local p-refinement can be achieved straightforwardly. Keywords Scaled boundary finite element method · Octree meshes · Transition elements · Transfinite mapping · Local mesh refinement
1 Introduction The scaled boundary finite element method (SBFEM) is a semi-analytical technique—loosely based on finite elements— that involves only a boundary discretization of the computational (sub-)domains. Roughly speaking, this method aims at transforming a partial differential equation (PDE) in two or three spatial coordinates into a set of ordinary differential equations (ODE) in one coordinate by discretizing all but this one coordinate. In order to apply this idea effectively, a particular coordinate system is generally chosen in which one coordinate ξ points from the origin1 to the boundary while the remaining one or two coordinate(s) (η, ζ ) describe
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In the context of the SBFEM, the origin of the coordinate system is usually positioned inside the domain and referred to as scaling center.
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Hauke Gravenkamp [email protected] Department of Civil Engineering, University of Duisburg-Essen, Universitätsstraße 15, 45141 Essen, Germany School of Civil and Environmental Engineering, The University of New South Wales, Sydney, NSW 2052, Australia
a parametrization of the boundary. The ‘radial’ coordinate ξ is typically set to unity everywhere on the boundary. The SBFEM was originally developed to model large and unbounded domains in the context of soil-structure interaction and was inspired by concepts such as similarity [64], cloning [65], as well as the thin layer method [30,32]. There, it was assumed that the entire computational domain was enclosed by a simply connected boundary which was discretized by finite elements. The analytical solution of the resulting ODE can then be applied to describe either a bounded (ξ ≤ 1) or an unbounded domain (ξ ≥ 1). A detailed description of the underlying formulation can be found in the early papers [57,58,66],
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