Construction of Parent Elements in 3D Based on an Isoparametric Formulation of the Scaled Boundary Finite Element Method
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Abstract This paper presents a technique for constructing 3D parent elements within the scaled boundary finite element method (SBFEM). The parent elements are arbitrary faceted star-convex polyhhedra and formulated in a parametric space using the SBFEM local coordinate system. Shape functions are constructed by solving Laplace’s equation in the SBFEM coordinate system. Polyhedral elements in physical space are mapped to their corresponding parent elements by an isoparametric transformation analogous to that used in the finite element method (FEM). The resulting formulation is shown to preserve linear completeness. An integration scheme is developed to accurately integrate each pyramidal sector forming a polyhedral element. The isoparametric concept enables the use of standard techniques of the FEM, including its ability to address geometric and material nonlinearities. To deploy the formulation, an augmented octree mesh generator employed. A domain is discretised using a set of unique cells consisting of standard octree patterns and augmented octree cells obtained by trimming with pre-defined templates. These cells define the parent element library used in the analysis. The application of the developed formulation is demonstrated for finite strain problems. Numerical benchmark problems validate the feasibility and demonstrate the advantages of the proposed method.
