A scaled boundary polyhedral element for three-dimensional analyses

Abstract

Computational structural analysis, primarily the finite element method (FEM), has been widely applied in engineering practice. The FEM for three-dimensional (3D) analysis employs simple element shapes such as tetrahedron and hexahedron, and polynomial functions to interpolate nodal variables so that element formulations can be easily developed. Nonetheless, laborious interventions of a numerical analyst are often required to ensure the mesh complies with complex boundary. The method is also inefficient to capture the stress singularities as occurring at crack tips. Additionally, the applications of digital imaging technology to structural and material engineering pose difficulties to the current computational modelling technology, which has mostly been developed for treating computer-aided design (CAD) models.

This thesis presents the development of a novel polyhedral element and its application in fully automatic 3D structural analyses. The polyhedral element is formulated based on the scaled boundary finite element method. It can have an arbitrary number of faces and edges. Only its surfaces are discretised and it can be extended to include higher-order approximations.

The polyhedral element can be efficiently used in fracture analyses. Its semi-analytical solutions are able to accurately represent various types of singularities without special treatment. The singular stress solution can be separated which allows direct extraction of fracture parameters.

The developed element also allows meshing flexibility. An automatic mesh generation for CAD models is developed using the polyhedral elements and an octree algorithm. Each octree cell is modelled as a polyhedral element eliminating the issue of hanging nodes encountered in FEM. Some polyhedra that intersect the domain boundary are trimmed using a robust cutting algorithm based on level sets. The cut polyhedra are again modelled as polyhedral elements.

Moreover, an automatic 3D image-based analysis is achieved using the octree mesh created based on an image’s colour intensities. This results in a mesh containing a limited number of octree cell patterns. Thus, each cell's solutions for each material present can be precomputed in a linear elastic analysis to improve computational efficiency.

The automatic meshing techniques using polyhedral elements presented can be used efficiently by practicing engineers to analyse problems with changing mesh topology such as crack propagation, moving boundary and damage.