The development of a polygon based numerical technique for structural analyses: The Scaled Boundary Polygons

A novel polygon based numerical technique is formulated using the scaled boundary finite element method. Advantages of polygon elements include greater flexibility in mesh generation, higher order approximations and improved numerical accuracy. Arbitrary polygon elements of three or more edges are successfully developed, and polygon based interpolants, the scaled boundary polygon shape functions, are formulated. These shape functions are smooth and continuous within the element, and are proven to be linearly complete. Notably, when modelling fracture, singularities are naturally included in the derivatives of these functions.The capabilities of this technique are demonstrated in two key applications: the analysis of fracture in functionally graded materials, and elastoplastic limit analysis. The application of the scaled boundary polygons to modelling both static and dynamic fracture in functionally graded materials (a new class of composite where the mineral constituents are tailored in a predetermined profile) is demonstrated first. In the present model, the material variations are approximated locally in each polygon using polynomial functions. The resulting semi-analytical expressions for both the stiffness and the mass matrices can be integrated in a straight forward manner. Stress intensity factors are obtained using the unified definition of generalised stress intensity factors, based on the scaled boundary solution of the singular stress field. Next, a scaled boundary polygon based formulation for the limit analysis of elastoplastic materials is formulated. Using mathematical programming, this technique is a powerful tool that can be used to evaluate limit loads without step-by-step evolutive analysis. The efficiencies of these formulations are demonstrated using numerical benchmarks and the method generally outperforms the available reference solutions. Coarser meshes with significantly fewer degrees of freedom are used to achieve the same accuracy as in the referenced studies. The scaled boundary polygons have been successfully implemented by other researchers in applications including crack propagation and elastoplastic analyses which further highlights the utility and significance of the present work