Construction of generalized shape functions over arbitrary polytopes based on scaled boundary finite element method's solution of Poisson's equation

A general technique to develop arbitrary-sided polygonal elements based on the scaled boundary finite element method is presented. Shape functions are derived from the solution of the Poisson's equation in contrast to the well-known Laplace shape functions that are only linearly complete. The application of the Poisson shape functions can be complete up to any specific order. The shape functions retain the advantage of the scaled boundary finite element method allowing direct formulation on polygons with arbitrary number of sides and quadtree meshes. The resulting formulation is similar to the finite element method where each field variable is interpolated by the same set of shape functions in parametric space and differs only in the integration of the stiffness and mass matrices. Well-established finite element procedures can be applied with the developed shape functions, to solve a variety of engineering problems including, for example, coupled field problems, phase field fracture, and addressing volumetric locking in the near-incompressibility limit by adopting a mixed formulation. Application of the formulation is demonstrated in several engineering problems. Optimal convergence rates are observed.