New elements with harmonic shape functions in adaptive mesh refinement

New elements are proposed in the present work based on the harmonic coordinates in which the shape functions satisfy the Laplace equation. The harmonic functions have some appealing characteristics that made it possible to define elements with arbitrary shape functions on the element boundaries and arbitrary node arrangement. In the present work, without loss of generality, we formulate a set of quadrilateral elements with midside nodes and piecewise linear shape functions along the element boundaries. The analytical solution of Laplace equation in rectangular domains are used here to represent the harmonic shape function as Fourier series. To derive the shape functions of different elements with different node arrangements a systematic approach is proposed to represent all of the shape functions using a small set of harmonic functions. Two model boundary value problems, Poisson equation and linear elasticity are used here to evaluate the proposed method. Patch test and convergence analysis are done for each model problems via some examples. The elements with midside nodes have great potential to be used in mesh adaptive solutions. Therefore, some adaptive mesh refinement examples are solved based on the ZZ error estimation and quadtree grid structure as the mesh refinement technique.