Abstract
In this paper, the accuracy and the convergence properties of Trefftz finite element method over arbitrary polygons are studied. Within this approach, the unknown displacement field within the polygon is represented by the homogeneous solution to the governing differential equations, also called as the T-complete set. While on the boundary of the polygon, a conforming displacement field is independently defined to enforce the continuity of the field variables across the element boundary. An optimal number of T-complete functions are chosen based on the number of nodes of the polygon and the degrees of freedom per node. The stiffness matrix is computed by the hybrid formulation with auxiliary displacement frame. Results from the numerical studies presented for a few benchmark problems in the context of linear elasticity show that the proposed method yields highly accurate results with optimal convergence rates.
Highlights
Wachspress [1] introduced the concept of defining basis functions on any wedge form, which yields interpolants on polytopes of any convex shapes
Until recently elements with arbitrary number of sides did not find their applications in the computational mechanics, partly because of the associated difficulties with mesh generation and numerical integration
Overview of hybrid Trefftz finite element method The basic idea in the Trefftz FEM is to employ the series of the homogeneous solution to the governing differential equation as the approximation function to model the displacement field within the domain and an independent set of functions to represent the boundary and to satisfy inter-element compatibility
Summary
Wachspress [1] introduced the concept of defining basis functions on any wedge form, which yields interpolants on polytopes of any convex shapes. As the approximation functions over arbitrary polygonal elements are usually non-polynomial (in particular, rational polynomials) which introduces difficulties in the numerical integration, improving numerical integration over polytopes has gained increasing attention [3, 13,14,15]. The salient features of the approach are (a) only the boundary of the element is discretized with 1D finite elements, and (b) explicit form of the shape functions and special numerical integration scheme are not required to compute the stiffness matrix. Overview of hybrid Trefftz finite element method The basic idea in the Trefftz FEM is to employ the series of the homogeneous solution to the governing differential equation (see Eq 1) as the approximation function to model the displacement field within the domain and an independent set of functions to represent the boundary and to satisfy inter-element compatibility (see Fig. 3). For linear elastostatics, based on the Mushelishvili’s complex variable formulation, the NI and the corresponding stress fields are given by [29]
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