Shape functions for polygonal domains with interior nodes

Abstract

SUMMARYThe presented formulation follows in a series of publications which outline a method for constructing test functions which satisfy essential edge conditions exactly. The method promises a complete solution, satisfying all of the requirements of a Ritz coordinate function. The influence of interior points on the domain solution is included in this construction. Similar to conformal bubble functions, the test functions are zero along the boundary and single valued only at the points they describe. Unlike the bubble function construction, the interior points can be located at any desired point in the domain. KEY WORDS: test function; distribution; finite element; rational polynomial; bubble function; meshless methods DOMAIN-DEPENDENT TEST FUNCTIONSFor simple domains where a domain dependent Green's function or eigenfunction solution exists, interior points, corresponding to external forces can be added at any point in the domain. The distributional solution satisfies the governing field equation and essential edge conditions exactly. The presented method approximates the ideal Green's function solution on any arbitrary polygonal domain using the Rayleigh-Ritz approximation method. It satisfies the essential edge conditions exactly, but not necessarily the governing field equation. Using such a test function, the solution to any elliptic boundary value problem can be approximated [1]. Similar to the exact domain dependent Green's function, the test functions are continuous but not smooth at the interior nodes. Nevertheless, the energy of the derivatives is finite and the method satisfies the continuity requirements [2, pp. 73-74]. Established methods for approximating the influence of a value at an interior point on the behaviour of the domain include: the finite element method, bubble function constructions and meshless methods. Bubble functions are not always used to represent interior points, instead

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Polytope finite elements

“…This makes it possible for natural and technical structures to be modelled using the geometry of their real elements, which is relevant, for example, in biomechanics and for crystal structures. In recent years, a number of researchers have proposed strategies to develop and extend the barycentric coordinates [3][4][5][6][7].…”

Three-dimensional variable-node elements based upon CS-FEM for elastic–plastic analysis

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