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
A smooth and bounded interpolant can be constructed in explicit algebraic form within any polygon, convex or concave. The resulting function is not unique and accordingly can be adjusted to satisfy desired global conditions, such as linear fields. The closed-form representation is obtained by combining simple geometric descriptions, such as the side lengths and areas. The interpolant distributes values given at discrete nodes smoothly over the interior of the domain. On a convex polygon, the interpolant is a rational function of the product of areas. On a concave or multiply connected polygon, the interpolant is a function of areas and edge lengths, which introduces a square root term.
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