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The natural element method in solid mechanics. (English) Zbl 0940.74078
The authors present applications of the natural element method (NEM) to boundary value problems in two-dimensional small displacement elastostatics. The discrete model of the domain \(\Omega\) consists of a set of distinct nodes \(N\) and a polygonal description of the boundary \(\delta\Omega\). In the NEM, the trial and test functions are constructed using natural neighbour interpolants. These interpolants are based on the Voronoi tesselation of the set of nodes \(N\). The interpolants are \(C^\infty\) smooth everywhere, except at the nodes where they are \(C^0\). The NEM interpolant is strictly linear between adjacent nodes on the boundary of the convex hull. The authors also describe a methodology to model material discontinuities and nonconvex bodies (cracks) using NEM. A standard displacement-based Galerkin procedure is used to obtain the discrete system of linear equations. The authors present applications of NEM to various problems in solid mechanics, which include the patch test, gradient problems, bimaterial interface and a static crack problem. Excellent agreement with exact (analytical) solutions is obtained. The potential applications to other classes of problems – crack growth, plates and large deformations – are possible.

MSC:
74S30 Other numerical methods in solid mechanics (MSC2010)
74S05 Finite element methods applied to problems in solid mechanics
Software:
Triangle
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