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Overview and recent advances in natural neighbour Galerkin methods. (English) Zbl 1050.74001

Summary: We give a survey of the most relevant advances in natural neighbour Galerkin methods. In these methods (also known as natural element methods, NEM), the Sibson and Laplace (non-Sibsonian) interpolation schemes are used as trial and test functions in a Galerkin procedure. Natural neighbour-based methods have certain unique features among the wide family of the so-called meshless methods: a well-defined and robust approximation with no user-defined parameters on non-uniform grids, and the ability to exactly impose essential (Dirichlet) boundary conditions.
A comprehensive review of the method is conducted, including a description of Sibson and Laplace interpolants in two and three dimensions. Application of the NEM to linear and nonlinear problems in solid as well as in fluid mechanics is studied. Other issues that are pertinent to the vast majority of meshless methods, such as numerical quadrature, imposing essential boundary conditions, and the handling of secondary variables are also addressed. The paper is concluded with some benchmark computations that demonstrate the accuracy and the key advantages of these numerical methods.

MSC:

74-02 Research exposition (monographs, survey articles) pertaining to mechanics of deformable solids
74S30 Other numerical methods in solid mechanics (MSC2010)
74S05 Finite element methods applied to problems in solid mechanics
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