Weight Function Shape Parameter Optimization in Meshless Methods for Non-uniform Grids
J. Perko; and B. S arler;

Source CMES: Computer Modeling in Engineering & Sciences, Vol. 19, No. 1, pp. 55-68, 2007
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Keywords Optimization, shape parameters, meshless methods, Diffuse Approximate Method.
Abstract This work introduces a procedure for automated determination of weight function free parameters in moving least squares (MLS) based meshless methods for non-uniform grids. The meshless method used in present work is Diffuse Approximate Method (DAM). The DAM is structured in 2D with the one or two parameter Gaussian weigh function, 6 polynomial basis and 9 noded domain of influence. The procedure consists of three main elements. The first is definition of the reference quality function which measures the difference between the MLS approximation on non-uniform and hypothetic uniform node arrangements. The second is the construction of the object function from the reference quality function which has to be minimized for optimum performance of the method on non-uniform node arrangement. The third is the optimization procedure for obtaining the minimum of the object function. The main idea of this paper is demonstrated on solution of the transient Burgers equation on stretched non-uniform grid and three types of random non-uniform grids. The inverse Gauss function is used as a reference quality function, object function is built from the second partial derivatives, and the$k$-ary heap like tree procedure is used for optimization. A substantial improvement of the accuracy of the method is achieved with the locally optimized values of the weight function compared to the fixed value that was exclusively used in previous DAM literature. With statistical comparison it was shown that in addition to improvement of accuracy also the stability of simulation is substantially improved.
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