Abstract
Abstract In recent years, the Meshless Local Petrov-Galerkin (MLPG) Method has attracted the attention of many researchers in solving several types of boundary value problems. This method is based on a local weak form, evaluated in local subdomains and does not require any mesh, either in the construction of the test and shape functions or in the integration process. However, the shape functions used in MLPG have complicated forms, which makes their computation and their derivative's computation costly. In this work, using the Moving Least Square (MLS) Method, we dissociate the point where the approximating polynomial's coefficients are optimized, from the points where its derivatives are computed. We argue that this approach not only is consistent with the underlying approximation hypothesis, but also makes computation of derivatives simpler. We apply our approach to a two-point boundary value problem and perform several tests to support our claim. The results show that the proposed model is efficient, achieves good precision, and is attractive to be applied to other higher-dimension problems.
Highlights
Engineers and scientists often need to solve and simulate physical problems for which analytical solutions do not exist
Several tests demonstrate the advantages of our Least-Square-Consistent Meshless Local PetrovGalerkin formulation over the traditional MLPG1 formulation
We use specific values of αα and ββ based on the results shown in Section 3.1 and compare both our solution and MLPG1's solution against the exact solution given in Equation (43)
Summary
Engineers and scientists often need to solve and simulate physical problems for which analytical solutions do not exist. The MLPG has already been used to solve various types of boundary value problems (Amini et al, 2018; Han and Atluri, 2004; Hu and Sun, 2011; Kamranian et al, 2017; Liu et al, 2011; Sheikhi et al, 2019; Zhang et al, 2006) In developing those formulations, the authors broke the underlying consistency with the Moving Least-square assumptions, which, in our view, led to shape functions that have unduly complex forms, making their computation and their derivatives' computation quite costly (Liu, 2009; Mirzaei and Schaback, 2013). We consider a two-point boundary value problem
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