A localized MAPS using polynomial basis functions for the fourth-order complex-shape plate bending problems
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O R I G I NA L
Zhuo-Chao Tang · Zhuo-Jia Fu
· C. S. Chen
A localized MAPS using polynomial basis functions for the fourth-order complex-shape plate bending problems
Received: 31 January 2020 / Accepted: 3 June 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract In this paper, the localized method of approximate particular solutions using polynomial basis functions is proposed to solve plate bending problems with complex domains. The closed-form particular solutions of fourth-order differential equations can be reduced to the linear combination of the particular solutions of Helmholtz and modified Helmholtz equations. To alleviate the difficulty of solving overdetermined fourth-order plate bending problems using the localized collocation method, additional ghost points outside the computational domain are introduced to improve the stability and accuracy. Three examples are illustrated to validate the feasibility of the proposed localized MAPS. Keywords Localized · Method of approximate particular solutions (MAPS) · Helmholtz-type equations · Fourth-order plate bending problems
1 Introduction The traditional regular-shape plates have been used for many years in different engineering areas [24]. In recent years, irregular-shape plates are taken into consideration to satisfy the needs of different working conditions [19]. The plate bending problems are always one of the indispensable problems in engineering, and great effort has been devoted to developing novel algorithms to find efficient solution methods. In popular mesh-based methods such as finite element method (FEM) [1,2,20], finite difference method (FDM) [28] and finite volume method (FVM) [11,15], mesh generation and numerical integration [34] are undesirable, but essential. In contrast to the FEM, collocation methods which are also called mesh-free methods are only point-related and divided into two families including boundary-type and domain-type approaches. In boundary-type families, the method of fundamental solution [21,26,30] is one of the most famous methods so that it has been applied to various engineering problems. The drawback of this method is that the fundamental Z.-C. Tang · Z.-J. Fu (B) Key Laboratory of Ministry of Education for Coastal Disaster and Protection, Hohai University, Nanjing 210098, China E-mail: [email protected] Z.-C. Tang · Z.-J. Fu Center for Numerical Simulation Software in Engineering and Sciences, College of Mechanics and Materials, Hohai University, Nanjing 211100, China Z.-J. Fu Institute of Continuum Mechanics, Leibniz University Hannover, 30167 Hannover, Germany C. S. Chen School of Mathematics and Natural Sciences, University of Southern Mississippi, Hattiesburg, MS 39406, USA
Z.-C. Tang et al.
solution of the governing equation must exist to get the approximate solutions, as with other boundary-type methods such the boundary knot method [9], singular boundary method [22,27], boundary particle method [29] and etc [16]. On the other side, the domain-type methods usually select a basis f
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