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# Quasilinear Hybrid Boundary Node Method for Solving Nonlinear Problems

F. Yan1,2, Y. Miao2,3, Q. N. Yang2
State Key Laboratory of Geomechanics and Geotechnical Engineering, Institute of Rock and SoilMechanics, Chinese Academy of Science, Wuhan 430071, China
School of Civil Engineering and Mechanics, Huazhong University of Science and Technology, Wuhan 430074, China
Corresponding auther, Tel. +86 27 87540172. Fax: +86 27 87542231; E-mail: my_miaoyu@163.com
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Computer Modeling in Engineering & Sciences 2009, 46(1), 21-50. https://doi.org/10.3970/cmes.2009.046.021

### Abstract

A novel boundary type meshless method called Quasilinear Hybrid Boundary Node Method (QHBNM), which combines quasilinearization method, dual reciprocity method (DRM) and hybrid boundary node method (HBNM), is developed to solving a class of nonlinear problems. The nonlinear term of the governing equation is linearized by the generated quasilinearization method, in which the solution of the linearized equation can exactly converge to the solution of original equation at a very wide range initial value, and the convergence rate is quadratic. Then dual hybrid boundary node method is applied to solving the linearized equation, in which DRM is introduced into HBNM to deal with the integral for the inhomogeneous terms of the governing equations. The solution in present method is divided into two parts, i. e., the complementary solution and the particular solution. The complementary solution is solved by HBNM, and the particular one is obtained by DRM. In order to get a generated use, the basis form of particular solution is presented in this paper. So a boundary type truly meshless method QHBNM is proposed, which retains all the advantages of BEM of linear problems. It does not require the 'boundary element mesh', either for interpolation of the variables, or for the integration of the 'energy'. The convergence of present iteration scheme is quadratic, and the initial values can be widely chosen. The computation is small in this method, in which only several matrixes are needed to update on each iteration. The numerical examples are presented for several nonlinear problems, for which accurate results, quadratic convergence and high stability can be available. It is shown that present method is effective and can be widely applied in practical engineering.

### Keywords

meshless method, dual reciprocity method, hybrid boundary node method, quasilinearization method, particular solution, nonlinear problem

Yan, F., Miao, Y., Yang, Q. N. (2009). Quasilinear Hybrid Boundary Node Method for Solving Nonlinear Problems. CMES-Computer Modeling in Engineering & Sciences, 46(1), 21–50. This work is licensed under a Creative Commons Attribution 4.0 International License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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