Strong-form framework for solving boundary value problems with geometric nonlinearity

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• 作者：J. P. Yang ; W. T. Su
• 关键词：Key wordsgeometric nonlinearity ; incremental ; iterative algorithm ; radial basis collocation method (RBCM) ; strong form
• 刊名：Applied Mathematics and Mechanics
• 出版年：2016
• 出版时间：December 2016
• 年：2016
• 卷：37
• 期：12
• 页码：1707-1720
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Applications of Mathematics; Classical Mechanics; Mathematical Modeling and Industrial Mathematics;
• 出版者：Shanghai University
• ISSN：1573-2754
• 卷排序：37

文摘

In this paper, we present a strong-form framework for solving the boundary value problems with geometric nonlinearity, in which an incremental theory is developed for the problem based on the Newton-Raphson scheme. Conventionally, the finite element methods (FEMs) or weak-form based meshfree methods have often been adopted to solve geometric nonlinear problems. However, issues, such as the mesh dependency, the numerical integration, and the boundary imposition, make these approaches computationally inefficient. Recently, strong-form collocation methods have been called on to solve the boundary value problems. The feasibility of the collocation method with the nodal discretization such as the radial basis collocation method (RBCM) motivates the present study. Due to the limited application to the nonlinear analysis in a strong form, we formulate the equation of equilibrium, along with the boundary conditions, in an incremental-iterative sense using the RBCM. The efficacy of the proposed framework is numerically demonstrated with the solution of two benchmark problems involving the geometric nonlinearity. Compared with the conventional weak-form formulation, the proposed framework is advantageous as no quadrature rule is needed in constructing the governing equation, and no mesh limitation exists with the deformed geometry in the incremental-iterative process.