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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2015, Vol. 32 ›› Issue (3): 348-358.doi: 10.3969/j.issn.1005-3085.2015.03.004

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基于多边形支持域的无单元Galerkin方法研究

张金凤,   王晓东,   欧阳洁,  冯  昭   

  1. 西北工业大学应用数学系，西安 710129
  • 收稿日期:2012-11-06 接受日期:2013-05-07 出版日期:2015-06-15 发布日期:2015-08-15
  • 基金资助:
    国家重点基础研究发展项目 (2012CB025903)；2011-2012 西北工业大学本科毕业设计重点扶持项目.

Study on Element-free Galerkin Method Based on Polygon Support Domain

ZHANG Jin-feng,   WANG Xiao-dong,   OUYANG Jie,   FENG Zhao   

  1. Department of Applied Mathematics, Northwestern Polytechnical University, Xi'an 710129
  • Received:2012-11-06 Accepted:2013-05-07 Online:2015-06-15 Published:2015-08-15
  • Supported by:
    The National Basic Research Program of China (2012CB025903); the Key Support Project on Undergraduate Graduation Design of Northwestern Polytechnical University (2011-2012).

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摘要： 本文针对传统无单元Galerkin方法不能直接施加本质边界条件的缺点，提出了基于多边形支持域的无单元Galerkin方法．该方法将计算点的支持域由矩形或圆形扩展为多边形，使得移动最小二乘形函数满足Kronecker函数性质，进而使无单元Galerkin方法可以直接施加本质边界条件．此外，该方法将积分背景网格与多边形支持域关联，可以避免重复的节点搜索，提高了无单元Galerkin方法的计算效率．数值结果表明，基于多边形支持域的无单元Galerkin方法不但具有较高的计算效率，且与稳定化方案耦合，可以成功克服对流占优引起的数值不稳定问题．

关键词: 多边形支持域, 无单元方法, 本质边界条件, Galerkin方法

Abstract:

In this paper, we propose the element-free Galerkin method based on a polygon support domain since the traditional method cannot enforce the essential boundary condition directly. The proposed method extends the support domain of the computing point to a polygon instead of a rectangle or circular domain, so that the moving least squares shape functions satisfy the property of the Kronecker function, which makes it available for an essential boundary condition to be enforced directly. In addition, the background cells are associated with support domains in the proposed method, which avoids searching nodes tautologically in the traditional method and increases the computational efficiency compared to the traditional method. Numerical examples show that the proposed method not only possesses a higher computational efficiency, but also can successfully conquer numerical instability problems introduced by the dominated convection, when combined with stabilization scheme.

Key words: polygon support domain, element-free method, essential boundary condition, Galerkin method

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