等几何分析中Dirichlet边界条件的配点施加方法  

Imposing Dirichlet Boundary Conditions with Point Collocation Method in Isogeometric Analysis

作　　者：陈涛[1] 莫蓉[1] 万能[1] 

机构地区：[1]西北工业大学现代设计与集成制造技术教育部重点实验室,西安710072

出　　处：《机械工程学报》2012年第5期157-164,共8页Journal of Mechanical Engineering

基　　金：国家高技术研究发展计划资助项目(863计划;2007AA04Z184)

摘　　要：相对于传统有限元,等几何分析使用NURBS基函数统一表示几何和分析模型,它能消除传统有限元的网格离散误差,容易构造高阶协调单元。但是由于NURBS基函数缺乏插值性,控制顶点也不一定位于几何边界上,使得难以直接施加Dirichlet边界条件。针对这一问题,提出一种基于样条拟合的Dirichlet边界条件施加方法,通过引入一组边界配点来拟合边界条件。注意到不合适的配点策略会使得边界插值方程组奇异或者条件数过大,详细讨论配点选取的基本准则,提出两种鲁棒的配点方案:Greville横标和Chebyshev插值点法。并且将配点方法扩展到最小二乘形式,设计一种快速的场变量消去算法。通过实例验证上述方法的可行性和有效性。试验结果表明,各种配点策略的收敛率基本保持在一个量级,因此配点法的关键是选择稳定的配点方案。Contrasting to the conventional finite element method,isogeometric analysis uses the exact geometric representations for the modeling and numerical simulations by NURBS basis functions.It eliminates the geometric approximation errors during the mesh discretization,and the high-order conforming elements can be conveniently constructed.Due to the lack of the interpolation properties for the NURBS basis functions,it＇s difficult to impose the Dirichlet boundary condition directly.In order to solve this problem,the point collocation method is proposed to impose the boundary condition basing on the spline approximation theory.It approximates the constraints by introducing an array of distributed points along with boundary edges.The coefficient matrix of the resulting boundary systems will be singular or ill-condition while the unsuitable collocation schemes are adopted.Therefore,the distribution criterions of points are discussed in detail,and two robust collocation schemes that are Greville abscissa and Chebyshev sites are proposed.Meanwhile,the collocation method is extended to the least-squares forms.A simple and effective reduction approach of the degree of freedom is also devised.The numerical examples demonstrate the feasibility and effectiveness of the proposed methods.The results also show that almost all schemes remain the same magnitude of the convergence rate and the key issue of the collocation method is to ensure the numerical stability.

关 键 词：等几何分析 非插值性 Dirichlet条件 边界配点法 Greville横标 Chebyshev插值点 

分 类 号：O241[理学—计算数学]