基于非结构化T样条的薄壳等几何分析

Unstructured T-spline-based Isogeometric Analysis for Thin Shells

等几何分析方法采用几何的样条基函数来构建分析模型,从而避免传统有限元法的网格离散过程.采用非结构化T样条构建复杂模型,结合Kirchhoff-Love薄壳理论构造基于非结构化T样条的薄壳单元,并研究其在模态计算和弹性变形分析中的应用.由于样条几何基函数缺乏插值性,等几何分析中单元所受载荷无法像传统有限元法一样直接均分到单元各个节点上,针对这一问题,将任意载荷作用区域分为两种基本形式,通过一定的映射关系将这两种形式由规则形状表示,从而将规则形状的高斯积分点同样映射到不规则区域上.数值算例结果表明,和传统有限元相比,基于非结构化T样条的薄壳等几何分析能够以更少的系统自由度获得精确解,而任意区域的积分映射方法也有效地解决了等几何分析中载荷施加的问题.

Isogeometric analysis(IGA) is a method aimed at avoiding meshing procedure of the traditional finite element analysis by using the same basis for analysis as is used to describe the geometry, thus integrating CAE into a general CAD modeling system. The unstructured T-splines are adopted to construct complex geometric models. By integrating the Kirchhoff-Love theory, an IGA thin-shell element is established for the modal analysis and elastic analysis. Due to the non-interpolatory nature of the geometric basis functions, the loads of IGA elements cannot be evenly distributed to the control points as with the traditional finite element. The loading areas are then classified to two basic types and an integration mapping method is proposed to map the Gauss quadrature points of the regular domain to the basic types. Numerical examples illustrated that the unstructured T-splines-based thin-shell element could achieve the convergence with much less degrees of freedom than the traditional finite element method and the integration mapping method solves the imposition problem of external loading effectively.