同几何分析研究进展

ADVANCES IN ISOGEOMETRIC ANALYSIS

同几何分析(isogeometric analysis)是当前工程分析的一种发展趋势,有可能对计算机辅助工程(CAE)产生重大影响.同几何分析的思想是采用计算机辅助设计(CAD)的几何语言,如NURBS(non-uniform rational B-spline)几何替代拉格朗日插值作为分析计算的基础.这种看似简单的几何语言变化,消除了困扰CAE多年的瓶颈问题,开启了一条紧密联系分析、设计和优化的新途径.本文论述了同几何分析的产生背景、理论、优点及其在各个领域的应用.系统总结了同几何分析在NURBS,T样条基函数构建,非结构化网格构建,有效积分方法,曲面修剪技术,网格细化等基础理论方面的进展,以及在板壳问题、大变形问题、流固耦合、结构优化、接触问题、生物力学、温度场和电磁场等领域的应用,展示了同几何分析相对于标准多项式插值有限元法的优势.

Isogeometric analysis （IGA） is a current trend in engineering analysis that is likely to leave a significantly impact on Computer-Aided Engineering （CAE）. The basic idea of IGA is to utilize CAD geometry to facilitate analysis. This seemingly simple change of geometric language eliminates many bottle-neck issues that plagued CAE for years, and opens a pathway for a tighter integration of design, analysis, and optimization. In this review paper, the background, theory, advantages, and applications of isogeometric analysis are discussed. Topics of algorithmic development, including NURBS bases, T-spline bases, unstructured meshes, efficient quadrature methods, trimmed surface technologies, re^nements and so on are reviewed. Recent advances of IGA in shell problems, large deformation, fluid-structure interaction, structural optimization, contact problems, biomechanics, thermal analysis, and electromagnetics are also summarized.