计算机辅助工程中的设计—分析—再设计的反复过程,蕴含着计算机辅助设计(CAD)与计算机辅助工程分析(CAE)相融合的迫切需求。提出基于CAD几何的数值流形方法:按照CAE真实物理场分布的复杂程度来布置数学网格和设置近似函数阶次,比等几何分析方法更合理;只需引入自动且快速的切割操作,就能实现CAD模型进入CAE后无需修改而直接建模;针对以往的流形法需要将曲线边界离散成折线的问题,给出了曲线边界与网格直边的切割算法,实现了几何模型在CAE建模和网格细化中的保形性;针对流形法通常使用的多项式近似函数,推导了曲线“近似”单纯形的解析积分公式并应用于带有曲线边界的流形元的精确积分运算;最后通过平板内的圆孔算例验证了方法的可行性。该方法对CAD和CAE的融合提出了全新的思路,为实现CAD进入CAE后的自动化分析打下了基础。

The iterative process of design-analysis-redesign implies urgent requests of the integration of computer aided design (CAD) and computer aided engineering (CAE). In this paper, we present a numerical manifold method (NMM) based on CAD geometry: we can arrange mathematical meshes and set order of approximation functions according to the complexity degree of physical field distribution in CAE, which is more reasonable than isogeometric analysis (IGA) method. Through introducing automatic and fast cutting operations, we can realize direct modeling from CAD model to CAE model without modifications. Moreover, to solve the problem that curves on geometric boundaries are required to be discretized into line segments in present NMM, we put forward algorithms to cut the curves of geometric boundary with the lines of mesh boundary, hence preserving the shape of the geometric model in the procedures of CAE modeling and mesh refinement. Furthermore, as for the polynomial approximation functions usually used in NMM, we deduce analytical integral formula of “approximate” simplex with a curved edge and use it to obtain precise integral calculations of manifold elements with curved boundaries. Finally, we verify the feasibility of the method through an example of a circular hole in a plate. The research offers new thinking for the integration of CAD and CAE, and lays foundation for the automatic analysis from CAD models to CAE.