基于全局相容最优角的平面网格优化

Planar mesh optimization based on globally-compatible optimal angles

为了改善平面三角网格的单元质量,提高有限元分析的精度和收敛性,提出了一种基于全局相容最优角的优化平面网格的方法。该方法首先直接以网格所有三角形的角度为优化变量,以网格顶点的拓扑度来定义理想角度,根据平面可嵌入条件构造全局相容性约束,并在该约束空间中拟合理想角度;然后采用高效的序列线性约束规划法数值求解最优角;其次运用最小二乘共形映射,将该角度及其拓扑关系嵌入平面欧氏空间;最后为改善网格的最小角度量,进一步采用自适应权重的相对角度误差能量进行局部优化。经过大量实验并与主流平面网格优化方法进行比较,结果表明:该方法与网格初始几何坐标无关,且其优化结果的最大角度量在大多数情况下优于已有方法,同时能够获得在平均意义上更好的三角形形状。

In order to optimize the element quality of planar triangle mesh and improve the accuracy and convergence of finite element analysis,this paper proposes a new planar mesh optimization method based on globally-compatible optimal angles.The proposed method first directly takes angles of all triangles of the mesh as the variants,defines the ideal angle by the topological degree of the vertices of the mesh,builds globally-compatible constraints according to the planar embedding conditions and fits the ideal angle in the constraint space.The optimal angle is then solved by using the efficient sequential linearly constrained programming.Secondly,this angle and its topological relationship are embedded into the planar Euclidean space via least square conformal mapping.Finally,in order to improve the minimal angle of the mesh,a local optimization method is further adopted based on the adaptive weighted relative angle error energy.After a lot of experiments and comparison with the mainstream planar mesh optimization methods,the results show that the proposed method is independent of the initial geometric coordinates of the mesh.In most cases,the maximum angle obtained by the proposed optimization method is superior to that of existing methods,and can achieve better triangle shapes on average.