包含复杂边界的三维非结构混合网格生成与优化方法研究

The unstructured hybrid grid technology could achieve the best trade-off between ease of use and computational accuracy. However, the application of the dynamic hybrid grid technology in moving boundary simulations needs to resolve three challenging issues. These issues correspond to three different gridding procedures, respectively. Firstly, the advancing layer technique for the generation of layered structured grids is geometrically heuristic and may thus fail to manage some complicated geometric configurations. Secondly, to ensure the grid validity, more powerful deformation algorithms are necessary for stretched structured grids. However, the computing efficiency of powerful grid deformation algorithms is usually low such that the speedup of such powerful algorithms must be considered when applying them in real applications. Thirdly, topology optimization is necessary when grid deformation results in low-quality and inverted grid elements. The local topology update technique for layered structured grids needs to maintain not only grid orthogonality and alignment, but also grid conformity. The later requirement could make it necessary to propagate the local update of one structured element to a sequence of neighboring grid elements. In this proposal, we suggest three techniques to resolve the above issues accordingly. Firstly, a PDE-based technique is proposed to overcome the robustness issues of the traditional advancing layer technique due to invalid normal, local intersections and global intersections. Secondly, a new mesh deformation technique will be developed upon the analogy of large-strain analysis of solids. Two types of interpolation schemes, coupled with the parallel solution technique, will be incorporated to speed up the mesh deformation procedure. Thirdly, a unified adaption approach will be investigated for the optimization of both unstructured and boundary layer parts of the hybrid grid. Meanwhile, surface adaption is allowed following the similar approach for the volume grid. It is highly hopeful that the above techniques, once developed successfully as expected, would enable the application of the dynamic hybrid grid technology in moving boundary simulations such as multi-medium flow simulations.

非结构混合网格能取得易用性和计算精度的最佳平衡，将相应动网格方法应用于动边界计算时，在初始网格生成、小变形后的网格变形与大变形后的网格优化等方面均面临尚未完全解决的难题: 1）生成结构化单元的前沿层进法存在鲁棒性问题；2）结构化单元对变形算法的变形能力要求苛刻，如何同时平衡计算效率的要求极具挑战；3）结构化单元的拓扑优化需满足贴边、正交等要求，且需扩散到相邻区域，算法设计更为复杂。针对上述难题，本项目：1）提出基于PDE方程的结构化网格生成新方法，解决前沿层进法的法向无效、局部相交和全局相交等问题；2）研究基于固体力学大应变分析类比的网格变形算法，结合并行有限元求解、沿空间方向插值和沿时间方向插值等技术提高变形算法的效率；3）基于网格局部编辑技术为非结构混合网格的自适应优化建立统一的方法体系。上述研究可促进非结构混合网格技术在多介质流动计算等动边界问题中的应用，有重要的理论和实践意义。

针对包含复杂边界流体计算问题，研究解决其中的非结构混合网格生成与优化难题，在混合网格生成、动网格以及大规模网格并行生成等专题上取得4项创新性成果：.1）提出基于拉普拉斯方程求解的边界层单元生长法向计算新方法，保证了法向的光滑过渡，有效减少了已有法向计算方法引起的的单元质量低、法向相交等情形。提出采用边界元法求解拉普方程，推导了法向精确计算的边界积分方程，开发了方程的快速多级边界元解算器，避免了有限元类方法基于插值计算得到的法向不够精确，需先构建覆盖问题区域的实体背景网格等缺点。.2）提出先在遗失边界与网格相交处加点实现保形恢复，然后利用点分裂及边删除操作移除Steiner点实现约束恢复的边界恢复新算法，发展了具有理论收敛保证的Delaunay网格生成方法。提出新的网格递归拓扑变换技术（壳变换），减少了复杂输入情形下边界恢复所需Steiner点数量，解决了因加点过多引起的浮点计算不稳定难题。该工作被成功拓展应用于解决动边界计算的网格局部重构、三角表面网格的实体布尔计算等难题。.3）提出了可靠高效的并行非结构网格重构方法，发展了带动边界流场的计算技术，解决了由于强非线性以及网格变形和拓扑改变导致的计算精度、稳定性和效率低下的难题。发展了面向动边界流场分析动网格生成以及全过程并行模拟等创新方法，建立了运用超级计算机进行飞行器空气动力设计分析的有效途径，成功应用该成果于多型飞行器气动分离特性计算。.4）提出以消除不理想形状特征为目标的网格简化技术及基于该技术的区域分解方法，减少了子区域边界对单元质量的负面影响，实现了高质量、高可扩展的并行网格生成方法与流程，开发了集并行曲面网格生成、并行实体网格生成以及并行体网格优化于一体的全过程并行网格生成方法体系，解决了十亿以上自由度精细数值模拟与分析面临的计算网格生成难题。