SBIR Phase I: A Unified Coordinates Approach to Gridless Computation

这个小企业创新研究（SBIR）第一阶段研究项目将开发一种创新的统一坐标方法来进行无网格计算。与大多数计算流体动力学（CFD）计算中使用空间固定的欧拉坐标系不同，本文提出的统一坐标方法是基于一个广义坐标系，该坐标系以q为流体粒子的速度，h为自由函数，运动速度为hq。它包括h=0时的欧拉坐标法和h=1时的拉格朗日坐标法。选择h来保持网格正交性可以得到一个坐标系，它不仅可以避免像欧拉坐标系那样滑动线上过多的数值扩散，还可以避免像拉格朗日坐标系那样严重的网格变形。与任意拉格朗日-欧拉（ALE）方法不同，不需要从扭曲的拉格朗日网格到空间固定的欧拉网格的重新映射。所有的计算完全在变换后的空间中完成，没有交错的网格。更重要的是，使用统一的坐标，计算网格是在计算流场的同时由流动（更准确地说是由伪粒子的运动）生成的。因此，计算网格不再是CFD计算的必需输入，无网格CFD计算成为可能。传统的CFD方法是基于空间固定的欧拉坐标方法，需要先验的体拟合计算网格来精确实现表面边界条件。在实际应用中，围绕复杂几何形状生成贴体网格是一个非常繁琐的过程，需要大量的人为干预，因此需要经验和专业培训，这使得CFD在某种程度上成为一门艺术。提出的SBIR将解决这一问题，并显著促进CFD在制造业作为设计工具的使用。

This Small Business Innovation Research (SBIR) Phase I research project will develop an innovative unified coordinates approach to gridless computation. Instead of using the spatially fixed Eulerian coordinate system as in most Computional Fluid Dynamics (CFD) computation, the proposed unified coordinates approach is based on a generalized coordinate system which moves with velocity hq with q as the velocity of fluid particles and h as a free function. It includes the Eulerian coordinates approach as a special case when h=0 and the Lagrangian when h=1. Choosing h to preserve grid orthogonality can result in a coordinate system which avoids not only excessive numerical diffusion across slip lines as in the Eulerian coordinates but also severe grid deformation as in the Lagrangian coordinates. Unlike Arbitrary Lagrangian-Eulerian (ALE) approach, no remapping from the distorted Lagrangian grid onto the spatially fixed Eulerian grid is required. All computations are done entirely in the transformed space without a staggered grid. More importantly, using the unified coordinates, the computational grid is generated simultaneously by the flow (more precisely by the movement of the pseudo particles) while computing the flow field. Therefore, the computational grid is no longer a required input for CFD computation and gridless CFD computation becomes feasible. The traditional CFD methods are based on the spatially fixed Eulerian coordinates approach, in which body-fitted computational grids are needed a priori to accurately implement surface boundary conditions. The generation of body-fitted grids around complicated geometries in real applications is a very tedious process, which requires substantial human intervention and hence experience and specialized training, making CFD as somewhat of an art. The proposed SBIR effort will resolve this problem and significantly facilitate the use of CFD in manufacturing industry as a design tool.