Castelo Antonio, Afonso Alexandre M, De Souza Bezerra Wesley
Departamento de Matemática Aplicada e Estatística, Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, Cx.P. 668, São Carlos 13560-970, SP, Brazil.
Centro de Estudos de Fenómenos de Transporte, Departamento de Engenharia Mecânica, Faculdade de Engenharia da Universidade do Porto, 4200-465 Porto, Portugal.
Polymers (Basel). 2021 Sep 18;13(18):3168. doi: 10.3390/polym13183168.
Tree-based grids bring the advantage of using fast Cartesian discretizations, such as finite differences, and the flexibility and accuracy of local mesh refinement. The main challenge is how to adapt the discretization stencil near the interfaces between grid elements of different sizes, which is usually solved by local high-order geometrical interpolations. Most methods usually avoid this by limiting the mesh configuration (usually to graded quadtree/octree grids), reducing the number of cases to be treated locally. In this work, we employ a moving least squares meshless interpolation technique, allowing for more complex mesh configurations, still keeping the overall order of accuracy. This technique was implemented in the HiG-Flow code to simulate Newtonian, generalized Newtonian and viscoelastic fluids flows. Numerical tests and application to viscoelastic fluid flow simulations were performed to illustrate the flexibility and robustness of this new approach.
基于树的网格带来了使用快速笛卡尔离散化方法（如有限差分法）的优势，以及局部网格细化的灵活性和准确性。主要挑战在于如何在不同大小的网格单元之间的界面附近调整离散模板，这通常通过局部高阶几何插值来解决。大多数方法通常通过限制网格配置（通常为分级四叉树/八叉树网格）来避免这一问题，从而减少需要局部处理的情况数量。在这项工作中，我们采用了移动最小二乘无网格插值技术，它允许更复杂的网格配置，同时仍保持整体精度阶数。该技术已在HiG-Flow代码中实现，用于模拟牛顿流体、广义牛顿流体和粘弹性流体流动。进行了数值测试并将其应用于粘弹性流体流动模拟，以说明这种新方法的灵活性和稳健性。
