Meshless Methods for Computational Fluid Dynamics a Dissertation Submitted to the Department of Aeronautics and Astronautics and the Committee on Graduate Studies of Stanford University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy

While the generation of meshes has always posed challenges for computational scientists, the problem has become more acute in recent years. Increased computational power has enabled scientists to tackle problems of increasing size and complexity. While algorithms have seen great advances, mesh generation has lagged behind, creating a computational bottleneck. For industry and government looking to impact current and future products with simulation technology, mesh generation imposes great challenges. Many generation procedures often lack automation, requiring many man-hours, which are becoming far more expensive than computer hardware. More automated methods are less reliable for complex geometry with sharp corners, concavity, or otherwise complex features. Most mesh generation methods to date require a great deal of user expertise to obtain accurate simulation results. Since the application of computational methods to real world problems appears to be paced by mesh generation, alleviating this bottleneck potentially impacts an enormous field of problems. Meshless methods applied to computational fluid dynamics is a relatively new area of research designed to help alleviate the burden of mesh generation. Despite their recent inception, there exists no shortage of formulations and algorithms for meshless schemes in the literature. A brief survey of the field reveals varied approaches arising from diverse mathematical backgrounds applied to a wide variety of applications. All meshless schemes attempt to bypass the use of a conventional mesh entirely or in part by discretizing governing partial differential equations on scattered clouds of points. A goal of the present thesis is to develop a meshless scheme for computational fluid dynamics and evaluate its performance compared with conventional methods. v The meshless schemes developed in this work compare favorably with conventional finite volume methods in terms of accuracy and efficiency for the Euler and NavierStokes equations. The success of these schemes may be largely attributeed their sound mathematical foundation based on a local extremum diminishing property, which has been generalized to handle local clouds of points instead of mesh-based topologies. In addition, powerful algorithms are developed to accelerate convergence for meshless schemes, which also apply to mesh based schemes in a mesh transparent manner. The convergence acceleration technique, termed “multicloud,” produces schemes with convergence rates rivaling structured multigrid. However, the advantage of multicloud is that it makes no assumptions regarding mesh topology or discretization used on the finest level. Thus, multicloud is extrememly general and widely applicable. Finally, a unique application of meshless methods is demonstrated for overset grids in which a meshless method is used to seamlessly connect different types of grids. It is shown that meshless methods provide significant advantages over conventional interpolation procedures for overset grids. This application serves to highlight the practical utility of meshless schemes for computational fluid dynamics.