Localized method of approximate particular solution for solving incompressible Navier-Stokes equations

The localized method of approximate particular solutions (LMAPS) is a developing meshless numerical method. The LMAPS as introduced in Chapter 1, is developed from the method of particular solutions (MPS) and the method of approximate particular solutions (MAPS). Localization technique, along with normalization technique for radial basis functions and modified cross-shaped selection for local influence area allows the LMAPS to be more efficient than the MAPS for large scale computations or real time simulations. This research also provides generalized derivation applicable for various operators or equations in detail. The goal of this research is to use the LMAPS to solve multi-dimensional incompressible Navier-Stokes equations. In order to apply Chorin’s approach for solving incompressible Navier-Stokes equations, Poisson equation and Burgers equations have been tested prior to incompressible Navier-Stokes equations, respectively in Chapter 2 and Chapter 3. Chapter 2 uses Poisson equation to demonstrate the difference of applying the LMAPS with or without certain amount of global points, reasonable local influence area, or sufficient normalization technique. While in Chapter 3, an alternative numerical approach for obtaining solutions of Burgers equations was provided, via Cole-Hopf transformation, although it still has difficulties in dealing problems with Neumann boundary conditions. A generalized derivation of Cole-Hopf transformation is also demonstrated, including the derivation for numerical implementations of the essential conditions. Applying Cole-Hopf transformation transforms the governing equations into diffusion equation, it requires singular value decomposition (SVD) to solve the initial conditions, and least squares method (LSM) for solving multi-dimensional problems. Four experiments have been carried out to demonstrate capabilities of the proposed scheme with different number of global points, different number of local points, different time interval, irregular domain, and unstructured point distribution. After introducing Poisson equation and Burgers equations, Chapter 4 and 5 focus on the main task of solving multi-dimensional incompressible Navier-Stokes equations. The numerical solutions of incompressible Navier-Stokes equations is solved by the LMAPS with implementation of Chorin’s projection method, both chapter provide experiments on lid-driven cavity flow and backward-facing step flow. The experiments of two-dimensional viscous flow in Chapter 4 involve close investigations in matching the details with the results in literature or by other numerical methods, while Chapter 5 tries to solve three-dimensional problems with more efficiency and less consuming comparing with the experiments of Chapter 4. The optimal range for shape parameter has been determined while applying the proposed scheme to cases with different domain geometry, different temporal discretization, different point distribution, different governing equations, and different flow characteristics. This research proves the proposed scheme is capable of finding the same stable optimal range of shape parameter for all experiments given, can possibly give some accurate solutions, and can get accurate solution stably for different experiments. Finally, this research verifies the capability of the LMAPS to be able to solve multi-dimensional incompressible Navier-Stokes equations, and some possible approach for improving the proposed scheme is mentioned in future works.