The Least-squares Finite Element Method For Grid Deformation And Meshfree Applications

Grid adaptation is often needed to improve the numerical solution of a Partial Differential Equation (PDE), due to, for example, shock waves and boundary layers. In moving boundary problems, the grid needs to be regenerated or adapted to fit the new domain. In this work, a LSFEM deformation method is developed for grid generation on fixed or moving domains. The LSFEM is a finite-elements method which seeks to minimize the PDE residual equation through the least-squares method. A new class of numerical methods currently being researched is the meshfree methods, in which the main goal is to numerically solve PDEs without the node connectivity. The LSFEM and the meshfree concept can be combined using ideas from current meshfree methods. In the LSFEM, it is important to have enough residual equations from the discretization of the variation equations to obtain an overdetermined system. In some cases, however, this requirement may not be satisfied, or if it is, the system may be extremely overdetermined. Using the meshfree concept, overlapping elements can be created to obtain enough residual equations to meet the right conditions.