Least squares based finite element formulations and their applications in fluid mechanics

Abstract

In this research, least-squares based finite element formulations and their applications in fluid mechanics are presented. Least-squares formulations offer several computational and theoretical advantages for Newtonian as well as non-Newtonian fluid flows. Most notably, these formulations circumvent the inf-sup condition of Ladyzhenskaya-Babuska- Brezzi (LBB) such that the choice of approximating space is not subject to any compatibility condition. Also, the resulting coefficient matrix is symmetric and positive-definite. It has been observed that pressure and velocities are not strongly coupled in traditional leastsquares based finite element formulations. Penalty based least-squares formulations that fix the pressure-velocity coupling problem are proposed, implemented in a computational scheme, and evaluated in this study. The continuity equation is treated as a constraint on the velocity field and the constraint is enforced using the penalty method. These penalty based formulations produce accurate results for even low penalty parameters (in the range of 10-50 penalty parameter). A stress based least-squares formulation is also being proposed to couple pressure and velocities. Stress components are introduced as independent variables to make the system first order. The continuity equation is eliminated from the system with suitable modifications. Least-squares formulations are also developed for viscoelastic flows and moving boundary flows. All the formulations developed in this study are tested using several benchmark problems. All of the finite element models developed in this study performed well in all cases. A method to exploit orthogonality of modal bases to avoid numerical integration and have a fast computation is also developed during this study. The entries of the coefficient matrix are calculated analytically. The properties of Jacobi polynomials are used and most of the entries of the coefficient matrix are recast so that they can be evaluated analytically.