Development of K-Version of the Finite Element Method: A Robust Mathematical and Computational Procedure

摘要

This report summarizes the research carried out under Grant F49620- 03-1-0201 on the development of least-squares based finite element models of viscous compressible and incompressible flows as well as shear deformable plates and shells. The main objective of this research was to develop a robust and accurate computational methodology based on least-squares variational principles for the numerical solution of the equations governing plates and shells and viscous incompressible and compressible fluid flows. The use of least-squares principles leads to a variationally unconstrained minimization problem, where compatibility conditions between approximation spaces -such as inf-sup conditions -- never arise. Furthermore, the resulting linear algebraic problem will always have a symmetric positive definite (SPD) coefficient matrix, allowing the use of robust and fast preconditioned conjugate gradient methods for its solution. In this research, the basic theory of least-squares finite element formulations of the equations governing viscous incompressible flows and shear deformable theories of plate and shell structures was carried out and their application through a variety of benchmark problems was illustrated. In the case of fluid flows, penalty least-squares finite element models using high p-levels and low penalty parameters were developed as a good alternative to mixed least-squares finite element models, also developed in the research.