Research on a Finite Element Numerical Algorithm for the Three-Dimensional Navier-Stokes Equations

摘要

The objective of this research project was to derive and evaluate accurate and efficient numerical solution algorithms for the three-dimensional Navier-Stokes equations. As a consequence of this objective, a generalized coordinates, implicit finite element numerical algorithm has been established for the problem class. The theoretical basis utilizes a Galerkin-Weighted Residuals statement, rendering the semi-discrete approximation error orthogonal to the finite element subspace, augmented with a penalty constraint forcing orthogonality of the gradient of this error as well. As a consequence, the algorithm possesses highly phase selective dissipation mechanisms permitting accurate resolution of solutions exhibiting a high degree of non-smoothness. A Fourier stability analysis yields an estimate of the dissipation parameter set, which is then refined to enhance the accuracy of a shocked flow prediction. Multiple factors affecting solution accuracy, convergence and efficiency have been examined. The generalized coordinates framework directly facilitates matching of arbitrary surface descriptions of the solution domain for complete geometric versatility.