Hybrid numerical methods for convection-diffusion problems in arbitrary geometries

摘要

The hybrid nodal-integral/finite element method (NI-FEM) and the hybrid nodal-integral/finite analytic method (NI-FAM) are developed to solve the steady-state, two-dimensional convection-diffusion equation (CDL). The hybrid NI-FAM for the steady-state problem is then extended to solve the more general time-dependent, two-dimensional, CDL. These hybrid coarse mesh methods, unlike the conventional nodal-integral approach, are applicable in arbitrary geometries and maintain the high efficiency of the conventional nodal-integral method (NIM). In steady-state problems, the computational domain for both hybrid methods is discretized using rectangular nodes in the interior of the domain and along vertical and horizontal boundaries, while triangular nodes are used along the boundaries that are not parallel to the x or y axes. In time-dependent problems, the rectangular and triangular nodes become space-time parallelepiped and wedge-shaped nodes, respectively. The difference schemes for the variables on the interfaces of adjacent rectangular/parallelepiped nodes are developed using the conventional NIM. For the triangular nodes in the hybrid NI-FEM, a trial function is written in terms of the edge-averaged concentration of the three edges and made to satisfy the CDL in an integral sense. In the hybrid NI-FAM, the concentration over the triangular/wedge-shaped nodes is represented using a finite analytic approximation, which is based on the analytic solution of the one-dimensional CDL. The difference schemes for both hybrid methods are then developed for the interfaces between the rectangular/parallelepiped and triangular/wedge-shaped nodes by imposing continuity of the flux across the interfaces. A formal derivation of these hybrid methods and numerical results for several test problems are presented and discussed.