Finite element grids based on a localized truncation error analysis
With the exponential increase in computing power, modelers of coastal and oceanic regions are capable of simulating larger domains with increased resolution. Typically, these models use graded meshes. However, the meshes are generated using criteria that neither optimize placement of the node points, with regards to solution accuracy, nor properly incorporate the physics, as represented by discrete equations, underlying flow to the mesh generation process. Consequently, the model user must manually adjust such meshes based on knowledge of local flow and topographical features--a time consuming proposition at best. This dissertation develops a method using localized truncation error analysis (LTEA) as a means to efficiently generate meshes that incorporate estimates of flow variables and their derivatives.
Three different LTEA methodologies are examined in a one-dimensional setting and include: (1) LTEA-based node spacing requirements that use solutions from a fine grid; (2) use of an imposed maximum multiple of change; (3) a more practical method that utilizes coarse grid solutions. Each method is compared to two common algorithms: the wavelength to $\Delta x$ ratio criterion and the topographical length scale criterion. It is shown that solutions based on LTEA meshes are, in general, more accurate (locally and globally) and more efficient.
Analyses and results from the one-dimensional study lay the groundwork for developing an efficient, physics-based mesh generating algorithm suitable for two-dimensional models. The LTEA methods listed above progress from complex to simple (in terms of the amount of information needed to generate the mesh). Two general steps are followed to produce an LTEA-based finite element mesh for the Gulf of Mexico. First, a grid is developed that maintains the wavelength to $\Delta x$ ratio between 80 and 140. Dynamic, steady-state results, forced with a single tidal constituent (M2), are attained from a two-dimensional, depth-averaged simulation. Second, a localized truncation error analysis is performed using these results and an LTEA-based finite element mesh is generated. Use of a localized truncation error analysis is shown to produce an accurate two-dimensional grid that incorporates estimates of flow variables and their derivatives in nodal placement.
0547: Ocean engineering