A Mass Conservation Algorithm for Adaptive Unrefinement Meshes Used by Finite Element Methods

Published in Procedia Computer Science, v9, p727-736, 2 June 2012. Presented at International Conference on Computational Science, ICCS 2012, Omaha, Nebraska, June 4-6, 2012. The original document contains color images.

The Adaptive Hydraulics (ADH) model is an adaptive finite element method to simulate three-dimensional Navier-Stokes flow, unsaturated and saturated groundwater flow, overland flow, and two- or three-dimensional shallow-water flow and transport. In the shallow-water flow and transport, especially involving multispecies transport, the water depth (h), the product of water depth and velocities (uh and vh), as well as water depth and chemical concentration (hc) are dependent variables of fluid-motion simulations and are often solved at various times. It is important for the numerical model to predict accurate water depth, velocity fields, and chemical distribution, as well as conserve mass, especially for water quality applications. Solution accuracy depends highly on mesh resolution. Adaptive mesh refinement (AMR), particularly the h-refinement, is often used to add new nodes in the region where they are needed and to remove others where they are no longer required during the simulation. The AMR is proven to optimize the performance of a computed solution. However, mass with gain or loss can occur when elements are merged due to removing a node at mesh coarsening. Therefore, we develop and implement the mass-conservative unrefinement algorithm to ensure the mass conserved in a merged element in which a node has been removed. This study describes the use of the Galerkin finite element method to redistribute mass to nodes comprising a merged element. The algorithm was incorporated into the ADH code. This algorithm minimizes mass error during the unrefinement process to conserve mass during the simulation for two-dimensional shallow-water flow and transport. The implementation neither significantly increases the computational time nor memory usage. The simulation was run with various numbers of processors. The results showed good scaling of solution time as the number of processors increases.