A Finite-Element Coarse-Grid Projection Method: A Dual Acceleration/Mesh Refinement Tool for Incompressible Flows

TL;DR

This paper introduces a finite-element version of the coarse grid projection (CGP) method, significantly accelerating incompressible flow simulations while maintaining velocity accuracy, and explores its effects on pressure fields and boundary conditions.

Contribution

The paper presents the first finite-element CGP framework with semi-implicit time integration, enabling efficient pressure-velocity coupling in incompressible flows with minimal computational cost.

Findings

CGP achieves speedup factors of 2 to 30 in flow simulations.

Pressure gradient accuracy is preserved despite reduced pressure field accuracy.

Boundary conditions influence the extent of computational speedup.

Abstract

Coarse grid projection (CGP) methodology is a novel multigrid method for systems involving decoupled nonlinear evolution equations and linear elliptic Poisson equations. The nonlinear equations are solved on a fine grid and the linear equations are solved on a corresponding coarsened grid. Mapping operators execute data transfer between the grids. The CGP framework is constructed upon spatial and temporal discretization schemes. This framework has been established for finite volume/difference discretizations as well as explicit time integration methods. In this article we present for the first time a version of CGP for finite element discretizations, which uses a semi-implicit time integration scheme. The mapping functions correspond to the finite-element shape functions. With the novel data structure introduced, the mapping computational cost becomes insignificant. We apply CGP to pressure-correction schemes used for the incompressible Navier-Stokes flow computations. This version is validated on standard test cases with realistic boundary conditions using unstructured triangular meshes. We also pioneer investigations of the effects of CGP on the accuracy of the pressure field. It is found that although CGP reduces the pressure field accuracy, it preserves the accuracy of the pressure gradient and thus the velocity field, while achieving speedup factors ranging from approximately 2 to 30. Exploring the influence of boundary conditions on CGP, the minimum speedup occurs for velocity Dirichlet boundary conditions, while the maximum speedup occurs for open boundary conditions. We discuss the CGP method as a guide for partial mesh refinement of incompressible flow computations and show its application for simulations of flow over a backward-facing step and flow past a cylinder.