Jarno Verkaik

Site of the project:

TNO Technisch Physische Dienst

Stieltjesweg 1 (Post box 155)

2600 AD Delft

start of the project: October 2002

In February 2003 the Interim Thesis has been appeared.

The Master project has been finished in August 2003 ( Masters Thesis). For working address etc. we refer to our alumnipage.

At the Department of Process Physics at TNO TPD transport phenomena are investigated. For the glass industry a large CFD simulation package called X-stream is developed to simulate flows in glass furnaces. The involving equations are the incompressible Navier-Stokes equations, the energy equation, and other equations arising from additional physical models related to the process of glass melting. These equations are discretised with the Finite Volume Method on a colocated grid.

Within X-stream a domain decomposition (DD) approach is used. A DD algorithm (or multi-block algorithm) is an iterative method in which the spatial domain is decomposed into subdomains (blocks) for which the equation is solved. In X-stream an additive Schwarz DD method with minimal overlap is used with inaccurate subdomain solution. Different equations on different subdomains can be solved and local grid refinement is possible at subdomain level. The DD algorithm in X-stream can be run in parallel.

Solving the incompressible Navier-Stokes equations is time consuming because they are nonlinear. In X-stream the Semi-Implicit Method for Pressure-Linked Equations (SIMPLE) is used to solve the nonlinear system of equations. SIMPLE is an iterative method where the system in each iteration is split up into linear systems for the various unknowns. With a DD approach these systems contain couplings between the subdomains. In each SIMPLE iteration the systems for the velocities and the pressure are solved with the Schwarz method. This we will refer to as Schwarz in the SIMPLE method.

There is not much experience with the DD algorithm in X-stream. The aim of the Master's project is to improve the DD algorithm used for solving the incompressible Navier-Stokes equations. It can be observed that more iterations are needed when the pressure is solved with the Schwarz method for an increasing number of subdomains. This is due to the elliptic nature of discretised equations for the pressure. To overcome this problem, and to make the DD algorithm scalable, a deflation coarse grid correction will be applied. Besides this, another aim in the project is to accelerate Schwarz in the SIMPLE method. This method can be seen as nested basic iterative methods which lend themselves well for Krylov subspace or multigrid acceleration. Acceleration will be focussed on the Schwarz method for the pressure and the SIMPLE method because they are most time-consuming. This will be done with the GCR Krylov subspace method. Also parallellization aspects will be considered in the Master's project.