Keywords: finite element modelling, incompressible Navier-Stokes, edge based data structure, fractional-step method, parallel computing.

The objective of this paper is to present a computational system capable of numerically simulating three-dimensional laminar incompressible Navier-Stokes fluid flow problems written in primitive variable form, using parallel computing. A formulation using a stabilized finite element method and fractional step method is adopted, and a second order stable monolithic formulation is obtained [1,2,3]. An edge-based data structure is used. The main advantage of this data structure is the reduction of the computational time necessary for simulations as all coefficients are calculated in a pre-processing step. The local conservation and symmetry are enforced at the discrete level calculating only the non-diagonal terms and by computing the diagonal one as the negative value of the summation of the same-row non-diagonal terms. The final edge based terms are obtained replacing the standard loop over the elements by edge loops. This formulation has stability properties for the pressure and advective terms based on their projections in the finite element spaces. The parallel communication is managed using the MPI (Message Passing Interface). The domain decomposition is obtained using the ParMetis (Parallel Graph Partitioning and Fill-reducing Matrix Ordering) and the distributed arrays and the final algebraic system of equations uses the PETSc (Portable, Extensible Toolkit for Scientific Computation) toolkit. The computational program developed was verified using several model problems and the results obtained are in agreement with the experimental, theoretical and numerical data available in the literature for laminar fluid flow. A preliminary performance study was completed and good speed-up was obtained. Further studies are required for exploiting the whole system capacity and to properly verify the computational efficiency of the parallel program.

1

R. Codina, "Pressure stability in fractional step finite element methods for incompressible flows", Journal of Computational Physics, 170, 112-140, 2001. doi:10.1006/jcph.2001.6725

2

R. Codina, "Stability finite element approximation of transient flows using orthogonal subscales", Compt. Meth. Applied Mechanics and Engineering, 191, 4295-4321, 2002. doi:10.1016/S0045-7825(02)00337-7

3

O. Soto, R. Löhner, J. Cebral, F. Camelli, "A stabilized edge-based implicit incompressible flow formulation", Comp. Meth. Applied Mechanics and Engineering, 193, 2139-2154, 2004. doi:10.1016/j.cma.2004.01.018

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