Development of a scalable finite element solution to the Navier?Stokes equations

Abstract

A scalable numerical model to solve the unsteady incompressible Navier-Stokes equations is developed using the Galerkin finite element method. The coupled equations are decoupled by the fractional-step method and the systems of equations are inverted by the Krylov subspace iterations. The data structure makes use of a domain decomposition of which each processor stores the parameters in its subdomain, while the linear equations solvers and matrices constructions are parallelized by a data parallel approach. The accuracy of the model is tested by modeling laminar flow inside a two-dimensional square lid-driven cavity for Reynolds numbers at 1,000 as well as three-dimensional turbulent plane and wavy Couette flow and heat transfer at high Reynolds numbers. The parallel performance of the code is assessed by measuring the CPU time taken on an IBM SP2 supercomputer. The speed up factor and parallel efficiency show a satisfactory computational performance. IntroductionDue to the relatively inexpensive high speed computers, numerical simulation approach, such as computational fluid dynamics (CFD), is widely adopted for investigating realistic and research problems. Numerical simulation has full control on computing the parameters of problems of different complexities. Therefore, it is able to provide a compromising solution among cost, efficiency and complexity to engineering problems.Although high speed computers and robust numerical techniques have been developed rapidly, the computation of turbulence at high Reynolds number using direct numerical simulation (DNS) is too expensive for practical problems. The large-eddy simulation (LES) is an alternative that demands relatively less computational load. However, it still requires huge amount of computation resources for simulations conducted on sequential computers. The recent advance of supercomputers provides a possibility for conducting these large scale computations. Sequential computer codes could be parallelized directly by compilers but it is unable to fully utilize supercomputers. Therefore, innovative parallel solution techniques are necessary for exploring the power of parallel computing.To facilitate parallel computation the domain is usually divided into several subdomains according to the structure of the mesh. This method is known as domain decomposition or the Schwarz method that is commonly adopted by CFD analysts. The discretized information is distributed to each processor which is responsible for the calculation in the corresponding subdomain. The boundary conditions are obtained from the neighboring subdomains during computations. Bjørstad et al. [1] show a parallel numerical solution by using nonoverlapping subdomains for elliptic problems. Liu and Leung [2] and Akal et al. [3] demonstrate the feasibility of overlapping domain decomposition for solving scalar transport equations and the incompressible Navier-Stokes equations, respectively.Analogous to domain decomposition, data parallelism is an alternative parallel algorithm. The basic concept is ...

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