PhamSy, Nam (2015) Parallel computations based on domain decompositions and integrated radial basis functions for fluid flow problems. [Thesis (PhD/Research)]

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Abstract
The thesis reports a contribution to the development of parallel algorithms based on Domain Decomposition (DD) method and Compact Local Integrated Radial Basis Function (CLIRBF) method. This development aims to solve large scale
fluid flow problems more efficiently by using parallel high performance computing (HPC). With the help of the DD method, one big problem can be separated into subproblems and solved on parallel machines. In terms of numerical analysis, for each subproblem, the overall condition number of the system matrix is significantly reduced. This is one of the main reasons for the stability, high
accuracy and efficiency of parallel algorithms. The developed methods have been successfully applied to solve several benchmark problems with both rectangular
and nonrectangular boundaries.
In parallel computation, there is a challenge called Distributed Termination Detection (DTD) problem. DTD concerns the discovery whether all processes in a
distributed system have finished their job. In a distributed system, this problem is not a trivial problem because there is neither a global synchronised clock nor
a shared memory. Taking into account the specific requirement of parallel algorithms, a new algorithm is proposed and called the Bitmap DTD. This algorithm
is designed to work with DD method for solving Partial Differential Equations (PDEs). The Bitmap DTD algorithm is inspired by the Credit/Recovery DTD class (or weightthrow). The distinguishing feature of this algorithm is the use of a bitmap to carry the snapshot of the system from process to process. The proposed algorithm possesses characteristics as follows. (i) It allows any process to
detect termination (symmetry); (ii) it does not require any central control agent (decentralisation); (iii) termination detection delay is of the order of the diameter of the network; and (iv) the message complexity of the proposed algorithm is optimal.
In the first attempt, the combination of the DD method and CLIRBF based collocation approach yields an effective parallel algorithm to solve PDEs. This approach has enabled not only the problem to be solved separately in each subdomain by a Central Processing Unit (CPU) but also compact local stencils to be independently treated. The present algorithm has achieved high throughput
in solving large scale problems. The procedure is illustrated by several numerical examples including the benchmark liddriven cavity flow problem.
A new parallel algorithm is developed using the Control Volume Method (CVM) for the solution of PDEs. The goal is to develop an efficient parallel algorithm
especially for problems with nonrectangular domains. When combined with CLIRBF approach, the resultant method can produce highorder accuracy and economical solution for problems with complex boundary. The algorithm is verified
by solving two benchmark problems including the square liddriven cavity flow and the triangular liddriven cavity flow. In both cases, the accuracy is in great agreement with benchmark values. In terms of efficiency, the results show that the method has a very high efficiency profile and for some specific cases a superlinear speedup is achieved.
Although overlapping method yields a straightforward implementation and stable convergence, overlapping of subdomains makes it less applicable for complex
domains. The method even generates more computing overhead for each subdomain as the overlapping area grows. Hence, a parallel algorithm based on nonoverlapping DD and CLIRBF has been developed for solving NavierStokes
equations where a CLIRBF scheme is used to solve the problem in each subdomain. A relaxation factor is employed for the transmission conditions at the interface of subdomains to ensure the convergence of the iterative method while the Bitmap DTD algorithm is used to achieve the global termination. The parallel algorithm is demonstrated through two fluid flow problems, namely the natural
convection in concentric annuli (Boussinesq fluids) and the liddriven cavity flow (viscous fluids). The results confirm the high efficiency of the present method in
comparison with a sequential algorithm. A superlinear efficiency is also observed for a range of numbers of CPUs.
Finally, when comparing the overlapping and nonoverlapping parallel algorithms, it is found that the nonoverlapping one is less stable. The numerical results show that the nonoverlapping method is not able to converge for high Reynolds number while overlapping method reaches the same convergence profile as the sequential CLIRBF method. Thus, in this research when dealing with nonNewtonian
fluids and large scale problems, the overlapping method is preferred to the nonoverlapping one. The flow of OldroydB fluid through a planar contraction was considered as a benchmark problem. In this problem, the singularity of stress at the reentrant corners always poses difficulty to numerical methods in obtaining stable solutions at high Weissenberg numbers. In this work, a high resolution
simulation of the flow is obtained and the contour of streamline is shown to be in great agreement with other results.
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Item Type:  Thesis (PhD/Research) 

Item Status:  Live Archive 
Additional Information:  Doctor of Philosophy (PhD) thesis, 
Faculty/School / Institute/Centre:  Current  Faculty of Health, Engineering and Sciences  School of Mechanical and Electrical Engineering (1 Jul 2013 ) 
Faculty/School / Institute/Centre:  Current  Faculty of Health, Engineering and Sciences  School of Mechanical and Electrical Engineering (1 Jul 2013 ) 
Supervisors:  Tran, CanhDung; MaiDuy, Nam; TranCong, Thanh 
Date Deposited:  16 Feb 2016 05:50 
Last Modified:  12 Jul 2016 04:19 
Uncontrolled Keywords:  domain decomposition; compact local integrated radial basis function; fluid flow; parallel high performance computing 
Fields of Research (2008):  09 Engineering > 0913 Mechanical Engineering > 091307 Numerical Modelling and Mechanical Characterisation 
URI:  http://eprints.usq.edu.au/id/eprint/28771 
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