Least Squares Finite Element Methods for Large Scale Incompressible Flows

Transcription

1 Least Squares Finite Element Methods for Large Scale Incompressible Flows by Tate T. H. Tsang Department of Chemical & Materials Engineering University of Kentucky Lexington, KY A presentation to honor Prof. Thomas F. Edgar on his 65 th birthday in the AIChE Annual Meeting, Salt Lake City, 2010.

2 A Transport Equation has 4 terms, Accumulation + Convection = Diffusion + Source/Sink It is relatively easy to obtain numerical solution for Diffusion/Conduction terms (leading to Sparse, Symmetric Linear System) It is quite challenging to deal with the Convection terms (leading to Sparse, Non symmetric Linear System)

3 C t C + = x Example: u GFEM creates Spurious Oscillations Upwind Differencing creates Numerical Diffusion Conc X

4 Conc X

5 No perfect numerical method for Convection Choose Least Squares Finite Element Method (LSFEM) as a compromise between the Galerkin Finite Element Method and Upwind Differencing Prof. Graham Carey and his former student (UT Austin), Dr. Bonan Jiang developed the LSFEM in 80 Dr. Jiang introduced LSFEM to me in 1990

6 Applications of LSFEM 8 2D Stokes Flows 8 2D Lid Driven Cavity Flows 8 2D Flows over an Obstacle 8 2D Flows over a Backward Facing Steps 8 2D Von Karman Vortex Shedding behind a Cylinder 8 2D Thermally Stratified Flows 8 2D Natural Convection 8 2D Rayleigh Benard Convection Cells 8 2D Doubly Diffusive Flows 8 2D Atmospheric Transport and Chemistry for Air Pollution Modeling 8 3D Lid Driven Cavity Flows 8 3D Natural Convection 8 3D Thermocapillary Flows 8 3D Atmospheric Transport and Chemistry for Air Pollution Modeling 8 Large Eddy Simulations of Turbulent Flows 8 Large Eddy Simulations of Pollutant Dispersion in the Atmospheric Convective Boundary Layers 8 Domain Decomposition based LSFEM for Large Scale Parallel Computations

7 LSFEM FORMULATIONS FOR THE NAVIER STOKES EQUATIONS (1) Velocity Vorticity Pressure Formulation: 7 unknowns, 8 equations ui ui P 1 k u ω + j = ε ijk t x x Re x u x ω i j j ω x = j j = 0 ε ijk = 0 u x k j j i j

8 LSFEM FORMULATIONS (2) Velocity Stress Pressure Formulation: 10 unknowns, 10 equations. ui ui P 2 S + u j = + t x x Re x S ij j i j u x j j = 0 1 u i = + 2 x j u x i j ij

9 LSFEM FORMULATION { V} = { uvwpω ω ω } x y z Time Discretization (n th time level) and linearization (m th Newton s step) Leads to, ( n) ( n+ 1, m) { } { } { } { } ( n+!, m+ 1 R = f + g LV ) {} { } ( n+!, m+ 1 b LV ) = I = R R dω ( Objective Function: { } n + 1, m+ 1) T V { } { } Minimization leads to, e Ω e T T { } { } { V} { } { } L Φ L Φ dω = L Φ b dω A x = b e Ω Ω e

10 Least Squares Finite Element Methods (LSFEM) 8 First Order Formulations Tang and Tsang, Int. J. Numerical Methods Fluids, 21(1995), Ding and Tsang, Int. J. Comp. Fluid Dynamics, 17 (2003), LSFEM leads to Symmetric Positive Definite Linear System of Equations A x = b 8 Robust Preconditioned Conjugate Gradient Methods (iterative methods for 3D problems) can be used to obtain Numerical Solution for the above SPD Linear System 8 Matrix free Method (no need to assemble A) can be used to greatly reduce Memory Requirement. This allows us to simulate very large problems 8 LSFEM has been used Successfully for a variety of Laminar and Turbulent Flows Ding and Tsang, Int. J. Numerical Methods Fluids, 37(2001),

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12 Application : Lid driven Cavity Flow (LDCF) 8 Re = 1000; 500,000 elements; 3,500,000 unknowns Ding and Tsang, International Journal of Computational Fluid Dynamics, 17(2003), 183.

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16 Application : 3 D Rayleigh Benard Convection Ra = 8000; 50,400 elements; 613,965 unknowns Tang and Tsang, Computer Methods in Applied Mechanics & Engineering, 140 (1997)

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18 Colorful Fluid Dynamics

19 Application : Large Eddy Simulation of Turbulent Flows

20 Subgrid Scale Modeling 8Smagorinsky Model 8Dynamic Subgrid Scale Model (Germano, Lilly) υ = t ( C Δ) 2 s S

21 Application : Transitional LDCF, use LES 8Re = 3,200; 216,000 elements; 2,269,810 unknowns

22 Application : Turbulent Channel Flow

23 Application : Turbulent Channel Flows on Cruncher 8Re = 3,240; 0 < t < 12; 65,536 elements; 707,850 unknowns 8Large Eddy Simulation (LSFEM), Dynamic Subgrid Scale Model 8This simulation takes about 1,454 sec. on 8 Processors Application : Turbulent Channel Flows on Cruncher 8Re = 3,240; 2,097,152 elements; 21,466,890 unknowns 8This simulation takes about 3 hr. on 16 Processors

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26 Our Cluster Building Experience 8 Cruncher (a 16 node AMD 1.2/1.33 GHz, DDR Cluster)

27 Domain Decomposition based Least Squares Finite Element Method for Large Scale Parallel Computations 8Non Overlapping Domain Decomposition 8Each Processor uses LSFEM to Simulate Fluid Flow in each Subdomain Ding, Jiang and Tsang, Ind & Eng Chem Research (2010)

28 Parallel Computations: Lid driven Cavity Flow (LDCF) 8 Case 1: Re = 400; tf = 40, 64x64x32, 131,072 elements; 975,975 unknowns 8 Case 2: Re = 400; tf = 40, 96x96x48, 442,368 elements; 3,227,287 unknowns 8 Case 3: Re = 400; tf = 40, 128x128x64, 1,048,576 elements; 7,571,655 unknowns 8 Case 4: Re = 1000; tf = 50, 128x128x64, 1,048,576 elements; 7,571,655 unknowns 8 Case 5: Re = 1000; tf = 50, 192x192x96, 3,538,944 elements; 25,292,071 unknowns IBM Intel EM64T Linux Cluster, 2 Dual Core Intel Xeon 5160 CPUs (3GHz) per Blade IB SDX 4X Interconnect between Blades

29 CPU times in seconds, Speedups and Efficiencies based on the # of CPUs # CPU Case 1 Case 2 Case 3 Case 4 Case (1.00/100) 4838(1.00/100) 12926(1.00/100) 17356(1.00/100) 79193(1.00/100) 2 917(1.65/83) 3088(1.57/78) 7954(1.63/81) 10857(1.6/80) 50346(1.57/78) 4 441(3.43/86) 1600(3.02/75) 4148(3.11/78) 5665(3.06/77) 25866(3.06/77) 8 217(6.98/87) 837(5.78/72) 2176(5.94/74) 3049(5.69/71) 13798(5.74/72) (12.8/80) 453(10.7/67) 1225(10.6/66) 1642(10.6/66) 7604(10.4/65) The Speedup and the efficiency (in percentage) values are given in parentheses

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31 CPU times in seconds, Speedups and Efficiencies based on the # of Blades # Blades Case 1 Case 2 Case 3 Case 4 Case (1.00/100) 3088(1.00/100) 7954(1.00/100) 10857(1.00/100) 50346(1.00/100) 2 441(2.08/104) 1600(1.93/97) 4148(1.92/96) 5665(1.92/96) 25866(1.95/97) 4 217(4.22/106) 837(3.69/92) 2176(3.66/91) 3049(3.56/89) 13798(3.65/91) 8 119(7.71/96) 453(6.82/85) 1225(6.49/81) 1642(6.61/83) 7604(6.62/83) The Speedup and the efficiency (in percentage) values are given in parentheses

32 8 7 6 Sppedups based on the # of Blades Linear Case 1 Case 2 Case 3 Case 5 Speedup # of Blades

33 Conclusions LSFEM leads to SPD linear systems of equations The large SPD system can be solved efficiently by Matrix free Conjugated Gradient Method LSFEM does not use any adjusting parameter for its numerical solutions Non overlapping, Domain Decomposition technique allows LSFEM to solve larger flow problems

34 Acknowledgement 8 National Science Foundation 8 U. S. Environmental Protection Agency Laura Burrell Lynne Fosberry Jamie Wright L. Q. Tang Biswanath Chowdhury X. Ding Q. Y. Jiang

35 Last but far from the least, Dear Professor Edgar, as a practical way to honor you, I am going to use your new book for my Process Control course. Congratulation on your 65 th Birthday. May you live ten thousands years long, and ten thousands times ten thousands years long.