Adaptive Solution Techniques for fluid-structure interaction and multiphase flow

1
Adaptive Solution Techniques for fluid-structure
interaction and multiphase flow
Mark Sussman Department of Mathematics Florida
State University Samet Kadioglu Multiphysics
Methods Group Advanced Nuclear Energy Systems
Dept. Idaho National Laboratory Viorel
Mihalef Center for Biological Imaging in
Medicine, Department of Computer Science Rutgers
University
Interface Problems Workshop SAMSI program on
Random Media Thursday, November 15
2
Atomization
3
Atomization
4
Atomization
5
Flow Past a Whale
coarse
medium
6
(No Transcript)
7
Scalability of Pressure Projection Step?
Precond. Uniform? iter CPU
GSRB Yes 26 12.4
Line Yes 14 14.8
ILU Yes 12 9.0
GSRB No 95 37.7
Line No 26 25.0
ILU No 22 14.0
8
Level Set Equations for Multiphase
Flow Y.C. Chang, T.Y. Hou, B. Merriman, and S.
Osher, A Level Set Formulation of Eulerian
Interface Capturing Methods for Incompressible
Fluid Flows,J.Comput.Phys.,124 (1996), pp.
449-464
9

1. Advection
2. Diffusion
3. Pressure Projection Step

1. Nonlinear Advection
10
2. Diffusion
Li, J. Renardy, Y. Renardy, M. (2000)
Numerical simulation of breakup of a viscous drop
in simple shear flow through a volume-of-fluid
method. Physics of Fluids, vol. 12(2), pp.
269282.
11
3. Pressure projection step
12

• State-of-the-art
• Tanguy, Menard and Berlemont, A Level Set Method
  for vaporizing two-phase flows, JCP, 2007.
• Herrmann, A Eulerian level set/vortex sheet
  method for two-phase interface dynamics, JCP,
  2005.
• Quan and Schmidt, A moving mesh interface
  tracking method for 3D incompressible two-phase
  flows, JCP, 2007.
• Al-Rawahi and Tryggvason, Numerical simulation
  of dendritic solidification with convection
  Three-dimensional flow, JCP, 2004.
• Arienti, Madabushi, Van Slooten and Soteriou,
  Numerical simulation of liquid jet
  characteristics in a gaseous crossflow, ILASS
  Americas, 2005.
• Sussman, Smith, Hussaini, Ohta, Zhi-Wei, A sharp
  interface method for incompressible two-phase
  flows, JCP, 2007

13

• Marella, Krishnan, Liu and Udaykumar, Sharp
  interface Cartesian grid method I An easily
  implemented technique for 3D moving boundary
  computations, JCP, 2005.
• Francois, Cummins, Dendy, Kothe, Sicilian and
  Williams, A balanced-force algorithm for
  continuous and sharp interfacial surface tension
  models within a volume tracking framework, JCP,
  2006.
• Losasso, Fedkiw and Osher, Spatially Adaptive
  Techniques for Level Set Methods and
  Incomopressible Flow, Computers and Fluids,
  2006.
• Marchandise, Geuzaine, Chevaugeon, and Remacle,
  A stabilized finite element method using a
  discontinuous level set approach for the
  computation of bubble dynamics, JCP, to appear.
• Yang, James, Lowengrub, Zheng, and Cristini, An
  adaptive coupled level-set/volume-of-fluid
  interface capturing method for unstructured
  triangular grids, JCP, 2006.
• Yu, Sakai, Sethian, A coupled quadrilateral grid
  level set projection method applied to ink jet
  simulation, JCP, 2005.

14
Matrix solver for data organized on an adaptive
hierarchy of grids (Block Structured Adaptive
Mesh Refinement).
How to treat singular source terms, discontinuous
coefficients, grid stretching, and complex
geometries in a scalable way?
M. Sussman, A parallelized, adaptive algorithm
for multiphase flows in general geometries,
Journal of Computers and Structures, Volume 83,
Issues 6-7, February 2005, Pages 435-444.
15
Discretized Equation to be solved on hierarchy of
rectangular grids
16
Example in one dimension
17
Matrix Associated with the previous example.
18
Condition number dependence on density ratio
Density Ratio Condition Number
1 4.4E2
10 3.9E3
100 3.8E4
1000 3.8E5
19
Background of Matrix Iterative approaches used in
the context of incompressible two-phase flow.
Mark Sussman, Peter Smereka and Stanley Osher, A
Level Set Approach for Computing Solutions to
Incompressible Two-Phase Flow, Journal of
Computational Physics, Volume 114, Issue 1,
September 1994, Pages 146-159. (ILU
PCG) Elbridge Gerry Puckett, Ann S. Almgren,
John B. Bell, Daniel L. Marcus and William J.
Rider, A High-Order Projection Method for
Tracking Fluid Interfaces in Variable Density
Incompressible Flows, Journal of Computational
Physics, Volume 130, Issue 2, 15 January 1997,
Pages 269-282. (MGPCG 10015 speed-up over
MG) Mark Sussman, Ann S. Almgren, John B. Bell,
Phillip Colella, Louis H. Howell and Michael L.
Welcome, An Adaptive Level Set Approach for
Incompressible Two-Phase Flows, Journal of
Computational Physics, Volume 148, Issue 1, 1
January 1999, Pages 81-124 (single level
MGPCG) Frank Losasso, Ronald Fedkiw and Stanley
Osher, Spatially adaptive techniques for level
set methods and incompressible flow, Computers
Fluids, Volume 35, Issue 10, December 2006, Pages
995-1010. (PCG)
20
Algebraic Multigrid (AMG) for Ground Water Flow
and Oil Reservoir Simulation Klaus Stüben1,
Patrick Delaney2, Serguei Chmakov3 1Fraunhofer
Institute SCAI, Klaus.Stueben_at_scai.fhg.de, St.
Augustin, Germany 2Waterloo Hydrogeologic Inc.,
PDelaney_at_flowpath.com, Waterloo, Ontario, Canada
3Waterloo Hydrogeologic Inc., Schmakov_at_flowpath.c
om, Waterloo, Ontario, Canada
Ruge, J.W., Stüben, K., 1986. Algebraic Multigrid
(AMG), in .Multigrid Methods. (S. McCormick,
ed.), Frontiers in Applied Mathematics, Vol 5,
SIAM, Philadelphia.
The Black Box Multigrid Numerical Homogenization
Algorithm J. David Moulton, Joel E. Dendy Jr.,
and James M. Hyman JOURNAL OF COMPUTATIONAL
PHYSICS 142, 80108 (1998) Theoretical Division,
Los Alamos National Laboratory, Los Alamos, New
Mexico 87545

• Wan and Liu, A boundary condition capturing
  multigrid approach to irregular boundary
  problems, SIAM J. Sci. Comput., 2004.
• Mayo, The fast solution of poissons and the
  biharmonic equations in irregular domains, SIAM
  J. Numer. Anal., 1984.
• Howell and Bell, An adaptive-mesh projection
  method for viscous incomopressible flow, SIAM
  Journal on Scientific Computing, 1997.

21
Tatebe (1993) reported results that indicated
that MGPCG is superior to ILU-PCG and MG for
treating the matrix that arises from solving
Poissons equation with Discontinuous
Coefficients. Tatebe reports a 51 speed-up
over ILU-PCG and a 121 speed-up for MG for
problems with discontinuous coefficients. MGPCG
is guaranteed to converge for symmetric positive
definite matrix systems.
O. Tatebe, The multigrid preconditioned
conjugate gradient method, in, 6th Copper
Mountain Conference on Multigrid Methods, Copper
Mountain, CO, April 49, 1993.
22
Recent Improvements/Implementations of MGPCG

• Gilles, Vogel, Ellerbroek, A multigrid
  preconditioned conjugate gradient method for
  large scale wavefront reconstruction, J. Opt.
  Soc. Am. A, Vol. 19, Issue 9, 1817-1822, 2002.
• Ashby, Falgout, Smith, Fogwell, Multigrid
  Preconditioned Conjugate Gradients for the
  numerical simulation of groundwater flow on the
  CRAY T3D.
• Oosterlee, Washio, An evaluation of parallel
  multigrid as a solver and a preconditioner for
  singularly perturbed problems, SIAM J. Sci.
  Comput., Vol. 19, No. 1, pp. 87-110, 1998.

These methods hint at using line relaxation or
ILU preconditioning as replacement smoothers when
using multigrid as a preconditioner.
23
Multigrid Preconditioned Conjugate Gradient
Method on a Single, Fixed, Uniform Rectangular
Grid
Grid with 3 levels, h, 2h,4h
24
Multigrid Preconditioned Conjugate Gradient Method
25
The Multigrid Preconditioner
26
V-Cycle for multigrid preconditioner.
Briggs WL, Henson Van Emden, McCormick SF.
A multigrid tutorial. 2nd ed. Philadelphia SIAM
2000.
27
Multigrid for Block Structured Adaptive Mesh
Refinement
Level 0
Level 1
Level 2
28
Outer Multigrid Algorithm
1. Place equations in residual correction form
2. Call recursive routine MV(L)
29
Recursive Routine MV(L)
30
Multigrid invokes MGPCG on each adaptive level.
31
One-dimensional Test of multi-level algorithm
32
Rate of convergence of outside multigrid iteration
Outside iterate Residual
1 40
2 5.7
3 8.1E-1
4 1.2E-1
5 1.7E-2
6 2.4E-3
7 3.3E-4
33

• Remarks
• A standard Multigrid algorithm for an adaptive
  hierarchy of grids is not guaranteed to converge,
  especially for large density ratio problems.
• Convergence of the conjugate gradient method
  relies on the overall matrix system for the
  hierarchical grids to be symmetric. This
  constrains the order of accuracy at coarse/fine
  borders to be zeroeth order accurate.
• Our algorithm for solving elliptic equations on a
  hierarchy of adaptive grids is scalable with
  respect to increasing number of adaptive levels
  and increasing number of processors.
• At the very least, MGPCG is guaranteed faster
  than PCG since one can always drop back to PCG at
  the bottom level of the V-cycle-preconditioner.

34
Drop Collision problem with grid stretching and
sharp interface method (220 points per diameter
effective resolution at the collision point).
Reference for drop collision Pan and Suga,
numerical simulation of binary liquid droplet
collision, Physics of Fluids, 17, 082105 (2005).
35
Drop Collision Without Grid Stretching sharp
interface method, 80 points along the diameter.
36
Pressure Projection Step at time step100
tolerance1.0e-10 ILU MGPCG is scalable GSRB
MGPCG is not.
Method Precond. Uniform? CPU time iter.
MGPCG ILU No 0.117 6
MGPCG GSRB No 0.959 107
MGPCG GSRB Yes 0.104 6
MGPCG ILU Yes 0.096 4
PCG ILU No 0.290 58
37
Newtonian and Non-Newtonian bubbles and drops
Jimenez, Sussman, Ohta (Fluid Dynamics and
Materials Processing)
38
Bubble Formation.(work with Ohta)
First bubble R4.85E-3 m Second bubble R4.90E-3
m Experiment R4.99E-3m
39
Drop Deformation Sussman and Ohta (Fluid
Dynamics and Materials Processing)
40
Drop Deformation (continued)
Drop length/a
41
(No Transcript)
42
Nucleate Boiling (work with Mihalef, Unlusu,
Hussaini, Metaxas)
43
Ship Waves
44
Ship waves
M. Sussman, M.Y. Hussaini, K. Smith, R.-Z. Wei,
and V. Mihalef, A second order adaptive sharp
interface method for incompressible multiphase
flow To appear in the Proceedings of the 3rd
international conference on Computational Fluid
Dynamics, Toronto, Canada (2004). D.G.
Dommermuth, M. Sussman, R. Beck, T.T. O'Shea and
D.C. Wyatt, The Numerical Simulation of Ship
Waves using Cartesian Grid Methods with adaptive
mesh refinement To appear in the proceedings of
the Twenty Fifth Symposium on Naval Hydro., St.
John's, New Foundland and Labrador, Canada
(2004).
No. of Levels Grid Blocks Cells
Delta Processors CPU time/step
Cells/(cpu s) 1 64
2097152 1/128 32
376 174
2 64148 5324800 1/256
32 1300
128 3 64116475
17940480 1/512 64
3000 93
45
Comparison of ship wave computations with
experiment
M. Sussman, M.Y. Hussaini, K. Smith, R.-Z. Wei,
and V. Mihalef, A second order adaptive sharp
interface method for incompressible multiphase
flow To appear in the Proceedings of the 3rd
international conference on Computational Fluid
Dynamics, Toronto, Canada (2004). D.G.
Dommermuth, M. Sussman, R. Beck, T.T. O'Shea and
D.C. Wyatt, The Numerical Simulation of Ship
Waves using Cartesian Grid Methods with adaptive
mesh refinement To appear in the proceedings of
the Twenty Fifth Symposium on Naval Hydro., St.
John's, New Foundland and Labrador, Canada
(2004).
Details of experiment http//www.dt.navy.mil/hyd/
sur-shi-mod/index.html
46
Solid-fluid interaction (work with Kadioglu and
Mihalef)
47
Solid Fluid Interaction and Boiling (work with
Kadioglu, Unlusu and Mihalef)