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**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

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Atomization

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Atomization

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Atomization

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Flow Past a Whale coarse medium

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**Precond. Uniform? #iter CPU GSRB Yes 26 12.4 Line 14 14.8 ILU 12 9.0**

Scalability of Pressure Projection Step? Precond. Uniform? #iter CPU GSRB Yes 26 12.4 Line 14 14.8 ILU 12 9.0 No 95 37.7 25.0 22 14.0

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**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

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Advection Diffusion Pressure Projection Step 1. Nonlinear Advection:

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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. 269–282.

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**3. Pressure projection step:**

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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

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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.

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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

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**Discretized Equation to be solved on hierarchy of rectangular grids:**

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**Example in one dimension**

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**Matrix Associated with the previous example.**

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**Condition number dependence on density ratio**

1 4.4E+2 10 3.9E+3 100 3.8E+4 1000 3.8E+5

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**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 (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 (MGPCG 100:15 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 (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 (PCG)

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**Algebraic Multigrid (AMG) for **

Ground Water Flow and Oil Reservoir Simulation Klaus Stüben1, Patrick Delaney2, Serguei Chmakov3 1Fraunhofer Institute SCAI, St. Augustin, Germany 2Waterloo Hydrogeologic Inc., Waterloo, Ontario, Canada 3Waterloo Hydrogeologic Inc., Waterloo, Ontario, Canada Ruge, J.W., Stüben, K., 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, 80–108 (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 poisson’s 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.

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Tatebe (1993) reported results that indicated that MGPCG is superior to ILU-PCG and MG for treating the matrix that arises from solving Poisson’s equation with Discontinuous Coefficients. Tatebe reports a 5:1 speed-up over ILU-PCG and a 12:1 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 4–9, 1993.

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**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, , 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 , 1998. These methods hint at using line relaxation or ILU preconditioning as replacement smoothers when using multigrid as a preconditioner.

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**Multigrid Preconditioned Conjugate Gradient Method on a Single, Fixed, Uniform Rectangular Grid**

Grid with 3 levels, h, 2h,4h

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**Multigrid Preconditioned Conjugate Gradient Method**

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**The Multigrid Preconditioner**

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**V-Cycle for multigrid preconditioner.**

Briggs WL, Henson Van Emden, McCormick SF. A multigrid tutorial. 2nd ed. Philadelphia: SIAM; 2000.

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**Multigrid for Block Structured Adaptive Mesh Refinement**

Level 0 Level 1 Level 2

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**Outer Multigrid Algorithm:**

1. Place equations in residual correction form: 2. Call recursive routine MV(L)

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**Recursive Routine MV(L):**

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**Multigrid invokes MGPCG on each adaptive level.**

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**One-dimensional Test of multi-level algorithm**

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**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

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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.

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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, (2005).

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**Drop Collision Without Grid Stretching: sharp interface method, 80 points along the diameter.**

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**Method Precond. Uniform? CPU time #iter. MGPCG ILU No 0.117 6 GSRB**

Pressure Projection Step at time step=100; tolerance=1.0e-10 ILU MGPCG is scalable; GSRB MGPCG is not. Method Precond. Uniform? CPU time #iter. MGPCG ILU No 0.117 6 GSRB 0.959 107 Yes 0.104 0.096 4 PCG 0.290 58

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Newtonian and Non-Newtonian bubbles and drops: Jimenez, Sussman, Ohta (Fluid Dynamics and Materials Processing)

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**Bubble Formation.(work with Ohta)**

First bubble R=4.85E-3 m Second bubble R=4.90E-3 m Experiment R=4.99E-3m

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**Drop Deformation: Sussman and Ohta (Fluid Dynamics and Materials Processing)**

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**Drop Deformation (continued)**

Drop length/a

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**Nucleate Boiling (work with Mihalef, Unlusu, Hussaini, Metaxas)**

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Ship Waves

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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). Recently speed up bu factor of 100? Cells per cpu per second indicates efficiency of adaptive mesh and prallaelization. No. of Levels Grid Blocks Cells Delta Processors CPU time/step Cells/(cpu s) / / /

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**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:

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**Solid-fluid interaction (work with Kadioglu and Mihalef)**

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**Solid Fluid Interaction and Boiling (work with Kadioglu, Unlusu and Mihalef)**

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