Fast Multipole Methods for Incompressible Flow Simulation

Transcription

1 Fast Multipole Methods for Incompressible Flow Simulation Nail A. Gumerov & Ramani Duraiswami Institute for Advanced Computer Studies University of Maryland, College Park Support of NSF awards and is gratefully acknowledged

2 Fast Multipole Methods Complex geometric features (e.g., aircraft, submarine, or turbine geometries), physics (high <, wake-structure interactions etc.) all tend to increase problem sizes Many simulations involve several million variables Most large problems boil down to solution of linear systems or performing several matrix-vector products Regular product requires O(N 2 ) time and O(N 2 ) memory The FMM is a way to accelerate the products of particular dense matrices with vectors Do this using O(N) memory FMM achieves product in O(N) or O(N log N) time and memory Combined with iterative solution methods, can allow solution of problems hitherto unsolvable

3 Fast Multipole Methods To speed up a matrix vector product i Φ ij u i = v j Key idea: Let Φ ij =φ( x i, y j ) Translate elements corresponding to different x i to common location, x * φ (x i,y j )= l=1 p β l ψ l (x i -x * ) Ψ l (y j -x * ) Achieves a separation of variables by translation The number of terms retained, p, is only related to accuracy v j = j=1n u i l=1 p β l ψ l (x i ) Ψ l (y j ) = l=1 p β l Ψ l (y j ) i=1n u i ψ l (x i ) Can evaluate p sums over i A l = j=1n u i ψ l (x i ) Requires Np operations Then evaluate an expression of the type S(x i )= l=1 p A l β l ψ l (x i ) i=1,,m

4 Outline Incompressible Fluid problems where FMM can be used Boundary integral and particle formulations Potential flow Stokes Flow Vortex Element Methods Component of Navier-Stokes Solvers via Generalized Helmholtz decomposition Multi-sphere simulations of multiphase flow Laplace Stokes Helmholtz Numerical Analysis issues in developing efficient FMM Towards black-box FMM solvers

5 Basic Equations FMM & Fluid Mechanics n 2 1

6 Helmholtz Decomposition Key to integral equation and particle methods

7 Potential Flow Crum, 1979 Knowledge of the potential is sufficient to compute velocity and pressure Need a fast solver for the Laplace equation Applications panel methods for subsonic flow, water waves, bubble dynamics, Boschitsch et al, 1999

8 BEM/FMM Solution Laplace s Equation Lohse, 2002 Jaswon/Symm (60s) Hess & Smith (70s), Korsmeyer et al 1993, Epton & Dembart 1998, Boschitsch & Epstein 1999

9 Stokes Flow Green s function (Ladyzhenskaya 1969, Pozrikidis 1992) Motion of spermatozoa Cummins et al 1988 Integral equation formulation Stokes flow simulations remain a very important area of research MEMS, bio-fluids, emulsions, etc. BEM formulations (Tran-Cong & Phan-Thien 1989, Pozrikidis 1992) FMM (Kropinski 2000 (2D), Power 2000 (3D)) Cherax quadricarinatus. MEMS force calculations (Aluru & White, 1998

10 Rotational Flows and VEM For rotational flows Vorticity released at boundary layer or trailing edge and advected with the flow Simulated with vortex particles Especially useful where flow is mostly irrotational Fast calculation of Biot-Savart integrals (x 2,y 2,z 2 ) l n Γ (circulation strength) x Evaluation point (x 1,y 1,z 1 ) (Far field) (where y is the mid point of the filament)

11 Vorticity formulations of NSE Problems with boundary conditions for this equation (see e.g., Gresho, 1991) Divergence free and curl-free components are linked only by boundary conditions Splitting is invalid unless potentials are consistent on boundary Recently resolved by using the generalized Helmholtz decomposition (Kempka et al, 1997; Ingber & Kempka, 2001) This formulation uses a kinematically consistent Helmholtz decomposition in terms of boundary integrals When widely adopted will need use of boundary integrals, and hence the FMM Preliminary results in Ingber & Kempka, 2001

12 Generalized Helmholtz Decomposition Helmholtz decomposition leaves too many degrees of freedom Way to achieve decomposition valid on boundary and in domain, with consistent values is to use the GHD D is the domain dilatation (zero for incompressible flow) Requires solution of a boundary integral equation as part of the solution => role for the FMM in such formulations

13 Multi Sphere Problems So far we have seen FMM for boundary integral problems Can also use the FMM in problems involving many spheres Simulations of multiphase flow, effective media (porous media), dusty gases, slurries, and the like Key is a way to enforce proper boundary/continuity conditions on the spheres Translation theorems provide an ideal way to do this With FMM one can easily simulate particles on desktops and particles on supercomputers Sangani & Bo (1996), Gumerov & Duraiswami (2002,2003) Stokesian Dynamics (Brady/Bossis 1988; Kim/Karilla 1991)

14 Fast Multipole Methods Matrix-Vector Multiplication Middleman and Single Level Methods Multilevel FMM (MLFMM) Adaptive MLFMM Fast Translations Some computational results Conclusions

15 Iterative Methods To solve linear systems of equations; Simple iteration methods; Conjugate gradient or similar methods; We use Krylov subspace methods (GMRES): Preconditioners; Research is ongoing. Efficiency of these methods depends on efficiency of the matrix-vector multiplication.

16 Matrix-Vector Multiplication

17 FMM Works with Influence Matrices

18 Examples of Influence Matrices exp

19 Complexity of Standard Method Standard algorithm Sources N M Evaluation Points Total number of operations: O(NM)

20 Goal of FMM Reduce complexity of matrixvector multiplication (or field evaluation) by trading exactness for speed Evaluate to arbitrary accuracy

21 Five Key Stones of FMM Factorization Error Bound Translation Space Partitioning Data Structure

22 Degenerate Kernel: Factorization O(pN) operations: O(pM) operations: Total Complexity: O(p(N+M))

23 Factorization (Example) Scalar Product in d-dimensional space: O(dM) operations: O(dN) operations Total Complexity: O(d(N+M))

24 Middleman Algorithm Standard algorithm Sources Evaluation Points Middleman algorithm Sources Evaluation Points N M N M Total number of operations: O(NM) Total number of operations: O(N+M)

25 Non-Degenerate Kernel: Factorization Truncation Number Error Bound: Middleman Algorithm Applicability:

26 Factorization Problem: Usually there is no factorization available that provides a uniform approximation of the kernel in the entire computational domain.

27 Far and Near Field Expansions Far Field: Near Field: Far Field R c x i - x * r c x i - x * x i Ω Ω Near Field R S x * y y S: Singular (also Multipole, Outer Far Field ), R: Regular (also Local, Inner Near Field )

28 Example of S and R expansions (3D Laplace) z O θ φ r r y x Spherical Coordinates: Spherical Harmonics:

29 S and R Expansions (3D Helmholtz) z O θ φ r r y x Spherical Coordinates: Spherical Bessel Functions Spherical Hankel Functions

30 Idea of a Single Level FMM Standard algorithm Sources Evaluation Points SLFMM Sources Evaluation L groups Points K groups N M N M Total number of operations: O(NM) Needs Translation! Total number of operations: O(N+M+KL)

31 Space Partitioning Potentials due to sources in these spatial domains Φ (n) 1 (y) Φ (n) 2 (y) Φ (n) 3 (y) Ε 1 Ε 2 Ε 3 n n n

32 Single Level Algorithm 1. For each box build Far Field representation of the potential due to all sources in the box. 2. For each box build Near Field representation of the potential due to all sources outside the neighborhood. 3. Total potential for evaluation points belonging to this box is a direct sum of potentials due to sources in its neighborhood and the Near Field expansion of other sources near the box center.

33 The SLFMM requires S R-translation:

34 S R-translation Also Far-to-Local, Outer-to-Inner, Multipole-to-Local Ω 2i x *2 y R R 2 S R c x i - x *1 (S R) x *1 Ω 1 i x i Ω 1i R 2 = min{ x *2 - x *1 -R c x i - x *1,r c x i - x *2 }

35 S R-translation Operator S R-Translation Coefficients S R-Translation Matrix

36 S R-translation Operators for 3D Laplace and Helmholtz equations

37 Complexity of SLFMM We have p 2 terms and P(p) translation cost: F(K) For M~N: 0 K opt Additional Middleman complexity Κ

38 Idea of Multilevel FMM Source Data Hierarchy Evaluation Data Hierarchy S S S R N R R M Level 5 Level 2 Level 3 Level 4 Level 2 Level 3 Level 4 Level 5

39 Complexity of MLFMM Definitions: Upward Pass: Going Up on SOURCE Hierarchy Downward Pass: Going Down on EVALUATION Hierarchy Not factorial!

40 Hierarchical Spatial Domains Ε 1 Ε 2 Ε 3 Ε 4

41 Upward Pass. Step 1. S-expansion S-expansion valid in Ω Ε 3 Ω y y Ω x i x c (n,l) x * R x i S-expansion valid in E 3 (n,l)

42 Upward Pass. Step 2. S S-translation. Build potential for the parent box (find its S-expansion).

43 Downward Pass. Step 1. S R-translation Level 2: Level 3:

44 Downward Pass. Step 2. R R-translation

45 Final Summation Contribution of near sources (calculated directly) Contribution of far sources (represented by R-expansion) y j

46 Adaptive MLFMM Goal of computations:

47 Idea of Adaptation Source Point Evaluation Point Box Level Each evaluation box in this picture contains not more than 3 sources in the neighborhood. Very important for particle methods with concentrations of particles 2

48 We have implemented MLFMM General MLFMM shell software: Arbitrary dimensionality; Variable sizes of neighborhoods; Variable clustering parameter; Regular and Adaptive versions; User Specified basis functions, and translation operators; Efficient data structures using bit interleaving; Technical Report #1 is available online (visit the authors home pages).

49 Optimal choice of the clustering parameter is important 100 Max Level= CPU Time (s) N= Regular Mesh, d= Number of Points in the Smallest Box

50 Speed of computation 1000 Straightforward CPU Time (s) y=cx 2 y=bx FMM (s=4) Setting FMM FMM/(a*log(N)) 0.1 Middleman Regular Mesh, d=2, k=1, Reduced S R Number of Points

51 Error analysis

52 Complexity of Translation For 3D Laplace and Helmholtz series have p 2 terms; Translation matrices have p 4 elements; Translation performed by direct matrix-vector multiplication has complexity O(p 4 ); Can be reduced to O(p 3 ); Can be reduced to O(p 2 log 2 p); Can be reduced to O(p 2 ) (?).

53 Rotation-Coaxial Translation Decomposition Yields O(p 3 ) Method z z p 4 y x y x p3 p 3 p 3 y y z x x Coaxial Translation Rotation z

54 FFT-based Translation for 3D Laplace Equation Yields O(p 2 log 2 p) Method Translation Matrices are Toeplitz and Hankel Multiplication can be performed using FFT (Elliot & Board 1996; Tang et al 2003)

55 Sparse Matrix Decompositions Can Result in O(p 2 ) Methods Laplace and Helmholtz: Helmholtz 3D: D is a sparse matrix

56 Interaction of Multiple Spherical Particles Helmholtz (acoustical scattering) Laplace (potential incompressible flow) k 0

57 Multipole Solution 1) Reexpand solution near the center of each sphere, and satisfy boundary conditions 2) Solve linear system to determine the expansion coefficients

58 Computational Methods Used Computable on 1 GHz, 1 GB RAM Desktop PC MLFMM Method Multipole Iterative Multipole Straightforward BEM Number of Spheres 10 6

59 Comparisons with BEM HRTF (db) o 30 o θ 1 = 0 o 90 o o BEM MultisphereHelmholtz 30 o o 90 o 120 o o 150 o o Three Spheres, ka 1 = Angle φ 1 (deg) BEM discretization with 5400 elements

60 Convergence of the Iterative Procedure M ax Absolute Error 1.E+02 1.E+01 1.E+00 1.E-01 1.E-02 Iterations with Reflection Method 3D Helmholtz Equation, MLFMM 100 Spheres ka = E E Iteration # For Laplace equation convergence is much faster

61 Multiple Scattering from 100 spheres ka=1.6 ka=2.8 ka=4.8 FMM also used here for visualization

62 Various Configurations 343 spheres in a regular grid 1000 randomly placed spheres

63 Truncation Errors

64 Some Observations for Laplace Equation (limit at small k) Relative errors below 1% can be achieved even for p~1; Reflection method converges very fast; Number of iterations can be 2-10 for very high accuracy. CPU time on 1GHz, 1GB RAM PC for 1000 spheres ~ 1 min.

65 Some projects of students in our course related to incompressible flow 1). Jun Shen (Mechanical Engineering, UMD): 2D simulation of vortex flow (particle method + MLFMM)

66 Project 2- Calculate vortex induced forces 2). Jayanarayanan Sitaraman (Alfred Gessow Rotorcraft Center, UMD): Fast Multipole Methods For 3-D Biot-Savart Law Calculations (free vortex methods) Error, M=3200, N= p 6 7 Break-even point: MLFMM is faster then straightforward for N=M>1300

67 Conclusions We developed a shell and framework for MLFMM which can be used for many problems, including simulations of potential and vortex flows and multiparticle motion. MLFMM shows itself as very efficient method for solution of fluid dynamics problems; MLFMM enables computation of large problems (sometimes even on desktop PC); Research is ongoing. Problems: Efficient choice of parameters for MLFMM; Efficient translation algorithms; Efficient iterative procedures; More work on error bounds is needed; Etc. CSCAMM Meeting on FMM in November 2003

68 Thank You!