Multigrid Methods and their application in CFD

Transcription

1 Multigrid Methods and their application in CFD Michael Wurst TU München

2 Multigrid Methods Definition Multigrid (MG) methods in numerical analysis are a group of algorithms for solving differential equations They are among the fastest solution techniques known today

3 Outline 1. Typical design of CFD solvers 2. Methods for Solving Linear Systems of Equations 3. Geometric Multigrid 4. Algebraic Multigrid 5. Examples

4 Different CFD solvers Typical design of CFD solver CFD solver Coupled solver Segregated solver

5 SIMPLE Algorithm Typical design of CFD solver Preprocessing segregated, sequential solution of decoupled transport equations Momentum Equations pressure correction equation: a tight tolerance for guaranteeing mass conservation time loop Non-linear loop (outer iterations) Pressure Correction Equation Turbulence Equations other equations... Convergence Multigrid methods Last timestep Postprocessing

6 Coupled Solution Algorithm Typical design of CFD solver Preprocessing momentum equations and pressure correction equation are such discretized that one gets a big coupled block equation system this equation system becomes very large fast solver necessary time loop Non-linear loop (outer iterations) Momentum Equations Pressure Correction Equation Turbulence Equations other equations... Convergence Multigrid methods Last timestep Postprocessing

7 Coupled Solution Algorithm Typical design of CFD solver Big coefficient matrix consisting of the momentum matrixes, the pressure correction matrix and coupling matrixes The solution vector contains velocity componentes and pressure r r A 0 0 A uu up u qu 0 A 0 A r v r q vv vp v r = r 0 0 Aww A w q wp w A A A A r r p q Pu Pv Pw PP p

8 Basic Definitions Methods for Solving Linear Systems of Equations Linear System of Equation: A: sparse matrix of size n n, symmetric, pos. diagonal elements, non-positive off diagonal elements (M-Matrix) u: exact solution Au = f aiju j = fi v: approximation to the exact solution Two measures of v as an approximation to u: (Absolute) error: e = u - v Residual: r = f Av Measured by norms: L norm: e max e L 2 - norm: = j 1 j n e 2 = n 2 ej j =

9 Direct vs. Iterative Methods Methods for Solving Linear Systems of Equations Direct methods i.g. Gauss elimination / LU decomposition solve the problem to the computational accuracy high computational power Iterative methods / Relaxation methods Gauss-Seidel / Jacobi relaxation Solve the problem only by an approximation could be sufficient and so be less time consuming

10 Iterative methods Methods for Solving Linear Systems of Equations aiju j = fi Jacobi relaxation: 1 u = f a u ( n+ 1) ( n) i i ij j aii j i Gauss-Seidel relaxation: 1 u = f - a u - a u aii j<i j>i (n+1) (n+1) (n) i i ij j ij j

11 Properties of Iterative methods Methods for Solving Linear Systems of Equations Example: Poisson equation -u'' = 0 u(0) = u(n)= 0 Discretisation: -u j-1 + 2u j - u j+1 = 0 2 h -u + 2u - u = 0 1 j n+1 j -1 j j+1 u 0 = u n = 0 Exact solution: u = 0 error e = u v = -v

12 Properties of Iterative methods Methods for Solving Linear Systems of Equations Different starting values: Sinus waves: v j jkπ = sin n Fourier modes

13 Properties of Iterative methods Methods for Solving Linear Systems of Equations Error vs. Number of iteration

14 Properties of Iterative methods Methods for Solving Linear Systems of Equations Realistic starting value: 1 jπ 6jΠ 32jΠ v j = sin + sin + sin 3 n n n k = 1: low frequency wave k = 6: medium frequency wave k = 32: high frequency wave

15 Properties of Iterative methods Methods for Solving Linear Systems of Equations Error: written in eigenvectors of A: (0) e = n 1 ckw k k= 1 Eigenvectors correspond to the modes of the problem Our problem: 1 k n 2 w k,j jkπ = sin n 1 k n 1 n 2 k n 1 Low frequency modes Do not dissappear Smoother High frequency modes Disappear

16 Improvements of iterative solvers Geometric Multigrid Idea: Have a good initial guess How? Do some preliminary iterations on a coarse grid (grid with less points) Good, because iterations need less computational time How does an error look like on a coarse grid? It looks more oscillatory!

17 Improvements of iterative solvers Geometric Multigrid How does an error look like on a coarse grid? If error is smooth on fine grid, maybe good to move to coarse grid

18 Possible schemes for improvement Geometric Multigrid Nested iteration: Relax on Au = f on a very coarse grid to obtain an initial guess for the next finer grid... Ω h Ω 2 h Ω Ω h Relax on Au = f on 4 to obtain an initial guess for Relax on Au = f on to obtain an initial guess for Relax on Au = f on Ω h to obtain a final approximation to the solution. Ω 2h Problems: Relax on Au = f on? Last iteration: Error still smooth?

19 2nd possibility: Use of the residual equation Au = f Au Av = f Av Ae = r

20 Possible schemes for improvement Geometric Multigrid Correction scheme: Ω h Relax on Au = f on to obtain an approximation v h Compute the residual r = f - A v h Relax on the residual equation Ae = r on to obtain an approximation to the error e 2h Correct the approximation obtained on with the error estimate obtained on 2 : v v + e Ω h Ω h Ω 2h h h 2 h 2 Problems: Relax on Ae = r on? Ω h Ω 2h Ω h Transfer from to?

21 Transfer operators Geometric Multigrid Transfer from coarse to fine grids: Interpolation / Prolongation Ω Ω 2h h Transfer from fine to coarse grids: Restriction h Ω Ω 2h

22 Transfer operators Interpolation / Prolongation Geometric Multigrid Interpolation / Prolongation: from coarse to fine grid Points on fine and on coarse grid: Points only on the fine grid: h v = v 2 2 j h j h 1 2h 2h v 2 j + 1 = ( v j + v j + 1 )

23 Transfer operators Restriction Geometric Multigrid Restriction: from fine to coarse grid 2h h h h Full weigthening: v j = ( v 2 j 1 + 2v 2 j + v 2 j + 1)

24 Properties of transfer operators Geometric Multigrid Interpolation / Prolongation h I 2h = h I 2 h Restriction = Variational property: h ( h ) h I = c I 2 2h T

25 Properties of transfer operators Geometric Multigrid Transfer of vectors: Transfer of matrix A: h A A 2 h Geometric answer: A 2h is discretisation of the problem on the coarse grid 2h 2h h h Algebraic answer: A = I A I (Galerkin condition) h 2h

26 Recapitulation Geometric Multigrid Iterative methods can effectively reduce high-oscillating errors until only a smooth error remains Smooth errors look less smooth on coarse grids Transfer of vectors and matrices from coarse to fine grids possible with two conditions: Galerkin condition 2 2 A = I A I h h h h h 2h Variational property h ( h ) h I = c I 2 2h T How can we put this in a good solution algorithm?

27 V-Cycle Geometric Multigrid h h h Relax on A u = f ν times with initial guess v h 1 2h 2h h Compute f = I h r h h h Relax on A 2 u 2 = f 2 ν times with initial guess v 2h 1 Compute f 4h = I 4h 2h 2h r h h h Relax on A 4 u 4 = f 4 ν times with initial guess v 4h 1 Compute f 8h = I 8h r 4h Correct Solve A Lh u Lh = f Lh 4 Correct h 4 h 4 + h 8 v v I h 8hv Relax A 4h u 4h = f 4h times with initial guess v 4h Relax A 2h u 2h = f 2h times with initial guess v 2h Correct h h h Relax A u = f ν times with initial guess v h 2 4h v v + I v ν 2 2 h 2 h 2 h 4 h 4h ν 2 v v + I v h h h 2 h 2h

28 V-Cycle Geometric Multigrid Relax h Restricition 2h 4h 8h Prolongation 16h

29 Other cycles W Cycle Geometric Multigrid h 2h 4h 8h

30 Other cycles Full Multigrid Cycle (FMG) Geometric Multigrid ν 0 -times ν 0 -times h 2h ν 0 -times 4h 8h

31 Geometric vs. Algebraic multigrid Algebraic Multigrid Geometric Multigrid: structured meshes Problem: unstructured meshes, no mesh at all Algebraic Multigrid (AMG) Questions: 1) What is meant by grid now? 2) How to define coarse grids? 3) Can we use the same smoothers (Jacobi, Gauss-Seidel) 4) When is an error on a grid smooth? 5) How can we transfer data from fine grids to coarse grids or vice versa?

32 Grid Algebraic Multigrid GMG: known locations of grid points well-defined subset of the grid points define coarse grid AMG: subset of solution variables form coarse grid Au = f u u1 u = 2 M un

33 Smooth error Algebraic Multigrid Defined as an error which is not effectively reduced by an iterative method Jacobi method: e i+1 = ( I D 1 A) e i Measurement of the error with the A-inner product: e A = ( Ae,e) 1 2 Smooth error: ( I D A)e A e e D Ae A e A A 1 D A e A e A

34 Smooth error Algebraic Multigrid Smooth error: 1 D A e A e A ( D 1 A e, A e) ( e, A e) ( D 1 r, r ) ( e, r ) n 2 r a n i i = 1 ii i = 1 r e i i r i a ii e i Ae

35 Implications of smooth error Algebraic Multigrid ii i ij j j i Ae 0 a e + a e 0 a e a e ii i ij j j i

36 Selecting the coarse grid - requirements Algebraic Multigrid Smooth error can be approximated accurately. Good interpolation to the fine grid. Should have substantially fewer points, so the problem on coarse grid can be solved with little expense

37 Selecting the coarse grid Influence and Dependence Algebraic Multigrid Definition 1: Given a threshold value 0 < θ 1, the variable (point) ui strongly depends on the variable (point) if: u j a θ max ij k i { a } ik Definition 2: u i u j u i If the variable strongly depends on the variable, then the variable strongly influences the variable. u j

38 Selecting the coarse grid definitions Algebraic Multigrid Two important sets: S i : set of points that strongly influence i, that is the points on which the point i strongly depends. { : θ max{ }} S = j a a i ij ik k i T S i : set of points that strongly depend on the point i. { : j} T S = j i S i

39 Selecting the coarse grid - Example Algebraic Multigrid Poisson equation: u = 0 -u i ui - u -u i+1 j-1 + 2ui - u j = 0 h h 1 2 ( -u i -1 - u i+1 + 4u i - u j-1 - u j+1 ) = 0 h i j

40 Selecting the coarse grid - Example Algebraic Multigrid Discretisation on 5x5 grid: For example, Point 12: 1 2 ( -u 7 - u u 12 - u 13 - u 17 ) = 0 h a a a a a 12,7 12,11 12,13 12,17 12,12 = 1 = 1 = 1 = 1 = { : θ max { }} S 12 = { 7, 11, 13, 17} S = j a a i ij ik k i { j} T S = j i S i T : S 12 = { 7, 11, 13, 17}

41 Selecting the coarse grid - Example Algebraic Multigrid 1) Define a measure to each point of its potential quality as a coarse (C) T point: amount of members of λ i S i

42 Selecting the coarse grid - Example Algebraic Multigrid 2) Assign point with maximum to C-point S i T 3) All points in become fine (F) points λ i 4) For each new F point j: increase the measeure for all each unassigned point k that strongly influence j: k S λ k j F F C F F ) Do 2)-4) until all points are assigned

43 Selecting the coarse grid - Example Algebraic Multigrid F F 3 C F C F C 3 5 F C F 3 5 F C F F C F C F F F 5 C F C F C F C F F C F C F F 3 C F C F C

44 Definition of transfer operators Algebraic Multigrid Interpolation: from coarse to fine grids e if i C = e if i F i h ( I2he) ω i ij j j C i Each fine grid point i can have three different types of neighboring points: The neighboring coarse grid points that strongly influence i The neighboring fine grid points that strongly influence i Points that do not strongly influence i, can be fine and coarse grid points This information is contained in ω ij

45 Example Algebraic Multigrid Example: -au -cu + bu = xx yy xy Discretised with a two-dimensional mesh, divided into 4 parts; a=1000 c=1 b=0 a=1 c=1 b=2 a=1 c=1 b=0 a=1000 c=1 b=

46 Example Algebraic Multigrid Grid 2h 'Y' 'X'

47 Example Algebraic Multigrid Grid 4h 'Y' 'X'

48 Example Algebraic Multigrid Grid 8h 'Y' 'X'

49 Advantages & Disadvantages of AMG Algebraic Multigrid Normalized Residual SIP (Iterations: 2410) V-Cycle (3 Grid-Levels) W-Cycle (3 Grid-Level) FMG-Cycle (3 Grid-Levels) Number of Iterations

50 Advantages & Disadvantages of AMG Algebraic Multigrid Advantages Fast and robust Good for segregated solvers (SIMPLE) Disadvantages The Galerkin Operation is a very expensive step Diffucult to parallelize High setup-phase High storage requirements Not for coupled solvers A cure are the aggregation based AMGs

51 Aggregation based AMG Algebraic Multigrid In the simplest case strongly connected coefficient are simply summed up Example: II I cell 7 influences strongly cell 1 cell 2 influences strongly cell 1 build a new cell I from cell 1,2,7 do the same to get the new cell II

52 Aggregation based AMG Algebraic Multigrid II I To get the coefficients of the new coarse linear equation system sum up A I,II =A 7,13 +A 7,8 +A 2,8 A I,II =A 13,7 +A 8,7 +A 8,2 A I,I =A 1,1 +A 2,2 +A 7,7 A II,II =A 8,8 +A 9,9 +A 13,13 +A 14,14 +A 15,15 +A 18,18 +A 19,19 +A 20,

53 Aggregation based AMG Algebraic Multigrid Advantages The Galerkin operation becomes a simple summation of coefficients The setup-phase becomes very fast The procedure is easy to parallelize Through giving maximum and minimum size of cells on coarser grids, one can pre-estimate memory effort in a finite volume method, the coefficients are representing flux sizes from one cell to another, through summation on keeps the conservativness of the discretized system over all coarser levels Disadvantages The convergence rate becomes small compared to original AMG, but in the case of solution of the non-linear Navier-Stokes equation the reduction of the residual within one outer iteration has not to be very tight, reducing of about one to two orders of magnitude suffices The Agglomeration AMG is ideally applicable to the coupled solution of Navier-Stokes Equation System

54 Thank you! Discussion