Krylov methods for the solution of parameterized linear systems in the simulation of structures and vibrations: theory, applications and challenges

Transcription

1 Krylov methods for the solution of parameterized linear systems in the simulation of structures and vibrations: theory, applications and challenges Karl Meerbergen K.U. Leuven Autumn School on Model Order Reduction September 21 25, 2009

2 Outline 1 Motivation 2 Overview of methods Modal truncation Vector-Padé approximation Frequency sweeping Input/output MOR 3 Lanczos method 4 Rayleigh damping 5 Nonproportional damping 6 Multiple right-hand sides 7 Software 8 Future work 9 Conclusions Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

3 Collaborators Zhaojun Bai Yao Yue Maryam Saadvandi Jeroen De Vlieger Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

4 Outline 1 Motivation 2 Overview of methods Modal truncation Vector-Padé approximation Frequency sweeping Input/output MOR 3 Lanczos method 4 Rayleigh damping 5 Nonproportional damping 6 Multiple right-hand sides 7 Software 8 Future work 9 Conclusions Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

5 Examples of vibrating systems Car tyres Windscreens Structural damping Choice of connection (glue) to the car Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

6 Examples of vibrating systems Planes Bridge vibrating under footsteps and Thames wind Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

7 Examples of vibrating systems Maxwell-equation electrical circuits micro-gyroscope for navigation systems Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

8 Finite element analysis Numerical simulation of vibration problems. Spatial (finite element) discretization: with initial values x(0) and ẋ(0) Mẍ(t) + Cẋ(t) + Kx(t) = f (t) f and x : vectors of length n K, C and M : n n sparse matrices. In real applications n varies from 10 3 to over Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

9 Fourier analysis If f (t) = f e iωt, then (under certain conditions) for t, x(t) = xe iωt where (K + iωc ω 2 M) x = f The engineer is usually interested in the periodic regime solution, i.e. after a long integration time. Material properties are often frequency dependent. Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

10 Fourier analysis (K + iωc ω 2 M) x = f x is called the frequency response function. Compute x for ω = ω 1,...,ω p. Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

11 Acoustic industrial applications : vibro-acoustics vibrating structure (modelized by structural modes) acoustic domain (finite elements) acoustic radiation towards infinity (infinite elements) structure is modelized by modes (eigen functions) Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

12 Acoustic industrial applications : vibro-acoustics Linear system has three parts Fluid (finite elements) radiation to infinity (infinite elements) structure (modes) Modes FE IFE = 0 0 Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

13 Acoustic industrial applications : aero-acoustics airplane nacelle turning engine modelized by rotating modes acoustic domain with modeling of flow (finite elements) acoustic radiation towards infinity (infinite elements) Usually a few frequencies only: MOR not required. Often large linear systems: 1M dofs or more. Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

14 Infinite elements Acoustic radiation towards infinity: no finite elements, but infinite elements. Index 1 Differential algebraic equation (DAE) with M = [ ] M Mẍ(t) + Cẋ(t) + Kx(t) = f (t) [ ] C1 C, C = 1,2 C 2,1 C 2 [ ] K1 K and K = 1,2 K 2,1 K 2 Some models are unstable (zero blocks become nonzero) But those correspond to high frequency unphysical modes Unsuitable for time integration!. Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

15 Traditional frequency response computation 1. For ω = ω 1,...,ω p 1.1. Solve the linear system (K + iωc ω 2 M)x = f for x For each frequency, a large system of algebraic equations needs to be solved. This requires a linear solver for a large sparse matrix. For a direct solver (based on LU factorization): a sparse matrix factorization LU = K ω 2 M + iωc (expensive) and a backward solve LUx = f (relatively cheap). Note: no output The goal is to reduce the number of matrix factorizations. Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

16 Linear system solvers Discretization error depends on largest frequency: larger frequency means finer mesh Direct linear system solver: up to 1M dofs: no problem For a complex valued system of 3D volume discretization with 100,000 dofs, direct method solution time is of the order of 10 seconds. Iterative linear system solver The last ten years effective preconditioners for the Helmholtz equation have been developed. Iterative methods can be seen as validation of model AMLS: automated level substructuring Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

17 Damping models No damping: C = 0 Most general situation: C is frequency dependent (e.g. porous material, foams): Properties are meausured for different frequencies Large amount of uncertainty about damping parameters Stochastic methods may be appropriate Often a constant C is fine for a large frequency range as long as the tendency is right Proportional damping: Eigenvectors do not change with the damping Simple model often used for a zero order analysis Is often a result of measurements Valid for small damping Only valid for specific materials (glass, concrete, steel,... ) Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

18 Outline 1 Motivation 2 Overview of methods Modal truncation Vector-Padé approximation Frequency sweeping Input/output MOR 3 Lanczos method 4 Rayleigh damping 5 Nonproportional damping 6 Multiple right-hand sides 7 Software 8 Future work 9 Conclusions Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

19 Overview of methods Consider (K ω 2 M)x = f with K and M large sparse, real symmetric matrices M positive definite f independent of ω: typically point loads Three basic methods: Modal truncation Padé approximation Mixed direct iterative procedure Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

20 Modal truncation Consider the eigendecomposition Ku j = λ j Mu j The solution of (K ω 2 M)x = f is x = n j=1 u j u T j f λ j ω 2 Rational function with poles λ j. Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

21 Modal superposition, cont. x = n j=1 u j uj T f λ j ω 2 k j=1 u j u T j f λ j ω "undamped" "undamped10" "undamped7" Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

22 Vector-Padé approximation Approximation of x = (K αm) 1 f by x = x 0 + αx α k 1 x k 1 (α λ 1 ) (α λ k ) This is a rational function with k poles. Determine the coefficients so that the first k derivatives in σ match Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

23 Frequency sweeping For each ω precondition (K ω 2 M)x = f into (K σm) 1 (K ω 2 M)x = (K σm) 1 f and solve by an iterative method. Use linear system solver for applying (K σm) 1 For the AMLS method, K σm is a diagonal matrix. Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

24 Input-output system SISO (K ω 2 M)x = b y = d T x Compute y accurately and fast Use MOR as fast solver Often many outputs (100 s or 1000 s) Twosided methods (MOR) are not often used in this case Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

25 Summary Modal truncation: Padé approximation: x = k j=1 u j u T j f λ j α x = x 0 + αx α k 1 x k 1 (α µ 1 ) (α µ k ) Frequency sweeping Solve (M ω 2 M)x = f by an iterative method MOR: find reduced model for linear system (K ω 2 M)x = b y = d T x Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

26 Notation Assume σ = 0 and define α = ω 2 A = (K σm) 1 M and b = (K σm) 1 f then we solve or (K αm)x = f (I αa)x = b Eigenvalue problem: Ku j = λ j Mu j Assume A symmetric. Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

27 Outline 1 Motivation 2 Overview of methods Modal truncation Vector-Padé approximation Frequency sweeping Input/output MOR 3 Lanczos method 4 Rayleigh damping 5 Nonproportional damping 6 Multiple right-hand sides 7 Software 8 Future work 9 Conclusions Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

28 Lanczos method Krylov space: span{b, Ab,...,A k 1 b} Lanczos method builds orthogonal basis V k = [v 1,...,v k ]. Range(V k ) = span{b, Ab,...,A k 1 b} and a tridiagonal matrix T k = V T k AV k major cost: k matrix vector products with A : w = Av small cost when k is small Also called Ritz vector technique (mechanical engineering) Recall Rixen s talk yesterday Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

29 Lanczos method Transform a large size matrix into a small size matrix T k = V T k A V k Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

30 Shift-invariance property Krylov spaces and {v, Av, A 2 v,...} {v, (A + αi)v, (A + αi) 2 v,...} are equal, since (A + αi)v = Av + αv Applying the Lanczos method to A, applies it for free to A + αi for all α. Similarly, the Lanczos method applied to A produces the same Krylov space as the Lanczos method applied to I αa provided α 0. Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

31 Shifted or parameterized linear systems Analyzed in the context of model reduction methods (Connection with rational approximation) [Gallivan, Grimme, Van Dooren 1994], [Feldman, Freund 1995], [Gallivan, Grimme, Van Dooren 1996], [Grimme, Sorensen, Van Dooren 1996], [Ruhe & Skoogh 1998], [Bai & Freund 2000], [Bai & Freund 2001] [Bai & Su 2006] in the context of parameterized linear systems [Freund 1993], [Frommer & Glässner, 1993], [Simoncini & Gallopoulos 1998], [Simoncini, 1999], [Simoncini & Perotti 2002], [M. 2003], [Edema, Vuik 2008] Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

32 Undamped vibration problem When A = K 1 M, A is non-symmetric. However, x MAy = y MAx for all x, y. So, A is self-adjoint with the M inner product Use the Lanczos method with M orthogonalization: Matrix vector products with A: V k MV k = I One matrix factorization of K = LDL T k solves of the form LDL T w = Mv Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

33 Iterative solver connection For the solution of (I αa)x = b build a Krylov space of dimension k with matrix A and starting vector b (i.e. independent of α) Compute a solution of the form x = V k z = k j=1 v jζ j so that the residual is orthogonal to the Krylov space: V T k (b (I αa) x) = 0 V T k (b (I αa) x) = 0 Vk T (v 1 b (I αa)v k z) = 0 e 1 b (I αt k )z = 0 i.e. (I αt k )z = e 1 b Conjugate gradients or Lanczos Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

34 Lanczos convergence Let Ku j = λ j Mu j Eigenvalues of K 1 (K ω 2 M) are θ j = λ j ω 2 Eigenvalues are clustered around one. λ j λ j θ j 0 ω When there are no eigenvalues λ between 0 and ω 2, then we have a positive definite linear system Fast convergence when most eigenvalues are clustered around one: ω close to 0 Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

35 MINRES versus Lanczos Lanczos: Vertical asymptotes MINRES: x = x = y j α µ j y j (α) α µ j (α) Denominator is never zero No vertical asymptotes Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

36 Example 10 1 Lanczos MINRES Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

37 Padé connection Recall (I αa)x = f The solution computed by the Lanczos method can be written as x = x 0 + αx α k 1 x k 1 (α µ 1 ) (α µ k ) where x (j) (0) = x (j) (0) for j = 0,...,k 1 Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

38 Eigenvalue connection Lanczos method produces eigenvalue estimates in a similar way as the linear solves. Let Au = θu Then choose ũ = V k z so that the residual is orthogonal to the Krylov space: V T k (Aû θû) = 0 V T k (AV kz θv k z) = 0 T k z θz = 0 For the Ku = λmu problem: is small for the small λ s. Kũ λũm Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

39 Eigenvalue connection As for modal truncation, we can project K, M and f on the Ritz vectors. We can show that the Lanczos method computes x = k j=1 ũ j w T j f λ j α where ũ j is a Ritz vector. There are k terms, so we can only compute k vertical asymptotes in the function The number of eigenvalues in the frequency range should be smaller than k. Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

40 Eigenvalue connection: example Hard problem: 10,000 eigenvalues more than Easy problem: less than 20 eigenvalues Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

41 Numerical example: BMW Windscreen Glaverbel-BMW windscreen grid : 3 layers of HEX08 elements (n = 22, 692) unit point force at one of the corners wanted : displacement for ω = [0.5Hz, 200Hz]. Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

42 Numerical example: BMW Windscreen k = 10 vectors k = 20 vectors e-06 1e-06 1e-08 1e-08 1e-10 1e-10 1e e Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

43 Industrial example with NASTRAN Traditional computation For each frequency, perform factorization of K ω 2 M and solve Lanczos computation One matrix factorization of K σm and solve k solves. Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

44 Outline 1 Motivation 2 Overview of methods Modal truncation Vector-Padé approximation Frequency sweeping Input/output MOR 3 Lanczos method 4 Rayleigh damping 5 Nonproportional damping 6 Multiple right-hand sides 7 Software 8 Future work 9 Conclusions Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

45 Damping Damping often introduces a C term in the equation: (K ω 2 M + iωc)x = f If damping is global, i.e. in the fluid or structure itself, we often have Rayleigh damping, i.e. structural damping in a windscreen in order to reduce vibrations. We make the damping ω dependent: D(ω) (K ω 2 M + D(ω))x = f Rayleigh damping : D = γk + δm f is independent of ω Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

46 No damping D(ω) 0 Linear system: (K ω 2 M)x = f Corresponding eigenvalue problem: Ku = λmu Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

47 Structural Rayleigh damping D(ω) = iγk Linear system: ((1 + iγ)k ω 2 M)x = f Corresponding eigenvalue problem: (1 + iγ)ku = λmu Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

48 Fluid Rayleigh damping D(ω) = iω(α 0 M + α 1 K) Linear system: (K + iω(α 0 M + α 1 K) ω 2 M)x = f Corresponding eigenvalue problem: (K + iλ(α 0 M + α 1 K) λ 2 M)u = 0 Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

49 Modal superposition Define U = [u 1,...,u n ] and Λ = diag(λ 1,...,λ n ) KU = MUΛ D = βk + γm U T MU = I U T KU = Λ U T DU = βi + γλ Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

50 Modal superposition, cont. Without damping: The solution of (K ω 2 M)x = f With damping x = n j=1 u j u T j f λ j ω 2 Simultaneous diagonalization of K, M, and D The solution of (K ω 2 M + D)x = f x = n j=1 u j u T j f λ j ω 2 + ζ j (ω) Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

51 Modal superposition, cont. x = n j=1 u j uj T f λ j ω 2 + ζ j k j=1 u j u T j f λ j ω 2 + ζ j 10 "damped" "damped10" "damped7" Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

52 Lanczos method For the similar reasons, we can use the Lanczos method [M. 2008]. Compute V k, T k for A = K 1 M with starting vector b = K 1 f (real arithmetic) For each α, solve the k k tridiagonal system (complex arithmetic) V T k MK 1 (K ω 2 M + D(ω))V k z = e 1 b Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

53 Numerical example: BMW Windscreen Glaverbel-BMW windscreen with 10% structural damping Direct method : 2653 seconds (complex arithmetic) Lanczos method : 14 seconds (mostly real arithmetic) e Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

54 Outline 1 Motivation 2 Overview of methods Modal truncation Vector-Padé approximation Frequency sweeping Input/output MOR 3 Lanczos method 4 Rayleigh damping 5 Nonproportional damping 6 Multiple right-hand sides 7 Software 8 Future work 9 Conclusions Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

55 Nonproportional damping (K + iωc ω 2 M)x = f Damping is not proportional: K, C and M cannot be diagonalized simultaneously. Linearization : Define matrices A and B [ ] [ ] K ic M A = B = I I so that ( x (A ωb) ωx ) = ( f 0 This is called a linearization, a similar trick as the solution of second order ODE s. ) Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

56 Linearizations Linearizations have been studied for the solution of the quadratic eigenvalue problem (K + λc + λ 2 M)u = 0 [Gohberg, Lancaster, Rodman, 1982] [Tisseur, M. 2001] Suppose that K, C and M are symmetric, then we can choose [ ] [ ] K C M A = B = M M symmetric, but both indefinite, so the (symmetric) Lanczos method cannot be used. Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

57 Methods [Parlett & Chen 1990] Pseudo Lanczos method (pretends B is positive definite) [Simoncini & al, 2005] similar [Freund, 2005]: analysis of Krylov spaces [Bai & Su, 2005] SOAR: based on Arnoldi s method Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

58 Structure preserving? Krylov space for linerizations of the form [Mackey,Mackey,Mehl,Mehrmann,2006] [ ] [ η1 K A A = 1 A1 η B = 1 C η 1 M η 2 K A 2 A 2 η 2 C η 2 M ] Arnoldi method for this linearization produces Krylov space with [ K 1 C K 1 ] M I 0 Structure preserving is not so easy from the point of view of the Krylov method It is possible on the level of projection (SOAR) Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

59 Outline 1 Motivation 2 Overview of methods Modal truncation Vector-Padé approximation Frequency sweeping Input/output MOR 3 Lanczos method 4 Rayleigh damping 5 Nonproportional damping 6 Multiple right-hand sides 7 Software 8 Future work 9 Conclusions Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

60 Multiple right-hand sides (K ω 2 M)[x 1,...,x s ] = [f 1,...,f s ] for ω Ω = [ω min, ω max ]. Methods: Use Lanczos method for each f j separately Low memory cost The cost is proportional to s Use block-lanczos method for each all f j together Fast method High memory cost Recycling Ritz vectors in Krylov methods [Giraud,Ruiz & Touhami, 2006] [Kilmer & de Sturler 2006] [Darnell, Morgan, Wilcox 2007] [Stathopoulos & Orginos, 2009][Bai & M. 2008] Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

61 Frequency sweeping For ω j solve (K ω 2 j M)x j = f iteratively Speed-up by preconditioning into (K σm) 1 (K ω 2 j M)x j = (K σm) 1 f and by using x j 1 as starting vector. Assume that a number of eigenvectors is given: speed up iterative process (See Daniel Rixen yesterday) Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

62 Frequency sweeping with modal acceleration The solution of for ω Ω is split into two parts. (K ω 2 M)x = f (1) Let U p = [u 1,...,u p ] be the eigenvectors corresponding to the eigenvalues in Ω 2. Compute x p = p j=1 u j u j f λ j ω 2 Solve (1) iteratively using starting vector x p, i.e. x = x p + y with y the solution of (K ω 2 M)y = f (K ω 2 M)x p = (I U p U pm)f Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

63 Preconditioning Preconditioner for the remainder system: (K σm) 1 Preconditioned system is with Ay = b b = (I U p U pm)(k σm) 1 f A = (K σm) 1 (K ω 2 M) y is also the solution of with By = b B = (I U p U pm)(k σm) 1 (K ω 2 M)(I U p U pm) Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

64 Spectral analysis of Lanczos method Eigenvalues of Kx = λmx Eigenvalues of A ω 2 min ω 2 max 0 1 Most eigenvalues of A lie near one. The number of required iterations is the number of isolated eigenvalues of A away from one. Convergence for all ω 2 Ω 2 requires the number of iterations, k, to be at least the number of eigenvalues in Ω 2. Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

65 Spectral analysis of deflated iterative method Eigenvalues of Kx = λmx Eigenvalues of A ω 2 min ω 2 max 0 1 Let B = A {Up} M Black eigenvalues only: B is positive definite Eigenvalues of B clustered around 1 spectral radius of I B is smaller than one Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

66 Inexact deflation Be careful for deflation with Ritz vectors [Darnell, Morgan, Wilcox 2007] Reason is that the system s residual need not be small and the direction may depend on ω r(ω) = R p z p (ω) + v k+1 ζ k (ω) with R p the residual vectors of the deflated Ritz vectors and v k+1 the k + 1st Lanczos vector. Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

67 Eigenvalue solver Use Ritz pairs of the (spectral transformation) Lanczos method [Grimes, Lewis, Simon 94] No exact eigenpairs But interesting properties as we now see: Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

68 Convergence of Ritz vectors Eigenvalues in Ω 2 are computed fairly accurately If Aˆx b M γ x M for all ω Ω then ρ j = Aû j û j ˆθj M with ρ j γ λ j σ Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

69 Padé via Lanczos Lanczos method: ˆx = k j=1 û j ûj f ˆλ j ω 2 First k moments of ˆx and x match. With deflation of simple eigenvalues: ˆx = p j=1 u j uj f λ j ω 2 + k j=p+1 û j ûj f ˆλ j ω 2 First k p moments of ˆx and x match. Interpolation in the p deflated eigenvalues. Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

70 Applications AMLS frequency sweeping Multiple right-hand sides: Parameterized Lanczos for right-hand side 1 Keep Ritz vectors Recycle Ritz vectors for coming right-hand sides Changing σ: recycle Ritz vectors for new pole. Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

71 Windscreen Glaverbel-BMW windscreen grid : 3 layers of HEX08 elements (n = 22, 692) Ω = [0, 100] First run: unit point force at one of the corners Use Lanczos method with k = 20 vectors. We keep the Ritz values in [0, ] : p = 14 Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

72 Windscreen Second run with other right-hand side Perform 6 additional Lanczos steps The largest κ(b) is Six iterations reduce the error in the M ˆB norm by Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

73 Acoustic cavity n = 48, 158 Frequency range : [0, 10000] 202 right-hand sides matrix factorization: 8 seconds Lanczos method with 40 vectors: 6 seconds Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

74 Acoustic cavity (cont.) 2nd right-hand side: keep the 31 Ritz values in [0, ]. 9 additional Lanczos iterations recycling 31 Ritz vectors: 2 seconds 1000 Exact recycling 1000 Exact k= Exact k= With recycling Lanczos k = 50 Lanczos k = 19 Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

75 Acoustic cavity n = 140, 228 Frequency range : [0, 10000] 202 right-hand sides matrix factorization: 13 seconds Lanczos method with 50 vectors: 15 seconds Recycling 36 vectors: only 4 seconds For 201 right-hand sides: 800 instead of 3000 seconds. Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

76 Multiple eigenvalues 3D Laplacian on a cube. 30 Lanczos iterations with first right-hand side Recycle 22 Ritz pairs Run 8 iterations with the second right-hand side Exact Recycling 10 Exact k= Lanczos for f 1 Recylcing for f 2 Lanczos k = 8 for f 2 Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

77 Conclusions Solving parameterized linear systems with multiple right-hand sides can benefit from recycling Ritz vectors Does not work well when eigenvalues are multiple Also works for Rayleigh damping Extension for block methods [Robbé & Sadkane] is straightforward Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

78 Outline 1 Motivation 2 Overview of methods Modal truncation Vector-Padé approximation Frequency sweeping Input/output MOR 3 Lanczos method 4 Rayleigh damping 5 Nonproportional damping 6 Multiple right-hand sides 7 Software 8 Future work 9 Conclusions Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

79 Software Software developed for FFT.Actran and MSC.Nastran: Shifting of K, C and M for faster convergence, Multiple right-hand sides, Arnoldi and Lanczos, possibility for out-of-core storage of iteration vectors, Error estimation Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

80 Software x is computed for a discrete number of ω s. We assume that ω 1 < ω 2 < < ω m. 1 Build Krylov subspace of dimension k for pole σ = ω 1. 2 For i = 1,...,m: 1 Compute x(ω i ) from the Krylov subspace. 2 If the solution is not accurate enough, pick a new σ and build a new Krylov subspace 3 Choice of k depends on the ratio of the cost of the sparse matrix factorization and the backward solves Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

81 Pole selection First pole σ = ω 1 If x(ω l ) did not converge, pick a new pole: σ j 1 ω l 1 ω l σ j Not too close to ω l, and not too far: σ j = σ j 1 + ω l + τ σ (ω l 1 σ j 1 ) Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

82 More on pole selection Similar to eigenvalue computations by Krylov methods: Pole close to eigenvalue: one eigenvalue converges Pole away from eigenvalues: slow but steady convergence The eigenvalues that matter are computed. Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

83 Example on pole selection Mushroom model Comparison between poles for 100 iterations Residual for σ = 300 Residual for σ = i Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

84 Storage Evaluation of x(ω): Storage of v j k is fixed (determined by available memory and execution time) k iteration vectors need to be stored ωi can be selected in a flexible way Storage of x(ω i ) Update x(ωi ) for i = 1,...,m at each iteration of the Krylov method Number of Krylov steps need not be selected beforehand All x(ωi ) need to be stored, sometimes additional vectors too. Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

85 Reorthogonalization The Lanczos process builds an M orthogonal basis in exact arithmetic Influence of reorthogonalization: 1 "exact" "noreorth" "reorth" e-06 1e-08 1e-10 1e-12 1e With reorthogonalization, we can solve for more frequencies Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

86 Finite precision Similar to eigenvalue computations: Solution of linear systems: choice of σ Not close to an eigenvalue: blows up the error Lanczos method: orthogonalization Error estimation of the solution On iteration j: (K σm)w j = Mv j + f j with f j 2 u( Mv j + K σm w j ) For the recurrence relation that implies with E k e j = (K σm) 1 f j. So E k 2 uκ(k σm). (K σm) 1 MV k V k+1 T k = E k Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

87 Example Figure shows the error norm "exact" "error-10" "ERR4-10" "ERR330-10" Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

88 Acoustic box ( m m 0.55m) with walls covered with carpet. K, C and M have order n = 13, 623. Wanted frequencies: ω {600, 605, 610,...,1500} k = 40 Arnoldi Lanczos direct factorizations time Speed-up of Arnoldi is 14. Loss of orthogonality of Lanczos vectors. Reorthogonalization Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

89 Example from vibro-acoustics Cube filled with air (source: Free Field technologies) with a steel plate inside The faces have infinite elements for radiation to infinity Point load on the plate Dimension is n = 36, 816. Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

90 For each frequency: (K + iωc ω 2 M)x = f The solution for 250 frequencies by the direct method costs 186 min. With Arnoldi s method, we only need 4.6 min. Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

91 3D head phone model Coupling structure acoustics. Problem of dimension 63, iteration vectors in Arnoldi s method with out-of-core linear solver. For ω = 10,...,335Hz Arnoldi Lanczos direct factorizations 1 (breakdown) 326 time (min.) Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

92 Outline 1 Motivation 2 Overview of methods Modal truncation Vector-Padé approximation Frequency sweeping Input/output MOR 3 Lanczos method 4 Rayleigh damping 5 Nonproportional damping 6 Multiple right-hand sides 7 Software 8 Future work 9 Conclusions Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

93 Future work Increasing model sizes Use of iterative linear system solvers Substructuring No Krylov methods POD type methods Other MOR methods needed Many (nonlinear) parameters Uncertainties Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

94 Many non-linear parameters Increase of computing power: larger models, but also more model parameters that need to be determined Can be an optimization loop fully automatically or tuned by hand Bottom line: model reduction helps to work with these models Making a reduced model for the entire parameter space is very unlikely to be possible. Therefore: Now reduced model is fed to a post-processing algorithm With many parameters, post-processing and modelreduction will be mixed: this may lead to new post-processing algorithms (related to talk by Yao Yue). Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

95 Parameter uncertainties Currently a hot topic in mechanical engineering, but many computational challenges. Optimal choice of parameters does not necessarily give information about how good the optimal model is. A sensitivity analysis around the optimum may be wanted. New is that the pertubrations can be large (so computing derivatives is not sufficient). When there are many parameters, practical approaches reduce the number of parameters. One idea is to compute the worst case scenario. Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

96 Uncertainties Traditional approach: fix all parameters and perform a deterministic analysis Variability (irreducible uncertainty): probability distribution stochastic methods compute the probability of the result performed when design is finished and ready for production Uncertainty (reducible uncertainty): early in the design stage interval analysis or fuzzy numbers possibilistic methods for evaluating the impact of the uncertainty Example: thickness of a plate. Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

97 Uncertain parameters: worst case Consider a vector of uncertain parameters x R m. We have a core value of x: x 0 We want to compute output y = f (x) with x [x] α with [x] α = {x : x x 0 α} for increasing values of α. [x] α is a hypercube. y α = min x [x] α f (x) y + α = max x [x] α f (x) This is an analysis that takes into account order α perturbations for all parameters = worst case scenario Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

98 Computing bounds Three approaches: Interval arithmetic: Use interval arithmetic in all operations (e.g. eigenvalue computation) Usually, it produces an overestimation (too large intervals) Design of experiments (DOE): Montecarlo approach Usually, it produces an underestimation (too small intervals) [Donders, 2008] Optimization problem: Global optimization method y α = min x [x] α f (x) y + α = max x [x] α f (x) Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

99 Computing bounds Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

100 Vertex methods If f is monotonous in all x j, then the minimum and maximum of f are attained at opposite corners of the α- cut. Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

101 Perturbed eigenvalues Let B, and E j, j = 1,...,m be real symmetric matrices. Let x R m. Define m A = B + x j E j. Assume eigenvalues ordered such that λ 1 (x) λ 2 (x) λ n (x) for all x. Denote by v j, v j 2 = 1, an eigenvector associated with λ j (x 0 ). j=1 Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

102 Smallest (or largest) eigenvalue Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

103 Derivatives λ i (x) = vi E j v i x j 2 λ i (x) = x 2 j 2 v i (A(x) λ i I) v i x j x j 2 λ 1 (x) = x 2 j 2 v 1 x j (A(x) λ 1 I) v 1 x j 0 Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

104 Tangent plane as lower bound t x0 (x) = λ 1 (x 0 ) + (x x 0 )(v Ev) where v, v 2 = 1, is an eigenvector associated with λ 1. Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

105 Optimization problem Since λ 1 is a convex function in x and x lives on a square domain, the maximum of λ 1 is attained at one of the corner points of the [x] α domain, and there is one and only one local minimum on each edge (face) of [x] α. Therefore, we have to compute the eigenvalues for x on the 2 m corners (for the maximum) and solve a convex optimization problems. Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

106 Local minimization problem When we have multiple eigenvalue problem, the optimization problem is non-smooth. For finding the local minimum, we can use line search methods, where the search direction is based on the tangent plane since the gradient might not exist. Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

107 Outline 1 Motivation 2 Overview of methods Modal truncation Vector-Padé approximation Frequency sweeping Input/output MOR 3 Lanczos method 4 Rayleigh damping 5 Nonproportional damping 6 Multiple right-hand sides 7 Software 8 Future work 9 Conclusions Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

108 Conclusions Krylov methods usually work well for acoustic simulation Recycling Ritz vectors is a reliable and efficient method for the solution with multiple right-hand sides Parametrized models with many parameters are current challenges Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109

109 Bibliography (Also: Z. Bai and K. Meerbergen. The Lanczos method for parameterized symmetric linear systems with multiple right-hand sides. Technical Report TW527, Department of Computer Science KULeuven, Celestijnenlaan 200A, 3001 Heverlee, Belgium, J. De Vlieger and K. Meerbergen. Analysis and computation of eigenvalues of symmetric fuzzy matrices. In T. Simos, editor, Proceedings of the ICNAAM09 Conference, K. Meerbergen. The solution of parametrized symmetric linear systems. SIAM J. Matrix Anal. Appl., 24(4): , K. Meerbergen. Fast frequency response computation for Rayleigh damping. International Journal of Numerical Methods in Engineering, 73(1):96 106, K. Meerbergen. The Quadratic Arnoldi method for the solution of the quadratic eigenvalue problem. SIAM Journal on Matrix Analysis and Applications, 30(4): , K. Meerbergen and J.P. Coyette. Connection and comparison between frequency shift time integration and a spectral transformation preconditioner. Numerical Linear Algebra with Applications, 16:1 17, F. Tisseur and K. Meerbergen. The quadratic eigenvalue problem. SIAM Review, 43(2): , Karl Meerbergen (K.U. Leuven) Parameterized linear systems MOR - September / 109