Krylov methods for fast frequency response computations Karl - - PowerPoint PPT Presentation

Outline Motivation Rayleigh damping Rational approximation Lanczos’ method Numerical example Conclusions

Krylov methods for fast frequency response computations

Karl Meerbergen January 8, 2006

Karl Meerbergen Krylov methods for fast frequency response computations

Outline Motivation Rayleigh damping Rational approximation Lanczos’ method Numerical example Conclusions

Outline

1 Motivation 2 Rayleigh damping 3 Rational approximation 4 Lanczos’ method 5 Numerical example 6 Conclusions

Karl Meerbergen Krylov methods for fast frequency response computations

Outline Motivation Rayleigh damping Rational approximation Lanczos’ method Numerical example Conclusions

Vibration problems

A car window is subjected to vibrations from outside, including

• wind. Glass manufacturers want to compute the transmission of

noise through windscreens.

Karl Meerbergen Krylov methods for fast frequency response computations

Outline Motivation Rayleigh damping Rational approximation Lanczos’ method Numerical example Conclusions

Finite element analysis

Numerical simulation of vibration problems. Spatial (finite element) discretization: M¨ x(t) + C ˙ x(t) + Kx(t) = f (t) with initial values x(0) and ˙ x(0) f and x : vectors of length n K, C and M : n × n sparse matrices. In real applications n varies from 103 to over 106.

Karl Meerbergen Krylov methods for fast frequency response computations

Outline Motivation Rayleigh damping Rational approximation Lanczos’ method Numerical example Conclusions

Fourier analysis

If f (t) = ˜ f eiωt, then (under certain conditions) for t → ∞, x(t) = ˜ xeiωt where (K + iωC − ω2M)˜ 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 Krylov methods for fast frequency response computations

Outline Motivation Rayleigh damping Rational approximation Lanczos’ method Numerical example Conclusions

Fourier analysis

(K + iωC − ω2M)˜ x = ˜ f ˜ x is called the frequency response function.

Karl Meerbergen Krylov methods for fast frequency response computations

Outline Motivation Rayleigh damping Rational approximation Lanczos’ method Numerical example Conclusions

Traditional frequency response computation

1. For ω = ω1, . . . , ωp 1.1. Solve the linear system (K + iωC − ω2M)x = f for x For each frequency, a large system of algebraic equations needs to be solved. This requires

a sparse matrix factorization LU = K − ω2M + iωC (expensive) and a backward solve LUx = f (relatively cheap).

The goal is to reduce the number of matrix factorizations. Important is speed, not good reduction.

Karl Meerbergen Krylov methods for fast frequency response computations

Outline Motivation Rayleigh damping Rational approximation Lanczos’ method Numerical example Conclusions

Damping

We make the damping ω dependent: D(ω) (K − ω2M + D(ω))x = f Rayleigh damping : D = γK + δM f is independent of ω

Karl Meerbergen Krylov methods for fast frequency response computations

Outline Motivation Rayleigh damping Rational approximation Lanczos’ method Numerical example Conclusions

No damping

D(ω) ≡ 0 Linear system: (K − ω2M)x = f Corresponding eigenvalue problem: Ku = λ2Mu

Karl Meerbergen Krylov methods for fast frequency response computations

Outline Motivation Rayleigh damping Rational approximation Lanczos’ method Numerical example Conclusions

Structural Rayleigh damping

D(ω) = iγK Linear system: ((1 + iγ)K − ω2M)x = f Corresponding eigenvalue problem: (1 + iγ)Ku = λ2Mu

Karl Meerbergen Krylov methods for fast frequency response computations

Outline Motivation Rayleigh damping Rational approximation Lanczos’ method Numerical example Conclusions

Fluid Rayleigh damping

D(ω) = iω(α0M + α1K) Linear system: (K + iω(α0M + α1K) − ω2M)x = f Corresponding eigenvalue problem: (K + iλ(α0M + α1K) − λ2M)u = 0

Karl Meerbergen Krylov methods for fast frequency response computations

Outline Motivation Rayleigh damping Rational approximation Lanczos’ method Numerical example Conclusions

Undamped problem

Consider the eigendecomposition Kuj = λ2

j Muj

The solution of (K − ω2M)x = f is x =

n

• j=1

uj u∗

j f

λ2

j − ω2

Rational function with poles λ2

j .

Karl Meerbergen Krylov methods for fast frequency response computations

Outline Motivation Rayleigh damping Rational approximation Lanczos’ method Numerical example Conclusions

Rayleigh damping

Simulteneous diagionalization of K, M, and D: u∗

j Mui = u∗ j Kui = u∗ j Dui = 0 iff i = j.

u∗

j Muj = 1

u∗

j Kuj = λ2 j

u∗

j D(ω)uj = ζj(ω)

The solution of (K − ω2M + D)x = f x =

n

• j=1

uj u∗

j f

λ2

j − ω2 + ζj(ω)

Karl Meerbergen Krylov methods for fast frequency response computations

Outline Motivation Rayleigh damping Rational approximation Lanczos’ method Numerical example Conclusions

Truncation

x =

n

• j=1

uj u∗

j f

λ2

j − ω2 ≈ k

• j=1

uj u∗

j f

λ2

j − ω2

0.01 0.1 1 10 2 4 6 8 10 12 "damped" "damped10" "damped7"

Karl Meerbergen Krylov methods for fast frequency response computations

Outline Motivation Rayleigh damping Rational approximation Lanczos’ method Numerical example Conclusions

Lanczos process

Lanczos method: 1. Compute the initial vector v1 = K −1f . 2. For j = 1, . . . , k 2.1. Compute Krylov vector vj+1 = K −1Mvj. 2.2. Orthogonalize vj+1 against v1, . . . , vj so that v ∗

j+1MVj = 0.

Krylov space: {K −1f , K −1MK −1f , . . . , (K −1M)jK −1f , . . .} Lanczos vectors Vk = [v1, . . . , vk].

Karl Meerbergen Krylov methods for fast frequency response computations

Outline Motivation Rayleigh damping Rational approximation Lanczos’ method Numerical example Conclusions

Eigenvalue solver

Lanczos vectors Vk = [v1, . . . , vk]. Projection Tk = V ∗

k MK −1MVk (k × k tridiagonal matrix)

Ritz values: Tkz = θz Ritz vectors: ˜ u = Vkz Form an approximate eigenpair of K −1M: K −1M˜ u − θ˜ uM is small for the large θ’s.

Karl Meerbergen Krylov methods for fast frequency response computations

Outline Motivation Rayleigh damping Rational approximation Lanczos’ method Numerical example Conclusions

Convergence

Lanczos method computes the dominant eigenvalues of K −1M Ku = λ2Mu λ−2u = K −1Mu The Lanczos method computes large λ−2 i.e. small λ’s.

Karl Meerbergen Krylov methods for fast frequency response computations

Outline Motivation Rayleigh damping Rational approximation Lanczos’ method Numerical example Conclusions

Shifted linear systems

Analysed in the context of model reduction methods Feldman, Freund, Bai, Grimme, Sorensen, Van Dooren, Ruhe, Skoogh, Olsson, Simoncini, M., ... Connection with eigendecomposition Connection with iterartive linear solvers Connection with rational approximation (Pad´ e)

Karl Meerbergen Krylov methods for fast frequency response computations

Outline Motivation Rayleigh damping Rational approximation Lanczos’ method Numerical example Conclusions

Iterative solver connection

Linear system (K − ω2M)x = f Precondition: K −1(K − ω2M)x = K −1f Apply Lanczos: Vk, Tk Same Vk as the Lanczos method applied to K −1M.

Karl Meerbergen Krylov methods for fast frequency response computations

Outline Motivation Rayleigh damping Rational approximation Lanczos’ method Numerical example Conclusions

Pad´ e connection

Suppose we solve (K − ω2M)x = f The solution can be written as a Taylor series: x = x0 + ω2x2 + ω4x4 + · · · The Lanczos method uses starting vector K −1f Let the solution computed by the projection on the Ritz vectors be ˜ x = ˜ x0 + ω2˜ x2 + ω4˜ x4 + · · · then ˜ x2j = x2j for j = 0, . . . , k − 1

Karl Meerbergen Krylov methods for fast frequency response computations

Outline Motivation Rayleigh damping Rational approximation Lanczos’ method Numerical example Conclusions

Implementation

For (K − ω2M)x = f : Apply Lanczos to K −1M: Vk, Tk For each ω, solve (I − ω2Tk)z = e1K −1f M and compute the solution x = Vkz.

Karl Meerbergen Krylov methods for fast frequency response computations

Outline Motivation Rayleigh damping Rational approximation Lanczos’ method Numerical example Conclusions

Windscreen

Glaverbel-BMW windscreen with 10% structural damping grid : 3 layers of 60 × 30 HEX08 elements (n = 22, 692) unit point force at one of the corners Direct method : 2653 seconds Lanczos method : 14 seconds wanted : displacement for ω = [0.5Hz, 200Hz].

Karl Meerbergen Krylov methods for fast frequency response computations

Outline Motivation Rayleigh damping Rational approximation Lanczos’ method Numerical example Conclusions

1e-16 1e-14 1e-12 1e-10 1e-08 1e-06 1e-04 0.01 1 100 20 40 60 80 100 120 140 160 180 200 "direct.exact" "kry.err"

Karl Meerbergen Krylov methods for fast frequency response computations

Outline Motivation Rayleigh damping Rational approximation Lanczos’ method Numerical example Conclusions

Conclusions

Large reduction in computation time possible with model reduction methods Rayleigh damping is special structure that can be exploited using the Lanczos method

Karl Meerbergen Krylov methods for fast frequency response computations