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 10 3 to over 10 6 . Karl Meerbergen Krylov methods for fast frequency response computations
Outline Motivation Rayleigh damping Rational approximation Lanczos’ method Numerical example Conclusions Fourier analysis f ei ω t , then (under certain conditions) for t → ∞ , If f ( t ) = ˜ xei ω t x ( t ) = ˜ where x = ˜ ( K + i ω C − ω 2 M )˜ 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 x = ˜ ( K + i ω C − ω 2 M )˜ 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 = ω 1 , . . . , ω p Solve the linear system ( K + i ω C − ω 2 M ) x = f for x 1.1. For each frequency, a large system of algebraic equations needs to be solved. This requires a sparse matrix factorization LU = K − ω 2 M + i ω C (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 − ω 2 M + D ( ω )) x = f Rayleigh damping : D = γ K + δ M 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 − ω 2 M ) x = f Corresponding eigenvalue problem: Ku = λ 2 Mu 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 ) = i γ K Linear system: ((1 + i γ ) K − ω 2 M ) x = f Corresponding eigenvalue problem: (1 + i γ ) Ku = λ 2 Mu 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 ) = i ω ( α 0 M + α 1 K ) Linear system: ( K + i ω ( α 0 M + α 1 K ) − ω 2 M ) x = f Corresponding eigenvalue problem: ( K + i λ ( α 0 M + α 1 K ) − λ 2 M ) 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 Ku j = λ 2 j Mu j The solution of ( K − ω 2 M ) x = f is n u ∗ j f � x = u j λ 2 j − ω 2 j =1 Rational function with poles λ 2 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 Mu i = u ∗ j Ku i = u ∗ j Du i = 0 iff i � = j . j Mu j = 1 u ∗ j Ku j = λ 2 u ∗ j u ∗ j D ( ω ) u j = ζ j ( ω ) The solution of ( K − ω 2 M + D ) x = f n u ∗ j f � x = u j j − ω 2 + ζ j ( ω ) λ 2 j =1 Karl Meerbergen Krylov methods for fast frequency response computations
Outline Motivation Rayleigh damping Rational approximation Lanczos’ method Numerical example Conclusions Truncation n k u ∗ j f u ∗ j f � � x = j − ω 2 ≈ u j u j λ 2 λ 2 j − ω 2 j =1 j =1 10 "damped" "damped10" "damped7" 1 0.1 0.01 0 2 4 6 8 10 12 Karl Meerbergen Krylov methods for fast frequency response computations
Outline Motivation Rayleigh damping Rational approximation Lanczos’ method Numerical example Conclusions Lanczos process Lanczos method: Compute the initial vector v 1 = K − 1 f . 1. 2. For j = 1 , . . . , k Compute Krylov vector v j +1 = K − 1 Mv j . 2.1. 2.2. Orthogonalize v j +1 against v 1 , . . . , v j so that v ∗ j +1 MV j = 0. Krylov space: { K − 1 f , K − 1 MK − 1 f , . . . , ( K − 1 M ) j K − 1 f , . . . } Lanczos vectors V k = [ v 1 , . . . , v k ]. Karl Meerbergen Krylov methods for fast frequency response computations
Outline Motivation Rayleigh damping Rational approximation Lanczos’ method Numerical example Conclusions Eigenvalue solver Lanczos vectors V k = [ v 1 , . . . , v k ]. k MK − 1 MV k ( k × k tridiagonal matrix) Projection T k = V ∗ Ritz values: T k z = θ z Ritz vectors: ˜ u = V k z Form an approximate eigenpair of K − 1 M : � K − 1 M ˜ u − θ ˜ u � M 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 − 1 M Ku = λ 2 Mu λ − 2 u = K − 1 Mu 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 − ω 2 M ) x = f Precondition: K − 1 ( K − ω 2 M ) x = K − 1 f Apply Lanczos: V k , T k Same V k as the Lanczos method applied to K − 1 M . 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 − ω 2 M ) x = f The solution can be written as a Taylor series: x = x 0 + ω 2 x 2 + ω 4 x 4 + · · · The Lanczos method uses starting vector K − 1 f Let the solution computed by the projection on the Ritz vectors be x 0 + ω 2 ˜ x 2 + ω 4 ˜ ˜ x = ˜ x 4 + · · · then x 2 j = x 2 j ˜ 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 − ω 2 M ) x = f : Apply Lanczos to K − 1 M : V k , T k For each ω , solve ( I − ω 2 T k ) z = e 1 � K − 1 f � M and compute the solution x = V k z . 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 100 "direct.exact" "kry.err" 1 0.01 1e-04 1e-06 1e-08 1e-10 1e-12 1e-14 1e-16 0 20 40 60 80 100 120 140 160 180 200 Karl Meerbergen Krylov methods for fast frequency response computations
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