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

Outline Motivation Rayleigh damping Modal truncation Lanczos’ method Numerical example Conclusions Krylov methods for fast frequency response computations Karl Meerbergen February 28, 2007 Karl Meerbergen Krylov methods for fast frequency response computations

Outline Motivation Rayleigh damping Modal truncation Lanczos’ method Numerical example Conclusions Outline 1 Motivation 2 Rayleigh damping 3 Modal truncation 4 Lanczos’ method 5 Numerical example 6 Conclusions Karl Meerbergen Krylov methods for fast frequency response computations

Outline Motivation Rayleigh damping Modal truncation Lanczos’ method Numerical example Conclusions Vibration problems Bridge vibrating under footsteps and Thames wind Karl Meerbergen Krylov methods for fast frequency response computations

Outline Motivation Rayleigh damping Modal truncation 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 Modal truncation 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 Modal truncation 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 Modal truncation 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 Modal truncation 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 Modal truncation 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 Modal truncation Lanczos’ method Numerical example Conclusions No damping D ( ω ) ≡ 0 Linear system: ( K − ω 2 M ) x = f Corresponding eigenvalue problem: Ku = λ Mu Karl Meerbergen Krylov methods for fast frequency response computations

Outline Motivation Rayleigh damping Modal truncation 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 Mu Karl Meerbergen Krylov methods for fast frequency response computations

Outline Motivation Rayleigh damping Modal truncation 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 Modal truncation Lanczos’ method Numerical example Conclusions Undamped problem Consider the eigendecomposition Ku j = λ j Mu j The solution of ( K − ω 2 M ) x = f is n u ∗ j f � x = u j λ j − ω 2 j =1 Rational function with poles λ j . Karl Meerbergen Krylov methods for fast frequency response computations

Outline Motivation Rayleigh damping Modal truncation Lanczos’ method Numerical example Conclusions Rayleigh damping Define U = [ u 1 , . . . , u n ] and Λ = diag( λ 1 , . . . , λ n ) KU = MU Λ D = β K + γ M DU = MU ( β I + γ Λ) U ∗ MU = I U ∗ KU = Λ U ∗ DU = β I + γ Λ Karl Meerbergen Krylov methods for fast frequency response computations

Outline Motivation Rayleigh damping Modal truncation Lanczos’ method Numerical example Conclusions Rayleigh damping Simultaneous diagionalization of K , M , and D : j Mu i = u ∗ j Ku i = u ∗ j Du i = 0 iff i � = j . u ∗ u ∗ j Mu j = 1 j Ku j = λ j u ∗ 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 λ j − ω 2 + ζ j ( ω ) j =1 Karl Meerbergen Krylov methods for fast frequency response computations

Outline Motivation Rayleigh damping Modal truncation Lanczos’ method Numerical example Conclusions Truncation n k u ∗ j f u ∗ j f � � x = ≈ u j u j λ j − ω 2 + ζ j λ j − ω 2 + ζ j 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 Modal truncation Lanczos’ method Numerical example Conclusions Lanczos process Krylov space: { K − 1 f , K − 1 MK − 1 f , . . . , ( K − 1 M ) j K − 1 f , . . . } 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. Lanczos vectors V k = [ v 1 , . . . , v k ]. Karl Meerbergen Krylov methods for fast frequency response computations

Outline Motivation Rayleigh damping Modal truncation 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 Modal truncation Lanczos’ method Numerical example Conclusions Convergence Lanczos method computes the dominant eigenvalues of K − 1 M Ku = λ Mu λ − 1 u = K − 1 Mu The Lanczos method computes large λ − 1 i.e. small λ ’s. Karl Meerbergen Krylov methods for fast frequency response computations

Outline Motivation Rayleigh damping Modal truncation Lanczos’ method Numerical example Conclusions Shifted linear systems Analyzed in the context of model reduction methods Feldman, Freund, Bai, Grimme, Sorensen, Van Dooren, Ruhe, Skoogh, Olsson, Simoncini, M., ... Connection with eigendecomposition Connection with iterative linear solvers Connection with rational approximation (Pad´ e) Karl Meerbergen Krylov methods for fast frequency response computations

Outline Motivation Rayleigh damping Modal truncation 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 Use Lanczos (Conjugate Gradients) : fast convergence when ω small Karl Meerbergen Krylov methods for fast frequency response computations

Outline Motivation Rayleigh damping Modal truncation Lanczos’ method Numerical example Conclusions Apply Lanczos to K − 1 ( K − ω 2 M ) − → V k ( ω ), T k ( ω ) If we apply Lanczos to K − 1 M − → V k , T k : V k ( ω ) = V k T k ( ω ) = I + ω 2 T k So, there is no need to recompute the Krylov space for each ω separately. Karl Meerbergen Krylov methods for fast frequency response computations

Outline Motivation Rayleigh damping Modal truncation Lanczos’ method Numerical example Conclusions Conjugate gradient connection eigenvalues of K − 1 ( K − ω 2 M ) are λ j − ω 2 λ j Eigenvalues are clustered around one. spectrum of K − 1 ( K − ω M ) spectrum of K − λ M ω 2 0 0 1 When there are no eigenvalues λ between 0 and ω 2 , then we have a positive definite linear system Karl Meerbergen Krylov methods for fast frequency response computations

Outline Motivation Rayleigh damping Modal truncation Lanczos’ method Numerical example Conclusions MINRES versus Lanczos Define the error e ( ω ) = y ( ω ) − ˜ y ( ω ) MINRES minimizes the residual, i.e. n � 2 � λ j − ω 2 � j Me ( ω 2 )) 2 ( u ∗ λ j j =1 Karl Meerbergen Krylov methods for fast frequency response computations