AA242B: MECHANICAL VIBRATIONS 1 / 17 AA242B: MECHANICAL VIBRATIONS Solution Methods for the Generalized Eigenvalue Problem These slides are based on the recommended textbook: M. Géradin and D. Rixen, Mechanical Vibrations: Theory and Applications to Structural Dynamics, Second Edition, Wiley, John & Sons, Incorporated, ISBN-13:9780471975465 1 / 17
AA242B: MECHANICAL VIBRATIONS 2 / 17 Outline 1 Eigenvector Iteration Methods 2 Subspace Construction Methods 2 / 17
AA242B: MECHANICAL VIBRATIONS 3 / 17 Eigenvector Iteration Methods Introduction Generalized eigenvalue problem associated with an n-degree-of-freedom system Kx = ω 2 Mx n eigenvalues 0 ω1 2 ω2 2 ωn 2 and n associated eigenvectors x 1, x 2,, x n 3 / 17
AA242B: MECHANICAL VIBRATIONS 4 / 17 Eigenvector Iteration Methods Introduction Dynamic flexibility matrix consider a system that has no rigid-body modes K is non-singular construct the dynamic flexibility matrix defined as D = K 1 M the associated eigenvalue problem is Kx = ω 2 Mx Dx = λx with λ = 1 ω 2 eigenvectors are same as for the generalized eigenproblem Kx = ω 2 Mx eigenvalues are given by λ i = 1 ω 2 i λ 1 λ 2 λ n drawbacks of working with D building cost is O(n 3 ) non-symmetric matrix and as such must be stored completely 4 / 17
AA242B: MECHANICAL VIBRATIONS 5 / 17 Eigenvector Iteration Methods The Power Algorithm Determination of the fundamental eigenmode an iterative solution to Dx = λx can be obtained by considering the iteration z p+1 = Dz p proof: expand z 0 in the basis of the eigenmodes of the system z 0 = n α i x i i=1 Then, at the p-th iteration, z p = D p z 0 = n α i D p x i = i=1 ( n α i λ p i x i = λ p 1 α 1x 1 + i=1 Recall that λ 1 λ 2 λ n and assume that λ 1 > λ 2 n i=2 = z p λ p zp+1 1α1x1 λ1 as p z p ( ) ) p λi α i x i λ 1 5 / 17
AA242B: MECHANICAL VIBRATIONS 6 / 17 Eigenvector Iteration Methods The Power Algorithm Convergence analysis p, z p = λ p 1 x1 + λp 2 x2 + (recall that x i is defined up to a multiplicative constant) = zt p+1 e j z T p e j where r = λ2 λ 1 define a j = xt 2 e j x T 1 e j = λp+1 1 x T 1 e j + λ p+1 2 x T 2 e j + λ p 1 xt 1 e j + λ p 2 xt 2 e j + = λ1(x1 + r p+1 x 2 + ) T e j (x 1 + r p x 2 + ) T e j and assume that λ 2 < λ 1 = λp+1 1 (x 1 + r p+1 x 2 + ) T e j λ p 1 (x1 + r p x 2 + ) T e j λ (p) 1 = zt p+1 e j z T p e = λ1(1 + a j r p+1 + ) j 1 + a j r p + p λ 1(1 + a j r p+1 )(1 + a j r p ) 1 p λ 1 ( 1 + aj r p (r 1) ) number of iterations N after which λ (N) 1 is stabilized with t significant digits λ (N+1) 1 λ (N) 1 λ (N) 10 t a j r N 10 t t N ( ) λ1 1 log λ 2 = N and therefore the convergence rate of the power algorithm depends only on the ratio of the first two eigenvalues 6 / 17
AA242B: MECHANICAL VIBRATIONS 7 / 17 Eigenvector Iteration Methods The Power Algorithm Determination of the higher modes by orthogonal deflation once x 1 is determined, the following orthogonal projection operator can be formed (orthogonal projection P 2 1 = P 1 and Ker(P 1) is M-orthogonal to span (P 1)) P 1 = I x1xt 1 M x T 1 Mx1 ( P 1x 1 = 0) next, the following starting vector can be constructed n z 0 = P 1z 0 = α i x i this vector does not have any component in the x 1 direction the power method equipped with the starting vector z 0 converges in principle to (λ 2, x 2) in practice, the mode x 1 reappears in the iterations because of roundoff errors; to remediate this issue, the following algorithm can be used instead { z p = P 1z p i=2 z p+1 = Dz p this approach can be generalized to extract higher modes 7 / 17
AA242B: MECHANICAL VIBRATIONS 8 / 17 Eigenvector Iteration Methods The Inverse Iteration Method Power iteration in terms of the generalized eigenproblem Kx = ω 2 Mx { yp = Mz p Kz p+1 = y p incurs the solution of an algebraic problem of the form Kq = g factorization-based algorithm 1 factorization of K (once) 2 forward and backward backward substitutions (repeated) Converges to ω 2 instead of λ (inverse power iteration) 8 / 17
AA242B: MECHANICAL VIBRATIONS 9 / 17 Eigenvector Iteration Methods Inverse Iteration with Spectral Shifting Spectral shifting (by µ) of the eigenvalue problem (K µm)x = (ω 2 µ)mx associated inverse iteration scheme { yp = Mz p (K µm)z p+1 = y p iterations n z 0 = α i x i z p = i=1 n p α i ((K µm) M) 1 xi = i=1 n i=1 α i (ω 2 i µ) p x i converge toward the eigenmode x r whose associated eigenvalue ωr 2 satisfies { } ωr 2 µ = min ωj 2 µ j = convergence accelerates as µ approaches ω 2 r 9 / 17
AA242B: MECHANICAL VIBRATIONS 10 / 17 Subspace Construction Methods The Subspace Iteration Method The subspace concept let X denote the M-normalized solution of the generalized eigenvalue problem Kx = ω 2 Mx X T MX = I n, X T KX = Λ = diag(ω 2 1,, ω 2 n) then span(x) = E of dimension n consider the first p eigenmodes X = [ x 1 x p ] span(x ) = E of dimension p; E E property 1: a R p z = X a E property 2: if A R p p and det(a) 0 span(x A) = span(x ) 10 / 17
AA242B: MECHANICAL VIBRATIONS 11 / 17 Subspace Construction Methods The Subspace Iteration Method Interaction problem once any set Z of p linearly independent vectors in E is found, the first p eigensolutions are determined by considering the reduced eigenproblem K y = ω 2 M y where K = Z T KZ and M = Z T MZ, then forming X = Z Y = Z [ y 1 y p ] 11 / 17
AA242B: MECHANICAL VIBRATIONS 12 / 17 Subspace Construction Methods The Subspace Iteration Method Using the inverse iteration algorithm, the subspace method generates a sequence of matrices Z 0, Z 1,, Z k R n p spanning the subspaces E 0, E 1,, E k that converge toward E = E = span(x ) (iterative method on the Grassmann manifold G(p, n)) Algorithm iteration kernel { Vk = MZ k KZ k+1 = V k aim is convergence toward E of the subspace as a whole, and not of the individual vectors in Z k to make sure that the reduced-order bases Z i span subspaces of dimension p, orthogonality of the column vectors is ensured by one of two ways Gram-Schmidt orthogonalization re-orthogonalization with respect to the M-induced norm 12 / 17
AA242B: MECHANICAL VIBRATIONS 13 / 17 Subspace Construction Methods The Subspace Iteration Method Convergence analysis the subspaces spanned by the matrices Z k converge toward the subspace spanned by X provided that Z 0 is not orthogonal to any of the first p eigenvectors global convergence of the subspace iteration algorithm guarantees that subspace E k E converges toward E the convergence rate toward each mode x j is of the order of O(rj k ) where r j = ω2 j ω 2 p+1 measures the ratio of the eigenvalue ωj 2 and that which is immediately outside the subspace E. This indicates that a higher convergence rate is obtained by using q > p iteration vectors. A practical trade-off suggests to use q = min(2p, p + 8). considering the convergence rate mentioned above, multiple eigenvalues do not decrease the rate of convergence provided that ωq+1 2 > ωp 2 13 / 17
AA242B: MECHANICAL VIBRATIONS 14 / 17 Subspace Construction Methods The Lanczos Method Principle of the method generate a subspace that includes the fundamental eigensolutions by applying the inverse iteration algorithm to a starting vector z 0 construct the following Krylov sequence { ( } 2 z 0, K 1 Mz 0, K M) 1 z0, orthogonalize the terms of the sequence Properties the search subspace is built column by column: its dimension is increased at each iteration fast convergence however, orthonormality rapidly deteriorates (and re-orthogonalization procedures can be computationally expensive) inability to detect multiple eigenvalues 14 / 17
AA242B: MECHANICAL VIBRATIONS 15 / 17 Subspace Construction Methods The Lanczos Method Algorithm iteration γ p+1x p+1 = K 1 Mx p α px p β p 1x p 1 where β p 1, α p, γ p+1 are determined such that { x T p+1 Mx j = 0 j < p + 1 x T p+1mx p+1 = 1 requiring M-orthogonality leads to α p = x T p MK 1 Mx p β p 1 = x T p MK 1 Mx p 1 γ p = β p 1 X = [ x 0 x p ] satisfies K 1 MX = XT + S = [ 0 0 γ p+1x p+1 ] = γp+1x p+1e T p where T = tridiag ([ γ 1 γ p ], [ α0 α p ], [ γ1 γ p ]) 15 / 17
AA242B: MECHANICAL VIBRATIONS 16 / 17 Subspace Construction Methods The Lanczos Method Interaction problem start with K 1 MX = XT + S premultiply by Y T = (MX) T associated to the Lanczos vectors X note that Y T S = 0 Y T K 1 MX = Y T XT = T hence, the tridiagonal matrix of the Lanczos coefficients results from the projection of D = K 1 M onto the subspace of Lanczos vectors X. Thus, it has the same eigenvalues as the projection of D onto the reduced-order basis X. 16 / 17
AA242B: MECHANICAL VIBRATIONS 17 / 17 Subspace Construction Methods The Lanczos Method Difficulties for the Lanczos method deterioration of the orthogonalization process detection of multiple eigenvalues appearance of parasitic solutions missing eigensolutions 17 / 17