Dimension reduction for damping optimization in linear vibrating systems

Benner, Peter; Tomljanović, Zoran; Truhar, Ninoslav

doi:10.1002/zamm.201000077

Cited by 31 publications

(67 citation statements)

References 16 publications

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“…It is obtained by modal analysis, linearization and applying a perfect shuffle permutation [2]. The rank of the initial perturbation U CV T is exactly the number of added viscous dampers.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…The numerical experiments are carried out in MATLAB 8.3.0.532 (R2014a) on an Intel Core i5-2520M CPU @ 2.50GHz with 4 GB RAM. We consider two vibrational systems -a triple chain oscillator with dimensions n = 6002 and n = 9002, where a viscous damper is placed in the middle of the first chain [2] and an FEM model of a viscously damped beam with dimension n = 6000 and n = 8000 [3]. While in the left plot of Figure 1 the running times of the structure preserving sign function method w.r.t.…”

Section: Numerical Resultsmentioning

confidence: 99%

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Numerical solution to low rank perturbed Lyapunov equations by the sign function method

Benner

Denißen

2016

Proc Appl Math and Mech

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We investigate the numerical solution to a low rank perturbed Lyapunov equation A T X + XA = W via the sign function method (SFM). The sign function method has been proposed to solve Lyapunov equations, see e.g. [1], but here we focus on a framework where the matrix A has a special structure, i.e. A = B + U CV T , where B is a blockdiagonal matrix and U CV T is a low rank perturbation. We show that this structure can be kept throughout the sign function iteration but the rank of the perturbation doubles per iteration. Therefore, we apply a low rank approximation to the perturbation in order to keep its numerical rank small. We compare the standard SFM with its structure preserving variant presented in this paper by means of numerical examples from viscously damped mechanical systems. The algebraic Lyapunov equationwhere A, W = W T ∈ R n×n with its solution X ∈ R n×n plays a fundamental role e.g. in the stability analysis of linear systems. We focus on a structured algebraic Lyapunov equation, where , c k being some scaling for k = 0, 1, . . . with the solution X = 1 2 lim k→∞ W k [1]. The given structure can be exploited by this iteration as the following theorem shows.r×r and r n. Then the next sign function iterate A k+1 can be expressed as, whereHence, A k . In the k-th step of the sign function method, the complexity of computing A −1 k can be reduced from O(n 3 ) to O(r 3 ) by using its structure. Nevertheless, the complexity grows per sign function iteration since the rank r of the perturbation doubles per iteration (cf. Theorem 1.1). In order to save workspace and reduce the complexity, we propose to perform rank revealing QR factorizations (RR-QR): Q U R U = U k+1 and Q V R V = V k+1 and a truncated SVDÛΣV T ≈ R U C k+1 R T V w.r.t. some predefined truncation tolerance ε > 0, i.e.,Σ = diag(σ 1 , . . . , σ p ), where σ i ≥ ε for i = 1, . . . , p. This results in Algorithm 1.

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“…It is obtained by modal analysis, linearization and applying a perfect shuffle permutation [2]. The rank of the initial perturbation U CV T is exactly the number of added viscous dampers.…”

Section: Numerical Resultsmentioning

confidence: 99%

Section: Numerical Resultsmentioning

confidence: 99%

Numerical solution to low rank perturbed Lyapunov equations by the sign function method

Benner

Denißen

2016

Proc Appl Math and Mech

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“…This result has been generalized in [5] to the non-modal damping, obtaining the same set of optimal matrices. The results for the optimal damping without restrictions on the structure of damping matrices usually are not very useful in practical applications, but there is some recent progress in treating such problems (see [2,18,19]). …”

Section: The Corresponding Matrix Differential Equationmentioning

confidence: 99%

Optimal damping of the infinite-dimensional vibrational systems: commutative case

Nakić¹

2013

Glas. Mat. Ser. III

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“…[21][22][23]16,18,24]) which is constructed using appropriate linearization of system (1). Within this category, one can consider the following three criteria: minimization of the trace the solution of the corresponding Lyapunov equation, minimization of the 2-norm or the Frobenius norm of the solution of the corresponding Lyapunov equation.…”

Section: Introductionmentioning

confidence: 99%