000013288 001__ 13288
000013288 005__ 20161114160335.0
000013288 04107 $$aeng
000013288 046__ $$k2009-06-22
000013288 100__ $$aPrabel, B.
000013288 24500 $$aIncremental inverse iteration for the nonlinear eigenvalue problem in structural dynamics

000013288 24630 $$n2.$$pComputational Methods in Structural Dynamics and Earhquake Engineering
000013288 260__ $$bNational Technical University of Athens, 2009
000013288 506__ $$arestricted
000013288 520__ $$2eng$$aCalculation of modes is of primal importance in structural dynamics. Most of the time discretization of the equation of motion leads to a quadratic eigenvalue problem to solve [1]. But when one wants to include more elaborate mechanical model (as viscoelasticity, feedback control, or fluid-structure interaction, ...), general nonlinear eigenvalue problem has to be solved. Nonlinearity leads to the loss of usual properties of modes and forces to adapt the procedures commonly used to solve linear eigenvalue problems, so that it constitutes a modern challenge in computational mechanics as noted by Mehrmann and Voss [2]. It is well known that direct application of Newton’s method to the linear eigenvalue problem leads to the Rayleigh quotient iteration (which can be identified as the inverse iteration method with a shift updated at every iterations). This explains why the idea to solve the nonlinear eigenvalue problem with the same Newton procedure (often called ”inverse iteration”) is so popular. To minimize the number of factorization of the operator used in inverse iteration (due to successive shifts), Neumaier [3] proposed to perform the iterative process without updating the shift, leading to the ”residual inverse iteration”. Because based on the Newton’s method, both above methods (inverse iteration and residual inverse iteration) converge at the conditions that initial guess is close enough to the sought-after mode, and that the functional considered is regular enough. In industrial applications, what is important is to determine the m first modes of the structure, and initial guess of the resonance frequencies are seldomly available with a sufficient precision to use directly Newton’s based procedures. What is proposed in this paper is to circumvent this problem, by using an incremental approach to localize successive eigenvalues λ from a starting point. Increment are driven by a monotonic and controlled shift λshif t . This enables to avoid problems inherent to Newton’s method when problem is genuinely nonlinear (failure by oscillating around local minima or by jumping roots). Numerical examples demonstrate the good behavior of the proposed method for Hermitian nonlinear eigenvalue problem.

000013288 540__ $$aText je chráněný podle autorského zákona č. 121/2000 Sb.
000013288 653__ $$aNonlinear eigenvalue problem, Inverse iteration, Newton method. Abstract. Calculation of modes is of primal importance in structural dynamics. Most of the time discretization of the equation of motion leads to a quadratic eigenvalue problem to solve [1]. But when one wants to include more elaborate mechanical model (as viscoelasticity, feedback control, or fluid-structure interaction, ...), general nonlinear eigenvalue problem has to be solved. Nonlinearity leads to the loss of usual properties of modes and forces to adapt the procedures commonly used to solve linear eigenvalue problems, so that it constitutes a modern challenge in computational mechanics as noted by Mehrmann and Voss [2]. It is well known that direct application of Newton’s method to the linear eigenvalue problem leads to the Rayleigh quotient iteration (which can be identified as the inverse iteration method with a shift updated at every iterations). This explains why the idea to solve the nonlinear eigenvalue problem with the same Newton procedure (often called ”inverse iteration”) is so popular. To minimize the number of factorization of the operator used in inverse iteration (due to successive shifts), Neumaier [3] proposed to perform the iterative process without updating the shift, leading to the ”residual inverse iteration”. Because based on the Newton’s method, both above methods (inverse iteration and residual inverse iteration) converge at the conditions that initial guess is close enough to the sought-after mode, and that the functional considered is regular enough. In industrial applications, what is important is to determine the m first modes of the structure, and initial guess of the resonance frequencies are seldomly available with a sufficient precision to use directly Newton’s based procedures. What is proposed in this paper is to circumvent this problem, by using an incremental approach to localize successive eigenvalues λ from a starting point. Increment are driven by a monotonic and controlled shift λshif t . This enables to avoid problems inherent to Newton’s method when problem is genuinely nonlinear (failure by oscillating around local minima or by jumping roots). Numerical examples demonstrate the good behavior of the proposed method for Hermitian nonlinear eigenvalue problem.

000013288 7112_ $$aCOMPDYN 2009 - 2nd International Thematic Conference$$cIsland of Rhodes (GR)$$d2009-06-22 / 2009-06-24$$gCOMPDYN2009
000013288 720__ $$aPrabel, B.
000013288 8560_ $$ffischerc@itam.cas.cz
000013288 8564_ $$s1327945$$uhttps://invenio.itam.cas.cz/record/13288/files/CD428.pdf$$yOriginal version of the author's contribution as presented on CD, section: Algorithms and computational tools in structural dynamics (MS).
000013288 962__ $$r13074
000013288 980__ $$aPAPER