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    On the distribution of real eigenvalues in linear viscoelastic oscillators
    (2019-03)
    Mohammadi, Seyyed Abbas
    ;
    Voß, Heinrich 
    © 2019 John Wiley & Sons, Ltd. In this paper, a linear viscoelastic system is considered where the viscoelastic force depends on the past history of motion via a convolution integral over an exponentially decaying kernel function. The free-motion equation of this nonviscous system yields a nonlinear eigenvalue problem that has a certain number of real eigenvalues corresponding to the nonoscillatory nature. The quality of the current numerical methods for deriving those eigenvalues is directly related to damping properties of the viscoelastic system. The main contribution of this paper is to explore the structure of the set of nonviscous eigenvalues of the system while the damping coefficient matrices are rank deficient and the damping level is changing. This problem will be investigated in the cases of low and high levels of damping, and a theorem that summarizes the possible distribution of real eigenvalues will be proved. Moreover, upper and lower bounds are provided for some of the eigenvalues regarding the damping properties of the system. Some physically realistic examples are provided, which give us insight into the behavior of the real eigenvalues while the damping level is changing.
    Publicationtype: Journal Article
      183
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    Optimal Shape Design for the p-Laplacian Eigenvalue Problem
    (2019-02-15)
    Mohammadi, Seyyed Abbas
    ;
    Bozorgnia, Farid
    ;
    Voß, Heinrich 
    In this paper, a shape optimization problem corresponding to the p-Laplacian operator is studied. Given a density function in a rearrangement class generated by a step function, find the density such that the principal eigenvalue is as small as possible. Considering a membrane of known fixed mass and with fixed boundary of prescribed shape consisting of two different materials, our results determine the way to distribute these materials such that the basic frequency of the membrane is minimal. We obtain some qualitative aspects of the optimizer and then we determine nearly optimal sets which are approximations of the minimizer for specific ranges of parameters values. A numerical algorithm is proposed to derive the optimal shape and it is proved that the numerical procedure converges to a local minimizer. Numerical illustrations are provided for different domains to show the efficiency and practical suitability of our approach.
    Publicationtype: Journal Article
      62
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    Variational characterization of real eigenvalues in linear viscoelastic oscillators
    (2018-10-01)
    Mohammadi, Seyyed Abbas
    ;
    Voß, Heinrich 
    This paper proposes a new approach for computing the real eigenvalues of a multiple-degrees-of-freedom viscoelastic system in which we assume an exponentially decaying damping. The free-motion equations lead to a nonlinear eigenvalue problem. If the system matrices are symmetric, the eigenvalues allow for a variational characterization of maxmin type, and the eigenvalues and eigenvectors can be determined very efficiently by the safeguarded iteration, which converges quadratically and, for extreme eigenvalues, monotonically. Numerical methods demonstrate the performance and the reliability of the approach. The method succeeds where some current approaches, with restrictive physical assumptions, fail.
    Publicationtype: Journal Article
      160
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    A nonlinear eigenvalue optimization problem: Optimal potential functions
    (2018-04)
    Antunes, Pedro R. S.
    ;
    Mohammadi, Seyyed Abbas
    ;
    Voß, Heinrich 
    In this paper we study the following optimal shape design problem: Given an open connected set Ω⊂RN and a positive number A∈(0,|Ω|), find a measurable subset D⊂Ω with |D|=A such that the minimal eigenvalue of −div(ζ(λ,x)∇u)+αχDu=λu in Ω, u=0 on ∂Ω, is as small as possible. This sort of nonlinear eigenvalue problems arises in the study of some quantum dots taking into account an electron effective mass. We establish the existence of a solution and we determine some qualitative aspects of the optimal configurations. For instance, we can get a nearly optimal set which is an approximation of the minimizer in ultra-high contrast regime. A numerical algorithm is proposed to obtain an approximate description of the optimizer.
    Publicationtype: Journal Article
      171
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    A minimization problem for an elliptic eigenvalue problem with nonlinear dependence on the eigenparameter
    (2016-10-01)
    Mohammadi, Seyyed Abbas
    ;
    Voß, Heinrich 
    In this paper we examine an eigenvalue optimization problem. Given two nonlinear functions α(λ) and β(λ), find a subset D of the unit ball of measure A for which the first Dirichlet eigenvalue of the operator -div((α(λ)χD+β(λ)χDc)∇ u)=λu is as small as possible. This sort of nonlinear eigenvalue problems arises in the study of some quantum dots taking into account an electron effective mass. We establish the existence of a solution, and we propose a numerical algorithm to obtain an approximate description of the optimizer.
    Publicationtype: Journal Article
      89