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Tower configuration: Computational domain; B indicates the location of the assumed manufacturing imperfections at x=0. (a) Shell-solid model; diameter of the base is random. (b) Computational domain. (c) Lowest mode; normalized transverse detection w; rainbow colour scale.

Tower configuration: Computational domain; B indicates the location of the assumed manufacturing imperfections at x=0. (a) Shell-solid model; diameter of the base is random. (b) Computational domain. (c) Lowest mode; normalized transverse detection w; rainbow colour scale.

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Numerical simulation of thin solids remains one of the challenges in computational mechanics. The 3D elasticity problems of shells of revolution are dimensionally reduced in different ways depending on the symmetries of the configurations resulting in corresponding 2D models. In this paper, we solve the multiparametric free vibration of complex she...

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Citations

... Recent literature has considered examples of mechanical vibration problems, where a parametrization of the uncertainties in either the physical coefficients or the geometry of the system results in a multiparametric eigenvalue problem, see e.g. [12,13,15,19,25,26]. ...
... is of the form (10) and satisifies (11) and (12). ...
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We consider computing eigenspaces of an elliptic self-adjoint operator depending on a countable number of parameters in an affine fashion. The eigenspaces of interest are assumed to be isolated in the sense that the corresponding eigenvalues are separated from the rest of the spectrum for all values of the parameters. We show that such eigenspaces can in fact be extended to complex-analytic functions of the parameters and quantify this analytic dependence in a way that leads to convergence of sparse polynomial approximations. A stochastic collocation method on an anisoptropic sparse grid in the parameter domain is proposed for computing a basis for the eigenspace of interest. The convergence of this method is verified in a series of numerical examples based on the eigenvalue problem of a stochastic diffusion operator.
... We study two structures, a cylindrical shell which is of practical importance, and an example of different shell geometry, a perforated hyperbolic shell (cooling tower), which is more of an academic interest but serves to underline the central issues. It follows naturally that inclusion of uncertainty in the simulations leads to distribution of frequency responses that require statistical analysis [9]. ...
... When the problem is cast into the stochastic setting, the eigenvalues are bound to cross within the parameter space, and therefore it makes sense to match eigenmodes according to their axial and angular waveunumbers K x and K y rather than by simple increasing enumeration. This has previously been illustrated in [8,9]. ...
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... Recent literature has considered examples of mechanical vibration problems, where a parametrization of the uncertainties in either the physical coefficients or the geometry of the system results in a multiparametric eigenvalue problem, see e.g. [23], [16], [10], [22], [9], [12]. ...
... An example of the associated eigenfunctions at y = 0 has been shown in Figure 7. We may again plug the calculated eigenvalues and an estimate of κ 1 (N) into (9), and see that Corollary 1 holds for both S = 2 and for S = 6. With this we conclude that the subspace U {3,4,5,6} is isolated. ...
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We consider computing eigenspaces of an elliptic self-adjoint operator depending on a countable number of parameters in an affine fashion. The eigenspaces of interest are assumed to be isolated in the sense that the corresponding eigenvalues are separated from the rest of the spectrum for all values of the parameters. We show that such eigenspaces can in fact be extended to complex-analytic functions of the parameters and quantify this analytic dependence in way that leads to convergence of sparse polynomial approximations. A stochastic collocation method on an anisoptropic sparse grid in the parameter domain is proposed for computing a basis for the eigenspace of interest.