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The paper is devoted to a review of recent achievements in the field of dynamic analysis of structures and structural elements, such as beams and plates, with embedded viscoelastic (VE) dampers and/or layers. The general characteristics of VE materials, their rheological models, and methods of parameters identification are discussed. New formulations of dynamic problems for systems with VE elements are also reviewed. The methods of determination of dynamic characteristics, together with the methods of analysis of steady-state and transient vibrations of such systems, are also discussed. Both linear and geometrically non-linear vibrations are considered. The paper ends with a review of the methods of sensitivity and uncertainty analysis, and the methods of optimization, for structures with VE elements.

We provide a priori error estimates for variational approximations of the ground state energy, eigenvalue and eigenvector of nonlinear elliptic eigenvalue problems of the form −div( A ∇ u )+ Vu + f ( u 2 ) u = λ u , . We focus in particular on the Fourier spectral approximation (for periodic problems) and on the ℙ 1 and ℙ 2 finite-element discretizations. Denoting by ( u δ , λ δ ) a variational approximation of the ground state eigenpair ( u , λ ), we are interested in the convergence rates of , , | λ δ − λ |, and the ground state energy, when the discretization parameter δ goes to zero. We prove in particular that if A , V and f satisfy certain conditions, | λ δ − λ | goes to zero as . We also show that under more restrictive assumptions on A , V and f , | λ δ − λ | converges to zero as , thus recovering a standard result for linear elliptic eigenvalue problems. For the latter analysis, we make use of estimates of the error u δ − u in negative Sobolev norms.

The problem of normal waves in an open metal-dielectric regular waveguide of arbitrary cross-section is considered. This problem is reduced to the boundary eigenvalue problem for longitudinal components of electromagnetic field in Sobolev spaces. To find the solution, we use the variational formulation of the problem. The variational problem is reduced to study of an operator-function. Discreteness of the spectrum is proved and distribution of the characteristic numbers of the operatorfunction on the complex plane is found.

We analyse the existence of multiple critical points for an even functional J:H→R in the following context: the Hilbert space H can be split into an orthogonal sum H=Y⊕Z in such a way that infJ(u):u∈Zand‖u‖=ρ≥α>J(0) and that here exits a point b∈H with ‖b‖>ρ and with J(b)≤J(0) . We develop a new variational characterization of multiple critical levels without an assumption on the dimension of Y. Our characterization is simple and natural: we can for example avoid the notion of pseudo-index and the definition of the activated levels does not substantially differs from the one used for the lowest critical level, giving us in this way a unified view of critical levels. We apply our results to a semi-linear Schrödinger equation of the form -Δu+V(x)u-q(x)|u|σu=λu,x∈RNu∈H1(RN)\\0 where λ is inside a spectral gap bounded on both sides by parts of the essential spectrum.

In modelling electrostatically actuated micro- and nano-electromechanical systems, researchers have typically relied on a small-aspect ratio to form a leading-order theory. In doing so, small gradient terms are dropped. Although this approximation has been fruitful, its consequences have not been investigated. Here, this approximation is re-examined, and a new theory which includes often neglected small curvature terms is presented. Furthermore, the solution set of the new theory is explored for the unit disk domain and compared to the standard theory. Also, the analytical results are compared to experimental data.

A new numerical technique for solving generalized Sturm--Liouville problem d2w/dx2 + q(x, λ )w = 0, bl[ λ ,w(a)] = br[ λ ,w(b)] = 0 is presented. In is presented. In particular, we consider the problems when the coefficient q(x, λ) or the boundary conditions depend on the spectral parameter λ in an arbitrary nonlinear manner. The method presented is based on mathematically modelling of physical response of a system to excitation over a range of frequencies. The response amplitudes are then used to determine the eigenvalues. The same technique can be applied to a very wide class of the eigenproblems: the Sturm--Liouville problems, the Schrodinger equation, the non-classical non-linear Sturm--Liouville problems, periodic problems. The results of the numerical experiments justifying the method are presented.

When a chemical sample composed of N elements is analyzed using sequential selective excitation by a tunable polyenergetic X-ray beam and selective measurement of the characteristic X-rays, the production of secondary fluorescence does not interfere with the measurements. This experimental situation leads to a particular case of the Sherman equations which can be written as a set of non-linear equations. The same kind of equations are also obtained when we excite a chemical sample with a polyenergetic X-ray beam and neglect the production of secondary fluorescence. This set of equations can be regarded as a non-linear eigenvalue problem. A non-linear extension of the Perron Frobenious theorem ensures that there is one and only one physically acceptable solution, and also leads to a method to obtaining it. The propagation off measurements errors of sample fluorescence to errors in the calculated sample concentrations, has been simulated, and the results show that the solution is well conditioned. The case of production of secondary fluorescence can not be treated, in general, as a nonlinear Perron eigenvalue problem, but it has been shown that it is rather plausible that Sherman equations corresponding to the actual chemical elements and that include the production of secondary fluorescence have one and only one physically acceptable solution. An exhaustive search could elucitate the existence and unicity of solutions for the equations corresponding to the actual chemical elements.

This work focuses on the computationally efficient detection of the buckling phenomena and bifurcation analysis of the parametric Von Kármán plate equations based on reduced order methods and spectral analysis. The computational complexity—due to the fourth order derivative terms, the non-linearity and the parameter dependence—provides an interesting benchmark to test the importance of the reduction strategies, during the construction of the bifurcation diagram by varying the parameter(s). To this end, together the state equations, we carry out also an analysis of the linearized eigenvalue problem, that allows us to better understand the physical behaviour near the bifurcation points, where we lose the uniqueness of solution. We test this automatic methodology also in the two parameter case, understanding the evolution of the first buckling mode.

This study examines the magnetic effect on Darcy Brinkman convection in a Bidispersive horizontal porous layer, considering the importance of convective motions of electrically conducting porous media accompanying a magnetic field in real-life applications such as geophysics, metallurgical field and solidification structures. In order to conduct a thorough study, the boundaries are classified as free-free, rigid-free, and rigid-rigid. The fluid motion is described using the Brinkman-Darcy equation with a single temperature in the macropores and micropores. The eigenvalue problem is solved analytically for the free-free case by employing linear stability theory. A non-linear analysis using the energy method is undertaken to prove that linear instability and global non-linear stability thresholds are the same. The eigenvalue problem for rigid-free and rigid-rigid boundaries is numerically solved with the bvp4c routine in MATLAB R2020 with the Rayleigh number as the eigenvalue. It is found that the Hartmann number M2, Darcy number Da, permeability ratio κr, and momentum transfer coefficient γ stabilize the system. Rigid-rigid boundaries are found to be the most stable ones, followed by rigid-free and free-free, which are the least stable boundaries.

We investigate the minimization problem where q = 2 N N - 2 , N ⩾ 4 , a and b are two continuous positive weight functions. We show the existence of solutions of the previous minimizing problem under some conditions on a ,  b , the dimension of the space and the parameter λ .