Shape sensitivity analysis of the eigenvalues of the Reissner-Mindlin system

We consider the eigenvalue problem for the Reissner-Mindlin system arising in the study of the free vibration modes of an elastic clamped plate. We provide quantitative estimates for the variation of the eigenvalues upon variation of the shape of the plate. We also prove analyticity results and establish Hadamard-type formulas. Finally, we address the problem of minimization of the eigenvalues in the case of isovolumetric domain perturbations. In the spirit of the Rayleigh conjecture for the biharmonic operator, we prove that balls are critical points with volume constraint for all simple eigenvalues and the elementary symmetric functions of multiple eigenvalues.Comment: Preprint version of a paper accepted for publication in SIAM Journal on Mathematical Analysi

On a classical spectral optimization problem in linear elasticity

We consider a classical shape optimization problem for the eigenvalues of elliptic operators with homogeneous boundary conditions on domains in the $N$-dimensional Euclidean space. We survey recent results concerning the analytic dependence of the elementary symmetric functions of the eigenvalues upon domain perturbation and the role of balls as critical points of such functions subject to volume constraint. Our discussion concerns Dirichlet and buckling-type problems for polyharmonic operators, the Neumann and the intermediate problems for the biharmonic operator, the Lam\'{e} and the Reissner-Mindlin systems.Comment: To appear in the proceedings of the workshop `New Trends in Shape Optimization', Friedrich-Alexander Universit\"{a}t Erlangen-Nuremberg, 23-27 September 201

Shape differentiability of the eigenvalues of elliptic systems

We consider second order elliptic systems of partial differential equations subject to Dirichlet and Neumann boundary conditions. We prove analyticity of the elementary symmetric functions of the eigenvalues, and compute Hadamard-type formulas for such functions. Then we provide a characterization of criticality of the domain under volume constraint, and prove that if the system is rotation invariant, then balls are critical domains for all those functions.Comment: Submitted for the IMSE 2014 Conference Proceeding

Discrete and Continuous Models for Static and Modal Analysis of Out of Plane Loaded Masonry

A critical review of analytical and numerical models for studying masonry out of plane behaviour is presented. One leaf historical masonry, composed by rigid blocks arranged regularly with dry or mortar joints, is considered. Discrete model with rigid blocks, Love-Kirchhoff and Reissner-Mindlin plate models and 3D heterogeneous FEM are adopted. An existing simple and effective discrete model is adopted and improved by applying matrix structural analysis techniques for static and modal analysis of masonry walls in the elastic field, but the formulation allows to account also for material nonlinearity. Elastic parameters of both plate models are based on an existing compatible identification between 3D discrete model and 2D plate models. Static and modal analysis of masonry walls with several boundary conditions are carried on, numerical tests account for in plane size of heterogeneity and structure thickness by means of in and out of plane scale factors. Results show that discrete model is simple and effective for representing masonry behaviour, especially when size of heterogeneity is smaller than overall panel size. Decreasing in plane scale factor, plate models converge to the discrete one, but the Reissner-Mindlin one shows a better convergence and also allows adopting a simple FE for performing numerical analysis

NOISE CONTROL OF AN ACOUSTIC CAVITY COUPLED WITH A VIBRATING PLATE TREATED WITH A SPATIALLY VAYING CONSTRAINED VISCOEALSTIC LAYER

Viscoelastic layers have long been recognized as an effective means of reducing the structural vibrations that can generate undesirably high pressure levels in a coupled acoustic cavity. Constraining the viscoelastic layer increases the effectiveness of the viscoelastic layer by adding transverse shear as a dissipation mechanism in the system. It is proposed in this dissertation to replace the traditionally homogeneous core of a constrained damping layer treatment by a non-homogeneous viscoelastic material in order to further improve the effectiveness of the treatment. A finite element model is developed to simulate the vibrations of plates treated with a non-homogeneous constrained layer treatment using Reissner-Mindlin plate theory. The predictions of the model are validated against the predictions of a commercially available finite element package (NASTRAN). The model of the plate/constraining layer treatment is then coupled with a finite element model of a coupled acoustic cavity. The integrated model is exercised to consider different material combinations and geometric layouts of the non-homogeneous damping treatment in order to determine general guidelines for producing the largest reduction in sound pressure levels inside an acoustic cavity that is being driven by a flexible boundary. The predictions of the integrated finite element model are validated through experimental and numerical work. Close agreements are found between theoretical predictions and experimental results. Generally, it is found that damping treatments with stiffer outer perimeters and softer cores are more effective in attenuating the sound pressure levels in the acoustic cavity than other configurations of the non-homogeneous treatment

A hierarchic optimisation approach towards locking-free shell finite elements

A hierarchic optimisation approach is presented for relieving inaccuracies in conforming shell elements arising from locking phenomena. This approach introduces two sets of strain modes: (i) objective strain modes, defined in the physical coordinate system, and (ii) corrective strain modes, representing conforming strains enhanced with hierarchic strain modes. This leads to two alternative families of element, objective and corrective, both arising from minimising the difference between objective and corrective strains. Importantly, the proposed approach not only alleviates shear and membrane locking, but it also addresses locking arising from element distortion. The application of the proposed optimisation approach is demonstrated for a 9-noded quadrilateral Lagrangian shell element, where the membrane, bending and transverse shear strains are separately optimised, all within a local co-rotational framework that extends the element application to geometric nonlinear analysis. Several numerical examples, including cases with geometric and material nonlinearity, are finally presented to illustrate the effectiveness of the optimised 9-noded shell element in relieving the various sources of locking

A hierarchic optimisation approach towards locking-free shell finite elements

A hierarchic optimisation approach is presented for relieving inaccuracies in conforming shell elements arising from locking phenomena. This approach introduces two sets of strain modes: (i) objective strain modes, defined in the physical coordinate system, and (ii) corrective strain modes, representing conforming strains enhanced with hierarchic strain modes. This leads to two alternative families of element, objective and corrective, both arising from minimising the difference between objective and corrective strains. Importantly, the proposed approach not only alleviates shear and membrane locking, but it also addresses locking arising from element distortion. The application of the proposed optimisation approach is demonstrated for a 9-noded quadrilateral Lagrangian shell element, where the membrane, bending and transverse shear strains are separately optimised, all within a local co-rotational framework that extends the element application to geometric nonlinear analysis. Several numerical examples, including cases with geometric and material nonlinearity, are finally presented to illustrate the effectiveness of the optimised 9-noded shell element in relieving the various sources of locking

Analytical investigation of squeeze film dampers

Squeeze film damping effects naturally occur if structures are subjected to loading situations such that a very thin film of fluid is trapped within structural joints, interfaces, etc. An accurate estimate of squeeze film effects is important to predict the performance of dynamic structures. Starting from linear Reynolds equation which governs the fluid behavior coupled with structure domain which is modeled by Kirchhoff plate equation, the effects of nondimensional parameters on the damped natural frequencies are presented using boundary characteristic orthogonal functions. For this purpose, the nondimensional coupled partial differential equations are obtained using Rayleigh-Ritz method and the weak formulation, are solved using polynomial and sinusoidal boundary characteristic orthogonal functions for structure and fluid domain respectively. In order to implement present approach to the complex geometries, a two dimensional isoparametric coupled finite element is developed based on Reissner-Mindlin plate theory and linearized Reynolds equation. The coupling between fluid and structure is handled by considering the pressure forces and structural surface velocities on the boundaries. The effects of the driving parameters on the frequency response functions are investigated. As the next logical step, an analytical method for solution of squeeze film damping based upon Green’s function to the nonlinear Reynolds equation considering elastic plate is studied. This allows calculating modal damping and stiffness force rapidly for various boundary conditions. The nonlinear Reynolds equation is divided into multiple linear non-homogeneous Helmholtz equations, which then can be solvable using the presented approach. Approximate mode shapes of a rectangular elastic plate are used, enabling calculation of damping ratio and frequency shift as well as complex resistant pressure. Moreover, the theoretical results are correlated and compared with experimental results both in the literature and in-house experimental procedures including comparison against viscoelastic dampers