A STUDY OF PENALTY FORMULATIONS USED IN THE NUMERICAL APPROXIMATION OF A RADIALLY SYMMETRIC ELASTICITY PROBLEM

We consider a class of two-dimensional problems in classical linear elasticity for which material overlapping occurs in the absence of singularities. Of course, material overlapping is not physically realistic, and one possible way to prevent it uses a constrained minimization theory. In this theory, a minimization problem consists of minimizing the total potential energy of a linear elastic body subject to the constraint that the deformation field must be locally invertible. Here, we use an interior and an exterior penalty formulation of the minimization problem together with both a standard finite element method and classical nonlinear programming techniques to compute the minimizers. We compare both formulations by solving a plane problem numerically in the context of the constrained minimization theory. The problem has a closed-form solution, which is used to validate the numerical results. This solution is regular everywhere, including the boundary. In particular, we show numerical results which indicate that, for a fixed finite element mesh, the sequences of numerical solutions obtained with both the interior and the exterior penalty formulations converge to the same limit function as the penalization is enforced. This limit function yields an approximate deformation field to the plane problem that is locally invertible at all points in the domain. As the mesh is refined, this field converges to the exact solution of the plane problem.

Enhanced aeroelastic energy harvesting by exploiting combined nonlinearities: theory and experiment

Converting aeroelastic vibrations into electricity for low power generation has received growing attention over the past few years. In addition to potential applications for aerospace structures, the goal is to develop alternative and scalable configurations for wind energy harvesting to use in wireless electronic systems. This paper presents modeling and experiments of aeroelastic energy harvesting using piezoelectric transduction with a focus on exploiting combined nonlinearities. An airfoil with plunge and pitch degrees of freedom (DOF) is investigated. Piezoelectric coupling is introduced to the plunge DOF while nonlinearities are introduced through the pitch DOF. A state-space model is presented and employed for the simulations of the piezoaeroelastic generator. A two-state approximation to Theodorsen aerodynamics is used in order to determine the unsteady aerodynamic loads. Three case studies are presented. First the interaction between piezoelectric power generation and linear aeroelastic behavior of a typical section is investigated for a set of resistive loads. Model predictions are compared to experimental data obtained from the wind tunnel tests at the flutter boundary. In the second case study, free play nonlinearity is added to the pitch DOF and it is shown that nonlinear limit-cycle oscillations can be obtained not only above but also below the linear flutter speed. The experimental results are successfully predicted by the model simulations. Finally, the combination of cubic hardening stiffness and free play nonlinearities is considered in the pitch DOF. The nonlinear piezoaeroelastic response is investigated for different values of the nonlinear-to-linear stiffness ratio. The free play nonlinearity reduces the cut-in speed while the hardening stiffness helps in obtaining persistent oscillations of acceptable amplitude over a wider range of airflow speeds. Such nonlinearities can be introduced to aeroelastic energy harvesters (exploiting piezoelectric or other transduction mechanisms) for performance enhancement.

On the convergence of successive Galerkin approximation for nonlinear output feedback H(infinity) control

Although the formulation of the nonlinear theory of H(infinity) control has been well developed, solving the Hamilton-Jacobi-Isaacs equation remains a challenge and is the major bottleneck for practical application of the theory. Several numerical methods have been proposed for its solution. In this paper, results on convergence and stability for a successive Galerkin approximation approach for nonlinear H(infinity) control via output feedback are presented. An example is presented illustrating the application of the algorithm.

Quantum mechanics as an approximation to classical mechanics in Hilbert space

Classical mechanics is formulated in complex Hilbert space with the introduction of a commutative product of operators, an antisymmetric bracket and a quasidensity operator that is not positive definite. These are analogues of the star product, the Moyal bracket, and the Wigner function in the phase space formulation of quantum mechanics. Quantum mechanics is then viewed as a limiting form of classical mechanics, as Planck's constant approaches zero, rather than the other way around. The forms of semiquantum approximations to classical mechanics, analogous to semiclassical approximations to quantum mechanics, are indicated.

Theory of multidimensional parametric band-gap simultons

Multidimensional spatiotemporal parametric simultons (simultaneous solitary waves) are possible in a nonlinear chi((2)) medium with a Bragg grating structure, where large effective dispersion occurs near two resonant band gaps for the carrier and second-harmonic field, respectively. The enhanced dispersion allows much reduced interaction lengths, as compared to bulk medium parametric simultons. The nonlinear parametric band-gap medium permits higher-dimensional stationary waves to form. In addition, solitons can occur with lower input powers than conventional nonlinear Schrodinger equation gap solitons. In this paper, the equations for electromagnetic propagation in a grating structure with a parametric nonlinearity are derived from Maxwell's equation using a coupled mode Hamiltonian analysis in one, two, and three spatial dimensions. Simultaneous solitary wave solutions are proved to exist by reducing the equations to the coupled equations describing a nonlinear parametric waveguide, using the effective-mass approximation (EMA). Exact one-dimensional numerical solutions in agreement with the EMA solutions are also given. Direct numerical simulations show that the solutions have similar types of stability properties to the bulk case, providing the carrier waves are tuned to the two Bragg resonances, and the pulses have a width in frequency space less than the band gap. In summary, these equations describe a physically accessible localized nonlinear wave that is stable in up to 3 + 1 dimensions. Possible applications include photonic logic and switching devices. [S1063-651X(98)06109-1].

A new diffusion and flow theory for activated carbon from low pressure to capillary condensation range

In this paper, we develop a theory for diffusion and flow of pure sub-critical adsorbates in microporous activated carbon over a wide range of pressure, ranging from very low to high pressure, where capillary condensation is occurring. This theory does not require any fitting parameter. The only information needed for the prediction is the complete pore size distribution of activated carbon. The various interesting behaviors of permeability versus loading are observed such as the maximum permeability at high loading (occurred at about 0.8-0.9 relative pressure). The theory is tested with diffusion and flow of benzene through a commercial activated carbon, and the agreement is found to be very good in the light that there is no fitting parameter in the model. (C) 2001 Elsevier Science B.V. All rights reserved.

Exact theory of sedimentation equilibrium made useful

The exact description of the thermodynamics of solutions has been used to describe, without approximation, the distribution of all the components of an incompressible solution in a centrifuge cell at sedimentation equilibrium. Thermodynamic parameters describing the interactions between solute components of known molar mass can be obtained by direct analysis of the experimental data. Interpretation of the measured thermodynamic parameters in terms of molecular interactions requires that an arbitrary distinction be made between nonassociative forces, like hard-sphere volume-exclusion and mean-field electrostatic repulsion or attraction, and specific short-range forces of association that give rise to the formation of molecular aggregates. Provided the former can be accounted for adequately, the effects of the latter can be elucidated in the form of good estimates of the equilibrium constants for the reactions of aggregation.

Simulation and theory of hybrid aligned liquid crystal films

We present a study of the effects of nanoconfinement on a system of hard Gaussian overlap particles interacting with planar substrates through the hard-needle-wall potential, extending earlier work by two of us [D. J. Cleaver and P. I. C. Teixeira, Chem. Phys. Lett. 338, 1 (2001)]. Here, we consider the case of hybrid films, where one of the substrates induces strongly homeotropic anchoring, while the other favors either weakly homeotropic or planar anchoring. These systems are investigated using both Monte Carlo simulation and density-functional theory, the latter implemented at the level of Onsager's second-virial approximation with Parsons-Lee rescaling. The orientational structure is found to change either continuously or discontinuously depending on substrate separation, in agreement with earlier predictions by others. The theory is seen to perform well in spite of its simplicity, predicting the positional and orientational structure seen in simulations even for small particle elongations.

Generic Nekhoroshev theory without small divisors

In this article, we present a new approach of Nekhoroshev theory for a generic unperturbed Hamiltonian which completely avoids small divisors problems. The proof is an extension of a method introduced by P. Lochak which combines averaging along periodic orbits with simultaneous Diophantine approximation and uses geometric arguments designed by the second author to handle generic integrable Hamiltonians. This method allows to deal with generic non-analytic Hamiltonians and to obtain new results of generic stability around linearly stable tori.

On the efficiency of recursive evaluations with applications to risk theory

On the efficiency of recursive evaluations with applications to risk theoryCette thèse est composée de trois essais qui portent sur l'efficacité des évaluations récursives de la distribution du montant total des sinistres d'un portefeuille de polices d'assurance au cours d'un période donnée. Le calcul de sa fonction de probabilité ou de quantités liées à cette distribution apparaît fréquemment dans la plupart des domaines de la pratique actuarielle.C'est le cas notamment pour le calcul du capital de solvabilité en Suisse ou pour modéliser la perte d'une assurance vie au cours d'une année. Le principal problème des évaluations récursives est que la propagation des erreurs provenant de la représentation des nombres réels par l'ordinateur peut être désastreuse. Mais, le gain de temps qu'elles procurent en réduisant le nombre d'opérations arithmétiques est substantiel par rapport à d'autres méthodes.Dans le premier essai, nous utilisons certaines propriétés d'un outil informatique performant afin d'optimiser le temps de calcul tout en garantissant une certaine qualité dans les résultats par rapport à la propagation de ces erreurs au cours de l'évaluation.Dans le second essai, nous dérivons des expressions exactes et des bornes pour les erreurs qui se produisent dans les fonctions de distribution cumulatives d'un ordre donné lorsque celles-ci sont évaluées récursivement à partir d'une approximation de la transformée de De Pril associée. Ces fonctions cumulatives permettent de calculer directement certaines quantités essentielles comme les primes stop-loss.Finalement, dans le troisième essai, nous étudions la stabilité des évaluations récursives de ces fonctions cumulatives par rapport à la propagation des erreurs citées ci-dessus et déterminons la précision nécessaire dans la représentation des nombres réels afin de garantir des résultats satisfaisants. Cette précision dépend en grande partie de la transformée de De Pril associée.

Ruin Theory with Excess of Loss Reinsurance and Reinstatements

The present paper studies the probability of ruin of an insurer, if excess of loss reinsurance with reinstatements is applied. In the setting of the classical Cramer-Lundberg risk model, piecewise deterministic Markov processes are used to describe the free surplus process in this more general situation. It is shown that the finite-time ruin probability is both the solution of a partial integro-differential equation and the fixed point of a contractive integral operator. We exploit the latter representation to develop and implement a recursive algorithm for numerical approximation of the ruin probability that involves high-dimensional integration. Furthermore we study the behavior of the finite-time ruin probability under various levels of initial surplus and security loadings and compare the efficiency of the numerical algorithm with the computational alternative of stochastic simulation of the risk process. (C) 2011 Elsevier Inc. All rights reserved.

Spin and density longitudinal response of quantum dots in the time-dependent local-spin-density approximation

The longitudinal dipole response of a quantum dot has been calculated in the far-infrared regime using local-spin-density-functional theory. We have studied the coupling between the collective spin and density modes as a function of the magnetic field. We have found that the spin dipole mode and single-particle excitations have a sizable overlap, and that the magnetoplasmon modes can be excited by the dipole spin operator if the dot is spin polarized. The frequency of the dipole spin edge mode presents an oscillation which is clearly filling factor (v) related. We have found that the spin dipole mode is especially soft for even-n values. Results for selected numbers of electrons and confining potentials are discussed.

Double folding with a density-dependent effective interaction and it s analytical approximation

The real part of the optical potential for heavy ion elastic scattering is obtained by double folding of the nuclear densities with a density-dependent nucleon-nucleon effective interaction which was successful in describing the binding, size, and nucleon separation energies in spherical nuclei. A simple analytical form is found to differ from the resulting potential considerably less than 1% all through the important region. This analytical potential is used so that only few points of the folding need to be computed. With an imaginary part of the Woods-Saxon type, this potential predicts the elastic scattering angular distribution in very good agreement with experimental data, and little renormalization (unity in most cases) is needed.

Classical solutions of the leading-logarithm approximation with non-trivial topology

Exact solutions of the classical equations corresponding to the leading-logarithm approximation are obtained. They are classified by an (integer) topological number.