Direct Variational Technique Research Articles

A Unified Variational Approach to Discontinuous Differential Equations

A direct variational technique involving Clarke generalized gradient is used to treat general boundary value problems with discontinuous nonlinearities. Based on the theory of positive definite symmetric operators it is established the nonsmooth variational form of the regularized inclusions which give the Filippov solutions of the discontinuous problems. These solutions reduce to classical solutions in case that a transversality condition on the set of discontinuities is satisfied. The results apply to a wide class of concrete boundary value problems of different orders. Two illustrative examples are given.

Size effects in ferroelectric nanocones

Ferroelectric size effects in cone-shaped nanoparticles (nanocones) are investigated for the first time. The Euler-Lagrange equations are solved using the direct variational technique. An approximate analytical expression is derived for the dependence of the ferroelectric phase transition temperature on the nanocone size. It is shown that the transition temperature for nanocones can be 2.5 times higher than the transition temperature for bulk materials.

Optical guiding of elliptical laser beam in nonuniform plasma

Guiding of laser beam in plasma channel formed by short ionizing laser pulse is studied in this paper. When a delayed pulse propagates through this channel diffraction, refraction and self-phase modulation phenomena results which are not balanced resulting in increase/decrease in beam width. These are studied using direct Variational Technique. In this paper cross-focussing phenomena is not observed. Parameters like beam width and longitudinal-phase delay of elliptical laser beam are also studied. It has been observed that the propagation of semi-major axis and semi-minor axis of elliptical beam are independent leading to oscillating guided beam.

Comparison of bootstrap current and plasma conductivity models applied in a self-consistent equilibrium calculation for tokamak plasmas

Different bootstrap current formulations are implemented in a self-consistent equilibrium calculation obtained from a direct variational technique in fixed boundary tokamak plasmas. The total plasma current profile is supposed to have contributions of the diamagnetic, Pfirsch–Schlüter, and the neoclassical Ohmic and bootstrap currents. The Ohmic component is calculated in terms of the neoclassical conductivity, compared here among different expressions, and the loop voltage determined consistently in order to give the prescribed value of the total plasma current. A comparison among several bootstrap current models for different viscosity coefficient calculations and distinct forms for the Coulomb collision operator is performed for a variety of plasma parameters of the small aspect ratio tokamak ETE (Experimento Tokamak Esférico) at the Associated Plasma Laboratory of INPE, in Brazil. We have performed this comparison for the ETE tokamak so that the differences among all the models reported here, mainly regarding plasma collisionality, can be better illustrated. The dependence of the bootstrap current ratio upon some plasma parameters in the frame of the self-consistent calculation is also analysed. We emphasize in this paper what we call the Hirshman–Sigmar/Shaing model, valid for all collisionality regimes and aspect ratios, and a fitted formulation proposed by Sauter, which has the same range of validity but is faster to compute than the previous one. The advantages or possible limitations of all these different formulations for the bootstrap current estimate are analysed throughout this work.

Propagation of ellipitic Gaussian laser beam in a higher order non-linear medium

Using a direct variational technique involving elliptic Gaussian laser beam trial function, the combined effect of nonlinearity and diffraction on wave propagation of optical beam in a homogeneous higher order nonlinear medium is presented. Particular emphasis is put to understand the variation of beam width and longitudinal phase delay with the distance of propagation in case of lossless and lossy medium. It is also observed that stationary self-trapping is possible in lossless medium at higher laser intensity where fifth order nonlinearity becomes comparable to third order nonlinearity. The phase is also seen to be always negative.

Self-consistent equilibrium calculation through a direct variationaltechnique in tokamak plasmas

A self-consistent equilibrium calculation, valid for arbitrary aspect ratiotokamaks, is obtained through a direct variational technique that reducesthe equilibrium solution, in general obtained from the 2D Grad-Shafranovequation, to a 1D problem in the radial flux coordinate ρ. The plasmacurrent profile is supposed to have contributions of the diamagnetic,Pfirsch-Schlüter and the neoclassical ohmic and bootstrap currents. Aniterative procedure is introduced into our code until the flux surfaceaveraged toroidal current density ⟨JT⟩, converges towithin a specified tolerance for a given pressure profile and prescribedboundary conditions. The convergence criterion is applied between the⟨JT⟩ profile used to calculate the equilibriumthrough the variational procedure and the one that results from theequilibrium and given by the sum of all current components. The ohmiccontribution is calculated from the neoclassical conductivity and from theself-consistently determined loop voltage in order to give the prescribedvalue of the total plasma current. The bootstrap current is estimatedthrough the full matrix Hirshman-Sigmar model with the viscositycoefficients as proposed by Shaing, which are valid in all plasmacollisionality regimes and arbitrary aspect ratios. The results of theself-consistent calculation are presented for the low aspect ratio tokamakExperimento Tokamak Esférico. A comparison among different modelsfor the bootstrap current estimate is also performed and their possiblelimitations to the self-consistent calculation is analysed.

Dynamics of self-focusing and self-phase modulation of elliptic Gaussian laser beam in a Kerr-medium

Using a direct variational technique involving elliptic Gaussian laser beam trial func- tion, the combined effect of non-linearity and diffraction on wave propagation of optical beam in a homogeneous bulk Kerr-medium is presented. Particular emphasis is put on the variation of beam width and longitudinal phase delay with the distance of propagation. It is observed that no stationary self-trapping is possible. The regularized phase is also seen to be always negative.

Super-Gaussian approximation of the fundamental radial mode in nonlinear parabolic-index optical fibers

Using a direct variational technique involving super-Gaussian trial functions, we find approximate solutions for the fundamental radial mode of an optical wave propagating in a nonlinear parabolic-index fiber. A detailed examination is made of the amplitude, width, radial profile, and longitudinal phase shift of the stationary solutions that represent a two-dimensional balance among diffraction, nonlinear self-focusing, and linear wave guiding. For pulse powers below the self-focusing power, the stationary solutions are stabilized by the parabolic index variation.

Hybrid analytical technique for the nonlinear analysis of curved beams

The application of a two-step hybrid technique to the geometrically nonlinear analysis of curved beams is used to demonstrate the potential of hybrid analytical techniques in nonlinear structural mechanics. The hybrid technique is based on successive use of the perturbation method and a classical direct variational procedure. The functions associated with the various-order terms in the perturbation expansion of the fundamental unknowns, and their sensitivity derivatives with respect to material and geometric parameters of the beam, are first obtained by using the perturbation method. These functions are selected as coordinate functions (or modes) and the classical direct variational technique is then used to compute their amplitudes. The potential of the proposed hybrid technique for nonlinear analysis of structures is discussed. The effectiveness of the hybrid technique is demonstrated by means of numerical examples. The symbolic computation system Mathematica is used in the present study. The tasks performed on Mathematica include: (1) generation of algebraic expressions for the perturbation functions of the different response quantities and their sensitivity derivatives; and (2) determination of the radius of convergence of the perturbation series.

Hybrid analytical technique for nonlinear analysis of structures

A two-step hybrid analytical technique is presented for predicting the nonlinear response of structures. The technique is based on the successive application of the regular perturbation expansion and either the method of weighted residuals or the classical direct variational technique. The functions associated with the various-order terms in the perturbation expansion of the fundamental unknowns are first obtained by using the regular perturbation method. These functions are selected as coordinate functions (or modes) and either the weighted residuals or the classical direct variational method of technique is then used to compute their amplitudes. The potential of the proposed hybrid technique for nonlinear analysis of structures is discussed. The effectiveness of this technique is demonstrated by means of three numerical examples. Results of the study indicate that the hybrid technique overcomes two major drawbacks of the parent (classical) techniques: 1) the requirement of using a small parameter in the regular perturbation method, and 2) the arbitrariness in the choice of the coordinate functions in both the weighted residual and the direct variational techniques. Therefore, the proposed technique extends the range of applicability of the regular perturbation method and enhances the effectiveness of both the method of weighted residuals and the direct variational technique.

Bending of glass fibre reinforced plastic (GFRP) plates on elastic supports

The behaviour of a fibreglass plate, subjected to a simulated hydraulic pressure and supported by a steel frame along the edges through a neoprene gasket, is investigated. The plate is analyzed theoretically as an orthotropic continuum using a direct variational technique in accordance with the Rayleigh-Ritz method. The load-deformation behaviour and the mode of failure is examined experimentally. Theoretical and experimental results are correlated empirically through Poisson’s ratio.