General Variational Principle Research Articles

Three-dimensional forward modeling for magnetotelluric sounding by finite element method

A finite element algorithm combined with divergence condition was presented for computing three-dimensional(3D) magnetotelluric forward modeling. The finite element equation of three-dimensional magnetotelluric forward modeling was derived from Maxwell’s equations using general variation principle. The divergence condition was added forcedly to the electric field boundary value problem, which made the solution correct. The system of equation of the finite element algorithm was a large sparse, banded, symmetric, ill-conditioned, non-Hermitian complex matrix equation, which can be solved using the Bi-CGSTAB method. In order to prove correctness of the three-dimensional magnetotelluric forward algorithm, the computed results and analytic results of one-dimensional geo-electrical model were compared. In addition, the three-dimensional magnetotelluric forward algorithm is given a further evaluation by computing COMMEMI model. The forward modeling results show that the algorithm is very efficient, and it has a lot of advantages, such as the high precision, the canonical process of solving problem, meeting the internal boundary condition automatically and adapting to all kinds of distribution of multi-substances.

Ambartsumian’s methods in the theory of radiative transfer

The purpose of this article is to provide some insight into Ambartsumian’s methods in the theory of radiative transfer, their applications, and further development. Two of these methods are emphasized--the invariance principle and the method of addition of layers, proposed by Ambartsumian in the 1940’s. The difference between these methods and the classical approach for solving radiative transfer problems is discussed. We discuss only a small portion of the subsequent work by others that we believe reveals, in a more intuitive way, the essence and significance of Ambartsumian’s methods and their efficiency for applications. Thus, for example, a separate section is devoted to applications of the Lagrangian formalism to radiative transfer and it is shown that the invariance principle is a special case of a more general variational principle that reflects an invariance with respect to translational transformation of the optical depth. Our discussion of the method of addition of layers points out its generality and the major role it has played in the later creation of such methods as Bellman’s invariant imbedding method and the method for solving radiative transfer problems in inhomogeneous media. The latter method has yielded a number of new analytic results. The concluding section is a brief summary of Ambartsumian’s results in the nonlinear theory of radiative transfer, where he was a pioneer in the study of the class of multilevel problems. This article also sets out to demonstrate the place and role of Ambartsumian’s methods in the theory of radiative transfer, which, to a great extent, set the path along which this theory developed for many years to come.

Existence of Solutions and Variational Principles for Generalized Vector Systems

By means of generalized KKM theory, we prove a result on the existence of solutions and we establish general variational principles, that is, vector optimization formulations of set-valued maps for vector generalized systems. A perturbation function is involved in general variational principles. We extend the theory of gap functions for vector variational inequalities to vector generalized systems and we prove that the solution sets of the related vector optimization problems of set-valued maps contain the solution sets of vector generalized systems. A further vector optimization problem is defined in such a way that its solution set coincides with the solution set of a weak vector generalized system.

COUPLING OF SURFACE AND VOLUME DIPOLE OSCILLATIONS INC60MOLECULES

We first give a short review of the "local-current approximation" (LCA), derived from a general variation principle, which serves as a semiclassical description of strongly collective excitations in finite fermion systems starting from their quantum-mechanical mean-field ground state. We illustrate it for the example of coupled translational and compressional dipole excitations in metal clusters. We then discuss collective electronic dipole excitations in C60molecules (Buckminster fullerenes). We show that the coupling of the pure translational mode ("surface plasmon") with compressional volume modes in the semiclasscial LCA yields semi-quantitative agreement with microscopic time-dependent density functional (TDLDA) calculations, while both theories yield qualitative agreement with the recent experimental observation of a "volume plasmon".

General variational principle for spherically symmetric perturbations in diffeomorphism covariant theories

We present a general method for the analysis of the stability of static, spherically symmetric solutions to spherically symmetric perturbations in an arbitrary diffeomorphism covariant Lagrangian field theory. Our method involves fixing the gauge and solving the linearized gravitational field equations to eliminate the metric perturbation variables in terms of the matter variables. In a wide class of cases---which include $f(R)$ gravity, the Einstein-\ae{}ther theory of Jacobson and Mattingly, and Bekenstein's TeVeS theory---the remaining perturbation equations for the matter fields are second order in time. We show how the symplectic current arising from the original Lagrangian gives rise to a symmetric bilinear form on the variables of the reduced theory. If this bilinear form is positive definite, it provides an inner product that puts the equations of motion of the reduced theory into a self-adjoint form. A variational principle can then be written down immediately, from which stability can be tested readily. We illustrate our method in the case of Einstein's equation with perfect fluid matter, thereby rederiving, in a systematic manner, Chandrasekhar's variational principle for radial oscillations of spherically symmetric stars. In a subsequent paper, we will apply our analysis to $f(R)$ gravity, the Einstein-\ae{}ther theory, and Bekenstein's TeVeS theory.

On Ekeland's Variational Principle for Set-Valued Mappings

In this paper, we derive a general vector Ekeland variational principle for set-valued mappings, which has a closed relation to ek 0-efficient points of set-valued optimization problems. The main result presented in this paper is a generalization of the corresponding result in [3].

Antisymmetric Hamiltonians: Variational resolutions for Navier‐Stokes and other nonlinear evolutions

AbstractThe theory of anti‐self‐dual (ASD) Lagrangians, introduced in [6], is developed further to allow for a variational resolution of nonlinear PDEs of the form Λu + Au + ∂φ(u) + f = 0 where φ is a convex lower‐semicontinuous function on a reflexive Banach space X, f ∈ X*, A : D(A) ⊂ X → X* is a positive linear operator, and where Λ : D(λ) ⊂ X → X* is a nonlinear operator that satisfies suitable continuity and antisymmetry properties. ASD Lagrangians on path spaces also yield variational resolutions for nonlinear evolution equations of the form u̇(t) + Λu(t) + Au(t) + ∂φ(u(t)) + f = 0 starting at u(0) = u0. In both stationary and dynamic cases, the equations associated to the proposed variational principles are not derived from the fact that they are critical points of the action functional, but because they are also zeroes of the Lagrangian itself. For that we establish a general nonlinear variational principle that has many applications, in particular to Navier‐Stokes‐type equations, to generalized Choquard‐Pekar Schrödinger equations with nonlocal terms, as well as to complex Ginsburg‐Landau‐type initial‐value problems. The case of Navier‐Stokes evolutions is more involved and will be dealt with in [9]. The general theory of antisymmetric Hamiltonians and its applications is developed in detail in an upcoming monograph (7). © 2007 Wiley Periodicals, Inc.

ON THE DIRICHLET PROBLEM FOR THE EQUATION $-\Delta u = g(x,u) + \lambda f(x,u)$ WITH NO GROWTH CONDITIONS ON $f$

In this paper we establish some results concerning the existence of nonzero nonnegative and nonzero nonpositive solutions for a Dirichlet problem via variational methods. In particular, a general variational principle of B.Ricceri is applied.

Shear correction for Mindlin type plate and shell elements

AbstractIn this paper, two shear correction methods, which can guarantee good accuracy, suitable for different materials and easy to implement, are proposed for the first‐order theory based plate and shell elements. The general variational principles based on the thermodynamic potentials are first derived, to provide a uniform frame for the different material non‐linearity. Then, the two shear correction methods are derived from the obtained general variational principles using the assumed strain and assumed stress methods, respectively. The shear correction factors (SCF) for homogeneous elastic materials are evaluated using the two methods, and the same result of 0.8 is obtained. The proposed shear correction method is then applied to an elastic‐linear hardening plastic deep beam for verification, and the conformable result is obtained as compared with the plane strain result. Copyright © 2006 John Wiley & Sons, Ltd.

UNIFIED EXPRESSIONS OF ALL INTEGRAL VARIATIONAL PRINCIPLES

In terms of the quantitative causal principle, this paper obtains a general variational principle, gives unified expressions of the general, Hamilton, Voss, Hölder, Maupertuis–Lagrange variational principles of integral style, the invariant quantities of the general, Voss, Hölder, Maupertuis–Lagrange variational principles are given, finally the Noether conservation charges of the general, Voss, Hölder, Maupertuis–Lagrange variational principles are deduced, and the intrinsic relations among the invariant quantities and the Noether conservation charges of all the integral variational principles are achieved.

Bifurcation and buckling analysis of a unilaterally confined self-rotating cantilever beam

A nonlinear dynamic model of a simple non-holonomic system comprising a self-rotating cantilever beam subjected to a unilateral locked or unlocked constraint is established by employing the general Hamilton's Variational Principle. The critical values, at which the trivial equilibrium loses its stability or the unilateral constraint is activated or a saddle-node bifurcation occurs, and the equilibria are investigated by approximately analytical and numerical methods. The results indicate that both the buckled equilibria and the bifurcation mode of the beam are different depending on whether the distance of the clearance of unilateral constraint equals zero or not and whether the unilateral constraint is locked or not. The unidirectional snap-through phenomenon (i.e. catastrophe phenomenon) is destined to occur in the system no matter whether the constraint is lockable or not. The saddle-node bifurcation can occur only on the condition that the unilateral constraint is lockable and its clearance is non-zero. The results obtained by two methods are consistent.

On general transformations and variational Principles of an ideal three-dimensional compressible

In this paper we have used the general theory of Arnold (1965,1966) and Vladimirov et al. (1997) to obtain sufficient conditions for linear stability of steady MHD flows of an ideal three-dimensional compressible gravitating flows.

On general transformations and variational principles of three-dimensional incompressible gravitating flows in ideal MHD

In this paper, we apply the general theory of Arnold (1965, 1966) and Moffatt et al. (1997). We search sufficient conditions for the linear stability of steady three-dimensional incompressible gravitating flows in ideal magnetohydrodynamics (MHD). The results suggest that the solar and the stellar convection zones must be sensitive to the density stratification.

Variational formulation of a rod under torsional vibration for crack identification

A variational formulation is developed for the torsional vibration of a cylindrical shaft with a circumferential crack. The work is compared with existing methods. The Hu–Washizu–Barr variational formulation was used to develop the differential equation and the boundary conditions of the cracked rod. The general variational principle and the independent assumptions about the displacement, the momentum, the strain and the stress fields of the cracked rod, and the equations of motion for a uniform rod in torsional vibration, are derived. The crack was modelled as a continuous flexibility using the displacement field in the vicinity of the crack, found with fracture mechanics methods. Rayleigh quotient was used to approximate the natural frequencies of the cracked rod. Independent evaluations of crack identification methods in rotating shafts are reported and compared with methods using the continuous crack flexibility theory.

Dynamic characteristic and stability analysis of a beam mounted on a moving rigid body

The phenomenon of dynamic stiffening has drawn general interest in flexible multi-body systems. In fact, approximately analytical, numerical and experimental research have proved that both dynamic stiffening and dynamic softening can occur in flexible multi-body systems. In this paper, the nonlinear dynamic model of a beam mounted on both the exterior and the interior of a rigid ring is established by adopting the general Hamilton's variational principle. The dynamic characteristics of the system are studied using a theoretical method when the rigid ring translates with constant acceleration or rotates steadily. The research proves theoretically that both dynamic stiffening and dynamic softening can occur in both the translation as well as the rotation state of multi-body systems. Furthermore, the approximate vibration frequency, critical value and post-buckling equilibria of the translational beam with constant acceleration are obtained by employing the assumed modes method, which validates the theoretical results. TheL2 norm stability of the trivial equilibrium of the system with the external beam and theL∞ norm stability of the bending of the external beam are proved by employing the energy-momentum method.

On Dynamic Behavior of a Cantilever Beam with Tip Mass in a Centrifugal Field*

The phenomenon of dynamic stiffening is a research field of general interest in flexible multi-body systems. In fact,there is dynamic stiffening as well as dynamic softening phenomenon in flexible multi-body systems. In this paper,a nonlinear dynamic model of a cantilever beam with tip mass in a centrifugal field is established by adopting the general Hamilton's Variational Principle. The dynamic behavior of the flexible multi-body system is studied by analytical,numerical,and experimental methods. The research indicates that the dynamic behavior of the flexible component can be affected by the overall motions of the system and the geometrical and physical parameters of the tip mass. The following conclusions are obtained. The overall motions can result in dynamic stiffening as well as dynamic softening of flexible components. The rigid body and the flexible components may bring about the coupling character of the mode. Dynamic softening may make the trivial equilibrium lose its stability through bifurcation and so on. Furthermore,the lower-order critical bifurcation values and the first buckled equilibria are investigated by employing approximately analytical and numerical methods. The result shows that the geometrical and physical parameters of the tip mass affect the critical bifurcation values and buckled equilibria significantly.

Lagrange multiplier method of the general variational principles of finite deformation theory of couple stress and its application in the strain gradient plasticity

Theory of couple stress was systematically studied by [A.C. Eringen, Basic Principles of Continuum Physics, Academic Press, 1975]. These theoretical results have many developments in the various areas of sciences. In this paper, one proposes a general variational principle of finite deformation theory of couple stress and uses it to the strain gradient plasticity. The general variational principle is a free variational principle. The variational functions need not satisfy any requirement (for example, when stress tensor is the variational function, it need not satisfy the requirements of equilibrium equation and boundary conditions); therefore, the variational functions are easy to choose.

Stable and unstable vortices attached to seamounts

We examine the properties of stationary, barotropic flows over isolated topographic features such as seamounts. According to a general variational principle, flows that maximize or minimize the energy in a set of isovortical flows are stationary and stable. Using this, it is shown that a large class of stable and stationary attached anticyclones exists at a seamount. Those with positive potential vorticity (PV) are maximum energy states, while those with negative PV are minimum energy states. If the seamount is circular, there are also stable attached cyclones, but these are destabilized by irregularities in the topographic shape, unlike the anticyclones. Numerical simulations broadly support these theoretical predictions, but also highlight the importance of time-dependent processes, particularly in cases in which the vortex collides with the seamount.

A sharp existence and localization theorem for a Neumann problem

We shall present a new version of a recently appeared theorem for the existence and localization of solutions of the Neumann problem associated to the equation $ -u^{\prime\prime} + u = \beta(t)g(u) $ , based on a general variational principle by Ricceri. Our study will be especially aimed to express a certain hypothesis regarding the function g in its sharpest form, and a ‘limit case’ is enquired by an approximation

A multiplicity theorem for critical points of functionals on reflexive Banach spaces

In this paper we establish a multiplicity theorem for critical points of functionals on reflexive Banach spaces. Precisely, we deduce the main result using a general variational principle proved by Ricceri. Moreover, we present an application to a Neumann problem which gives a positive answer to some questions formulated by the previous author.