Kinds Of Variational Principles Research Articles

Variational Principles and Solitary Wave Solutions of Generalized Nonlinear Schrödinger Equation in the Ocean

Internal solitary waves are very common physical phenomena in the ocean, which play an important role in the transport of marine matter, momentum and energy. Because the generalized nonlinear Schrodinger equation can well explain the effects of nonlinearity and dispersion in the ocean, it is more suitable for describing the deep-sea internal wave propagation and evolution than other mathematical models. At first, by designing skillfully the trial-Lagrange functional, different kinds of variational principles are successfully established for a generalized nonlinear Schrodinger equation by the semi-inverse method. Then, the constructed variational principles are proved correct by minimizing the functionals with the calculus of variations. Furthermore, some kinds of internal solitary wave solutions are obtained and demonstrated by semi-inverse variational principle for the generalized nonlinear Schrodinger equation.

Study on Parametric Variational Principles in Elasticity

A parametric variational principle is deduced according to the equivalent integral form of all the controlling equations and boundary conditions in elasticity, and by adjusting the parameters, all kinds of variational principles put forward past and some new variational principles can be gained, which means that the method above is more clear in concept, and more concise.

Plastic variational principle based on the least work consumption principle

Plastic variational principles are foundation to solve the boundary-value problems of plastic mechanics with the variational method (or energy method) and finite element method. The most convenient way of establishing different kinds of variational principles is to set up the extreme principle related to the studied problem. Based on a general new extreme principle-the Least work consumption principle, the variational principles of the rigid-plastic and rigid-viscoplastic material were derived. In comparison with existing methods, the method in this paper is more clear and direct, and the physical meaning is clear-cut. This method can offer a new way for establishing other kinds of variational principles.

A finite element model for numerical simulation of thermo-mechanical frictional contact problems

Two kinds of variational principles for numerical simulation of heat transfer and contact analysis are respectively presented. A finite element model for numerical simulation of the thermal contact problems is developed with a pressure dependent heat transfer constitutive model across the contact surface. The numerical algorithm for the finite element analysis of the thermomechanical contact problems is thus developed. Numerical examples are computed and the results demonstrate the validity of the model and algorithm developed.

An optimized mean first passage time approach for obtaining rates in activated processes

The mean first passage time has recently become a useful analytic and computational quantity for estimating reaction rates in many-dimensional activated processes. Unfortunately, the accuracy of this association is limited by the indeterminacy of the appropriate boundary surface with respect to which the first passage times are obtained. The standard choices for this boundary result in an overestimate of the rates in stochastic models using the Langevin equation in the low friction limit. We propose a boundary surface which is a subspace of phase space that results in rates that are accurate in the entire friction regime. It is to be contrasted with equally accurate mean-first-passage-time rates that are obtained using noninvariant subspaces of either the configuration space or phase space and hence are not amenable to nonnumerical analysis. The proposed boundary surface is also shown heuristically and numerically to result from a new kind of variational principle.

Linear constraints from generally covariant systems with quadratic constraints

How to make compatible both boundary and gauge conditions for generally covariant theories using the gauge symmetry generated by first class constraints is studied. This approach employs finite gauge transformations in contrast with previous works which use infinitesimal ones. Two kinds of variational principles are taken into account; the first one features non-gauge-invariant actions whereas the second includes fully gauge-invariant actions. Furthermore, it is shown that it is possible to rewrite fully gauge-invariant actions featuring first class constraints quadratic in the momenta into first class constraints linear in the momenta (and homogeneous in some cases) due to the full gauge invariance of their actions. This shows that the gauge symmetry present in generally covariant theories having first class constraints quadratic in the momenta is not of a different kind with respect to the one of theories with first class constraints linear in the momenta if fully gauge-invariant actions are taken into account for the former theories. These ideas are implemented for the parametrized relativistic free particle, parametrized harmonic oscillator, and the $\mathrm{SL}(2,R)$ model.

Splitting elastic modulus finite element method and its application

AbstractTo establish the best precision FEM, the proportion of potential and complementary energy in the functional of the variational principles must be changeable. A new kind of variational principle in linear theory of solid mechanics, called the splitting elastic modulus variational principle, is introduced. Its distinctive feature is that the functional contains one arbitrary additional parameter, called splitting factor; the proportion of potential and complementary energy in the functional can be changed by the splitting factor. Finite element method, which is based on the new principle, is established. It is called splitting modulus FEM, its stiffness can be adjusted by properly selecting the splitting factors, some ill‐conditioned problem can be conquered by it. The methods to choose the splitting factors, reduce the condition number of stiffness matrix and improve the precision of solutions are also discussed. The reason why the new method can transform the ill‐conditioned problems into well‐conditioned ones is analysed finally. Copyright © 2001 John Wiley & Sons, Ltd.

Deriving generalized variational principles in general mechanics by using Lagrangian multiplier method

By using the involutory transformations, the classical variational principle—Hamiltonian principle— of two kinds of variables in general mechanics is advanced and by using undetermined Lagrangian multiplier method, the generalized variational principles and generalized variational principles with subsidiary conditions are established. The stationary conditions of various kinds of variational principles are derived and the relational problems discussed.

Variational principles of fluid full-filled elastic solids

The generalized variational principles of isothermal quasi-static fluid full-filled elastic solids are established by using Variational Integral Method. Then by introducing constraints, several kinds of variational principles are worked out, including five-field variable, four-field variable, three-field variable and two-field variable formulations. Some new variational principles are presented besides the principles noted in the previous works. Based on variational principles, finite element models can be set up.

On the inverse problem in calculus of variations

The inverse problem in calculus of variations is studied. By introducing a new concept called Variational Integral, a new method to systematically study the inverse problem in calculus of variations is given. Using this new method to the elastodynamics and hydrodynamics of viscous fluids, some kinds of variational principles and generalized variational principles are obtained respectively.

Inverse variational principle in finite elasticity

Although there is a vast literature on the variational principles of the theory of nonlinear elasticity, some basic problems in this field are still open (cf. e.g. [1]). General speaking, almost all the variational principles were studied in the reference configuration ( i.e. undeformed configuration ). Since the loaded boundary is unknown, the well-known dead loading condition should be assumed in this kind of variational principle. Otherwise, the configuration-dependent loading should be considered [2]. In my opinion, the correct description of variational boundary value problem for finite deformation theory is a problem with a movable boundary under the true loading systems [3]. The complete solutions of this problem should include both field functions and deformed configuration. To solve this kind of problem, this paper establishes an Euler-type variational principle in the current configuration by introducing a new variational definition. The critical condition of this principle yields a complete solution of the boundary value problem and the optimal shape condition of the movable boundary. This principle can be applied to problems of optimal shape design.

Theory of connectivity: a systematic formulation of boundary element methods

A theory of connectivity recently developed by the author is applied to construct a systematic formulation of boundary element methods. The concept of complete connectivity condition is shown to supply an alternative to boundary integral equations. The general problem of connecting solutions defined in neighbouring regions R and E is shown to lead to complete connectivity conditions which permit the formulation of three kinds of variational principles; they involve, respectively, R ∪ E, R and the common boundary between R and E, only.

Variational Method in the Transport Theory of an Axisymmetric Toroidal Plasma

Neoclassical transport in an axisymmetric toroidal plasma is considered. By determining the poloidal dependence of ion mean velocity along the magnetic lines of force, the part of electron transport is separated from that of ions. Two kinds of variational principles, one being the maximum principle and the other being the minimum principle, are derived for each part of transport. The forms of variational functionals in the limits of large and small collision frequency are discussed.