Classical Variational Principle Research Articles

Effect of external prestressing on the special shear lag behavior of precast box girder bridges

The shear lag behavior exhibited by precast segmental box girders (PSBGs) under concentrated external prestressing differs significantly from that observed in box girders subjected to vertical loads. To study the special shear lag behavior of PSBGs under external prestressing, a theoretical analytical method was developed based on the classical energy variational principle. In this analysis, the effect of external prestressing was decomposed into axial compression and bending effects through the equivalent load method. For each of these effects, appropriate longitudinal warping displacement modes were assumed, and the total energy functional expressions of the system under these two effects were further established. The governing differential equations for the two effects were derived using the principle of energy variation. Analytical solutions for the normal stress of the beam section under axial compression and bending effects were deduced, and corresponding boundary conditions were obtained. Subsequently, the normal stress of the section under the two effects was superimposed to obtain the comprehensive normal stress distribution curve of the beam section under external prestressing. The reliability of the theoretical analysis method was validated through a comparison of the theoretical analysis results with experimental and numerical analysis results.

Variational Bayesian Method: Ritz Method in Stochastic System

Bayesian inference is to find posterior probabilities, but since it is difficult to find analytical solutions, it is often the case that approximate solutions are found. The variational Bayesian method is a powerful method for finding an approximate solution. It is a variational method in a stochastic system. Variational methods have been developed for the deterministic system since old times, and are one of the most powerful foundations for numerical solutions of a wide range of problems defined by partial differential equations. A detailed comparison and explanation of the classical variational principle and the variational Bayesian method are given, and the basic application examples of the variational Bayesian method are also given. Programming codes written in C are also shown to aid the readers’ understanding.

Noether’s theorem for fractional Birkhoffian system of Herglotz type with time delay

Abstract Noether symmetry theorem of Herglotz type for time-delayed fractional Birkhoffian system are studied. Firstly, based on the fractional derivative of Riemann-Liouville, the Herglotz variational principle of time-delayed fractional Birkhoffian system is established, and the time-delayed Birkhoff′s equation of Herglotz type is derived. Secondly, the definition and criterion of Herglotz type Noether symmetric transformation of time-delayed fractional Birkhoffian system are established. Thirdly, the Noether′s theorem of the system is proposed and proved, in addition, the inner relationship between Noether symmetries and conservation is accurately explored. Next, the special case of the theorem is discussed, in other words, when the Herglotz generalized variational principle is reduced to the classical variational principle, the result of this paper is degraded into the Noether symmetry theorem of the time-delayed fractional Birkhoffian system. Finally, an example is given.

A new type of adiabatic invariant for fractional order non-conservative Lagrangian systems

The Herglotz variational problem is also known as Herglotz generalized variational principle whose action functional is defined by differential equation. Unlike the classical variational principle, the Herglotz variational principle gives a variational description of a holonomic non-conservative system. The Herglotz variational principle can describe not only all physical processes that can be described by the classical variational principlen, but also the problems that the classical variational principle is not applicable for. If the Lagrangian or Hamiltonian does not depend on the action functional, the Herglotz variational principle reduces to the classical integral variational principle. In this work, in order to describe the dynamical behavior of complex non-conservative system more accurately, we extend the Herglotz variational principle to the fractional order model, and study the adiabatic invariant for fractional order non-conservative Lagrangian system. Firstly, based on the Herglotz variational problem, the differential variational principle of Herglotz type and the differential equations of motion of the fractional non-conservative Lagrangian system are derived. Secondly, according to the relationship between the isochronal variation and the nonisochronal variation, the transformation of invariance condition of Herglotz differential variational principle is established and the exact invariants of the system are derived. Thirdly, the effects of small perturbations on fractional non-conservative Lagrangian systems are studied, the conditions for the existence of adiabatic invariants for the Lagrangian systems of Herglotz type based on Caputo derivatives are established, and the adiabatic invariants of Herglotz type are obtained. In addition, the exact invariant and adiabatic invariant of fractional non-conservative Hamiltonian system can be obtained by Legendre transformation. When <inline-formula><tex-math id="M1">\begin{document}$ \alpha \to 1$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="22-20200488_M1.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="22-20200488_M1.png"/></alternatives></inline-formula>, the Herglotz differential variational principle for fractional non-conservative Lagrangian system degrades into classical Herglotz differential variational principle, and the corresponding exact invariants and adiabatic invariants also degenerate into the classical exact invariants and adiabatic invariants of Herglotz type. At the end of the paper, the fractional order damped oscillator of Herglotz type is discussed as an example to demonstrate the results.

Convergence analysis and error estimate of finite element method of a nonlinear fluid-structure interaction problem

In this paper, a semi-discrete finite element method for the nonlinear fluid-structure interaction problem interacts between the Navier-Stokes fluids and linear elastic solids, is studied and developed. A classical mixed variational principle of the weak formulation is given, and the corresponding finite element method is defined. As for the nonlinearity arising from the nonlinear interaction problem, we consider in time of a solution for suitably small data, and uniqueness hypothesis. This approach is fairly robust and adapts to the important case of interface with fractures or cracks. Convergence and estimate of the finite element method are also obtained for the nonlinear fluid-structure interaction problem. Finally, numerical experiments are presented to show the performance of the proposed method.

Finsler gravity action from variational completion

In the attempts to apply Finsler geometry to construct an extension of general relativity, the question about a suitable generalization of the Einstein equations is still under debate. Since Finsler geometry is based on a scalar function on the tangent bundle, the field equation which determines this function should also be a scalar equation. In the literature two such equations have been suggested: the one by Rutz and the one by one of the authors. Here we employ the method of canonical variational completion to show that Rutz equation can not be obtained from a variation of an action and that its variational completion yields the latter field equations. Moreover, to improve the mathematical rigor in the derivation of the Finsler gravity field equation, we formulate the Finsler gravity action on the positive projective tangent bundle. This has the advantage of allowing us to apply the classical variational principle, by choosing the domains of integration to be compact and independent of the dynamical variable. In particular in the pseudo-Riemannian case, the vacuum field equation becomes equivalent to the vanishing of the Ricci tensor.

Slow Entropy of Some Parabolic Flows

We study nontrivial entropy invariants in the class of parabolic flows on homogeneous spaces, quasi-unipotent flows. We show that topological complexity (i.e., slow entropy) can be computed directly from the Jordan block structure of the adjoint representation. Moreover using uniform polynomial shearing we are able to show that the metric orbit growth (i.e., slow entropy) coincides with the topological one for quasi-unipotent flows (this also applies to the non-compact case). Our results also apply to sequence entropy. We establish criterion for a system to have trivial topological complexity and give some examples in which the measure-theoretic and topological complexities do not coincide for uniquely ergodic systems, violating the intuition of the classical variational principle.

A Variational Approach to Solitary Gravity\u2013Capillary Interfacial Waves with Infinite Depth

We present an existence and stability theory for gravity–capillary solitary waves on the top surface of and interface between two perfect fluids of different densities, the lower one being of infinite depth. Exploiting a classical variational principle, we prove the existence of a minimiser of the wave energy {mathcal {E}} subject to the constraint {mathcal {I}}=2mu , where {mathcal {I}} is the wave momentum and 0< mu <mu _0, where mu _0 is chosen small enough for the validity of our calculations. Since {mathcal {E}} and {mathcal {I}} are both conserved quantities a standard argument asserts the stability of the set D_mu of minimisers: solutions starting near D_mu remain close to D_mu in a suitably defined energy space over their interval of existence. The solitary waves which we construct are of small amplitude and are to leading order described by the cubic nonlinear Schrödinger equation. They exist in a parameter region in which the ‘slow’ branch of the dispersion relation has a strict non-degenerate global minimum and the corresponding nonlinear Schrödinger equation is of focussing type. The waves detected by our variational method converge (after an appropriate rescaling) to solutions of the model equation as mu downarrow 0.

A Variational Reduction and the Existence of a Fully Localised Solitary Wave for the Three-Dimensional Water-Wave Problem with Weak Surface Tension

Fully localised solitary waves are travelling-wave solutions of the three-dimensional gravity-capillary water wave problem which decay to zero in every horizontal spatial direction. Their existence has been predicted on the basis of numerical simulations and model equations (in which context they are usually referred to as `lumps'), and a mathematically rigorous existence theory for strong surface tension (Bond number $\beta$ greater than $\frac{1}{3}$) has recently been given. In this article we present an existence theory for the physically more realistic case $0<\beta<\frac{1}{3}$. A classical variational principle for fully localised solitary waves is reduced to a locally equivalent variational principle featuring a perturbation of the functional associated with the Davey-Stewartson equation. A nontrivial critical point of the reduced functional is found by minimising it over its natural constraint set.

Hamilton’s principle for dynamical elasticity

There exists NO classical variational principle for continuum dynamics so far. The well-known Hamilton’s principle is valid only for the conditions prescribed at the beginning and at the end of the motion and is therefore useless to deal with the usual initial-boundary condition problems. In this paper an exact classical variational theory is established, all initial conditions are converted into natural ones, furthermore, all natural final conditions automatically meet the physical requirements, leading to a vital innovation of Hamilton’s principle.

Generalized variational principles for heat conduction models based on Laplace transforms

The classical variational principle does not exist for parabolic and hyperbolic heat conduction equations, which has led to the demand for special variational methods for heat conduction. O’Toole (1967) first used Laplace transforms for the variational principle only for Fourier’s law with the first type of boundary condition. In this paper, the Laplace transform strategy is extended to other parabolic and hyperbolic heat conduction models and other types of boundary conditions. Generalized variational principles are given for heat conduction models including Fourier’s law, the Cattaneo–Vernotte (CV) model, the Jeffrey model, the two-temperature model and the Guyer–Krumhansl (GK) model, based on Laplace transforms. The Laplace transform method transforms the heat conduction equations of these models into linear variational equations whose variational principles are already known. For the three standard types of boundary conditions, these generalized variational principles are strictly equivalent to the heat conduction equations for these models. The Laplace transform method has stronger convergence in infinite temporal domain problems. In physics, the Laplace transform method is understood as replacing the time dimension with the frequency of the temperature change and the rate of the entropy change.

Existence and conditional energetic stability of solitary gravity–capillary water waves with constant vorticity

We present an existence and stability theory for gravity–capillary solitary waves with constant vorticity on the surface of a body of water of finite depth. Exploiting a rotational version of the classical variational principle, we prove the existence of a minimizer of the wave energy𝓗subject to the constraint𝓘= 2µ, where𝓘is the wave momentum and 0 &lt;µ≪ 1. Since𝓗and𝓘are both conserved quantities, a standard argument asserts the stability of the setDµof minimizers: solutions starting nearDµremain close toDµin a suitably defined energy space over their interval of existence. In the applied mathematics literature solitary water waves of the present kind are described by solutions of a Korteweg–de Vries equation (for strong surface tension) or a nonlinear Schrödinger equation (for weak surface tension). We show that the waves detected by our variational method converge (after an appropriate rescaling) to solutions of the appropriate model equation asµ↓ 0.

Variational symmetries, conserved quantities and identities for several equations of mathematical physics

We find variational symmetries, conserved quantities and identities for several equations: envelope equation, Böcher equation, the propagation of sound waves with losses, flow of a gas with losses, and the nonlinear Schrödinger equation with losses or gains, and an electro-magnetic interaction. Most of these equations do not have a variational description with the classical variational principle and we find such a description with the generalized variational principle of Herglotz.

Homotopy Perturbation Method for Brachistochrone Problem

The most famous classical variational principle is the so-called Brachistochrone problem. In this work, Homotopy perturbation method (HPM) is applied to the Brachistochrone problem that arises invariational problems. The results reveal the efficiency and the accuracy of the proposed method. Homotopy perturbation method yields solutions in convergent series forms with easy computation.

Variational bounds of the effective moduli of piezoelectric composites

The classical Hashin-Shtrikman variational principle was re-generalized to the heterogeneous piezoelectric materials. The auxiliary problem is very much simplified by selecting the reference medium as a linearly isotropic elastic medium. The electromechanical fields in the inhomogeneous piezoelectrics are simulated by introducing into the homogeneous reference medium certain eigenstresses and eigen electric fields. A closed-form solution can be obtained for the disturbance fields, which is convenient for the manipulation of the energy functional. As an application, a two-phase piezoelectric composite with nonpiezoelectric matrix is considered. Expressions of upper and lower bounds for the overall electromechanical moduli of the composite can be developed. These bounds are shown better than the Voigt-Reuss type ones.

Interval oscillation criteria for a forced second-order nonlinear differential equations with damping

In this paper, we are concerned with the oscillations in a class of forced second-order differential equations with nonlinear damping terms. By using classical variational principle and averaging technique, several new interval oscillation criteria for the equations are established, which improve and extend some known results. Example is also given to illustrate the results.

Improved Harmony Search Methods to Replace Variational Principle in Geotechnical Problems

ABSTRACTVariational principle is an important principle in engineering discipline. This principle is suitable for simple problems where an analytical expression can be determined, but there are many practical problems where the classical variational principle is practically impossible to be applied. In this paper, the authors will try to demonstrate that the variational principle can be replaced by the use of modern artificial intelligence based optimization method (harmony search method) which can be applied to much more complicated problems. Two different improved harmony search algorithms are proposed in this paper. The new algorithms differ from the original algorithm in that: (1) The harmonies are rearranged into several pairs and the better pairs are used to develop several new harmonies; (2) Different probabilities are assigned to different harmonies. The robustness of the proposed methods is demonstrated by using three difficult examples, and the sensitivities of the related optimization parameters are investigated through statistical orthogonal analysis.

Natural vibrations and stability of shells of revolution interacting with an internal fluid flow

A finite-element algorithm is proposed to investigate the dynamic behavior of elastic shells of revolution containing a quiescent or a flowing inviscid fluid in the framework of linear theory. The fluid behavior is described using the perturbed velocity potential. The shell behavior is treated in the framework of the classical shell theory and variational principle of virtual displacements incorporating a linearized Bernoulli equation for calculation of hydrodynamic pressure acting on the shell. The problem reduces to evaluation and analysis of the eigenvalues in the connected system of equations obtained by coupling the equations for velocity perturbations with the equations for shell displacements. For cylindrical shells, the results of numerical simulations are compared with recently published experimental, analytical and numerical data. The paper also reports the results of studying the dynamic behavior of shells under various boundary conditions for the perturbed velocity potential. The investigation made for conical shells has shown that under certain conditions an increase in the cone angle can change a divergent type of instability to a flutter type.

On the Existence and Conditional Energetic Stability of Solitary Gravity-Capillary Surface Waves on Deep Water

This paper presents an existence and stability theory for gravity-capillary solitary waves on the surface of a body of water of infinite depth. Exploiting a classical variational principle, we prove the existence of a minimiser of the wave energy epsilon subject to the constraint I = root 2 mu, where I is the wave momentum and 0 < mu << 1. Since epsilon and I are both conserved quantities a standard argument asserts the stability of the set D-mu of minimisers: solutions starting near D-mu remain close to D-mu in a suitably defined energy space over their interval of existence. In the applied mathematics literature solitary water waves of the present kind are modelled as solutions of the nonlinear Schrodinger equation with cubic focussing nonlinearity. We show that the waves detected by our variational method converge (after an appropriate rescaling) to solutions of this model equation as mu down arrow 0.

Variational Principle for Fuzzy Gibbs Measures

In this paper we study a large class of renormalization transformations of measures on lattices. An image of a Gibbs measure under such transformation is called a fuzzy Gibbs measure. Transformations of this type and fuzzy Gibbs measures appear naturally in many fields. Examples include the hidden Markov processes (HMP), memory-less channels in information theory, continuous block factors of symbolic dynamical systems, and many renormalization transformations of statistical mechanics. The main result is the generalization of the classical variational principle of Dobrushin-Lanford-Ruelle for Gibbs measures to the class of fuzzy Gibbs measures.