THE STRESS-ENERGY TENSOR AND POHOZAEV'S IDENTITY FOR SYSTEMS

Autilizing stress-energy tensors which allow for a divergence-free formulation,we establish Pohozaev's identity for certain classes of quasilinear systems with variational structure.     

Formulation of Euler-Lagrange equations for fractional variational problems

This paper presents extensions to traditional calculus of variations for systems containing fractional derivatives. The fractional derivative is described in the Riemann-Liouville sense. Specifically, we consider two problems, the simplest fractional variational problem and the fractional variational problem of Lagrange. Results of the first problem are extended to problems containing multiple fractional derivatives and unknown functions. For the second problem, we also present a Lagrange type multiplier rule. For both problems, we develop the Euler-Lagrange type necessary conditions which must be satisfied for the given functional to be extremum. Two problems are considered to demonstrate the application of the formulation. The formulation presented and the resulting equations are very similar to those that appear in the field of classical calculus of variations.     

Variational integrators for electric circuits

Variational integrators are symplectic-momentum preserving integrators that are based on a discrete variational formulation of the underlying system. So far, variational integrators have been mainly developed and used for a wide variety of mechanical systems. In this work, we develop a variational integrator for the simulation of electric circuits. An appropriate variational formulation is presented to model the circuit from which the equations of motion are derived. Finally, a corresponding time-discrete variational formulation provides an iteration scheme for the simulation of the electric circuit. In this way, a variational integrator is constructed that gains several advantages. A comparison to standard integration techniques shows that even for simple LCR circuits a better long-time energy behavior and frequency preservation can be obtained. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Resolvent Methods for Solving System of General Variational Inclusions

In this paper, we introduce and consider some new systems of extended general variational inclusions involving seven different operators. Using the resolvent operator technique, we show that the new systems of extended general variational inclusions are equivalent to the fixed point problems. This equivalent formulation is used to suggest and analyze some new iterative methods for this system of extended general variational inclusions. We also study the convergence analysis of the new iterative method under certain mild conditions. Several special cases are also discussed. Results obtained in this paper can be viewed as pure mathematical contribution to variational analysis.     

Variational multirate integration in multi-body dynamics

For systems that contain slow and fast dynamics, variational multirate integration schemes are used. These schemes split the system into parts which are simulated using two time grids consisting of micro and macro nodes. This formulation can be extended for multi-body systems. The rigid multi-body system is described by the so called director formulation and constraints describing the joints connecting the bodies. With the Lagrange multiplier method, the constraints are introduced into the equations of motion. A way to implement the null space method into the variational multirate framework is shown and the influence on the number of unknowns is investigated. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nonsymmetric systems on nonsmooth planar domains

We study boundary value problems, in the sense of Dahlberg, for second order constant coefficient strongly elliptic systems. In this class are systems without a variational formulation, viz. the nonsymmetric systems. Various similarities and differences between this subclass and the symmetrizable systems are examined in nonsmooth domains.

     

Algorithms for solutions of extended general mixed variational inequalities and fixed points

In this paper, we introduce and study a new class of extended general nonlinear mixed variational inequalities and a new class of extended general resolvent equations and establish the equivalence between the extended general nonlinear mixed variational inequalities and implicit fixed point problems as well as the extended general resolvent equations. Then by using this equivalent formulation, we discuss the existence and uniqueness of solution of the problem of extended general nonlinear mixed variational inequalities. Applying the aforesaid equivalent alternative formulation and a nearly uniformly Lipschitzian mapping S, we construct some new resolvent iterative algorithms for finding an element of set of the fixed points of nearly uniformly Lipschitzian mapping S which is the unique solution of the problem of extended general nonlinear mixed variational inequalities. We study convergence analysis of the suggested iterative schemes under some suitable conditions. We also suggest and analyze a class of extended general resolvent dynamical systems associated with the extended general nonlinear mixed variational inequalities and show that the trajectory of the solution of the extended general resolvent dynamical system converges globally exponentially to the unique solution of the extended general nonlinear mixed variational inequalities. The results presented in this paper extend and improve some known results in the literature.     

Existence of Solutions of Systems of Generalized Implicit Vector Variational Inequalities

We consider five different types of systems of generalized vector variational inequalities and derive relationships among them. We introduce the concept of pseudomonotonicity for a family of multivalued maps and prove the existence of weak solutions of these problems under these pseudomonotonicity assumptions in the setting of Hausdorff topological vector spaces as well as real Banach spaces. We also establish the existence of a strong solution of our problems under lower semicontinuity for a family of multivalued maps involved in the formulation of the problems. By using a nonlinear scalar function, we introduce gap functions for our problems by which we can solve systems of generalized vector variational inequalities using optimization techniques. The first two authors were supported by SABIC and Fast Track Research Grants SAB-2006-05. They are grateful to the Department of Mathematical Sciences, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia for providing excellent research facilities.     

Extended Entropy Functional for Nonlinear Systems in Stochastic Dynamics

The maximum entropy approach is utilized for deriving the stationary probability density function of nonlinear stochastic systems to white noise excitation. To this aim a variational formulation is proposed where by means of the Lagrange multiplier methods the entropy functional is maximised constrained to the Fokker Planck equation. Some exact solutions in terms of Lagrange function of MDOF linear systems and for a class of SDOF nonlinear systems, are obtained.     

On a System of Equations of Evolution with a Non-Symmetrical Parabolic Part Occuring in the Analysis of Moisture and Heat Transfer in Porous Media

Most non-trivial existence and convergence results for systems of partial differential equations of evolution exclude or avoid the case of a non-symmetrical parabolic part. Therefore such systems, generated by the physical analysis of the processes of transfer of heat and moisture in porous media, cannot be analyzed easily using the standard results on the convergence of Rothe sequences (e.g. those of W. Jager and J. Kacur). In this paper the general variational formulation of the corresponding system is presented and its existence and convergence properties are verified; its application to one model problem (preserving the symmetry in the elliptic, but not in the parabolic part) is demonstrated.     

A variational inequality approach to constrained control problems for parabolic equations

A distributed optimal control problem for parabolic systems with constraints in state is considered. The problem is transformed to control problem without constraints but for systems governed by parabolic variational inequalities. The new formulation presented enables the efficient use of a standard gradient method for numerically solving the problem in question. Comparison with a standard penalty method as well as numerical examples are given.     

一类单边约束问题的数值方法

本文分别基于原始变分形式与对偶混合变分形式，对一类单边约束问题进行了数值求解，提出了求解离散对偶混合变分问题的Uzawa型算法，并用数值例子验证了算法的有效性．     

On General Mixed Variational Inequalities

In this paper, we introduce and consider a new class of mixed variational inequalities, which is called the general mixed variational inequality. Using the resolvent operator technique, we establish the equivalence between the general mixed variational inequalities and the fixed-point problems as well as resolvent equations. We use this alternative equivalent formulation to suggest and analyze some iterative methods for solving the general mixed variational inequalities. We study the convergence criteria of the suggested iterative methods under suitable conditions. Using the resolvent operator technique, we also consider the resolvent dynamical systems associated with the general mixed variational inequalities. We show that the trajectory of the dynamical system converges globally exponentially to the unique solution of the general mixed variational inequalities. Our methods of proofs are very simple as compared with others’ techniques. Results proved in this paper may be viewed as a refinement and important generalizations of the previous known results.     

An Evolutionary Boundary Value Problem

We consider a nonlinear initial boundary value problem in a two-dimensional rectangle. We derive variational formulation of the problem which is in the form of an evolutionary variational inequality in a product Hilbert space. Then, we establish the existence of a unique weak solution to the problem and prove the continuous dependence of the solution with respect to some parameters. Finally, we consider a second variational formulation of the problem, the so-called dual variational formulation, which is in a form of a history-dependent inequality associated with a time-dependent convex set. We study the link between the two variational formulations and establish existence, uniqueness, and equivalence results.

     

A unified theory of necessary conditions for nonlinear nonconvex control systems

We consider optimal problems for a general nonlinear nonconvex input-output relation for Banach space valued functions. A maximum principle is obtained using Ekeland's variational principle. The formulation applies to systems described by ordinary differential equations, functional differential equations, and partial differential equations (both for distributed and boundary control systems).This work was supported in part by the National Science Foundation under Grant No. DMS-8200645.     

A variational approach for large deflection of ends supported nanorod under a uniformly distributed load,using intrinsic coordinate finite elements

This paper presents a variational formulation that can be used for large deflection analysis of ends supported nanorod including the coupled effects of nonlocal elasticity and surface stress under a uniformly distributed load. The variational formulation involving the strain energy due to bending of nonlocal elasticity including the surface stress effect and virtual work done by a uniformly distributed load, is expressed in terms of the intrinsic coordinates. The Lagrange multiplier technique is applied to impose the boundary conditions which accomplished in the formulation. The validity of the variational approach is ensured by Euler's equation, which identical to the one derived by the force equilibrium consideration of an infinitesimal nanorod segment. The finite element method and Newton–Raphson iterative procedure based on the variational formulation are used to solve a system of nonlinear equations. Moreover, the very large deflection configurations of ends supported nanorod are highlighted in this study.     

Importance of the flexural and membrane stiffnesses in large deflection analysis of floating roofs

Applying integrated variational principles on fluid and deck plate to the large deflection analysis of floating roofs, this paper investigates the significance of the flexural and membrane components in the formulations of the deck plate. Integrated variational principles facilitate the treatment of the compatibility of deformation between floating roof and supporting liquid. Analysis results show that different assumptions about deck plate formulation commonly used in the literature, results in considerably different deflection and stress patterns on the floating roof. The results show that modeling of the deck plate as a flexural element rather than the membrane, by eliminating the need for nonlinear analysis, gives reasonable results for deflections and stresses in the deck plate. Finally, to check the results of the variational formulation, employing Bessel functions and ignoring membrane stiffness an approximate solution is derived and its results compared with those of the variational formulation. This comparison shows that the approximate solution closely follows the variational formulation.     

Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems

Whether or not the general asymmetric variational inequality problem can be formulated as a differentiable optimization problem has been an open question. This paper gives an affirmative answer to this question. We provide a new optimization problem formulation of the variational inequality problem, and show that its objective function is continuously differentiable whenever the mapping involved in the latter problem is continuously differentiable. We also show that under appropriate assumptions on the latter mapping, any stationary point of the optimization problem is a global optimal solution, and hence solves the variational inequality problem. We discuss descent methods for solving the equivalent optimization problem and comment on systems of nonlinear equations and nonlinear complementarity problems.     

A variational formulation for the torsional problem of piezoelastic beams

In this paper a variational formulation is presented for the torsional deformation of homogeneous, linear piezoelectric monoclinic beams. All results of the paper are based on a generalization of the Saint-Venant’s theory of uniform torsion of elastic beams to piezoelastic beams. Variational formulation uses the torsional and electric potential functions as the independent quantities of the considered variational functional. The mechanical meaning of the variational functional defined is also given. Examples illustrate the application of the presented variational formulation. Considered examples are the torsional problem of thin-walled piezoelastic beams with closed cross-section, and the torsion of hollow circular cylinders made of orthotropic piezoelectric material.     

The momentum of waves and their effect on the motion of lumped objects along one-dimensional elastic systems

Taking the example of the small longitudinal oscillations of a rod, it is shown that, in order to answer the question concerning wave momentum and its action on an obstacle, the problem of the wave motion in the medium has to be solved in a non-linear formulation. The variational formulation of problems in the dynamics of one-dimensional elastic systems with moving clampings and loads is improved taking account of non-linear factors. The equations of motion and the natural boundary conditions are obtained. The small longitudinal-transverse oscillations of a string and the motion of a bead sliding along it are considered.