Stationary Variational Principle Research Articles

Stationary variational principle of mixture unified gradient elasticity

The stress gradient, strain gradient, and classical elasticity theory are integrated within a consistent variational framework to conceive the mixture unified gradient theory of elasticity. The significant advantage of the established stationary variational principle lies in incorporating all the governing equations, viz. the boundary-value problem of the associated dynamic equilibrium along with the non-classical boundary conditions and the constitutive laws, into a single functional. The mixture unified gradient theory can effectively serve as a suitable counterpart for the two-phase local/nonlocal gradient theory with a noteworthy privilege; the conceived augmented elasticity theory can be efficiently adopted to examine various multi-dimensional structural problems of practical interest in Engineering Science. The well-posedness of the introduced generalized size-dependent elasticity theory is demonstrated via analytically examining the flexure mechanics of inflected elastic nano-beams. A viable approach is proposed for calibrating the gradient length-scale parameters associated with the mixture unified gradient elasticity, and accordingly, the size-dependent Young's modulus of carbon nanotubes can be inversely determined. The established size-dependent elasticity theory can be fruitfully invoked to address problems in nano-mechanics practice where the mechanical response is notably affected by nano-structural features. The developed mixture unified gradient theory of elasticity can, hence, pave way ahead in mechanics of ultra-small structures.

On the analytical and meshless numerical approaches to mixture stress gradient functionally graded nano-bar in tension

The mixture stress gradient theory of elasticity is conceived via consistent unification of the classical elasticity theory and the stress gradient theory within a stationary variational framework. The boundary-value problem associated with a functionally graded nano-bar is rigorously formulated. The constitutive law of the axial force field is determined and equipped with proper non-standard boundary conditions. Evidences of well-posedness of the mixture stress gradient problems, defined on finite structural domains, are demonstrated by analytical analysis of the axial displacement field of structural schemes of practical interest in nano-mechanics. An effective meshless numerical approach is, moreover, introduced based on the proposed stationary variational principle while employing autonomous series solution of the kinematic and kinetic field variables. Suitable mathematical forms of the coordinate functions are set forth in terms of the modified Chebyshev polynomials, satisfying the required classical and non-standard boundary conditions. An excellent agreement between the numerical results of the axial displacement field of the functionally graded nano-bar and the analytical solution counterpart is confirmed on the entire span of the nano-sized bar, in terms of the mixture parameter and the stress gradient characteristic parameter. The effectiveness of the established meshless numerical approach, demonstrating a fast convergence rate and an admissible convergence region, is hence ensured. The established mixture stress gradient theory can effectively characterize the peculiar size-dependent response of functionally graded structural elements of advanced ultra-small systems.

A mixed variational framework for higher-order unified gradient elasticity

The higher-order unified gradient elasticity theory is conceived in a mixed variational framework based on suitable functional space of kinetic test fields. The intrinsic form of the differential and boundary conditions of equilibrium along with the constitutive laws is consistently established. Various forms of the gradient elasticity theory, in the sense of stress or strain gradient models, can be retrieved as particular cases of the introduced generalized elasticity theory. The proposed stationary variational principle can effectively realize the nanoscopic structural effects while being exempt of restrictions typical of the nonlocal gradient elasticity model. The well-posed generalized gradient elasticity theory is invoked to study the mechanics of torsion and the torsional behavior of elastic nano-bars is analytically examined. The closed-form analytical formulae of the size-dependent shear modulus of nano-sized bar is determined and efficiently applied to reconstruct the shear modulus of SWCNTs with dissimilar chirality in comparison with the numerical simulation data. A practical approach to calibrate the characteristic lengths associated with the higher-order unified gradient elasticity theory is introduced. Numerical results associated with the torsion of higher-order unified gradient elastic bars are demonstrated and compared with the counterpart size-dependent elasticity theories. The conceived generalized gradient elasticity theory can beneficially characterize the nanoscopic response of advanced nano-materials.

Analytical and meshless numerical approaches to unified gradient elasticity theory

The unified gradient elasticity theory with applications to nano-mechanics of torsion is examined. The Reissner stationary variational principle is invoked to detect the differential and boundary conditions of equilibrium along with the consistent form of the constitutive laws. An efficient meshless numerical approach is established by making recourse to the Reissner variational functional wherein independent series solution of the kinematic and kinetic field variables are proposed. Suitable forms of the coordinate functions, in terms of the Chebyshev polynomials, are introduced to fulfill a set of kinematic and higher-order boundary conditions in the elastic torsion of nano-bars with practical kinematic constraints. Torsional behavior of the unified gradient elastic bar is studied for structural schemes of applicative interest. An excellent agreement between the torsional responses of the nano-bar detected based on the established meshless method and obtained exact analytical solution is realized. The proposed meshless numerical approach is confirmed to have a fast convergence rate and an admissible convergence region in determination of the torsional rotation field with high accuracy. The introduced meshless method is demonstrated to be highly efficacious in characterizing both the softening and stiffening structural behaviors at nano-scale. The presented numerical approach therefore paves the way ahead in mechanics of nano-structures.

Convolved energy variational principle in heat diffusion

One of the major, long-standing challenges in analytical mechanics involves the inability to address systems with dissipation in a rigorous manner. In this paper, we overcome that difficulty by formulating a novel temperature-based stationary variational principle for transient heat diffusion based upon a temporal convolution operator and fractional derivatives. The associated Euler-Lagrange equations provide the governing heat equation, along with the initial conditions on temperature and specified heat flux boundary conditions. A further integration-by-parts then leads to a formulation that is somewhat less symmetric but can be written without introducing fractional calculus. Finally, the resulting principle is used to solve two basic one-dimensional problems, as an illustration of a Ritz-type approach.

Variational principle and time-space finite element method for dynamic thermoelasticity based on mixed convolved action

Abstract A new variational formulation is proposed for time-domain analysis of initial/boundary value problems of dynamic thermoelasticity. By using the concept of mixed convolved action, the difficulties with dissipative phenomena and proper representations of the temporal end point conditions can be overcome to create a true stationary variational principle. After an elementary illustrative development for a lumped parameter thermoelastic model, the convolved action functional for linear thermoelastic continuum is written directly in terms of mixed variables, which include displacements and the impulses of stress, temperature and heat flux. Unlike previous variational approaches, based for example upon a generalization of Hamilton's principle , the present mixed convolved action formulation allows direct application of finite element methodology in both space and time. Here, simple linear shape functions are employed for the temporal representations. Meanwhile, standard three-noded triangular elements are used in the present two-dimensional numerical implementation. Several computational examples are considered to test this original approach and to investigate interesting aspects of coupled dynamic thermoelastic response.

Reissner stationary variational principle for nonlocal strain gradient theory of elasticity

Abstract The general form of Reissner stationary variational principle is established in the framework of the nonlocal strain gradient theory of elasticity. Including two size-dependent characteristic parameters, the nonlocal strain gradient elasticity theory can demonstrate the significance of the strain gradient as well as the nonlocal elastic stress field. Based on the Reissner functional, the governing differential and boundary conditions of dynamic equilibrium and differential constitutive equations of the classical and first-order nonlocal stress tensor are derived in the most general form. Additionally, the boundary congruence conditions are formulated and discussed for the nonlocal strain gradient theory. To exhibit the application value of Reissner variational principle, it is employed to examine the nonlinear vibrations of size-dependent Bernoulli-Euler and Timoshenko beams. In the case of immovable boundary conditions, employing the weighted residual Galerkin method, the homotopy analysis method is also utilized to determine the closed form analytical solutions of the geometrically nonlinear vibration equations. Consequently, the analytical expressions for the nonlinear natural frequencies of Bernoulli-Euler and Timoshenko nonlocal strain gradient beams are derived.

A symmetric fully optimized second‐order method for nonlinear homogenization

AbstractWe consider an alternate formulation of the recently developed ‘fully optimized second‐order’ (FOSO) nonlinear homogenization method, which is based on a stationary variational principle for the macroscopic energy function. In this method, the trial fields are the properties of a suitably designed linear comparison composite (LCC), thus allowing for the estimation of the effective response and field statistics of the nonlinear composite in terms of available estimates for the corresponding quantities in the LCC. The formulation considered in this paper makes use of an alternative choice for the linear comparison composite, leading to homogenization estimates that are more symmetric with respect to Legendre duality than earlier FOSO estimates. The new ‘symmetric’ FOSO method is applied to a class of two‐phase power‐law composites with fibrous microstructures subjected to plane strain loading. The resulting estimates for the effective response and field statistics are found to improve on earlier estimates, and to be in good agreement with full‐field numerical simulations for nonlinear composite cylinder assemblages, as well as with available results for sequentially layered composites.

Quadratic temporal finite element method for linear elastic structural dynamics based on mixed convolved action

A common approach for dynamic analysis in current practice is based on a discrete time-integration scheme. This approach can be largely attributed to the absence of a true variational framework for initial value problems. To resolve this problem, a new stationary variational principle was recently established for single-degree-of-freedom oscillating systems using mixed variables, fractional derivatives and convolutions of convolutions. In this mixed convolved action, all the governing differential equations and initial conditions are recovered from the stationarity of a single functional action. Thus, the entire description of linear elastic dynamical systems is encapsulated. For its practical application to structural dynamics, this variational formalism is systemically extended to linear elastic multidegree- of-freedom systems in this study, and a corresponding weak form is numerically implemented via a quadratic temporal finite element method. The developed numerical method is symplectic and unconditionally stable with respect to a time step for the underlying conservative system. For the forced-damped vibration, a three-story shear building is used as an example to investigate the performance of the developed numerical method, which provides accurate results with good convergence characteristics.

Stationary variational estimates for the effective response and field fluctuations in nonlinear composites

This paper presents a variational method for estimating the effective constitutive response of composite materials with nonlinear constitutive behavior. The method is based on a stationary variational principle for the macroscopic potential in terms of the corresponding potential of a linear comparison composite (LCC) whose properties are the trial fields in the variational principle. When used in combination with estimates for the LCC that are exact to second order in the heterogeneity contrast, the resulting estimates for the nonlinear composite are also guaranteed to be exact to second-order in the contrast. In addition, the new method allows full optimization with respect to the properties of the LCC, leading to estimates that are fully stationary and exhibit no duality gaps. As a result, the effective response and field statistics of the nonlinear composite can be estimated directly from the appropriately optimized linear comparison composite. By way of illustration, the method is applied to a porous, isotropic, power-law material, and the results are found to compare favorably with earlier bounds and estimates. However, the basic ideas of the method are expected to work for broad classes of composites materials, whose effective response can be given appropriate variational representations, including more general elasto-plastic and soft hyperelastic composites and polycrystals.

Fully optimized second-order variational estimates for the macroscopic response and field statistics in viscoplastic crystalline composites

A variational method is developed to estimate the macroscopic constitutive response of composite materials consisting of aggregates of viscoplastic single-crystal grains and other inhomogeneities. The method derives from a stationary variational principle for the macroscopic stress potential of the viscoplastic composite in terms of the corresponding potential of a linear comparison composite (LCC), whose viscosities and eigenstrain rates are the trial fields in the variational principle. The resulting estimates for the macroscopic response are guaranteed to be exact to second order in the heterogeneity contrast, and to satisfy known bounds. In addition, unlike earlier ‘second-order’ methods, the new method allows optimization with respect to both the viscosities and eigenstrain rates, leading to estimates that are fully stationary and exhibit no duality gaps. Consequently, the macroscopic response and field statistics of the nonlinear composite can be estimated directly from the suitably optimized LCC, without the need for difficult-to-compute correction terms. The method is applied to a simple example of a porous single crystal, and the results are found to be more accurate than earlier estimates.

Mixed variational principles for dynamic response of thermoelastic and poroelastic continua

The primary objective of the present work is to make further connections between variational methods on the one hand and reversible and irreversible thermodynamics on the other. This begins with the development of a new stationary principle, involving mixed field variables, for continuum problems in infinitesimal dynamic thermoelasticity. By defining Lagrangian and dissipation functions in terms of physically-relevant contributions and invoking the Rayleigh formalism for damped systems, we are able to recover the governing equations of thermoelasticity as the Euler–Lagrange equations. This includes the balance laws of linear momentum and entropy-energy, the constitutive models for elastic response and heat conduction, and the natural boundary conditions. By including energy contributions associated with second sound phenomena, one eliminates the paradox of infinite thermal propagation speeds and the resulting set of governing equations has an elegant symmetry, which is most easily seen in the Fourier wave number domain. A related formulation for dynamic poroelasticity yields two new stationary mixed variational principles. Depending upon the selection of primary field variables, these governing equations can also exhibit an elegant structure, which can deepen our understanding of the underlying phenomena and the thermoelastic–poroelastic analogy. In addition to the theoretical significance, the variational formulations developed here can provide the basis for a class of optimization-based methods for computational mechanics.

On the transition from nonadiabatic to adiabatic rate kernel: Schwinger’s stationary variational principle and Padé approximation

For a two state system coupled to each other by a nonzero matrix element Δ and to the bath arbitrarily, the generalized master equation is derived by applying the well-known projection operator techniques to the quantum Liouville equation. The time-dependent rate kernel is expressed by an infinite summation of the perturbative terms in Fourier–Laplace space. The Schwinger’s stationary variation principle in Hilbert space is extended to Liouville space and then applied to the resummation of the rate kernel. The Cini–Fubini-type trial state vector in Liouville space is used to calculate the variational parameters. It is found that the resulting stationary value for the rate kernel in Fourier–Laplace space is given by the [N,N−1]–Padé approximants, in the N-dimensional subspace constructed by the N perturbatively expanded Liouville space vectors. The (first-order) simplest approximation satisfying the variational principle turns out to be equal to the [1,0] Padé approximant instead of the second-order Fermi golden rule expression. Two well-known approximations, the noninteracting blip approximation (NIBA) and nonadiabatic approximation, are discussed in the context of the [1,0] Padé approximants, based on the variational principle. A higher-order approximation, [2,1] Padé approximant, is also briefly discussed.

An overview of variational methods

Variational methods provide a powerful method for determining the behavior of complex physical systems without needing to solve the equations of state or satisfy the boundary conditions exactly. This technique has been widely applied in the areas of structural vibration and scattering, but has not been used much in the areas of transduction and array modeling. The advantage of this method over many others is that the answers obtained from variational principles are always significantly better than the inputs to them. A totally general method for deriving stationary variational principles from the equations of state and the boundary/initial conditions will be shown. Techiniques for choosing the inputs, or trial functions, judiciously will be discussed. The method will be illustrated using two examples: Determining the resonance frequencies of a clamped vibrating string and of a string with an attached point mass. The results of the variational treatment will be compared with exact analytical solutions of the problem for both accuracy and simplicity of solution. A brief historical overview of variational methods in acoustics will also be given.

Self‐dual phase‐space representation of quantum mechanics and the variational principle

AbstractThe important duality problem of phase‐space representations of quantum mechanics is discussed and the class of self‐dual representations is established. It is shown that self‐dual phase‐space representations may be endowed with a Hilbert space structure resembling the standard picture of quantum mechanics. A stationary variational principle is then formulated, enabling one to solve eigenproblems of observables directly in phase space. A modified, more usable form of this variational principle is also given together with an example of practical application to the 1D Coulomb system. © 1994 John Wiley & Sons, Inc.

Variational principles for nonpotential operators

One presents numerous approaches for the construction of variational principles for equations with operators which, in general, are nonpotential. One considers separately linear and nonlinear ordinary differential equations, partial and integropartial differential equations. One constructs and investigates both extremal and stationary variational principles and one gives applications of these principles in theoretical physics and in analytic mechanics. A series of unsolved problems are indicated. The survey is intended for mathematicians, physicists, working in both theoretical and applied areas, as well as for graduate students of physics and mathematics.

A variational principle for resonances

A stationary variational principle for calculation of the complex poles of Green functions is given.

A variational principle for transport theory

A maximal variational principle is used to construct an infinite medium Green’s function for treating the boundary value problems of the linear transport theory (neutron and radiative). For the neutrons we consider the one-speed case and correspondingly for the radiative transfer the monochromatic case. The scattering properties of the medium are presumed to be dependent on the relaxation length. Thus, for the neutrons the secondary production function depends on the neutron’s relaxation length and for the radiative transfer the albedo for single scattering is dependent on the optical depth. These two functions are kept arbitrary so that a large class of problems can be covered. The basic principle involves a functional which is an absolute maximum when the trial function is an exact solution of an integral equation of the Fredholm type. The kernel of the integral equation is required to satisfy certain symmetry and boundedness properties. We also exhibit an interesting relation between the absolute maximal and Schwinger’s stationary variational principles, which in general is neither a maximum nor a minimum.

Subspace quantum mechanics and the variational principle

This work is a continuation and extension of the delineation of the properties of a quantum subspace—a region of the real space of a molecular system bounded by a surface through which the flux in the gradient of the (observable) charge density is zero. Such subspaces are of interest as they constitute a basis for theoretical definitions of chemical concepts as obtained through experiment. The subspace stationary state variational principle, which was previously used to define this particular class of subspaces, reveals a close interrelationship between the hypervirial theorem and the variational property of the energy. By exploiting this interrelationship, one may obtain a variational solution to Schrödinger’s equation with the zero-flux surface requirement serving as a variational constraint. The connection between the statement of the subspace variational principle and the result of perturbation theory is established at the level of the first-order correction to the total energy. The statement of the generalized time-dependent subspace variational principle is derived using a modified form of Hamilton’s principle. The principle is also a variational statement of a time-dependent hypervirial theorem, generalized to systems bounded by surfaces of zero-flux in the gradient of the charge density. It, therefore, enables one to describe the time dependence of subspace averaged properties. The use of virial sharing operators, which permit the definition of single particle potential energies, in a quantum analogue of Hamilton’s extended principle, leads to a unique variational definition of the electronic energy within the Born–Oppenheimer approximation. In terms of this extended principle, both the classical and the quantum equations of motion are obtained for systems in which forces of constraint are operative by demanding that the integral of the variation of the kinetic energy plus the virtual work involved in the variation be zero. As a result of this discussion, the total energy of a subspace of a molecular system is defined in terms of separate electronic and nuclear contributions. This definition of the energy of a subspace applies whether or not applied forces are acting on the nuclei of the system.

Existence and uniqueness of solutions to a stationary finite element model of the biharmonic equation

This paper describes the finite element approximations based on a stationary variational principle of the biharmonic equation. Independent approximations for the solution and its second derivatives are used in the element: the normal derivatives of the solution and its second derivatives are approximated independently on the element boundary. Thus, the present model is a ‘mixed-hybrid’ model. Existence and uniqueness of solutions to the exact weak (or variational) problem are established and the associated finite element approximations are described. Existence and uniqueness of the finite element solutions are proved and a priori error estimates are given.