Classical Variational Principles Research Articles

Вариационные принципы Экланда и Бишопа–Фелпса в частично упорядоченных пространствах

In this paper, an assertion about the minimum of the graph of a mapping acting in partially ordered spaces is obtained. The proof of this statement uses the theorem on the minimum of a mapping in a partially ordered space from [A.V. Arutyunov, E.S. Zhukovskiy, S.E. Zhukovskiy. Caristi-like condition and the existence of minima of mappings in partially ordered spaces // Journal of Optimization Theory and Applications. 2018. V. 180. Iss. 1, 48–61]. It is also shown that this statement is an analogue of the Eckland and Bishop-Phelps variational principles which are effective tools for studying extremal problems for functionals defined on metric spaces. Namely, the statement obtained in this paper and applied to a partially ordered space created from a metric space by introducing analogs of the Bishop-Phelps order relation, is equivalent to the classical Eckland and Bishop-Phelps variational principles.

Yield limited optimal topology design of elastoplastic structures

This study is devoted to a novel method for topology optimization of elastoplastic structures subjected to stress constraints. It should be noted that in spite of the classical solutions of the different type of elastoplastic topology problems are more than 70 years old, the integration of the Prandtl-Reuss constitutive equations into the topology optimization process is not very often investigated in the last three decades. In the presented methodology where the classical variational principles of plasticity and the functor-oriented programming technique are applied in topology design, the aim is to find a minimum weight structure which is able to carry a given load, fulfills the allowable stress limit, and is made of a linearly elastic, perfectly plastic material. The optimal structure is found in an iterative way using only a stress intensity distribution and a return mapping algorithm. The method determines representative stresses at every Gaussian point, averages them inside every finite element using the von Mises yield criterion, and removes material proportionally to the stress intensities in individual finite elements. The procedure is repeated until the limit load capacity is exceeded under a given loading. The effectiveness of the methodology is illustrated with three numerical examples. Additionally, different topologies are presented for a purely elastic and an elastoplastic material, respectively. It is also demonstrated that the proposed method is able to find the optimal elastoplastic topology for a problem with a computational mesh of the order of tens of thousands of finite elements.

Rational eigenvalue problems and applications to photonic crystals

We establish new analytic results for a general class of rational spectral problems. They arise e.g. in modelling photonic crystals whose capability to control the flow of light depends on specific features of the eigenvalues. Our results comprise a complete spectral analysis including variational principles and two-sided bounds for all eigenvalues, as well as numerical implementations. They apply to the eigenvalues between the poles where classical variational principles fail completely. In the application to multi-pole Lorentz models of permittivity functions we show, in particular, that our abstract two-sided eigenvalue estimates are optimal and we derive explicit bounds on the band gap above a Lorentz pole. A high order finite element method (FEM) is used to compute the two-sided bounds for a selection of eigenvalues for several concrete Lorentz models, e.g. polaritonic materials and multi-pole models.

On the correspondence between quantum and classical variational principles

Classical variational principles can be deduced from quantum variational principles via formal reparameterization of the latter. It is shown that such reparameterization is possible without invoking any assumptions other than classicality and without appealing to dynamical equations. As examples, first principle variational formulations of classical point-particle and cold-fluid motion are derived from their quantum counterparts for Schrödinger, Pauli, and Klein–Gordon particles.

Multivariate variational principles for topological pressure

We prove two variational principles for continuous real-valued functions defined on Cartesian products of a discrete dynamical system. They involve nonadditive topological pressure for a certain class of potentials, and can be seen as extension to the classical variational principles.

レーリーの原理の逆問題型定式化に基づくせん断型構造物の剛性決定法

The shape, boundary conditions of the loads and displacements, and distributions of mass and stiffness of a continuum are given in classical variational principles, whereas an inverse variational principle deals with a type of problems where boundary shape is unknown. Another type of inverse formulation of variational principle proposed in this paper is applicable to sizing optimization or a stiffness identification problem. This paper discusses an inverse formulation of Rayleigh's principle that is useful to identify unknown stiffness for a given eigenvalue and eigenmode. An analytical example is employed to illustrate that the inverse minimum principle of potential energy proposed in this paper is useful in identifying discretized stiffness distribution of a shear structure having designated eigenvalue and eigenmode.

Rigorous bounds on the effective moduli of composites and inhomogeneous bodies with negative-stiffness phases

We review the theoretical bounds on the effective properties of linear elastic inhomogeneous solids (including composite materials) in the presence of constituents having non-positive-definite elastic moduli (so-called negative-stiffness phases). Using arguments of Hill and Koiter, we show that for statically stable bodies the classical displacement-based variational principles for Dirichlet and Neumann boundary problems hold but that the dual variational principle for traction boundary problems does not apply. We illustrate our findings by the example of a coated spherical inclusion whose stability conditions are obtained from the variational principles. We further show that the classical Voigt upper bound on the linear elastic moduli in multi-phase inhomogeneous bodies and composites applies and that it imposes a stability condition: overall stability requires that the effective moduli do not surpass the Voigt upper bound. This particularly implies that, while the geometric constraints among constituents in a composite can stabilize negative-stiffness phases, the stabilization is insufficient to allow for extreme overall static elastic moduli (exceeding those of the constituents). Stronger bounds on the effective elastic moduli of isotropic composites can be obtained from the Hashin–Shtrikman variational inequalities, which are also shown to hold in the presence of negative stiffness.

Gyarmati’s Variational Principle of Dissipative Processes

Like in mechanics and electrodynamics, the fundamental laws of the thermodynamics of dissipative processes can be compressed into Gyarmati’s variational principle. This variational principle both in its differential (local) and in integral (global) forms was formulated by Gyarmati in 1965. The consistent application of both the local and the global forms of Gyarmati’s principle provides all the advantages throughout explicating the theory of irreversible thermodynamics that are provided in the study of mechanics and electrodynamics by the corresponding classical variational principles, e.g., Gauss’ differential principle of least constraint or Hamilton’s integral principle.

Review of stochastic mechanics

Stochastic mechanics is an interpretation of nonrelativistic quantum mechanics in which the trajectories of the configuration, described as a Markov stochastic process, are regarded as physically real. The natural stochastic generalization of classical variational principles leads to a derivation of the Schrödinger equation. A brief review of the successes and failures of the theory is given, with references.

Symmetry in variational principles and applications

We formulate symmetric versions of classical variational principles. Within the framework of nonsmooth critical point theory, we detect Palais–Smale sequences with additional second-order and symmetry information. We discuss applications to partial differential equations, fixed point theory and geometric analysis.

Sharp bounds on the volume fractions of two materials in a two-dimensional body from electrical boundary measurements: the translation method

We deal with the problem of estimating the volume of inclusions using a small number of boundary measurements in electrical impedance tomography. We derive upper and lower bounds on the volume fractions of inclusions, or more generally two phase mixtures, using two boundary measurements in two dimensions. These bounds are optimal in the sense that they are attained by certain configurations with some boundary data. We derive the bounds using the translation method which uses classical variational principles with a null Lagrangian. We then obtain necessary conditions for the bounds to be attained and prove that these bounds are attained by inclusions inside which the field is uniform. When special boundary conditions are imposed the bounds reduce to those obtained by Milton and these in turn are shown here to reduce to those of Capdeboscq–Vogelius in the limit when the volume fraction tends to zero. The bounds of this article, and those of Milton, work for inclusions of arbitrary volume fractions. We then perform some numerical experiments to demonstrate how good these bounds are.

Generalized Variational Principles for Uncertainty Quantification of Boundary Value Problems of Random Heterogeneous Materials

Asymptotic theories of classical micromechanics are built on a fundamental assumption of large separation of scales. For random heterogeneous materials the scale-decoupling assumption however is inapplicable in many circumstances from conventional failure problems to novel small-scale engineering systems. Development of new theories for scale-coupling mechanics and uncertainty quantification is considered to have significant impacts on diverse disciplines. Scale-coupling effects become crucial when size of boundary value problems (BVPs) dynamic wavelength is comparable to the characteristic length of heterogeneity or when local heterogeneity becomes crucial due to extreme sensitivity of local instabilities. Stochasticity, vanishing in deterministic homogenization, resurfaces amid multiscale interactions. Multiscale stochastic modeling is expected to play an increasingly important role in simulation and prediction of material failure and novel multiscale systems such as microelectronic-mechanical systems and metamaterials. In computational mechanics a prevalent issue is, while a fine mesh is desired to achieve high accuracy, a certain mesh size threshold exists below which deterministic finite elements become questionable. This work starts investigation of the scale-coupling problems by first looking at uncertainty of material responses due to randomness or incomplete information of microstructures. The classical variational principles are generalized from scale-decoupling problems to scale-coupling BVPs, which provides upper and lower variational bounds for probabilistic prediction of material responses. It is expected that the developed generalized variational principles will lead to novel computational methods for uncertainty quantification of random heterogeneous materials.

The method of integrodifferential relations for analysing the natural oscillations of membranes

An approach to solving of problems of the free oscillations of membranes, based on the method of integrodifferential relations, is proposed. The variational properties of the integrodifferential problems are investigated and their relation to classical variational principles is demonstrated. Integral and local criteria for the quality of the approximate solutions are proposed. A numerical-analytical algorithm is developed for finding the characteristic frequencies and modes of oscillations. Problems involving the free transverse motions of circular and elliptic membranes are considered as an example.

On variational principles of conformal mappings

Refinements and generalizations of the classical variational principles of conformal mappings are presented; mainly, they follow from potential theory and symmetrization. Part of the results can be viewed as properties of Robin functions and Robin capacities, and also as distortion theorems for univalent functions in finitely connected domains.

Applications of two critical point results for non-differentiable indefinite functionals

In this paper we study the existence of solution for two kinds of hemivariational inequalities: the first of them is of elliptic type, the second one of hamiltonian type. In those problems the energy functional is indefinite, so the classical variational principles can’t be used in a direct way. The results are an application of two theorems of existence of critical points for non-differentiable functionals recently obtained.

Applications of two critical point results for non-differentiable indefinite functionals

In this paper we study the existence of solution for two kinds of hemivariational inequalities: the first of them is of elliptic type, the second one of hamiltonian type. In those problems the energy functional is indefinite, so the classical variational principles can’t be used in a direct way. The results are an application of two theorems of existence of critical points for non-differentiable functionals recently obtained.

Inelastic Analysis by Symmetrization of the Constitutive Law

This paper deals with the transformation of the inelastic continuum problem from the differential to the variational form. Due to the lack of symmetry of the constitutive law, the variational formulation of the problem has been, above all, obtained using techniques of symmetrization of the overall problem operator (as with Tonti’s techniques). In the present paper, instead, the adopted method is based on the symmetrization of the constitutive law only. On this basis, taking into account the consequent transformations to be done on the equilibrium and on the compatibility boundary condition operators, a symmetric global operator for the original problem is achieved. This leads to a new ‘enlarged’ formulation and then, by the choice of a suitable operator, to a new wide class of variational principles and, by specialization, to classical and recent variational principles.

Dynamics, stability, and bifurcations

Dynamics, stability, and bifurcations

Mixed methods and flexibility approaches for nonlinear frame analysis

The principle of virtual forces has long been an attractive tool for linear matrix structural analysis because it provides a means to compute the exact flexibility matrix for a linear, elastic, non-prismatic Bernoulli–Euler beam element while the classical stiffness method, based on the principle of virtual displacements, does not. Nonlinear flexibility methods have recently appeared in the literature as a means of improving the accuracy of frame analysis when the curvature fields are more complex than the moment fields (e.g., when inelastic response is important). This paper surveys the basic methods of formulating nonlinear Bernoulli–Euler beam elements based on classical and mixed variational principles. We show that the variational structure provided by the Hellinger–Reissner and Hu–Washizu functionals gives a framework that reaps the benefits of the so-called nonlinear flexibility methods while allowing a standard treatment of non-linear problems by Newton's method.

A Unified Representation of Variational Principles in Non-Linear Elasticity

In elastomechanics there exist besides of the classical variational principles so-called generalized (extended, mixed) ones which are used mainly for the discretization of boundary and eigenvalue problems. These principles can be constructed together with their incremental versions systematically from the canonical basic equations characterized by four special operators (statics, kinematics, material, loading), and the boundary- and (eventually) transition conditions. Under certain assumptions they can be strengthened to complementary extremum principles. The proposed abstract and unified formulation holds true for all kinds of elastic structures and, additionally, for related problems and facilitates the construction of the corresponding variational principles. At last the relevance of drilling degrees of freedom is discussed.