A dual form of Noether's theorem with applications to continuum mechanics

Considering simultaneously the equations of motion of the physical system and of the non-physical adjoint system, we introduce a general form of Noether's theorem by constructing a “dual Lagrangian” functional with a corresponding invariant of motion which preserves its value along the trajectories of combined physical and unphysical systems. The statement of invariance of this functional reduces to the classical statement of Noether's theorem if the system is self-adjoint; some possible generalizations are indicated. Applications to continuum mechanics are discussed within the framework of Noble's dual variational formulation.     

Macro-Hybrid Variational Formulations of Constrained Boundary Value Problems

In the context of convex analysis, macro-hybrid variational formulations of constrained boundary value problems are presented. Monotone mixed variational inclusions are macro-hybridized on the basis of nonoverlapping domain decompositions, and corresponding three-field versions are derived. Then, for regularization purposes, augmented formulations are established via preconditioned exact penalizations and expressed in terms of proximation operators. Optimization interpretations are given for potential problems, recovering the classic two- and three-field augmented Lagrangian formulations. Furthermore, associated parallel two- and three-field proximal-point algorithms are discussed for numerical resolution of finite element discretizations. Applications to dual mixed variational formulations of problems from mechanics illustrate the theory.     

On different geometric formulations of Lagrangian formalism

We consider two geometric formulations of Lagrangian formalism on fibred manifolds: Krupka's theory of finite order variational sequences, and Vinogradov's infinite order variational sequence associated with the -spectral sequence. On one hand, we show that the direct limit of Krupka's variational bicomplex is a new infinite order variational bicomplex which yields a new infinite order variational sequence. On the other hand, by means of Vinogradov's -spectral sequence, we provide a new finite order variational sequence whose direct limit turns out to be the Vinogradov's infinite order variational sequence. Finally, we provide an equivalence of the two finite order and infinite order variational sequences modulo the space of Euler-Lagrange morphisms.     

Pseudo-Rigid Bodies: A Geometric Lagrangian Approach

The pseudo-rigid body model is viewed within the context of continuum mechanics and elasticity theory. A Lagrangian reduction, based on variational principles, is developed for both anisotropic and isotropic pseudo-rigid bodies. For isotropic Lagrangians, the reduced equations of motion for the pseudo-rigid body are a system of two (coupled) Lax equations on so(3)×so(3) and a second-order differential equation on the set of diagonal matrices with a positive determinant. Several examples of pseudo-rigid bodies such as stretching bodies, spinning gas cloud and Riemann ellipsoids are presented.     

Relations between symmetries and conservation laws for difference systems

This paper presents a relation between divergence variational symmetries for difference variational problems on lattices and conservation laws for the associated Euler–Lagrange system provided by Noether's theorem. This inspires us to define conservation laws related to symmetries for arbitrary difference equations with or without Lagrangian formulations. These conservation laws are constrained by partial differential equations obtained from the symmetries generators. It is shown that the orders of these partial differential equations have been reduced relative to those used in a general approach. Illustrative examples are presented.     

A non-conventional discontinuous Lagrangian for viscous flow

Although many attempts for finding a variational formulation of Navier-Stokes equations have been made, a Lagrangian for viscous flow has not been established yet. An auspicious suggestion was made by Scholle [1] by extending Seliger and Whitham's Lagrangian [2] with additional terms ending up with some partial success: on the one hand, the phenomenon ‘viscosity’ occurs in a qualitatively correct manner, on the other hand the equations of motion resulting from the variation of Hamilton's principle differ from Navier-Stokes equations and therefore, their solutions reveal noticeable quantitative differences to those of Navier-Stokes equations. In this paper the Lagrangian [1] is modified by applying an innovative idea by Anthony [3], motivated by the reformulation of the Lagrangian in terms of complex fields, which can also be understood as the inversion of Madelungs idea [4] of reformulating the complex Schrödinger's equation into a hydrodynamic form. The prize one has to pay is that the resulting Lagrangian is discontinuous and therefore the mathematical treatment of the related variational problem challenging. Furthermore, an additional parameter, ω₀, has to be introduced. However, it is demonstrated that Navier-Stokes equations are recovered by the limit ω₀ → ∞, whereas the case of finite ω₀ can be interpreted as a generalization towards non-equilibrium thermodynamics [3]. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Basis difference method for orthogonal systems on a surface

The basis operator method intended for constructing systems of difference approximations to differential operators in vector and tensor analysis is extended to orthogonal systems on a surface. A class of completely conservative differential-difference schemes for continuum mechanics in Lagrangian variables is constructed. Basis operators are constructed using the finite volume equation, consistency conditions for discrete operators of the first derivative, and consistent projection operators for grid functions. A system of differential-difference continuum mechanics equations on a surface is obtained, which implies all conservation laws typical of the continuum case, including additional ones. A stability estimate is derived for discrete equations of an incompressible viscous fluid.     

On a boundary variational inequality of the second kind modelling a friction problem

A scalar contact problem with friction governed by the Yukawa equation is reduced to a boundary variational inequality. The presence of the non‐differentiable friction functional causes some difficulties when approximated. We present two methods to overcome this difficulty. The first one is a regularization leading to a non‐linear boundary variational equation, for which we propose an iterative procedure, whereas the second method is based on the boundary mixed variational formulation involving Lagrange multipliers. We propose Uzawa's algorithm to compute the saddle point of the corresponding boundary Lagrangian and investigate the discretization of various formulations by the boundary element Galerkin method. Convergence of the boundary element solution is proved and a convergence order is obtained. Copyright © 2002 John Wiley & Sons, Ltd.     

On a Deformation-Driven Continuum-Related Parrinello-Rahman Molecular Dynamics

Investigations into the atomistic-to-continuum coupling are recently pursued in literature. A hierarchical modelling in terms of a macroscale treated by continuum mechanics and the microscale governed by statistical mechanics may be a very fruitful combination. If the microscale is simulated with the help of molecular dynamics, the isostress-isoenthalpic ensemble as proposed by Parrinello and Rahman presents a beneficial choice. This statistical ensemble is remarkable as the equations of motion are derived from a Lagrangian. Recently, this Lagrangian was situated into a continuum mechanics setting. This paper investigates the behavior of this continuum-related Lagrangian in a kinetics-driven setting (by imposing an external stress) and a kinematics-driven setting (by imposing the shape of the molecular dynamics cell) in terms of a numerical example. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Variational principles for systems with unilateral constraints

Derivations and formulations are given of the variational principles of analytical mechanics for systems with unilateral ideal smooth constraints, originally established for systems with bilateral constraints. The virtual work principle, the Fourier inequality, the d’Alembert–Lagrange principle, the Gauss principle of least constraint and its modification – the Chetayev principle of maximum work, the Jourdain principle, the Hamilton–Ostrogradskii principle, the principle of least action in Lagrangian and Jacobian forms, and the Suslov–Voronets principle are described.     

离散型固体力学及其间断型变分原理

本文对即将形成的一门新的学科——离散型固体力学的基本假设,微分形式的方程,及其间断型变分原理进行了论述.离散型固体力学是离散介质力学的一个分支,是近期以来力学的一个发展方向.它是基于固体体系具有离散性、可分性,待解函数在定义城内具有各种不同的间断性,以及定义域的边界可动性的基础上,为解决种种情况下的固体的应力,位移、应变所形成的力学系统.当待解函数在整个定义域内为充分光滑的函数类和略去边界可动性的影响时,则离散型固体力学就退化为连续介质力学范畴的古典固体力学.它所属的变分原理,在相应的情况下,也就退化为古典与非占典变分原理.     

A New Minimum Principle for Lagrangian Mechanics

We present a novel variational view at Lagrangian mechanics based on the minimization of weighted inertia-energy functionals on trajectories. In particular, we introduce a family of parameter-dependent global-in-time minimization problems whose respective minimizers converge to solutions of the system of Lagrange’s equations. The interest in this approach is that of reformulating Lagrangian dynamics as a (class of) minimization problem(s) plus a limiting procedure. The theory may be extended in order to include dissipative effects thus providing a unified framework for both dissipative and nondissipative situations. In particular, it allows for a rigorous connection between these two regimes by means of Γ-convergence. Moreover, the variational principle may serve as a selection criterion in case of nonuniqueness of solutions. Finally, this variational approach can be localized on a finite time-horizon resulting in some sharper convergence statements and can be combined with time-discretization.     

Variational and Geometric Structures of Discrete Dirac Mechanics

In this paper, we develop the theoretical foundations of discrete Dirac mechanics, that is, discrete mechanics of degenerate Lagrangian/Hamiltonian systems with constraints. We first construct discrete analogues of Tulczyjew’s triple and induced Dirac structures by considering the geometry of symplectic maps and their associated generating functions. We demonstrate that this framework provides a means of deriving discrete Lagrange–Dirac and nonholonomic Hamiltonian systems. In particular, this yields nonholonomic Lagrangian and Hamiltonian integrators. We also introduce discrete Lagrange–d’Alembert–Pontryagin and Hamilton–d’Alembert variational principles, which provide an alternative derivation of the same set of integration algorithms. The paper provides a unified treatment of discrete Lagrangian and Hamiltonian mechanics in the more general setting of discrete Dirac mechanics, as well as a generalization of symplectic and Poisson integrators to the broader category of Dirac integrators.     

Damage and delamination processes in thermo-viscoelastic materials

This contribution addresses several models for rate-independent damage and delamination processes in thermo-viscoelastic materials. In the spirit of continuum damage mechanics, both degradation phenomena are modeled by means of internal variables, governed by a rate-independent flow rule. The latter is coupled in a highly nonlinear way with the heat equation and the momentum balance for the displacements. We present a suitable weak formulation for this class of models, and discuss existence and approximation results in the framework of variational convergence. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Morse index and stability of elliptic Lagrangian solutions in the planar three-body problem

We illustrate a new way to study the stability problem in celestial mechanics. In this paper, using the variational nature of elliptic Lagrangian solutions in the planar three-body problem, we study the relation between Morse index and its stability via Maslov-type index theory of periodic solutions of Hamiltonian system. For elliptic Lagrangian solutions we get an estimate of the algebraic multiplicity of unit eigenvalues of its monodromy matrix in terms of the Morse index, which is the key to understand the stability problem. As a special case, we provide a criterion to spectral stability of relative equilibrium.     

Variational formulations for discontinuous displacement fields in problems of the deformation theory of plasticity without hardening

The problem of the construction and use of extended variational formulations which enable an explicit analysis to be made of discontinuous displacement fields for a wide class of problems of the deformation theory of plasticity is discussed. Three-dimensional, as well as plane problems with the Mises and Schleicher-Moreau criteria are investigated. In the case of a piecewise-continuous discontinuity line it is shown that the existence of a saddle point of an extended Lagrangian results in an integral inequality, which imposes certain conditions on the trace of the stress tensor on the line of discontinuity. Different arguments were used in [1–3] to obtain different versions of this condition for a number of problems of the theory of plasticity. When sufficient regularity of the stresses is assumed, then from the condition in question a simple algebraic relation follows connecting, at the line of discontinuity, the value of the stress tensor with the parameters determining the magnitude and direction of the discontinuity. Examples are given, which show that, generally speaking, only some of the stress states lying on the yield surface correspond to discontinuous solutions.     

Variational Formulations for Viscous Flow

For physical systems, the dynamics of which is formulated within the framework of Lagrange formalism the dynamics is completely defined by only one function, namely the Lagrangian. As well-known the whole conservative Newtonian mechanics has been successfully embedded into this methodical concept. Different from this, in continuum theories many open questions remain up to date, especially when considering dissipative processes. The viscous flow of a fluid, given by the Navier-Stokes equations is a typical example for this. In this contribution a special approach for finding a Lagrangian for viscous flow is suggested and discussed. The equations of motion resulting from the respective Lagrangian are compared to the Navier-Stokes equations and differences are discussed. For a simple flow example their solution is compared to the one resulting from Navier-Stokes equations. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)