Potential systems for PDEs having several conservation laws

A generalization of the usual procedure for constructing potential systems for systems of partial differential equations with multidimensional spaces of conservation laws is considered. More precisely, for the construction of potential systems with a multi-dimensional space of local conservation laws, instead of using only basis conservation laws, their arbitrary linear combinations are used that are inequivalent with respect to the equivalence group of the class of systems or symmetry group of the fixed system. It appears that the basis conservation laws can be equivalent with respect to groups of symmetry or equivalence transformations, or vice versa; in this sense the number of independent linear combinations of conservation laws can be grater than the dimension of the space of conservation laws. The first possibility leads to an unnecessary, often cumbersome, investigation of equivalent systems, the second one makes possible that a great number of inequivalent potential systems are missed. Examples of all these possibilities are given.     

Quasivelocities and symmetries in non-holonomic systems

This article is concerned with the theory of quasivelocities for non-holonomic systems. The equations of non-holonomic mechanics are derived using the Lagrange–d'Alembert principle written in an arbitrary configuration-dependent frame. The article also shows how quasivelocities may be used in the formulation of non-holonomic systems with symmetry. In particular, the use of quasivelocities in the analysis of symmetry that leads to unusual momentum conservation laws is investigated, as is the applications of these conservation laws and discrete symmetries to the qualitative analysis of non-holonomic dynamics. The relationship between asymptotic dynamics and discrete symmetries of the system is also elucidated.     

Conservation laws for 3-dimensional compressible Euler equations

The generators of infinitesimal symmetry transformations for the Euler equations, and the corresponding set of adjoint variables are derived. The associated conservation laws are then discussed. A detailed analysis of 1-dimensional flows brings into evidence the connections with current alternative approaches to conservation laws.     

A general approach to conservation laws in viscoelasticity

Summary The equation of motion and the associated Lagrangian density of a linear viscoelastic solid are considered in terms of the Laplace transform of the displacement vector. A generalized form of Noether's theorem is applied where invariance is meant up to a divergence. Then a systematic approach is developed which allows the derivation of conservation laws with non-vanishing densities. As a preliminary application, new conservation laws are found and previous results are recovered.     

Path-independent integral for the dilatation symmetry group in thermoplasticity

The aim of this study is to establish a fatigue criteria based on an energy threshold corresponding to crack initiation in the case of dilatational symmetry of a structure under thermomechanical loading. An extension of the M integral to thermoplasticity relating to the scaling symmetry is presented. Based on a discrete Lagrangian describing the thermo-inelastic system and Noether's theorem defining the conservation law, the null Lagrangian theorem is applied in order to introduce thermoplasticity. By the use of the divergence theorem, the modified M integral is obtained. Path-domain independence is discussed illustrated by a numerical example.     

On the duality principle of conservation laws in finite elastodynamics

The duality principle of conservation laws which holds in finite elastodynamics is studied using the two-point tensor method. Based on the general Noether's theorem, two basic equations of variational invariance are first derived, which correspond to the action integrals given, respectively, in Lagrangian and Eulerian representations for a finite motion of an elastic body. The dual relations between the conservation laws in both representations are given. The procedure for constructing these dual relations is to apply simultaneously the same infinitesimal transformation of either time or position coordinates as well as field variables to the dual equations of variational invariance, where the position coordinates could be taken either from the reference configuration or from the deformed configuration of the material body. Based on these dual relations it is shown that the conservation equations of material momentum and moment of material momentum possess the same structure as those of physical momentum and physical moment of momentum. Furthermore, three pairs of dual relations between stress tensors and material momentum tensors of various kinds are derived based on the duality principle by using the two-point tensor method. Finally, using the dual integral forms of conservation laws the concepts of dynamic material force and moment acting on defects are introduced and analyzed. The force and moment can be decomposed into a pure kinetic part and a pure deformation part, the latter corresponding to the path-independent integral as suggested in elastostatics.     

Relevance of symmetry methods in mechanics of materials

The interest and relevance of symmetry methods as a predictive and systematic methodology in the continuum mechanics of materials is analyzed, relying on a classification of the inherent aspects in terms of the direct, extended direct, and inverse methods. Although being interrelated, these three problems each have a specific argumentation which is separately exposed in the present contribution. The direct problem of finding invariants associated with a given constitutive law for materials, including dissipation, is first envisaged. The abstract formulation of constitutive laws in terms of the state laws and a dissipation potential expressing the evolution of internal state variables is considered, in the framework of irreversible thermodynamics. It is shown that a specific choice of the components of the symmetry vector acting in the space of independent and dependent variables leads to a local invariance condition of the constitutive law fully equivalent to the variational symmetry condition using the rate of the internal energy density. As a specific situation involving this methodology, a time–temperature equivalence principle of polymers is obtained from the requirement of group invariance of the field equations. A validation of this invariance principle is given by a comparison of the modelled master response and the master curve constructed from a set of experimental results at various temperatures. The extended direct method is next presented as a generalization of the direct method, in the sense that a classification of constitutive functions modelling the material behavior is achieved via a symmetry analysis. In the third part of the paper, the inverse problem of constructing a material’s constitutive law exploiting a postulated Lie-group structure is exposed. A constitutive model is then identified which satisfies the symmetries exhibited by the experimental data.     

Application of polynomial preconditioners to conservation laws

Polynomial preconditioners which are suitable in implicit time-stepping methods for conservation laws are reviewed and analyzed. The preconditioners considered are either based on a truncation of a Neumann series or on Chebyshev polynomials for the inverse of the system-matrix. The latter class of preconditioner is optimal in a space of polynomials of certain degree if the matrix has only real eigenvalues and a non-singular system of eigenvectors. The preconditioning can be applied to any convergent splitting of the system matrix, i.e. to any classical implicit time-stepping method for conservation laws that is based on a quasi-Newton iteration. An efficient implementation based on SSOR is presented and the approach is applied to simulations of the viscous unsteady Burgers equation and to inviscid steady flow around an airfoil in two spatial dimensions to illustrate the method in large-scale computations. For viscous flows the efficiency increase due to preconditioning is considerable.     

Theories of Linear Response in BCS Superfluids and How They Meet Fundamental Constraints

We address the importance of symmetry and symmetry breaking on linear response theories of fermionic BCS superfluids. The linear response theory of a noninteracting Fermi gas is reviewed and several consistency constraints are verified. The challenge to formulate linear response theories of BCS superfluids consistent with density and spin conservation laws comes from the presence of a broken U(1)_EM symmetry associated with electromagnetism (EM) and we discuss two routes for circumventing this. The first route follows Nambu’s integral-equation approach for the EM vertex function, but this method is not specific for BCS superfluids. We focus on the second route based on a consistent-fluctuation-of-the order-parameter (CFOP) approach where the gauge transformation and the fluctuations of the order parameter are treated on equal footing. The CFOP approach allows one to explicitly verify several important constraints: The EM vertex satisfies not only a Ward identity which guarantees charge conservation but also a Q-limit Ward identity associated with the compressibility sum rule. In contrast, the spin degrees of freedom associated with another U(1) _z symmetry are not affected by the Cooper-pair condensation that breaks only the U(1)_EM symmetry. As a consequence the collective modes from the fluctuations of the order parameter only couple to the density response function but decouple from the spin response function, which reflects the different fates of the two U(1) symmetries in the superfluid phase. Our formulation lays the ground work for applications to more general theories of BCS-Bose Einstein Condensation (BEC) crossover both above and below T _c .     

On conservation laws in geometrically nonlinear elasto-dynamic field of non-homogenous materials

By applying Noether’s theorem to the Lagrangian density of non-homogenous elastic materials in the so-called Lagrangian framework, conservation laws in geometrically nonlinear elasto-dynamic field have been studied, and a clear picture of relations between the conservation laws in material space and the material balance laws is given. It is found that the mass density and Lamé’s moduli have to satisfy a set of first-order linear partial differential equations, which contain all the symmetry-transformations of space–time based on Newtonian viewpoint of mechanics. The existence and existent forms of conservation laws in material space are governed by these equations. Especially, translation and rotation of coordinates are symmetry-transformations of the Lagrangian density for obtaining both the conservation laws of homogenous material and the material balance laws of non-homogenous material, but change of coordinate scale is not. However, if the mass density and Lamé’s moduli satisfy special equations simplified from those partial differential equations, change of coordinate scale becomes a symmetry-transformation of the Lagrangian density from which a conservation law follows, whereas the associated material balance law does not exist still. An insight into the usability of those equations for constructing conservation laws is presented, and all the non-trivial conservation laws of the functionally graded material (FGM) layer bonded to a substrate are given for mechanical analysis.     

Calculating the (Almost) Exact Control Limits for a C-Chart

A power transformation using an exponent of 2/3 for Poisson-distributed data, with a small constant added, achieves symmetry for improved statistical process control (SPC) applications whether it is for an individual, a cumulative sum, or an exponentially weighted moving average chart. Two simple equations are proposed for calculating the lower control limit (LCL) and the upper control limit (UCL) for Poisson type data. Agreement between the exact LCL and UCL, as determined by the lower and upper tail area, is excellent. The square-root transformation that stabilizes the variance produces a negatively skewed distribution and tends to give false SPC signals.     

Conservation laws and the material momentum tensor for the elastic dielectric

It is shown that a Langrangian formulation of continuum mechanics can provide not only the equations of motion, but the conservation laws related to the material symmetries in a perfect continuum interacting with an external electric field. These conservation laws in the presence of defects lead to the path-independent integrals widely used in fracture mechanics. They are basically related to the “material force” on a defect in a continuum. The quantity playing the role of the physical stress tensor in this formulation is the material momentum tensor. A material force in the form of a path-independent integral for the elastic dielectric is derived employing Toupin's [1] formulations.     

The use of central reflection in the formulation of unit cells for micromechanical FEA

Central reflection as a type of symmetry has been exploited in the context of structural analysis for the first time in this paper, especially in terms of applications to the formulation of unit cells for micromechanical FE analysis of materials of periodic microstructures. It reduces the size of the unit cell to be analysed without compromising useful features of the unit cells obtained purely from periodic conditions, viz. the unit cell can still be analysed with a single set of boundary conditions under all loading conditions and any combined loading condition. Such a feature cannot be preserved if a conventional plane reflectional symmetry or a rotational symmetry has been used in order to reduce the size of the unit cell to be analysed. Most of existing unit cells obtained purely from periodic conditions can benefit from this further symmetry. Boundary conditions for unit cells of reduced sizes resulting from the use of the centrally reflectional symmetry have been derived rationally in this paper. They have been validated systematically through examples of applications.     

Computation of fluxes of conservation laws

The direct method for the construction of local conservation laws of partial differential equations (PDE) is a systematic method applicable to a wide class of PDE systems (S. Anco and G. Bluman, Eur J Appl Math 13:567–585, 2002). According to the direct method one seeks multipliers, such that the linear combination of PDEs of a given system with these multipliers yields a divergence expression. Once local-conservation-law multipliers have been found, one needs to reconstruct the fluxes of the conservation law. In this review paper, common methods of flux computation are discussed, compared, and illustrated by examples. An implementation of these methods in symbolic software is also presented.     

Fracture Parameter for Thermoplasticity in the Case of Dilatation Symmetry

An extension of the M integral to thermoplasticity related to the dilatational or scaling symmetry is presented. Based on a discrete Lagrangian describing the thermoinelastic system andon Noether's theorem defining the conservation law, the null Lagrangian theorem is applied in order to introduce thermoplasticity. By the use of the divergence theorem, the modified M integral is obtained.     

Conservation laws for one-dimensional isentropic gas flows

The governing equations of one-dimensional isentropic gas flow are expressed in terms of a pair of exterior differential forms. By employing Cartan's theory and the classical Frobenius theorem linear partial differential equations are obtained to determine conservation laws which are conjectured to be the key to detect completely integrable systems. By using a similarity technique explicit expressions are provided for polynomial type of conservation laws in terms of Gegenbauer and Chebyshev polynomials.     

Lagrange identification of absolutely unstable regimes in wakes

Summary The recent theory of a nonlocal micropolar continuum is used to derive explicit expressions for the set of conservations laws. Using the Euclidean group of transformation, the equivalence between conservation laws and Euclidean invariance is demonstrated. It is shown that one of the integrals (J type integral) has a physical meaning of the energy release rate.     

反铁电晶体PbTaO3的对称性和序参量

根据室温下X射线衍射结果,仔细研究了反铁电晶体PbTaOa的晶体结构。确定PbTaO3在973K发生顺电一反铁电相变,反铁电相对称性所属点群是P2／m（C2i）。用轴矢量R作为序参量来描述反铁电相便是对称性的变化,对于已得到的晶体相变前后的对称点群,应用居里原理,得到了理想结果。     

Conservation laws of the J-integral type in micropolar elastostatics

Noether's theorem is used in an appropriate generalized form to deduce cnnservation laws of the J-integral type in the linear elastostatics of micropolar continua. The 1-parameter family of transformations associated with these conservation laws is determined on the basis of physically reasonable assumptions.