Elliptic systems of partial differential equations and the finite element method

An existence and uniqueness theorem is established for finite element solutions of elliptic systems of partial differential equations. To establish this result, an extension of Gårding's inequality is obtained which is valid for functions that do not necessarily vanish on the boundary of the region. To accomplish this extension, a stronger ellipticity condition, called very strong ellipticity, is defined with various necessary and sufficient conditions given.     

On the equivalence of strong ellipticity in the material and spatial settings of finite elasticity

It is known from Shield’s theorem that the equilibrium equations for the inverse deformation in finite elasticity are equivalent to those for the primary deformation. We review this result for compressible elastic solids and then use it to provide a simple proof that strong ellipticity in the material setting is equivalent to that in the spatial setting. The result is then extended to incompressible materials.     

时滞神经网络全局渐近稳定性条件

利用Liapunov泛函方法,结合矩阵不等式技巧,分析了时滞细胞神经网络(DCNNs)的平衡点存在的唯一性和全局渐近稳定性,保证DCNNs的全局稳定性的一个新的充分判据被得到．所得判据提供了一些参数来适当地弥补了反馈矩阵与时滞反馈矩阵之间所需要的平衡关系．这些判据可以容易被使用来设计和检验全局稳定的网络．此外,所得判据是与时滞参数无关,且比已有文献具有更少的限制．     

Improved global exponential stability for delay difference equations with impulses

In this paper, by means of constructing an impulsive delay difference inequality, we study the global exponential stability for a class of delay difference equations with impulses. A new criterion ensuring the global exponential stability of the equilibrium point is obtained, which is less restrictive and conservative than that given in the earlier reference. An illustrative example is given to demonstrate the effectiveness and advantage of the obtained result.     

The problem of the equilibrium of a helical spring in the non-linear three-dimensional theory of elasticity

The problem of the loading of a helical spring by an axial force and a torque is considered using the three-dimensional equations of the non-linear theory of elasticity. The problem is reduced to a two-dimensional boundary-value problem for a plane region in the form of the transverse cross section of the coil of the spring. The solution of the two-dimensional problem obtained enables the equations of equilibrium in the volume of the body and the boundary conditions on the side surface to be satisfied exactly. The boundary conditions at the ends of the spring are satisfied in the integral Saint-Venant sense. The problem of the equivalent prismatic beam in the theory of springs is discussed from the position of the solution of the non-linear Saint-Venant problem obtained. The results can be used for accurate calculations of springs in the non-linear strain region, and also when developing applied non-linear theories of elastic rods with curvature and twisting.     

Variational approaches to solving initial-boundary-value problems in the dynamics of linear elastic systems

Problems of the controlled motion of an elastic body are considered in the linear theory. Using the method of integrodifferential relations, a family of quadratic functionals is introduced, which define the state of the elastic body, and variational formulations of the initial-boundary-value problem of dynamics are given. Euler's equations and boundary and terminal relations corresponding to them are obtained from the condition for the functionals to be stationary. It is shown that there is a relation between the proposed formulations and the Hamilton variational principle in the case of boundary-value and time-periodic problems of dynamics. A numerical algorithm is developed for finding the motions of an elastic body, based on piecewise-polynomial approximations and a criterion is proposed for estimating the quality of the approximate solutions. An example of the calculation and analysis of the forced transverse motions of a rectilinear beam with a square cross section is given for the three-dimensional model.     

Dynamics of non-equilibrium phase boundaries in a heat conducting non-linearly elastic medium

The phase transformation of the first kind in a non-linearly elastic heat conducting medium is simulated by the relationships on a strong discontinuity. A generalization of the Stefan formulation is given. An existence condition for stationary flow, analogous to the Gibbs phase equilibrium condition, is obtained for non-equilibrium phase boundaries. A pure dilatational phase transition in a compressible fluid and pure shear transformation of the twinning type in non-linearly elastic crystals are considered as model examples. The problem of the structure is solved for closure of the system of relationships on the shock.

A phase transformation ordinarily turns out to be localized in a narrow domain of space and it can be simulated in terms of the conditions on a strong discontinuity /1/. Formulation of the problem of the static equilibrium of liquid phases as well as of liquid and (non-linearly elastic) solid phases was given by Gibbs, who proposed a phase equilibrium criterion and formulated appropriate conditions on the shock; the extension of the Gibbs conditions to the case of the equilibrium of two solid phases is known in both the linear /2/ and non-linear /3/ theories of elasticity. The dynamic problem of the propagation of the equilibrium phase boundary is considered in the Stefan formulation as a rule, including the assumption about the continuity of the density (the strain tensor component) on the shock; the thermal problem is here separated from the mechanical one. Simulating the interphasal surface on the shock the temperature fields are merged by using the well-known Stefan conditions as well as the phase equilibrium condition that reduces to giving the temperature on the front.

The purpose of this paper is to extend the Stefan-Gibbs formulation to the case of the motion of a coherent isothermal phase boundary in a non-linearly elastic heat conducting medium and to derive the dynamic analogue of the phase equilibrium condition (and the Stefan conditions) with possible dissipation at the transformation front. Two dissipative mechanisms are examined, viscous and kinetic. The case of equilibrium phase boundaries was investigated in /4–6/.     

On the constitutive inequalities and acceleration waves in micropolar media

Acceleration waves in nonlinear thermoelastic micropolar media are considered. We establish the kinematic and dynamic compatibility relations for a singular surface of order 2 in the media. An analogy to the Fresnel–Hadamard–Duhem theorem and an expression for the acoustic tensor are derived. The condition for acceleration wave's propagation is formulated as an algebraic spectral problem. It is shown that the condition coincides with the strong ellipticity of equilibrium equations. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Solvability of Linear and Quasilinear Elliptic Boundary Value Problems via the A-Proper Mapping Theory

ABSTRACT

In this paper we use the theory of A-proper mappings and their uniform limits to obtain in a simple way general variational approximation-solvability and/or existence theorems for quasilinear elliptic BVProblems in divergence form of order 2m. In Section 1 we first show how our simple approach is used to obtain constructive solvability of linear elliptic equations under a somewhat different definition of strong ellipticity condition which allows us to dispense with the usage of the Garding inequality. In view of this, the same approach is also used in Section 2 to establish the constructive solvability of nonlinear BVProblems when the leading part satisfies a nonlinear version of the strong ellipticity condition used here. This result is then used to establish a new existence result (see Theorem 2.3) for a not necessarily coercive quasilinear elliptic BVProblem which properly includes earlier results of Vi?ik, Browder, Poho?ayev, Leray and Lions and others. In Section 3 some special cases of our abstract results are stated. These are applicable to ODE's and PDE's which are not in divergence form.     

A criterion for the non-existence or non-uniqueness of solutions of self-similar problems in the mechanics of a continuous medium

Systems of hyperbolic partial differential equations expressing conservation laws are considered. A sufficient condition is formulated under which the self-similar problem of the disintegration of an arbitrary discontinuity (or the “piston” problem) either has no solution or the solution is not unique. This sufficient condition is determined by the existence of non-evolutionary discontinuities which may be considered as a sequence of two evolutionary discontinuities moving at the same velocity, if such a representation is unique. The condition is more general than that formulated previously, which was based on the existence of a non-proper Jouguet point. The new criterion is satisfied by weak quasitranverse shock waves in elastic media, whatever the sign of the coefficient of the non-linear deformation term. It also enables one to draw conclusions as to the non-existence or non-uniqueness of solutions of problems of the theory of elasticity in the case of finite-amplitude waves.     

Elastoplastic torsion of a cylindrical rod for finite deformations

A solution of the problem of the torsion of a cylindrical rod was obtained in /1/ for a general, isotropic, incompressible elastic material. The present paper gives an analytical solution of the elastoplastic torsion problem for finite deformations, written in terms of quadratures of elliptic functions. The non-linear kinematics of elastoplastic deformation is introduced into the defining equations with the help of a multiplicative decomposition of the deformation gradient into elastic and plastic components /2, 3/. The elastic deformation and rate of plastic deformation are related to the state of stress of the body, in accordance with the defining Mooney-Rivlin equations /4/ and the law of flow for finite deformations associated with the Tresca yield condition /5/. A non-linear first-order partial differential equation and the initial data at the elastoplastic boundary are obtained in order to determine the angle of rotation within the plastic zone of the basis formed from the eigenvectors of the stress tensor, relative to the radial direction. The integration of the resulting equation is reduced to determining the general integral of the Ricatti equation with right-hand side determined from the angular velocity of flow of the material within the plastic zone. It is shown that neglecting the finiteness of the deformation leads to too high an estimate of the rigidity of the rod.     

The existence of riemann invariants in the one-dimensional equations of the non-linear theory of elasticity

The conditions for the existence of Riemann invariants of a one-dimensional system of equations of the non-linear theory of elasticity are investigated. Haantjes' diagonalization criterion is used to determine the form of the elastic potential for which the system has six Riemann invariants or three Riemann invariants (for waves which propagate in one direction). In particular, it is shown that the Haantjes criterion is satisfied and there are three Riemann invariants in the case of the elastic potential for slightly-non-linear weakly-anisotropic elastic media [1–3]. A procedure for computing Riemann invariants is described. The Riemann invariants are computed approximately for a form of elastic potential which satisfies the Haantjes criterion.     

Some studies on mathematical models for general elastic multi-structures

The aim of this paper is to study the static problem about a general elastic multi-structure composed of an arbitrary number of elastic bodies, plates and rods. The mathematical model is derived by the variational principle and the principle of virtual work in a vector way. The unique solvability of the resulting problem is proved by the Lax-Milgram lemma after the presentation of a generalized Korn's inequality on general elastic multi-structures. The equilibrium equations are obtained rigorously by only assuming some reasonable regularity of the solution. An important identity is also given which is essential in the finite element analysis for the problem.     

An iterative penalty method for a nonlinear problem of equilibrium of a Timoshenko-type plate with a crack

A variational problem of equilibrium of an elastic Timoshenko-type plate in a domain with a slit is considered. A nonpenetration boundary condition in the form of an inequality is specified on the edges of the slit. A penalized equation and an iterative linear equation in integral and differential forms are constructed. Some results on solution convergence and an error estimate are obtained.     

Derivatives of the energy functional for 2D‐problems with a crack under Signorini and friction conditions

We consider the two‐dimensional elasticity problem for an elastic body with a crack under unilateral constraints imposed at the crack. We assume that both the Signorini condition for non‐penetration of the crack faces and the condition of given friction between them are fulfilled. The problem is non‐linear and can be described by a variational inequality. Varying the shape of the crack by a local coordinate transformation of the domain, the first derivative of the energy functional to the problem with respect to the crack length is obtained, which gives the criterion for the crack growing. The regularity of the solution is discussed and the singular solution is performed. Copyright © 2000 John Wiley & Sons, Ltd.     

Nonlinear measure approach for the robust exponential stability analysis of interval inertial Cohen–Grossberg neural networks

This article is concerned with the existence and robust stability of an equilibrium point that related to interval inertial Cohen–Grossberg neural networks. Such condition requires the existence of an equilibrium point to a given system, so the existence and uniqueness of the equilibrium point are emerged via nonlinear measure method. Furthermore, with the help of Halanay inequality lemma, differential mean value theorem as well as inequality technique, several sufficient criteria are derived to ascertain the robust stability of the equilibrium point for the addressed system. The results obtained in this article will be shown to be new and they can be considered alternative results to previously results. Finally, the effectiveness and computational issues of the two models for the analysis are discussed by two examples. © 2016 Wiley Periodicals, Inc. Complexity 21: 459–469, 2016     

A novel algorithm for area traffic capacity control with elastic travel demands

A non-linear area traffic control system with limited capacity is considered in this paper. Optimal signal settings and link capacity expansions can be determined while trip distribution and network flow are in equilibrium. This problem can be formulated as a non-linear mathematical program with equilibrium constraints. For the objective function a non-linear constrained optimization program for signal settings and link capacity expansion is determined. For the constraint set the elastic user equilibrium traffic assignment obeying Wardrop’s first principle can be formulated as a variational inequality. Since the constrained optimization problem is non-convex, only local optima can be obtained. In this paper, a novel algorithm using a non-smooth trust region approach is proposed. Numerical tests are performed using a real data city network and various example test networks in which the effectiveness and robustness of the proposed method are confirmed as compared to other well-known solution methods.     

Problems of the partial stability and detectability of dynamical systems

The conditions under which uniform stability (uniform asymptotic stability) with respect to a part of the variables of the zero equilibrium position of a non-linear non-stationary system of ordinary differential equations signifies uniform stability (uniform asymptotic stability) of this equilibrium position with respect the other, larger part of the variables, which include an additional group of coordinates of the phase vector, are established. These conditions include the condition for uniform asymptotic stability of the zero equilibrium position of the “reduced” subsystem of the original system with respect to the additional group of variables. Since within the conditions obtained the stability with respect to the remaining unmeasured coordinates of the phase vector remains undetermined or is investigated additionally, partial zero-detectability of the original system occurs in this case, and the conditions obtained supplement the series of known results from partial stability theory. The application of the results obtained to problems of the partial stabilization of non-linear controlled systems, particularly to the problem of stabilizing an asymmetric rigid body relative to an assigned direction in an inertial space, is considered. The partial detectability of linear systems with constant coefficients is also investigated.     

可压缩液晶系统强解的破裂准则

研究可压缩液晶方程组强解的破裂准则,建立了一种仅依据于速度梯度的破裂准则,此种准则类似于理想可压缩流情形的Beale-Kato-Majda准则和由Huang和Xin得到的可压缩Navier-Stokes方程组情形的准则.证明用到能量不等式和高阶能量不等式.主要困难是初始密度含有真空.     

New criteria for global asymptotic stability of direct-form digital filters utilizing single saturation nonlinearity

A new sufficient condition in the form of a state-space criterion for global asymptotic stability of direct-form digital filters utilizing single saturation nonlinearity is presented. An alternative sufficient condition in the form of a matrix frequency-domain inequality is also presented. Examples show the effectiveness of the present criteria.