一类Signorini问题的边界变分不等式

本文讨论了一类简化的Signorini问题。首先将原问题和一个边值问题建立联系，其次将原问题的解分解为不带不等边界条件的变分方程的解和一个变分不等式的解。然后利用边值问题的边界积分方程将变分不等式等价地化解为边界变分不等式。这样原求区域上的第一类椭圆变分不等式问题化解为求一个区域上的变分方程和一个边界变分不等式。最后说明了边界变分不等式解的存在唯一性。文末计算了柱面和半无限刚性基础的摩擦接触问题。结论表明文中方法具有较好的精度。     

第二类抛物型变分不等式中的MRM方法

本文讨论了第二类抛物型变分不等式中的MRM(多重互易方法)方法。首先采用时间项半离散和隐格式方法将抛物型变分不等式化解为一个椭圆变分不等式，然后利用MRM-边界积分方程，将其化解为MRM-边界混合变分不等式，并给出了MRM-边界混合变分不等式解的存在唯一性。说明了该MRM-边界混合变分不等式与常规边界积分方程得到的边界混合变分不等式是一致的，并且具有更容易编程实现。这为使用MRM边界元方法数值求解抛物型变分不等式提供了方法和理论依据。文末给出了数值算例。     

Existence and stability of solutions to inverse variational inequality problems

In this paper, two new existence theorems of solutions to inverse variational and quasi-variational inequality problems are proved using the Fan-Knaster-KuratowskiMazurkiewicz(KKM) theorem and the Kakutani-Fan-Glicksberg fixed point theorem.Upper semicontinuity and lower semicontinuity of the solution mapping and the approximate solution mapping to the parametric inverse variational inequality problem are also discussed under some suitable conditions. An application to a road pricing problem is given.     

A regularization method for a boundary variational inequality of the second kind associated to a friction problem

A scalar contact problem with friction is formulated as a boundary variational inequality of the second kind. The presence of the non-differentiable friction functional causes difficulties when approximating it. We present two approaches to overcome these difficulties: A regularization procedure leading to a non-linear boundary variational equation, for which we propose an iterative process and the second one is a boundary mixed variational formulation involving Lagrange multiplier. We reformulate our problem in terms of a saddle point problem for the corresponding boundary Lagrangian and describe Uzawa's algorithm to compute it.     

Existence and algorithm of solutions for general multivalued mixed implicit quasi-variational inequalities

A new class of general multivalued mixed implicit quasi-variational inequalities in a real Hilbert space was introduced, which includes the known class of generalized mixed implicit quasi-variational inequalities as a special case , introduced and studied by Ding Xie-ping . The auxiliary variational principle technique was applied to solve this class of general multivalued mixed implicit quasi-variational inequalities. Firstly, a new auxiliary variational inequality with a proper convex , lower semicontinuous , binary functional was defined and a suitable functional was chosen so that its unique minimum point is equivalent to the solution of such an auxiliary variational inequality . Secondly , this auxiliary variational inequality was utilized to construct a new iterative algorithm for computing approximate solutions to general multivalued mixed implicit quasi-variational inequalities . Here , the equivalence guarantees that the algorithm can generate a sequence of approximate solutions. Finally, the e     

Penalty finite element method for Navier–Stokes equations with nonlinear slip boundary conditions

The penalty finite element method for Navier–Stokes equations with nonlinear slip boundary conditions is investigated in this paper. Since this class of nonlinear slip boundary conditions include the subdifferential property, the weak variational formulation is a variational inequality problem of the second kind. Using the penalty finite element approximation, we obtain optimal error estimates between the exact solution and the finite element approximation solution. Finally, we show the numerical results which are in full agreement with the theoretical results. Copyright © 2011 John Wiley & Sons, Ltd.     

The Navier–Stokes Equations Under a Unilateral Boundary Condition of Signorini’s Type

We propose a new outflow boundary condition, a unilateral condition of Signorini’s type, for the incompressible Navier–Stokes equations. The condition is a generalization of the standard free-traction condition. Its variational formulation is given by a variational inequality. We also consider a penalty approximation, a kind of the Robin condition, to deduce a suitable formulation for numerical computations. Under those conditions, we can obtain energy inequalities that are key features for numerical computations. The main contribution of this paper is to establish the well-posedness of the Navier–Stokes equations under those boundary conditions. Particularly, we prove the unique existence of strong solutions of Ladyzhenskaya’s class using the standard Galerkin’s method. For the proof of the existence of pressures, however, we offer a new method of analysis.     

SINGULARLY PERTURBED BOUNDARY VALUE PROBLEMS FOR SEMI-LINEAR RETARDED DIFFERENTIAL EQUATIONS WITH NONLINEAR BOUNDARY CONDITIONS

IntroductionInthispaper,westudiedakindofboundaryvalueproblems (BVPs)forsemi_linearretardeddifferentialequationwithnonlinearboundarycondition :　　　　εx″(t) =f(t,x(t) ,x(t-ε) ,ε) ,　　t∈(0 ,1 ) ,(1 )　　　　x(t) =φ(t,ε) ,　t∈[-ε0 ,0 ] ,h(x(1 ) ,x′(1 ) ,ε) =A(ε) ,(2 )whereε>0isasmallparameterandε0 isasufficientlysmallpositiveconstant.ThereweremanyresultsofstudyingonsingularlyperturbedboundaryvalueproblemforretardeddifferentialequationinRefs.[1～5] .Butthosestudiespossessedanesse…     

Free boundary problem of non-steady state seepage flow

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弹性力学混合状态方程的弱形式及其边值问题

导出了弹性力学混合状态方程和边界条件弱形式的统一方程，此法使函数的选择无需事先完全满足边界条件，对于各种不同的边值问题可以用统一形式处理，这使得求解弹性力学问题的形式得以扩大和统一     

BOUNDARY ELEMENT METHOD FOR BUCKLING EIGENVALUE PROBLEM AND ITS CONVERGENCE ANALYSIS

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｂｕｃｋｌｉｎｇｅｉｇｅｎｖａｌｕｅｐｒｏｂｌｅｍｈａｓｉｍｐｏｒｔａｎｔｓｉｇｎｉｆｉｃａｎｃｅｉｎｔｈｅｓｔａｂｉｌｉｔｙａｎａｌｙｓｉｓｏｆｅｎｇｉｎｅｅｒｉｎｇｓｔｒｕｃｔｕｒｅ.Ｈｅｎｃｅｔｈｅｎｕｍｅｒｉｃａｌｃａｌｃｕｌａｔｉｏｎｆｏｒｔｈｅｓｅｐｒｏｂｌｅｍｓｉｓｅｘｔｒｅｍｅｌｙｍｅａｎｉｎｇｆｕｌｉｎｃｏｍｐｕｔａｔｉｏｎａｌｍｅｃｈａｎｉｃｓ.ＴｈｅｐｒｅｓｅｎｔｃｏｍｐｕｔａｔｉｏｎａｌｍｅｔｈｏｄｓｆｏｃｕｓｏｎＦＥＭ ,ｄｉｆｆｅｒｅｎｃｅｍ…     

Asymptotic approximation method for elliptic variational inequality of first kind简

An asymptotic approximation method is proposed to solve a particular elliptic variational inequality of first kind associated with unilateral obstacle problems. In this method, the free boundary is first captured, and then the method of the fundamental solution (MFS) is used to find the solution of the Dirichlet problem for Laplace’s equation in the non-coincidence set. Numerical examples are given to show the efficiency of the method.     

GENERALIZED VECTOR VARIATIONAL-TYPE INEQUALITIES IN FC-SPACES

A class of generalized vector variational-type inequality problems (GVVTIP) are studied in FC-spaces, which includes the most of vector equilibrium problems, vector variational inequality problems, generalized vector equilibrium problems and generalized vector variational inequality problem as special cases. By using F-KKM theorem, some new existence results for GVVTIP are established in noncompact FC-space. As consequences, some recent known results in literature are obtained under much weaker assumption.     

Auxiliary principle and three-step iterative algorithms for generalized set-valued strongly nonlinear mixed variational-like inequalities

An auxiliary principle technique to study a class of generalized set-valued strongly nonlinear mixed variational-like inequalities is extended. The existence and uniqueness of the solution of the auxiliary problem for the generalized set-valued strongly nonlinear mixed variational-like inequalities are proved, a novel and innovative three-step iterative algorithm to compute approximate solution is constructed, and the existence of the solution of the generalized set-valued strongly nonlinear mixed variational-like inequality is shown using the auxiliary principle technique. The convergence of three-step iterative sequences generated by the algorithm is also proved.     

Well-posedness of a model of strain gradient plasticity for plastically irrotational materials

The initial boundary value problem corresponding to a model of strain gradient plasticity due to [Gurtin, M., Anand, L., 2005. A theory of strain gradient plasticity for isotropic, plastically irrotational materials. Part I: Small deformations. J. Mech. Phys. Solids 53, 1624–1649] is formulated as a variational inequality, and analysed. The formulation is a primal one, in that the unknown variables are the displacement, plastic strain, and the hardening parameter. The focus of the analysis is on those properties of the problem that would ensure existence of a unique solution. It is shown that this is the case when hardening takes place. A similar property does not hold for the case of softening. The model is therefore extended by adding to it terms involving the divergence of plastic strain. For this extended model the desired property of coercivity holds, albeit only on the boundary of the set of admissible functions.     

轴对称层合圆柱壳混合状态方程的弱形式及其边值问题

从Ｈｅｌｉｎｇｅｒ－Ｒｅｉｓｓｎｅｒ变分原理出发，在柱坐标系中，导出圆柱壳轴对称问题的弱形式混合状态方程和边界条件，联用状态空间法给出强厚度叠层柱壳的解析解，此法使得求解该类问题的形式得以扩大和统一。     

Optimal control of structures governed by variational and hemiva-riational inequalities and applications

The optimal control problem for broad classes of structures is studied, including those structures having as state relations variational equalities, variational inequalities and hemivariational inequalities. The optimal control problem consists in the minimization of a functional (performance index) having the state relation, enlarged by the control actions, as side condition. Certain new results are given of the optimal control of structures governed by variational and hemivariational inequalities.Some propositions are proved on the existence and the approximation of the solution of the static optimal control problem of structures having a variational inequality as state relation. Then a regularization procedure is proposed for the treatment of corresponding dynamic problem, as well as for the case of hemivariational inequalities. The theory is illustrated by applications concerning convex elastoplasticity and convex and nonconvex unilateral contact problems.     

Variational analysis of thermomechanically coupled steady-state rolling problem

A steady-state, rigid-plastic rolling problem for temperature and strain-rate dependent materials with nonlocal friction is considered. A variational formulation is derived, coupling a nonlinear variational inequality for the velocity and a nonlinear vari- ational equation for the temperature. The existence and uniqueness results are obtained by a proposed fixed point method.     

Two-dimensional Hierarchical Models for Prismatic Shells with Thickness Vanishing at the Boundary

We construct variational hierarchical two-dimensional models for elastic, prismatic shells of variable thickness vanishing at boundary. With the help of variational methods, existence and uniqueness theorems for the corresponding two-dimensional boundary value problems are proved in appropriate weighted functional spaces. By means of the solutions of these two-dimensional boundary value problems, a sequence of approximate solutions in the corresponding three-dimensional region is constructed. We establish that this sequence converges in the Sobolev space H¹ to the solution of the original three-dimensional boundary value problem. Mathematics Subject Classifications (2000) 74K20, 74K25.     

Hybrid projection method for generalized mixed equilibrium problems,variational inequality problems,and fixed point problems in Banach spaces

A new hybrid projection iterative scheme is introduced to approximate a common element of the solution set of a generalized mixed equilibrium problem, the solution set of a variational inequality problem, and the set of fixed points of a relatively weak nonexpansive mapping in the Banach spaces. The obtained results generalize and improve the recent results announced by many other authors.