带摩擦的弹性接触问题广义变分不等原理的简化证明

在弹性摩擦接触问题中 ,从变分原理出发来研究接触问题 ,可以将摩擦力纳入问题的能量泛函 .为了得到摩擦约束弹性接触问题的能量泛函 ,日前大多是用拉格朗日乘子法 ,但拉格朗日方法用在变分不等问题中 ,要利用非线性泛函分析和凸分析来证明 ,证明复杂 .本文利用向量分析的工具及巧妙的变换 ,对带摩擦约束的弹性接触问题的广义变分不等原理进行了严格的证明 ,由于只用到向量分析 ,简化了证明 .     

A Dual-Primal FETI method for incompressible Stokes equations

In this paper, a dual-primal FETI method is developed for incompressible Stokes equations approximated by mixed finite elements with discontinuous pressures. The domain of the problem is decomposed into nonoverlapping subdomains, and the continuity of the velocity across the subdomain interface is enforced by introducing Lagrange multipliers. By a Schur complement procedure, the solution of an indefinite Stokes problem is reduced to solving a symmetric positive definite problem for the dual variables, i.e., the Lagrange multipliers. This dual problem is solved by the conjugate gradient method with a Dirichlet preconditioner. In each iteration step, both subdomain problems and a coarse level problem are solved by a direct method. It is proved that the condition number of this preconditioned dual problem is independent of the number of subdomains and bounded from above by the square of the product of the inverse of the inf-sup constant of the discrete problem and the logarithm of the number of unknowns in the individual subdomains. Numerical experiments demonstrate the scalability of this new method. This work is based on a doctoral dissertation completed at Courant Institute of Mathematical Sciences, New York University. This work was supported in part by the National Science Foundation under Grants NSF-CCR-9732208, and in part by the U.S. Department of Energy under contract DE-FG02-92ER25127.     

Pressure of a circular stamp on a composite layer

The method of dual integral equations is used to obtain a solution to the problem of a rigid circular stamp pressing on an elastic composite layer, with a cylindrical surface separating the materials. A large number of papers have already been published, dealing with the mechanics of multilayered media in which the surfaces separating the layers from each other do not intersect the outer boundary (see references in /1/). The formulation and methods of solution of the fundamental boundary value problems can be found for such media in the monographs /2,3/.

Considerably less attention has been given to the study of the boundary value problems for composite media in which the surfaces separating the layers do intersect the outer boundary. The authors of /4, 5/ call such media the regions with transverse (vertical) layer folding. Out of the publications dealing with the methods of solving contact problems for transversely layered regions, attention should be drawn to /4–11/.     

Application of the alternating direction method of multipliers to separable convex programming problems

This paper presents a decomposition algorithm for solving convex programming problems with separable structure. The algorithm is obtained through application of the alternating direction method of multipliers to the dual of the convex programming problem to be solved. In particular, the algorithm reduces to the ordinary method of multipliers when the problem is regarded as nonseparable. Under the assumption that both primal and dual problems have at least one solution and the solution set of the primal problem is bounded, global convergence of the algorithm is established.     

A mixed variational formulation for a class of contact problems in viscoelasticity

We consider a deformable body in frictionless unilateral contact with a moving rigid obstacle. The material is described by a viscoelastic law with short memory, and the contact is modeled by a Signorini condition with a time-dependent gap. The existence and uniqueness results for a weak formulation based on a Lagrange multipliers approach are provided. Furthermore, we discuss an efficient algorithm approximating the weak solution for the more general case of a two-body contact problem including friction. In order to illustrate the theory we present two numerical examples in 3D.     

A Smooth Embedding Domain Method Based On the Penalty Approach

It is well known that convergence of the fictitious domain formulation with boundary Lagrange multipliers is slow due to the lower global regularity of its solution. This article presents a smoothed variant of this approach which is based on a formulation in the form of a state constraint optimal control problem. The convergence rate is increased as seen from a model example.     

Staggered solution of fluid-porous-media interaction using the method of local Lagrange multipliers

Surface interaction among non-overlapping bulk-fluid and porous-medium bodies occurs in different situations, e. g., the interaction of blood with a blood vessel wall, a body of water with an earth dam structure, or acoustic waves with acoustic panels used in soundproofing. These are multi-field phenomena, comprising various surface- and volume-coupling mechanisms that should be reflected in the corresponding mathematical models. These models, together with appropriate initial and boundary values, assemble a coupled problem, the solution of which reveals the behaviour of the system under external excitations. The solution is commonly done numerically, following a monolithic or a decoupled approach. Here, the focus is on the latter. To design an efficient decoupled scheme, different types of coupling within the problem are addressed. These are the volume coupling between the degrees of freedom (DOF) within each subdomain, and the surface coupling between the DOF on the common boundaries. In particular, the latter constrains the feasible space of the solution of the problem. In this regard, local Lagrange multipliers (LLM) are employed to reformulate the problem in an unconstrained form. Unlike other domain decomposition methods which are based on using global Lagrange multipliers, the LLM method yields a complete separation of the subdomains and, consequently, facilitates parallel solution of the sub-problems. Moreover, within the subdomains, the penalty method is used to decouple pressure from other DOF. This procedure, on the one hand, reduces the size of the problem that should be solved at the interface and, on the other hand, removes the burden of using mixed finite elements within the subsystems. In the next step, the stability behaviour of the resulting staggered approach is analysed, and the unconditional stability of the method is established. Finally, the method is employed to solve a benchmark example, and using the numerical results, the reliability of the outcomes of the stability analysis is investigated. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Three-dimensional kirsch problem for a transtropic plate

Qualitative and quantitative study of the concentration of stresses in the Kirsch problem for isotropic plates is carried out in the three-dimensional formulation using the superposition method /1/ and the method of homogeneous solutions /2/. An asymptotic method of solving the Kirsch problem for transtropic plates is given in /3/. Below the problem in question is solved for the transtropic bodies of finite dimensions.     

Dispersion of internal waves by an obstacle floating on the boundary separating two liquids

The problem of the scattering of a wave, that propagates along the boundary between two liquids, by a semi-infinite obstacle floating on this boundary is solved in a two-dimensional formulation. The solution is constructed using the Wiener-Hopf method interpreted by Jones in the framework of linear potential theory /1/. The fundamental properties of the processes of scattering and reflection of a wave by the obstacle are stated and an asymptotic analysis of the field in a far zone is presented.     

Finite element and boundary element contact stress analysis with remeshing technique

The fundamental part of the contact stress problem solution using a finite element method is to locate possible contact areas reliably and efficiently. In this research, a remeshing technique is introduced to determine the contact region in a given accuracy. In the proposed iterative method, the meshes near the contact surface are modified so that the edge of the contact region is also an element’s edge. This approach overcomes the problem of surface representation at the transition point from contact to non-contact region. The remeshing technique is efficiently employed to adapt the mesh for more precise representation of the contact region. The method is applied to both finite element and boundary element methods. Overlapping of the meshes in the contact region is prevented by the inclusion of displacement and force constraints using the Lagrange multipliers technique. Since the method modifies the mesh only on the contacting and neighbouring region, the solution to the matrix system is very close to the previous one in each iteration. Both direct and iterative solver performances on BEM and FEM analyses are also investigated for the proposed incremental technique. The biconjugate gradient method and LU with Cholesky decomposition are used for solving the equation systems. Two numerical examples whose analytical solutions exist are used to illustrate the advantages of the proposed method. They show a significant improvement in accuracy compared to the solutions with fixed meshes.     

Solving dual problems using a coevolutionary optimization algorithm

In solving certain optimization problems, the corresponding Lagrangian dual problem is often solved simply because in these problems the dual problem is easier to solve than the original primal problem. Another reason for their solution is the implication of the weak duality theorem which suggests that under certain conditions the optimal dual function value is smaller than or equal to the optimal primal objective value. The dual problem is a special case of a bilevel programming problem involving Lagrange multipliers as upper-level variables and decision variables as lower-level variables. Another interesting aspect of dual problems is that both lower and upper-level optimization problems involve only box constraints and no other equality of inequality constraints. In this paper, we propose a coevolutionary dual optimization (CEDO) algorithm for co-evolving two populations—one involving Lagrange multipliers and other involving decision variables—to find the dual solution. On 11 test problems taken from the optimization literature, we demonstrate the efficacy of CEDO algorithm by comparing it with a couple of nested smooth and nonsmooth algorithms and a couple of previously suggested coevolutionary algorithms. The performance of CEDO algorithm is also compared with two classical methods involving nonsmooth (bundle) optimization methods. As a by-product, we analyze the test problems to find their associated duality gap and classify them into three categories having zero, finite or infinite duality gaps. The development of a coevolutionary approach, revealing the presence or absence of duality gap in a number of commonly-used test problems, and efficacy of the proposed coevolutionary algorithm compared to usual nested smooth and nonsmooth algorithms and other existing coevolutionary approaches remain as the hallmark of the current study.     

A mixed variational formulation of a contact problem with adhesion

A mathematical model describing the contact between a viscoplastic body and a deformable foundation is analyzed under small deformation hypotheses. The process is quasistatic and in normal direction the contact is with adhesion, normal compliance, memory effects and unilateral constraint. We derive a mixed-variational formulation of the problem using Lagrange multipliers. Finally, we prove the unique weak solvability of the contact problem.     

The Singular Function Boundary Integral Method for singular Laplacian problems over circular sections

The Singular Function Boundary Integral Method (SFBIM) for solving two-dimensional elliptic problems with boundary singularities is revisited. In this method the solution is approximated by the leading terms of the asymptotic expansion of the local solution, which are also used to weight the governing partial differential equation. The singular coefficients, i.e., the coefficients of the local asymptotic expansion, are thus primary unknowns. By means of the divergence theorem, the discretized equations are reduced to boundary integrals and integration is needed only far from the singularity. The Dirichlet boundary conditions are then weakly enforced by means of Lagrange multipliers, the discrete values of which are additional unknowns. In the case of two-dimensional Laplacian problems, the SFBIM converges exponentially with respect to the numbers of singular functions and Lagrange multipliers. In the present work the method is applied to Laplacian test problems over circular sectors, the analytical solution of which is known. The convergence of the method is studied for various values of the order p of the polynomial approximation of the Lagrange multipliers (i.e., constant, linear, quadratic, and cubic), and the exact approximation errors are calculated. These are compared to the theoretical results provided in the literature and their agreement is demonstrated.     

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.     

Parallel Uzawa Method for Large-Scale Minimization of Partially Separable Functions

A parallel Uzawa-type algorithm, for solving unconstrained minimization of large-scale partially separable functions, is presented. Using auxiliary unknowns, the unconstrained minimization problem is transformed into a (linearly) constrained minimization of a separable function.The augmented Lagrangian of this problem decomposes into a sum of partially separable augmented Lagrangian functions. To take advantage of this property, a Uzawa block relaxation is applied. In every iteration, unconstrained minimization subproblems are solved in parallel before updating Lagrange multipliers. Numerical experiments show that the speed-up factor gained using our algorithm is significant.     

Domaines fictifs, éléments finis H(div) et condition de Neumann: le problème de la condition inf-sup

Approximation of the Neumann problem for a second order elliptic operator by a fictitious domain method with a Lagrange multiplier on the boundary is considered. The problem is written in its vectorial dual formulation and H(div) mixed finite elements for the vector unknown and H^1/2 conforming elements for the multiplier are used. The uniform inf-sup condition is demonstrated under a compatibility condition between surface and volume meshes.     

Perturbation Lemma for the Newton Method with Application to the SQP Newton Method

We study the convergence of a general perturbation of the Newton method for solving a nonlinear system of equations. As an application, we show that the augmented Lagrangian successive quadratic programming is locally and q-quadratically convergent in the variable x to the solution of an equality constrained optimization problem, under a mild condition on the penalty parameter and the choice of the Lagrange multipliers.     

A surrogate and Lagrangian approach to constrained network problems

This paper presents the use of surrogate constraints and Lagrange multipliers to generate advanced starting solutions to constrained network problems. The surrogate constraint approach is used to generate a singly constrained network problem which is solved using the algorithm of Glover, Karney, Klingman and Russell [13]. In addition, we test the use of the Lagrangian function to generate advanced starting solutions. In the Lagrangian approach, the subproblems are capacitated network problems which can be solved using very efficient algorithms.The surrogate constraint approach is implemented using the multiplier update procedure of Held, Wolfe and Crowder [16]. The procedure is modified to include a search in a single direction to prevent periodic regression of the solution. We also introduce a reoptimization procedure which allows the solution from thekth subproblem to be used as the starting point for the next surrogate problem for which it is infeasible once the new surrogate constraint is adjoined.The algorithms are tested under a variety of conditions including: large-scale problems, number and structure of the non-network constraints, and the density of the non-network constraint coefficients.The testing clearly demonstrates that both the surrogate constraint and Langrange multipliers generate advanced starting solutions which greatly improve the computational effort required to generate an optimal solution to the constrained network problem. The testing demonstrates that the extra effort required to solve the singly constrained network subproblems of the surrogate constraints approach yields an improved advanced starting point as compared to the Lagrangian approach. It is further demonstrated that both of the relaxation approaches are much more computationally efficient than solving the problem from the beginning with a linear programming algorithm.     

Meshless solution of two-dimensional incompressible flow problems using the radial basis integral equation method

The two-dimensional incompressible fluid flow problems governed by the velocity–vorticity formulation of the Navier–Stokes equations were solved using the radial basis integral (RBIE) equation method. The RBIE is a meshless method based on the multi-domain boundary element method with overlapping subdomains. It solves at each node for the potential and its spatial derivatives. This feature of the RBIE is advantageous in solving the velocity–vorticity formulation of the Navier–Stokes equations since the calculated velocity gradients can be used to compute the vorticity that is prescribed as a boundary condition to the vorticity transport equation. The accuracy of the numerical solution was examined by solving the test problem with known analytical solution. Two benchmark problems, i.e. the lid driven cavity flow and the thermally driven cavity flow were also solved. The numerical results obtained using the RBIE showed very good agreement with the benchmark solutions.     

Pressure of a plane stamp of nearly circular cross section on an elastic half-space : PMM vol. 40, n 6, 1976, pp. 1143–1145

An approximate method of solving the contact problem of impressing a plane stamp of nearly circular cross section into an elastic half-space is suggested. The friction of the contact surface is neglected. A numerical algorithm for the method is produced. An elliptical and rectangular stamps are considered as examples.There is no general method of solving the problems for stamps of nearly circular cross section. Apart from the classical problem of a plane elliptical stamp, the literature gives solutions for the problems of polygonal stamps, with each problem however requiring a different approach. An approximate solution for the problem of impressing a stamp of nearly circular cross section into an elastic half-space is given in [1]. The method makes it possible to use the same approach to solve the contact problem for an arbitrary region of contact, and to construct an universal numerical algorithm. The program can be adapted to each particular case by making the corresponding changes in the procedure of computing the Fourier coefficients of the equation of the boundary of the area of contact. Below a numerical algorithm for the approximate method in question is given. A more effective formulation of the solution is given for the case of the elliptical stamp.