Uzawa block relaxation domain decomposition method for a two-body frictionless contact problem

We propose a Uzawa block relaxation domain decomposition method for a two-body frictionless contact problem. We introduce auxiliary variables to separate subdomains representing linear elastic bodies. Applying a Uzawa block relaxation algorithm to the corresponding augmented Lagrangian functional yields a domain decomposition algorithm in which we have to solve two uncoupled linear elasticity subproblems in each iteration while the auxiliary variables are computed explicitly using Kuhn–Tucker optimality conditions.     

Method of nonlinear boundary equations in problems of contact of several elastic bodies

The method of nonlinear boundary equations is applied to develop new formulations of contact problems with unknown contact regions. Our formulation is free from inequality constraints, which enter the method of variational inequalities and the standard formulations of contact problems. Methods of the theory of operator equations are applied to prove that the problems are well-posed.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 61, pp. 62–70, 1987.     

A domain decomposition method for two-body contact problems with Tresca friction

The paper analyzes a continuous and discrete version of the Neumann-Neumann domain decomposition algorithm for two-body contact problems with Tresca friction. Each iterative step consists of a linear elasticity problem for one body with displacements prescribed on a contact part of the boundary and a contact problem with Tresca friction for the second body. To ensure continuity of contact stresses, two auxiliary Neumann problems in each domain are solved. Numerical experiments illustrate the performace of the proposed approach.     

Convergence rate estimate for a domain decomposition method

Summary We provide a convergence rate analysis for a variant of the domain decomposition method introduced by Gropp and Keyes for solving the algebraic equations that arise from finite element discretization of nonsymmetric and indefinite elliptic problems with Dirichlet boundary conditions in ². We show that the convergence rate of the preconditioned GMRES method is nearly optimal in the sense that the rate of convergence depends only logarithmically on the mesh size and the number of substructures, if the global coarse mesh is fine enough.This author was supported by the National Science Foundation under contract numbers DCR-8521451 and ECS-8957475, by the IBM Corporation, and by the 3M Company, while in residence at Yale UniversityThis author was supported by the Applied Mathematical Sciences subprogram of the Office of Energy Research, U.S. Department of Energy under Contract W-31-109-Eng-38This author was supported by the National Science Foundation under contract number ECS-8957475, by the IBM Corporation, and by the 3M Company     

A parallel iterative nonoverlapping domain decomposition procedure for elliptic problems

E-mail: yangd{at}math.purdue.edu or yangd{at}cs.purdue.edu Present address: Department of Mathematics Wayne State University, Detroit, MI 48202, USA. A parallel iterative nonoverlapping domain decomposition methodis proposed and analyzed for elliptic problems. Each iterationin this method contains two steps. In the first step, at theinterface of two subdomains, one subdomain problem requiresthat Dirichlet data be passed to it from the previous iterationlevel, while the other subdomain problem requires that Neumamdata be passed to it. In the second step, we interchange thetypes of data passing at the interface of the two subdomains.This domain decomposition method is suitable for parallel processingwith coarse granularity. Convergence analysis is demonstratedat the differential level by Hilbert space techniques. Numericalresults are provided to confirm the convergence theory.     

Spatial contact problems for rough elastic bodies under elastoplastic deformations of the unevenness

On the basis of /1, 2/, a model is constructed for the contact between a rigid stamp and a rough body taking elastoplastic deformations of the unevenness into account. The contact model for rough bodies with elastic deformations of the unevenness is a special case. A classical approach utilizing boundary integral equations is applied in the mathematical formulation of the contact problem. Under quite general assumptions (for instance, the multiconnectedness of the contact domain desired), the uniqueness and existence of the solution are investigated. A method is developed to determine the contact pressure, the closure of the bodies, and also the contact area which consists of two parts in the general case, a zone of elastoplastic deformation of the unevenness and a zone of their elastic deformation. The efficiency of the method is shown in examples of new contact problems. The solution is represented in a convenient form for analysing the influence of the roughness. This is of considerable value for material testing by a contact method. A fairly complete survey of research on contact problems for rough bodies can be found in /1–4/.     

Convergence analysis of domain decomposition algorithms with full overlapping for the advection-diffusion problems

The aim of this paper is to study the convergence properties of a time marching algorithm solving advection-diffusion problems on two domains using incompatible discretizations. The basic algorithm is first described, and theoretical and numerical results that illustrate its convergence properties are then presented.

     

The domain decomposition method for parabolic transmission problems

An estimate of the rate of convergence is given for the domain decomposition method for the second-order parabolic transmission problem. A brief discussion of the method and some of its applications are presented.     

Convergence rates of Neumann problems for Stokes systems

In quantitative homogenization of the Neumann problems for Stokes systems with rapidly oscillating periodic coefficients, this paper studies the convergence rates of the velocity in $L^2$ and $H^1$ as well as those of the pressure term in $L^2$, without any smoothness assumptions on the coefficients.     

An effective asymptotic method in the axisymmetric frictionless contact problem for an elastic layer of finite thickness

A brief review of asymptotic methods to deal with frictionless unilateral contact problems for an elastic layer of finite thickness is presented. Under the assumption that the contact radius is small with respect to the layer thickness, an effective asymptotic method is suggested for solving the unilateral contact problem with a priori unknown contact radius. A specific feature of the method is that the construction of an asymptotic approximation is reduced to a linear algebraic system with respect to integral characteristics (polymoments) of the contact pressure. As an example, the sixth‐order asymptotic model has been written out. Copyright © 2015 John Wiley & Sons, Ltd.     

Differential quadrature domain decomposition method for problems on a triangular domain

The differential quadrature method (DQM) has been studied for years and it has been shown by many researchers that the DQM is an attractive numerical method with high efficiency and accuracy. The conventional DQM is mostly effective for one‐dimensional and multidimensional problems with geometrically regular domains. But to deal with problems on a triangular domain, we will meet difficulties. In this article we will study how to solve problems on a triangular domain by using DQM combined with the domain decomposition method (DDM). © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

Solution of boundary-value problems of the statics of layered elastic bodies with nonrigid contact between the layers

A technique is proposed for solving three-dimensional problems of the stress-strain state of cylinders, spheres, and shallow elastic bodies with a rectangular projection which are composed of laterally nonhomogeneous anisotropic layers with nonrigid contact between the layers. The solution of the corresponding many-point boundary-value problem is reduced to solving a number of two-point problems by a known numerical apparatus. Solution results are reported for the strain of a three-layer spherical shell with slipping layers.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 56, pp. 62–68, 1985.     

An efficient domain decomposition method for three-dimensional parabolic problems

A non-overlapping domain decomposition algorithm to solve three-dimensional parabolic partial differential equations is presented. It has been shown in this paper that the algorithm is unconditionally stable and efficient. Spectral radii for the interface and interior region are provided. Unlike two-dimensional problem, it has been found out that estimating the values of the points of the interface in three-dimensional problem is no longer negligible.     

Approximation method for the problems of mechanics of inhomogeneous hereditarily elastic bodies

We consider a boundary-value problem of mechanics of inhomogeneous hereditarily elastic bodies formulated as a linear equation with an operator of fractional integration, partial derivatives with respect to time and spatial variables, and polynomial-type coefficients of one of the variables. An approximate solution of this problem is constructed according to Dzyadyk's a-method combined with the use of the Laplace transformation. It is proved that the errors of the approximation of the required function and its derivatives decrease in geometric progression.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 9, pp. 1234–1245, September, 1994.     

A domain decomposition method for a kind of optimization problems

In this paper, we are concerned with the monotone convergence of a multiplicative method for solving a kind of optimization problems. We show that the iterate sequence produced by the method converges to the solution of the problem monotonically.     

On the convergence rate of a parallel nonoverlapping domain decomposition method

In recent years,a nonoverlapping domain decomposition iterative procedure,which is based on using Robin-type boundary conditions as information transmission conditions on the subdomain interfaces,has been developed and analyzed.It is known that the convergence rate of this method is 1-O(h),where h is mesh size.In this paper,the convergence rate is improved to be 1-O(h1/2 H-1/2)sometime by choosing suitable parameter,where H is the subdomain size.Counter examples are constructed to show that our convergence estimates are sharp,which means that the convergence rate cannot be better than 1-O(h1/2H-1/2)in a certain case no matter how parameter is chosen.     

A spatial domain decomposition method for parabolic optimal control problems

We present a non-overlapping spatial domain decomposition method for the solution of linear–quadratic parabolic optimal control problems. The spatial domain is decomposed into non-overlapping subdomains. The original parabolic optimal control problem is decomposed into smaller problems posed on space–time cylinder subdomains with auxiliary state and adjoint variables imposed as Dirichlet boundary conditions on the space–time interface boundary. The subdomain problems are coupled through Robin transmission conditions. This leads to a Schur complement equation in which the unknowns are the auxiliary state adjoint variables on the space-time interface boundary. The Schur complement operator is the sum of space–time subdomain Schur complement operators. The application of these subdomain Schur complement operators is equivalent to the solution of an subdomain parabolic optimal control problem. The subdomain Schur complement operators are shown to be invertible and the application of their inverses is equivalent to the solution of a related subdomain parabolic optimal control problem. We introduce a new family of Neumann–Neumann type preconditioners for the Schur complement system including several different coarse grid corrections. We compare the numerical performance of our preconditioners with an alternative approach recently introduced by Benamou.     

A domain decomposition method for linear exterior boundary value problems

In this paper, we present a domain decomposition method, based on the general theory of Steklov-Poincaré operators, for a class of linear exterior boundary value problems arising in potential theory and heat conductivity. We first use a Dirichlet-to-Neumann mapping, derived from boundary integral equation methods, to transform the exterior problem into an equivalent mixed boundary value problem on a bounded domain. This domain is decomposed into a finite number of annular subregions, and the Dirichlet data on the interfaces is introduced as the unknown of the associated Steklov-Poincaré problem. This problem is solved with the Richardson method by introducing a Dirichlet-Robin-type preconditioner, which yields an iteration-by-subdomains algorithm well suited for parallel computations. The corresponding analysis for the finite element approximations and some numerical experiments are also provided.     

A hybrid nonoverlapping domain decomposition scheme for advection dominated advection–diffusion problems

Two nonoverlapping domain decomposition algorithms are proposed for convection dominated convection–diffusion problems. In each subdomain, artificial boundary conditions are used on the inflow and outflow boundaries. If the flow is simple, each subdomain problem only needs to be solved once. If there are closed streamlines, an iterative algorithm is needed and the convergence is proved. Analysis and numerical tests reveal that the methods are advantageous when the diffusion parameter ɛ is small. In such cases, the error introduced by the domain decomposition methods is negligible in comparison with the error in the singular layers, and it allows easy and efficient grid refinement in the singular layers. This revised version was published online in August 2006 with corrections to the Cover Date.