Unilateral buckling of thin elastic plates by the boundary integral equation method

The unilateral buckling of thin elastic plates, according to Kirchhoff's theory, is studied by using a boundary integral method. A representation for the second member of the equation is given. In the matrix formulatiea, boundary unknowns are eliminated; therefore, the unilateral buckling problem reduces to compute the eigenvalues and the eigenvectors of a matrix depending on the contact zone with the rigid foundation. An iterative process allows this zone and the buckling load to be computed. The capacities of the proposed method are illustrated by four examples.     

Variational boundary-integral-equation approach to unilateral contact problems in elasticity

On the basis of the boundary integral equation method, three variational principles for the frictionless unilateral contact problem in elasticity are presented. Two of them are saddle-point principles for the boundary unknowns (including the contact displacements); a third one is a maximum principle for the unknown contact displacements only. A discretization by boundary elements leads to algebraic formulations in the shape either of quadratic programming problems, or of linear complementarity problems, all characterized by symmetry and sign definiteness of the coefficient matrices. The method is also applicable to contact problems between two uncompenetrable elastic solids, as well as to the crack problem of fracture mechanics.     

BIEM-based variational principles for elastoplasticity with unilateral contact boundary conditions

The structural step problem for elastic-plastic internal-variable materials is addressed in the presence of frictionless unilateral contact conditions. Basing on the BIEM (boundary integral equation method) and making use of deformation-theory plasticity (through the backward-difference method of computational plasticity), two variational principles are shown to characterize the solution to the step problem: one is a stationarity principle having as unknowns all the problem variables, the other is a saddle-point principle having as unknowns the increments of the boundary tractions and displacements, along with the plastic strain increments in the domain. The discretization by boundary and interior elements transforms the above principles into well-posed mathematical programming formulations belonging to the symmetric Galerkin BEM formulations (with features such as a symmetric sign-definite coefficient matrix, double integrations, and hypersingular integrals).     

An augmented Lagrangian method for discrete large-slip contact problems

An augmented Lagrangian formulation is proposed for large-slip frictionless contact problems between deformable discretized bodies in two dimensions. Starting from a finite element discretization of the two bodies, a node-on-facet element is defined. A non-linear gap vector and its first variation are derived in terms of the nodal displacements. The relevant action and reaction principle is stated. The gap distance is then related to the conjugate pressure by a (multivalued non-differentiable) unilateral contact law. The resulting inequality constrained minimization problem is transformed into an unconstrained saddle point problem using an augmented Lagrangian function. Large slip over several facets is possible and the effects of target convexity or concavity are investigated. A generalized Newton method is used to solve the resulting piecewise differentiable equations necessary for equilibrium and contact. The proper tangent (Jacobian) matrices are calculated. The primal (displacements) and dual (contact forces) unknowns are simultaneously updated at each iteration.     

Efficient boundary element analysis of sharp notched plates

The present paper further develops the boundary element singularity subtraction technique, to provide an efficient and accurate method of analysing the general mixed-mode deformation of two-dimensional linear elastic structures containing sharp notches. The elastic field around sharp notches is singular. Because of the convergence difficulties that arise in numerical modelling of elastostatic problems with singular fields, these singularities are subtracted out of the original elastic field, using the first term of the Williams series expansion. This regularization procedure introduces the stress intensity factors as additional unknowns in the problem; hence extra conditions are required to obtain a solution. Extra conditions are defined such that the local solution in the neighbourhood of the notch tip is identical to the Williams solution; the procedure can take into account any number of terms of the series expansion. The standard boundary element method is modified to handle additional unknowns and extra boundary conditions. Analysis of plates with symmetry boundary conditions is shown to be straightforward, with the modified boundary element method. In the case of non-symmetrical plates, the singular tip-tractions are not primary boundary element unknowns. The boundary element method must be further modified to introduce the boundary integral stress equations of an internal point, approaching the notch-tip, as primary unknowns in the formulation. The accuracy and efficiency of the method is demonstrated with some benchmark tests of mixed-mode problems. New results are presented for the mixed-mode analysis of a non-symmetrical configuration of a single edge notched plate.     

Numerical analysis of elastic contact problems using the boundary integral equation method. Part 1: Theory

A boundary integral equation formulation for the analysis of two-dimensional elastic contact problems with friction is developed. In this formulation, the contact equations are written explicitly with both tractions and displacements retained as unknowns. These equations are arranged such that a blocked coefficient matrix results. An incremental and iterative procedure is discussed which, in the case of proportional loading, modifies only the equations in the potential contact zone.     

Analysis of frictional contact problem using boundary element method and domain decomposition method

In this paper, we study the bilateral or unilateral contact with Coulomb friction between two elastic solids, using a domain decomposition method coupled with the boundary element method. The decomposition method we have selected is the Schur complement method, a non‐overlapping technique. It enables to reduce the solution of the global problem to the solution of a problem defined only on the contact surface. Moreover, its principal advantage is that computing is done separately on each solid. We have chosen to associate it with the boundary element method. Indeed, it only requires the discretization of the boundaries of solids. This technique of coupling reduces the number of unknowns and the time of computing. We have applied it to the study of indentation of an elastic foundation by an elastic flat punch and a sphere. In this last case, our results are in conformity with the Hertz theory and the analytical solution of Spence. Moreover, we have shown the influence of friction on the size of the contact radius and on the normal pressure at centre. Copyright © 1999 John Wiley & Sons, Ltd.     

Interior point surrogate dual algorithm for unilateral problems

Summary. A unilateral problem is one kind of variational inequality problems; it can be discretized with the finite element method and converted to an inequality constrained nonsmooth optimization problem with the potential energy of the structure acting as the objective and unilateralization as the constraints. One characteristic in the optimization problem is that the number of constraints is much smaller than the dimension of the problem; therefore dual methods are often adopted. This paper focuses on the more complicated unilateral problems, the frictional contact problems. First, it is investigated that the value of the Coulomb friction coefficient has great influence on the property of the optimization problem; the friction orientation constraint is accordingly introduced in the solution procedure for treating the problems resulting from the larger friction coefficient. Then, the primal optimization problem is converted to an explicit surrogate dual problem which can be solved by a Karmarkars interior point based method. Finally, the method is verified by two typical frictional contact examples.     

A Trefftz approach to computational mechanics

This paper presents a Trefftz method for solving structural elasticity problems and flow problems of incompressible viscous fluids. The problem of unilateral contact is also dealt with. For each type of problem, Trefftz polynomials and associated variational formulations are given. Complex structures are studied by a sub‐structuring technique. This method requires the resolution of a non‐symmetrical linear system. It is shown that it is possible to take advantage of this Trefftz approximation in two ways: (i) the approach presented can be considered as a simplified method which enables a solution to be evaluated quickly; (ii) this approach also makes it possible to obtain a good quality solution associated with high degree polynomial bases. This method is adapted to optimization processes because the discretization of the structure requires only very few sub‐domains to build a good approximation and offers a great flexibility in use. Copyright © 2003 John Wiley & Sons, Ltd.     

A multivalued boundary integral equation for adhesive contact problems

The present paper proposes a boundary integral equation (BIE) formulation for adhesive contact interface problems, i.e. problems involving interfaces glued by adhesives where unilateral contact conditions also hold. A non-monotonic, multi-valued law is assumed to describe the behaviour of the adhesive tangential to the interface direction, which leads to a hemivariational inequality problem. For the numerical treatment of the non-convex-non-smooth optimization problem, a new method is proposed which reduces the initial problem to a sequence of simple quadratic programming problems.     

A suitable computational strategy for the parametric analysis of problems with multiple contact

The aim of the present work is to develop an application of the LArge Time INcrement (LATIN) approach for the parametric analysis of static problems with multiple contacts. The methodology adopted was originally introduced to solve viscoplastic and large‐transformation problems. Here, the applications concern elastic, quasi‐static structural assemblies with local non‐linearities such as unilateral contact with friction. Our approach is based on a decomposition of the assembly into substructures and interfaces. The interfaces play the vital role of enabling the local non‐linearities, such as contact and friction, to be modelled easily and accurately. The problem on each substructure is solved by the finite element method and an iterative scheme based on the LATIN method is used for the global resolution. More specifically, the objective is to calculate a large number of design configurations. Each design configuration corresponds to a set of values of all the variable parameters (friction coefficients, prestress) which are introduced into the mechanical analysis. A full computation is needed for each set of parameters. Here we propose, as an alternative to carrying out these full computations, to use the capability of the LATIN method to re‐use the solution to a given problem (for one set of parameters) in order to solve similar problems (for the other sets of parameters). Copyright © 2003 John Wiley & Sons, Ltd.     

On the bifurcation and global energy approach in propagating plasticity

A bifurcation method is proposed to solve a class of steady-state propagation problems in plastic shells. The method is explained on a simple one-dimensional example of a collapsing underwater pipeline. The pipe is modeled as a rigid–plastic beam/string resting on a plastic strain-softening foundation and is subjected to uniform pressure loading. The unknowns in the problem are the critical pressure for the buckle to propagate and the length and shape of the so-called transition zone. Using the method of local equilibrium closed-form solutions are derived for three different structural models: beam, string, and beam/string.The global formulation is then presented in which the rate of energy associated with the change in the length of the transition zone is included in the balance equations. The solution is obtained as an intersection of the equilibrium paths corresponding to a constant and variable length of the deformation zone. An interpretation of the “Calladine paradox” is offered so that the global energy approach to propagating plasticity remains unchallenged.     

A LATIN computational strategy for multiphysics problems: application to poroelasticity

Multiphysics phenomena and coupled‐field problems usually lead to analyses which are computationally intensive. Strategies to keep the cost of these problems affordable are of special interest. For coupled fluid–structure problems, for instance, partitioned procedures and staggered algorithms are often preferred to direct analysis. In this paper, we describe a new strategy for solving coupled multiphysics problems which is built upon the LArge Time INcrement (LATIN) method. The proposed application concerns the consolidation of saturated porous soil, which is a strongly coupled fluid–solid problem. The goal of this paper is to discuss the efficiency of the proposed approach, especially when using an appropriate time‐space approximation of the unknowns for the iterative resolution of the uncoupled global problem. The use of a set of radial loads as an adaptive approximation of the solution during iterations will be validated and a strategy for limiting the number of global resolutions will be tested on multiphysics problems. Copyright © 2003 John Wiley & Sons, Ltd.     

Impact analysis of arbitrarily‐shaped bodies using a finite element method and a smoothed particle hydrodynamics method

We propose a new method to obtain contact forces under a non‐smoothed contact problem between arbitrarily‐shaped bodies which are discretized by finite element method. Contact forces are calculated by the specific contact algorithm between two particles of smoothed particle hydrodynamics, which is a meshfree method, and that are applied to each colliding body. This approach has advantages that accurate contact forces can be obtained within an accelerated collision without a jump problem in a discrete time increment. Also, this can be simply applied into any contact problems like a point‐to‐point, a point‐to‐line, and a point‐to‐surface contact for complex shaped and deformable bodies. In order to describe this method, an impulse based method, a unilateral contact method and smoothed particle hydrodynamics method are firstly introduced in this paper. Then, a procedure about the proposed method is handled in great detail. Finally, accuracy of the proposed method is verified by a conservation of momentum through three contact examples. Copyright © 2016 John Wiley & Sons, Ltd.     

Frictional contact analysis of composite materials

In this paper, the highly non-linear frictional contact problems of composite materials are analysed. A proportional loading, the potential contact zone method and finite element analysis are used to solve the problems. A tree-like searching method is used to obtain the solution of the parametric linear complementary problem, which may overcome the anisotropic properties of contact equations caused by composite materials. In the frictional contact analysis of composite materials, the distributions of normal contact pressures, tangential contact stresses and relative tangential displacements are presented for different contact material systems and different coefficients of friction. The results show that the solutions in the paper have good agreement with Hertzian solutions. The influence of different contact material systems and different coefficients of friction on the contact stresses and displacements is large. As a numerical example, ball-indentation tests of composite materials are modelled by the three-dimensional finite element method.     

Boundary element method applied to three dimensional thermoelastic contact

In this paper we present a boundary element method to analyze and solve three dimensional frictionless thermoelastic contact problems. Although many problems in engineering can be solved with one-dimensional or two-dimensional models, those simplifications there are not possible in many others, such as the design of microelectronics packages. We calculate the stresses, movements, temperatures and thermal gradients on 3D solids. A thermal resistance at the contact zone depends on the local pressure is considered. The problem is solved by a double iterative method, so that in the final solution do not appear tensions in the contact zone or penetrations between the two solids. The solutions are compared with other works, where possible, to validate the method.     

Buckling of multi-lamellae compression flanges of welded I-beams: A unilateral elasto-plastic plate-stability problem

Buckling of compression flanges of welded I-beams consisting of more than one lamella is characterized by partial loss of contact between the lamella welded to the web and the neighbouring lamella. The boundaries of the contact regions of the buckling modes of the two lamellae are originally unknown. Thus, the present elasto-plastic plate-stability problem is coupled with a contact problem. It represents a unilateral problem because eigenforms with penetrations of lamellae are physically impossible. This problem is solved first for the elastic material domain. The Rayleigh-Ritz method is used for determining symmetric eigenforms. A finite-strip technique is employed for determining unsymmetric eigenforms. The solution of the unilateral elastic plate-buckling problems serves as the starting point for the iterative solution of the corresponding elasto-plastic plate-stability problem. The buckling pressures obtained are compared with corresponding buckling stresses resulting from a classical design procedure disregarding the interaction of the lamellae at buckling.     

TOPOLOGY OPTIMIZATION OF SHEETS IN CONTACT BY A SUBGRADIENT METHOD

We consider the solution of finite element discretized optimum sheet problems by an iterative algorithm. The problem is that of maximizing the stiffness of a sheet subject to constraints on the admissible designs and unilateral contact conditions on the displacements. The model allows for zero design volumes, and thus constitutes a true topology optimization problem. We propose and evaluate a subgradient optimization algorithm for a reformulation into a non-differentiable, convex minimization problem in the displacement variables. The convergence of the method and its low computational complexity are established. An optimal design is derived through a simple averaging scheme which combines the solutions to the linear design problems solved within the subgradient method. To illustrate the efficiency of the algorithm and investigate the properties of the optimal designs, thealgorithm is numerically tested on some medium- and large-scale problems. © 1997 by John Wiley & Sons, Ltd.     

Use of the Constrained Optimization Algorithms in Some Problems of Fracture Mechanics

In this paper we consider an algorithm of constrained optimization which arises from boundary variational principles of elastodynamics for bodies with cracks and unilateral constraints on the cracks edges. Variational formulation of unilateral contact problems with friction was considered, boundary variational functionals used with boundary integral equations were obtained and algorithm for solution of the unilateral contact problem with friction was developed. Some numerical results for 3-D elastodynamic unilateral contact problem for bodies with cracks are presented.     

A two‐level nonoverlapping Schwarz algorithm for the Stokes problem without primal pressure unknowns

A two‐level nonoverlapping Schwarz algorithm is developed for the Stokes problem. The main feature of the algorithm is that a mixed problem with both velocity and pressure unknowns is solved with a balancing domain decomposition by constraints (BDDC)‐type preconditioner, which consists of solving local Stokes problems and one global coarse problem related to only primal velocity unknowns. Our preconditioner allows to use a smaller set of primal velocity unknowns than other BDDC preconditioners without much concern on certain flux conditions on the subdomain boundaries and the inf–sup stability of the coarse problem. In the two‐dimensional case, velocity unknowns at subdomain corners are selected as the primal unknowns. In addition to them, averages of each velocity component across common faces are employed as the primal unknowns for the three‐dimensional case. By using its close connection to the Dual–primal finite element tearing and interconnecting (FETI‐DP algorithm) (SIAM J Sci Comput 2010; 32 : 3301–3322; SIAM J Numer Anal 2010; 47 : 4142–4162], it is shown that the resulting matrix of our algorithm has the same eigenvalues as the FETI‐DP algorithm except zero and one. The maximum eigenvalue is determined by H/h, the number of elements across each subdomains, and the minimum eigenvalue is bounded below by a constant, which does not depend on any mesh parameters. Convergence of the method is analyzed and numerical results are included. Copyright © 2011 John Wiley & Sons, Ltd.