the regularization method for an obstacle problem we give a relatively complete analysis for the regularization method which is usually used in solving non differentiable minimization problems the model problem considered in the paper is an obstacle problem in addition to the usual convergence result and a priori error estimates we provide a posteriori error estimates which are highly desired for practical implementation of the regularization method

dynamic analysis of structures with unilateral constraints numerical integration and reduction of structural equations structural dynamic equations with unilateral constraints upon the displacements velocities and accelerations are employed in order to represent vibrating elastic structures with normal oblique impact and friction interaction points for obtaining a numerical integration scheme the lagrange multipliers and a minimum work approach are employed at each time step the algorithm is presented as an extension of the generalized newmark scheme it seems to retain the asymptotic features of the original one the reduction of the number of dynamic degrees of freedom of the unilaterally constrained structures is carried out by representing the equations of motion in modal co ordinates of the unconstrained structure and truncating the dynamic contributions of higher modes the presented techniques have been verified by investigating free longitudinal vibroimpact motion laws of an elastic vibroconverter and free longitudinal and bending vibration of a vibroconverter interacting with a moving rigid body by oblique impact and friction forces

mesh refinement processes based on the generalized bisection of simplices mesh refinement processes based on the generalized bisection of simplices are discussed and characterized in the context of the general theory of rheinboldt siam j numer anal 17 1980 pp 766 778 then it is proved that such processes allow the construction of sequences of naturally smooth conforming and nested nonuniform meshes in fact it is shown that there exists a generalized mesh refinement operator that for any conforming triangulation delta and for any v subset delta produces a nested smooth conforming triangulation delta containing successors of all elements of v and such that the minimum cell size of delta is bounded from below by half the minimum cell size of delta two conforming mesh refinement algorithms that allow the selective refinement of computational triangulations are explicitly given

qualitative and numerical analysis of quasi static problems in elastoplasticity the quasi static problem of elastoplasticity with combined kinematic isotropic hardening is formulated as a time dependent variational inequality vi of the mixed kind that is it is an inequality involving a nondifferentiable functional and is imposed on a subset of a space this vi differs from the standard parabolic vi in that time derivatives of the unknown variable occur in all of its terms the problem is shown to possess a unique solution we consider two types of approximations to the vi corresponding to the quasi static problem of elastoplasticity semidiscrete approximations in which only the spatial domain is discretized by finite elements and fully discrete approximations in which the spatial domain is again discretized by finite elements and the temporal domain is discretized and the time derivative appearing in the vi is approximated in an appropriate way estimates of the errors inherent in the above two types of approximations in suitable sobolev norms are obtained for the quasi static problem of elastoplasticity in particular these estimates express rates of convergence of successive finite element approximations to the solution of the variational inequality in terms of element size h and where appropriate of the time step size k a major difficulty in solving the problems is caused by the presence of the nondifferentiable terms we consider some regularization techniques for overcoming the difficulty besides the usual convergence estimates we also provide a posteriori error estimates which enable us to estimate the error by using only the solution of a regularized problem

boundary minimum principles for the unilateral contact problems abstract the aim of the present paper is the derivation of boundary variational principles for the unilateral contact problem using the inequality constrained principles of minimum potential and complementary energy and the equivalent variational inequality formulations we derive first saddle point formulations for the problems using appropriate langrangian functions an elimination technique allows the formulation of two minimum principles on the boundary with respect to the unknown normal displacements and reactions of the contact region respectively it is also shown that these two minimum problems are equivalent to multivalued boundary integral equations involving symmetric operators the theory is illustrated by numerical examples solved both by the fem and the bem

on an elastic plastic problem etude d un probleme elastoplastique de valeur limite quasi statique pour des corps pouvant etre discretement heterogenes et elastiquement anisotropes dont le comportement plastique est regi par des fonctions de deformation lineaires par morceaux dependantes de l histoire du materiau et par des lois de durcissement couplees existence unicite et regularite d une solution faible convergence des solutions d elements finis

pointwise a posteriori error estimates for elliptic problems on highly graded meshes pointwise a posteriori error estimates are derived for linear second order elliptic problems over general polygonal domains in 2d the analysis carries over regardless of convexity accounting even for slit domains and applies to highly graded unstructured meshes as well a key ingredient is a new asymptotic a priori estimate for regularized green s functions the estimators lead always to upper bounds for the error in the maximum norm along with lower bounds under very mild regularity and nondegeneracy assumptions the effect of both point and line singularities is examined three popular local estimators for the energy norm are shown to be equivalent when suitably interpreted to those introduced here

mixed fem of higher order for time dependent contact problems abstract in this paper mixed finite element methods of higher order for time dependent contact problems are discussed the mixed methods are based on resolving the contact conditions by the introduction of lagrange multipliers dynamic signorini problems with and without friction are considered involving thermomechanical and rolling contact rothe s method is used to provide a suitable time and space discretization to discretize in time a stabilized newmark method is applied as an adequate time stepping scheme the space discretization relies on finite elements of higher order in each time step the resulting problems are solved by uzawa s method or alternatively by methods of quadratic programming via a suitable formulation in terms of the lagrange multipliers numerical results are presented towards an application in production engineering the results illustrate the performance of the presented techniques for a variety of problem formulations

maximum norm a posteriori error estimates for singularly perturbed reaction diffusion problems on anisotropic meshes residual type a posteriori error estimates in the maximum norm are given for singularly perturbed semilinear reaction diffusion equations posed in polygonal domains linear finite elements are considered on anisotropic triangulations the error constants are independent of the diameters and the aspect ratios of mesh elements and of the small perturbation parameter

variational principles and convergence of finite element approximations of a holonomic elastic plastic problem it is shown that a boundary value problem based on a holonomic elastic plastic constitutive law may be formulated equivalently as a variational inequality of the second kind a regularised form of the problem is analysed and finite element approximations are considered it is shown that solutions based on finite element approximation of the regularised problem converge

a level set enhanced natural kernel contact algorithm for impact and penetration modeling summary a natural kernel contact nkc algorithm under the framework of the semi lagrangian reproducing kernel particle method semi lagrangian rkpm is proposed to model multi body contact with specific consideration for impact and penetration modeling the nkc algorithm utilizes the interaction of the semi lagrangian kernel functions associated with contacting bodies to serve as the non penetration condition the effects of friction are represented by introducing a layer of the friction like elasto plasticity material between contacting bodies this approach allows the frictional contact conditions and the associated kinematics to be naturally embedded in the semi lagrangian rkpm inter particle force calculation the equivalence in the karush kuhn tucker conditions between the proposed nkc algorithm and the conventional contact kinematic constraints as well as the associated state variable relationships are identified a level set method is further introduced in the nkc algorithm to represent the contact surfaces without pre defined potential contact surfaces the stability analysis performed in this work shows that temporal stability in the semi lagrangian rkpm with nkc algorithms is related to the velocity gradient between contacting bodies the proposed methods have been verified by several benchmark problems and applied to the simulation of impact and penetration processes copyright 2014 john wiley sons ltd

a quasistatic frictional problem with normal compliance a friction and contact problem will be considered for an elastic solid body which under the influence of volume and surface forces is in contact with a rigid foundation

reliable a posteriori error control for nonconforming finite element approximation of stokes flow we derive computable a posteriori error estimates for the lowest order nonconforming crouzeix raviart element applied to the approximation of incompressible stokes flow the estimator provides an explicit upper bound that is free of any unknown constants provided that a reasonable lower bound for the inf sup constant of the underlying problem is available in addition it is shown that the estimator provides an equivalent lower bound on the error up to a generic constant

theory and approximation of quasistatic frictional contact problems abstract a variational formulation of a class of quasistatic frictional contact problems is derived for which a compliant interface model of the contact surface is assumed proof of existence and uniqueness of solutions is given finite element approximations of a regularized version of the fractional problem are developed and a priori error estimates are derived the theory is implemented and used to solve representative test problems of frictional contact the numerical simulations show that frictional stress distributions can be very sensitive to load histories and surface properties of contact interfaces

maximum norm error estimators for three dimensional elliptic problems in this paper we define an a posteriori error estimator for finite element approximations of 3 d elliptic problems we prove that the estimator is equivalent up to logarithmic factors of the meshsize to the maximum norm of the error the results are valid for an arbitrary polyhedral domain and rather general meshes we also obtain analogous results for the nonconforming method of crouzeix raviart finally we present some numerical results comparing adaptive procedures based on controlling the error in different norms

error estimators for nonconforming finite element approximations of the stokes problem in this paper we define and analyze a posteriori error estimators for nonconforming approximations of the stokes equations we prove that these estimators are equivalent to an appropriate norm of the error for the case of piecewise linear elements we define two estimators both of them are easy to compute but the second is simpler because it can be computed using only the right hand side and the approximate velocity we show how the first estimator can be generalized to higher order elements finally we present several numerical examples in which one of our estimators is used for adaptive refinement

a posteriori pointwise error computation for 2 d transport equations based on the variational multiscale method abstract this article presents a general framework to estimate the pointwise error of linear partial differential equations the error estimator is based on the variational multiscale theory in which the error is decomposed in two components according to the nature of the residuals element interior residuals and inter element jumps the relationship between the residuals coarse scales and the error components fine scales is established yielding to a very simple model in particular the pointwise error is modeled as a linear combination of bubble functions and green s functions if residual free bubbles and the classical green s function are employed the technology leads to an exact explicit method for the pointwise error if bubble functions and free space green s functions are employed then a local projection problem must be solved within each element and a global boundary integral equation must be solved on the domain boundary as a consequence this gives a model for the so called fine scale green s functions the numerical error is studied for the standard galerkin and supg methods with application to the heat equation the reaction diffusion equation and the convection diffusion equation numerical results show that stabilized methods minimize the propagation of pollution errors which stay mostly locally

error estimates for space time discretizations of a rate independent variational inequality this paper deals with error estimates for space time discretizations in the context of evolutionary variational inequalities of rate independent type after introducing a general abstract evolution problem we address a fully discrete approximation and provide a priori error estimates the application of the abstract theory to a semilinear case is detailed in particular we provide explicit space time convergence rates for classical strain gradient plasticity and the isothermal souza auricchio model for shape memory alloys

regularized friction and continuation comparison with coulomb s law abstract periodic solutions of systems with friction are difficult to investigate because of the non smooth nature of friction laws this paper examines periodic solutions and most notably stick slip on a simple one degree of freedom system mass spring damper and belt with coulomb s friction law and with a regularized friction law i e the friction coefficient becomes a function of relative speed with a stiffness parameter with coulomb s law the stick slip solution is constructed step by step which gives a usable existence condition with the regularized law the asymptotic numerical method and the harmonic balance method provide bifurcation diagrams with respect to the belt speed or normal force and for several values of the regularization parameter formulations from the coulomb case give the means of a comparison between regularized solutions and a standard reference with an appropriate definition regularized stick slip motion exists its amplitude increases with respect to the belt speed and its pulsation decreases with respect to the normal force

a posteriori error estimators for nonconforming approximation of some quasi newtonian flows this paper deals with a posteriori error estimators for nonconforming approximations of quasi newtonian flows we consider the crouzeix raviart piecewise linear approximations of scalar elliptic problems and define an error estimator when omega is a simply connected domain the error is dominated by the estimator this estimator can be generalized to higher order elements we define a posteriori error estimators for the fortin soulie piecewise quadratic approximations of quasi newtonian flows and prove that the error is dominated by the estimator this estimator can be computed locally in terms of the approximate solution and is therefore suitable for adaptive refinement

algorithms for the analysis of 3d finite strain contact problems from the constraint imposition aspects in 3d to friction regularization various ideas are exposed in this paper a variation of the rockafellar lagrangian is proposed which results in continuous second order derivatives if lagrange multiplier estimates are greater or equal than one this fact allows the adoption of a full second order i e lagrange newton method avoiding sequential unconstrained minimization techniques an algorithm for global and local contact detection is presented which is developed for dealing with large step sizes typical of implicit methods a modified constraint definition to deal with non smooth situations is presented aspects of friction implementation including a regularization scheme which ensures stepwise objectivity are detailed finally several illustrative examples are carried out with success

pointwise a posteriori error control for elliptic obstacle problems we consider a finite element method for the elliptic obstacle problem over polyhedral domains in d which enforces the unilateral constraint solely at the nodes we derive novel optimal upper and lower a posteriori error bounds in the maximum norm irrespective of mesh fineness and the regularity of the obstacle which is just assumed to be holder continuous they exhibit optimal order and localization to the non contact set we illustrate these results with simulations in 2d and 3d showing the impact of localization in mesh grading within the contact set along with quasi optimal meshes

pointwise a posteriori error estimates for monotone semi linear equations we derive upper and lower a posteriori estimates for the maximum norm error in finite element solutions of monotone semi linear equations the estimates hold for lagrange elements of any fixed order non smooth nonlinearities and take numerical integration into account the proof hinges on constructing continuous barrier functions by correcting the discrete solution appropriately and then applying the continuous maximum principle no geometric mesh constraints are thus required numerical experiments illustrate reliability and efficiency properties of the corresponding estimators and investigate the performance of the resulting adaptive algorithms in terms of the polynomial order and quadrature

a modification of the delves lyness method for locating the zeros of analytic functions abstract a new formula is proposed for the calculation of the integrals involved in the application of the delves lyness method for locating the zeros of analytic functions inside an arbitrary simple smooth contour this formula does not contain the derivative of the analytic function whose zeros are sought and moreover is free from multivaluedness problems for the integrand

asymptotically exact a posteriori estimators for the pointwise gradient error on each element in irregular meshes part 1 a smooth problem and globally quasi uniform meshes a class of a posteriori estimators is studied for the error in the maximum norm of the gradient on single elements when the finite element method is used to approximate solutions of second order elliptic problems the meshes are unstructured and in particular it is not assumed that there are any known superconvergent points the estimators are based on averaging operators which are approximate gradients recovered gradients which are then compared to the actual gradient of the approximation on each element conditions are given under which they are asympotically exact or equivalent estimators on each single element of the underlying meshes asymptotic exactness is accomplished by letting the approximate gradient operator average over domains that are large in a controlled fashion to be detailed below compared to the size of the elements

a posteriori error estimators for the stokes equations we present two a posteriori error estimators for the mini element discretization of the stokes equations one is based on a suitable evaluation of the residual of the finite element solution the other one is based on the solution of suitable local stokes problems involving the residual of the finite element solution both estimators are globally upper and locally lower bounds for the error of the finite element discretization numerical examples show their efficiency both in estimating the error and in controlling an automatic self adaptive mesh refinement process the methods presented here can easily be generalized to the navier stokes equations and to other discretization schemes

a generalization of the siewert burniston method for the determination of zeros of analytic functions the siewert burniston method for the derivation of closed form formulas for the zeros of sectionally analytic functions with a discontinuity interval along the real axis based on the riemann hilbert boundary value problem in complex analysis is generalized to apply to the determination of the zeros of analytic functions without discontinuity intervals inside or outside simple smooth contours an application of this method to the closed form solution of the transcendental equation zez beb appearing in the theory of neutron moderation in nuclear reactors is also made

determination of poles of sectionally meromorphic functions abstract the numerical methods of abd ellal delves and reid for locating poles of meromorphic functions inside smooth closed contours c in the complex plane are generalized to apply to sectionally meromorphic functions satisfying a homogeneous or nonhomogeneous riemann hilbert boundary value problem on the simple closed contours of their discontinuity which may intersect c in the first case the zeros of the meromorphic functions can be determined as well two simple illustrations of the present generalization are also presented

united elements method for general contact impact problems abstract this work presents the details of a finite element based numerical method to analyse general contact impact problems called united elements method uem in contrast to usual methods that is lagrange multipliers and penalty function this method is conceptual suitable for computer programming and free from inherent deficiencies of those methods the key points of this procedure are elimination of the normal degree of freedom of the contactor node from the system energy potential functional in terms of target surface motion and proper calculation of target element unbalanced forces to incorporate the contact constraints after a review on classic formulations the mathematical and matrix presentation of the united element stiffness matrix are discussed solution of some examples adopting a newton raphson based solution together with a normal return scheme support the capability and accuracy of the procedure in static and dynamic problems

quasi optimal and robust a posteriori error estimates in for the approximation of allen cahn equations past singularities quasi optimal a posteriori error estimates in l 0 t l 2 are derived for the finite element approximation of allen cahn equations the estimates depend on the inverse of a small parameter only in a low order polynomial and are valid past topological changes of the evolving interface the error analysis employs an elliptic reconstruction of the approximate solution and applies to a large class of conforming nonconforming mixed and discontinuous galerkin methods numerical experiments illustrate the theoretical results

local a posteriori estimates on a nonconvex polygonal domain a class of a posteriori estimators is studied for the error in the maximum norm of the gradient on single elements when the finite element method is used to approximate solutions of second order elliptic problems on a nonconvex polygonal domain the results are extensions of previous results for smooth domains w hoffmann et al math comp 70 2001 pp 897 909 a h schatz and l b wahlbin math comp 73 2004 pp 517 523

a continuum based finite element formulation for the implicit solution of multibody large deformation frictional contact problems in this paper a formulation is presented for the finite element treatment of multibody large deformation frictional contact problems the term multibody is used to mean that when two bodies mechanically contact both may be deformable a novel aspect of the approach advocated is that the equations governing contact are developed in the continuum setting first before deriving the corresponding finite element equations this feature distinguishes the current work from many earlier treatments of contact problems and renders it considerably more general in particular the approach yields a characterization of the frictional constraint assuming a coulomb law suitable for arbitrary discretizations in either two or three dimensions a geometric framework is constructed within which both frictionless and frictional response are naturally described making subsequent finite element discretization a straightforward substitution of finite dimensional solution spaces for their continuum counterparts to our knowledge this general formulation and implementation of the frictional contact problem in a finite element setting has not been reported previously in the literature the development includes exact linearization of the statement of virtual work which enables optimal convergence properties for newton raphson solution strategies and which appears to be highly desirable if not essential for the general robustness of implicit finite element techniques since the theory and subsequent linearization require no limitations on the amount of deformation or relative sliding that can occur the resulting treatment of frictional contact is suitable for a wide range of examples displaying significant non linear behaviour this assertion is substantiated through presentation of a variety of examples in both two and three dimensions

analysis of dynamic nonlinear thermoviscoelastic beam problems abstract in this paper we study a dynamic contact problem between a nonlinear viscoelastic beam and two rigid obstacles thermal effects are also taken into account and the contact is modeled using the classical signorini condition the existence of solutions is proved by considering approximate problems from a penalization procedure proving some a priori estimates and passing to the limit an exponential decay property is also obtained then fully discrete approximations of the approximate problem are provided using the finite element method for the spatial approximation and the implicit euler scheme for the discretization of the time derivatives a stability property is obtained and a priori error estimates are proved from which under some additional regularity conditions the linear convergence of the algorithm is deduced fully discrete approximations of the signorini problem are then introduced proceeding in a similar way for which a stability property is obtained finally some numerical simulations are performed to demonstrate the accuracy of the approximation and the behavior of the solution

subdomain methods in structural mechanics the principle of subdomain methods has been introduced many years ago1 and such methods have been used with success in recent years in this paper we discuss some methods used to solve the condensed system resulting from the schur complement method we will present the hybrid method and a mechanical application the contact problem between two deformable solids in the case of such a problem we propose two methods one for the contact with friction and one for the contact without friction finally the results of validation tests for both proposed approaches are given

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

riemann solutions and spacetime discontinuous galerkin method for linear elastodynamic contact abstract we derive riemann solutions for stick slip and separation contact modes in a linear elastic material with an isotropic coulomb friction relation and explore their numerical implementation the riemann solutions preserve the characteristic structure of the underlying elastodynamic system and imply dynamic contact conditions that are distinct from the quasi static conditions used in some numerical models nonphysical discontinuities in the standard coulomb model at stick slip transitions can cause contact mode chatter in numerical simulations we restate the coulomb relation to remove these artificial discontinuities and eliminate the need for algorithmic remedies discontinuous response at abrupt separation to contact transitions is physically reasonable and we propose a regularization scheme to address this case we implement the riemann contact solutions within an adaptive spacetime discontinuous galerkin sdg code and report numerical results that demonstrate the model s efficacy

on the finite element method for mixed variational inequalities arising in elastoplasticity we analyze the finite element method for a class of mixed variational inequalities of the second kind which arises in elastoplastic problems an abstract variational inequality of which the elast

on geometrically nonlinear contact problems with friction abstract the analysis of quasi static contact problems applying an updated respectively a total lagrangian formulation is extended in a way that frictional effects according to coulomb s law are being incorporated into the incremental procedure which is closely related to the unit load method further novel aspects are to be found in the iterative computation of the tangential components of the stress distribution in the contact area due to various boundary conditions the second part of the paper deals with the application of the nonlinear programming technique to problems with unilateral constraints using standard finite element techniques an augmented lagrangian is defined for a predeformed compressed rod neglecting frictional effects the contact zones and the nodal contact forces between the deformed rod and lateral rigid objects are described by the kuhn tucker conditions

dynamic analysis of two dimensional frictional contact by linear complementarity problem formulation two dimensional dynamic contact is formulated as a linear complementarity problem using contact compatibility conditions and the coulomb friction law for the time integration of dynamic response displacements are approximated as second order polynomials and a new scheme of considering velocity discontinuities is presented the efficiency of the method is shown with three numerical examples

a variational principle for dynamic contact with large deformation abstract dynamic contact between a deformable body with either large or small deformation and a rigid obstacle can be modeled with signorini contact condition stated in terms of the normal component of the displacement as long as the relative positions are clearly defined the geometrical relation however is unpredictable when large motion or large deformation and complex surface geometry are involved the use of normal velocity is more suitable a variational inequality is proposed for the dynamic contact problem the penalty method is introduced and implemented in explicit finite element software numerical examples are presented to demonstrate the robustness of the algorithm

a one mesh method for the cell centered discretization of sliding abstract a new method is described to treat slide lines in cell centered lagrangian schemes for the modeling of sliding problems between two fluids in the framework of compressible hydrodynamics the method is an extension of the one proposed in clair et al 2013 1 and is conservative in momentum and total energy we illustrate our method which is based on the minimization of an objective function over a specific set that models the sliding constraint by several basic problems

localized pointwise a posteriori error estimates for gradients of piecewise linear finite element approximations to second order quasilinear elliptic problems two types of pointwise a posteriori error estimates are presented for gradients of finite element approximations of second order quasilinear elliptic dirichlet boundary value problems over convex polyhedral domains omega in space dimension n ge 2 we first give a residual estimator which is equivalent to nabla u u h l infty omega up to higher order terms the second type of residual estimator is designed to control nabla u u h locally over any subdomain of omega it is a novel a posteriori counterpart to the localized or weighted a priori estimates of sch98 this estimator is shown to be equivalent up to higher order terms to the error measured in a weighted global norm which depends on the subdomain of interest all estimates are proved for general shape regular meshes which may be highly graded and unstructured the constants in the estimates depend on the unknown solution u in the nonlinear case but in a fashion which places minimal restrictions on the regularity of u

local a posteriori estimates for pointwise gradient errors in finite element methods for elliptic problems we prove local a posteriori error estimates for pointwise gradient errors in finite element methods for a second order linear elliptic model problem first we split the local gradient error into a computable local residual term and a weaker global norm of the finite element error the pollution term using a mesh dependent weight the residual term is bounded in a sharply localized fashion in specific situations the pollution term may also be bounded by computable residual estimators on nonconvex polygonal and polyhedral domains in two and three space dimensions we may choose estimators for the pollution term which do not employ specific knowledge of corner singularities and which are valid on domains with cracks the finite element mesh is only required to be simpliciai and shape regular so that highly graded and unstructured meshes are allowed

convergence of approximations to the primal problem in plasticity under conditions of minimal regularity this work considers semi and fully discrete approximations to the primal problem in elastoplasticity the unknowns are displacement and internal variables and the problem takes the form of an evolution variational inequality strong convergence of time discrete as well as spatially and fully discrete approximations is established without making any assumptions of regularity over and above those established in the proof of well posedness of this problem

finite element analysis of a holonomic elastic plastic problem a thorough analysis of the finite element method is given for a holonomic elastic plastic problem an inequality of the cea s lemma type is proved which is the basis of error estimates for various finite element solutions difficulty caused by a non differentiable term in the problem can be overcome by using two convergent procedures an iterative procedure and a regularization procedure an a posteriori quantitative error estimate is derived for the regularized solution

finite elements in space and time for dynamic contact problems dynamic obstacle and signorini problems are discretized by continuous and discontinuous finite elements in space and time the resulting discrete problems are attributed to a standard implicit time stepping scheme through relaxation of impact phenomena and suitable numerical integration the method can cope with dynamic contact problems which is shown by an analysis of a model problem moreover numerical examples demonstrate that it is actually able to approximate the solution of dynamic contact problems which are not fully covered by the theory copyright 2008 john wiley sons ltd

adaptive finite element methods for an optimal control problem involving dirac measures the purpose of this work is the design and analysis of a reliable and efficient a posteriori error estimator for the so called pointwise tracking optimal control problem this linear quadratic optimal control problem entails the minimization of a cost functional that involves point evaluations of the state thus leading to an adjoint problem with dirac measures on the right hand side control constraints are also considered the proposed error estimator relies on a posteriori error estimates in the maximum norm for the state and in muckenhoupt weighted sobolev spaces for the adjoint state we present an analysis that is valid for two and three dimensional domains we conclude by presenting several numerical experiments which reveal the competitive performance of adaptive methods based on the devised error estimator

parametric quadratic programming method for dynamic contact problems with friction based on the parametric variational principle a general but effective parametric quadratic programming technique satisfying various contact conditions is established for dynamic analysis of contact problems with friction and damping the discretization with respect to time and space leads to a static linear complementary problem lcp for each time step which is solved by a quadratic optimization algorithm such as the lemke algorithm etc thus the convergence and numerical stability of the solution can be guaranteed the substructure condensation technique is implemented to handle the unknown contact boundary condition so that the computation effect is considerably reduced as an application of the method presented three dynamic contact examples and numerical results are given

a numerical analysis of a class of problems in elastodynamics with friction on utilise les methodes d elements finis pour analyser le comportement dynamique de corps lineairement elastiques ayant une partie de leurs frontieres soumise au frottement et a des tractions normales prescrites on etablit des estimations d erreur on presente un algorithme de resolution des problemes a deux dimensions et on decrit plusieurs essais numeriques

fully localized a posteriori error estimators and barrier sets for contact problems we derive novel pointwise a posteriori error estimators for elliptic obstacle problems which except for obstacle resolution completely vanish within the full contact set localization we then construct a posteriori barrier sets for free boundaries under a natural stability or nondegeneracy condition we illustrate localization properties as well as reliability and efficiency for both solutions and free boundaries via several simulations in 2 and 3 dimensions

an a posteriori error analysis of an elliptic optimal control problem in measure space abstract we propose an a posteriori error estimator for an optimal control problem with sparsity promoting cost the control variable lies in the space of regular borel measures we consider a solution technique that relies on the discretization of the control variable as a linear combination of dirac measures the proposed a posteriori error estimator can be decomposed into the sum of two contributions an error estimator in the maximum norm for the discretization of the adjoint equation and an estimator in the l 2 norm that accounts for the approximation of the state equation we prove that the designed error estimator is locally efficient and we explore its reliability properties the analysis is valid for two and three dimensional domains we illustrate the theory with numerical examples

a linear finite element approach to the solution of the variational inequalities arising in contact problems of structural dynamics the present paper deals with the theoretical and numerical treatment of dynamic unilateral problems the governing equations are formulated as an equivalent variational inequality expressing d alembert s principle in its inequality form the discretization with respect to time and space leads to a static nonlinear programming problem which is solved by an appropriate algorithm some properties of dynamic unilateral problems are outlined and the influence of several parameters on the solution is investigated by means of numerical examples

coupling of finite elements and boundary elements methods for study of the frictional contact problem this paper presents the numerical study of the elastic contact problem with coulomb friction between both deformable sphere and support using the finite element method we solve this problem governed by a variational inequality with a successive approximations method each iteration of which being constituted by the solution of an intermediate problem with normal pressure fixed to compute stresses in contact we couple the finite elements with the boundary elements our results are conform to the hertz theory and the analytical solution of spence moreover our study shows the influence of friction on parameters such as the contact radius and the normal pressure

an analysis of metal forming processes using large deformation elastic plastic formulations a concise survey of the literature related to the large deformation elasto plasticity problems including unilateral contact and friction is presented together with an extension of the friction law for large deformation analysis starting from the principle of virtual work the so called total lagrangian and updated lagrangian formulations are derived based on some fundamental assumptions in linearizing the nonlinear equations by introducing the zaremba jaumann co rotational increment to the cauchy stress tensor the classical prandtl reuss equations are generalized for describing the elastic plastic material behavior to allow a proper consideration of the contact conditions in the incremental analysis a general friction law with an associated isotropic coulomb sliding rule is obtained by the similarity between dry friction and plasticity finite element discretizations and approximations are applied to the resulting formulation of the updated lagrangian approach four example problems are solved to test the formulations developed in this paper the emphasis is made toward the numerical accuracy of the finite element solutions

a mixed eulerian lagrangian displacement model for large deformation analysis in solid mechanics abstract a new mixed kinematic description for large deformation solid mechanics is presented the novel aspect of the new model is that both the initial and current configurations are treated as unknown large deformation statements of the principles of virtual work and stationary potential energy are formulated with the new description and presented in incremental form suitable for iterative finite element solutions the well known total lagrangian and updated lagrangian descriptions are shown to be special cases of the mixed formulation the new description is advantageous in situations where it is difficult to select an appropriate material discretization prior to analysis applications to large deformation frictional contact and fracture mechanics are presented to demonstrate the utility of the new kinematic description

a dynamic thermoviscoelastic contact problem with the second sound effect abstract this paper deals with a contact problem describing the mechanical and thermal evolution of a damped extensible thermoviscoelastic beam under the cattaneo law relating the heat flux to the gradient of the temperature the beam is rigidly clamped at its left end whereas the right end of the beam moves vertically between reactive stops like a nonlinear spring existence and uniqueness of the solution is proved as well as the exponential decay of the related energy then fully discrete approximations are introduced by using the classical finite element method and the implicit euler scheme to approximate the spatial variable and to discretize the time derivatives respectively an a priori error estimates result is proved from which the linear convergence of the algorithm is deduced the case where the two stops are rigid is also studied from the point of view of the existence and longtime behavior of the solutions finally some numerical simulations are presented to demonstrate the accuracy of the approximation and the behavior of the solution

on the consideration of unilateral contact and friction in cracks the boundary integral method the scope of the present paper is the study of cracks having a given geometry by taking into account unilateral contact and friction phenomena between the two sides of the crack the arising problem is highly non classical because of the inequality relations describing the unilateral contact and friction interface conditions the indirect biem is extended appropriately in order to treat this problem numerical examples concerning the calculation of stress intensity factors under the unilateral contact and friction interface conditions illustrate the developed method

the variational formulation of viscoplastic constitutive updates we present a class of constitutive updates for general viscoplastic solids including such aspects of material behavior as finite elastic and plastic deformations non newtonian viscosity rate sensitivity and arbitrary flow and hardening rules the distinguishing characteristic of the proposed constitutive updates is that by construction the corresponding incremental stress strain relations derive from a pseudo elastic strain energy density this in turn confers the incremental boundary value problem a variational structure in particular the incremental deformation mapping follows from a minimum principle this minimum principle may conveniently be taken as a basis for error estimation and mesh adaption the accuracy and robustness of the variational constitutive updates is demonstrated with the aid of convergence tests involving the finitely deforming mises solid and ductile single crystals the ability of the updates to resolve the complex patterns of slip activity which arise in the latter application is particularly noteworthy

a posteriori error estimates in the maximum norm for parabolic problems we derive a posteriori error estimates in the l infty 0 t l infty omega norm for approximations of solutions to linear parabolic equations using the elliptic reconstruction technique introduced by makridakis and nochetto and heat kernel estimates for linear parabolic problems we first prove a posteriori bounds in the maximum norm for semidiscrete finite element approximations we then establish a posteriori bounds for a fully discrete backward euler finite element approximation the elliptic reconstruction technique greatly simplifies our development by allowing the straightforward combination of heat kernel estimates with existing elliptic maximum norm error estimators

locating a crack of arbitrary but known shape by the method of path independent integrals abstract the solution of a crack problem of an arbitrary but known shape inside an infinite plane isotropic elastic medium can be achieved in general by the method of complex singular integral equations and their numerical solution by using the gauss or the lobatto chebyshev methods in a few special cases like straight or circular arc shaped cracks this solution is available in closed form in this paper we will not contribute to the above methods of solution of crack problems but we will propose a method for the determination of the exact position of such a crack inside a closed contour in the elastic medium by gathering and using information along this contour only by experimental techniques and applying the method of complex path independent integrals for the location of the crack this paper constitutes a nontrivial generalization of relevant previous results by the author and it is of quite general applicability in fracture mechanics for nondestructive testing numerical results for the particular case of a straight crack are displayed for the illustration of the efficiency of the method the generalization of the present results to the determination of additional geometric and loading parameters of the crack is also suggested very briefly and related numerical results concerning the length of the crack and the pressure distribution on it in the aforementioned numerical application are also presented

maximum norm a posteriori error estimates for singularly perturbed elliptic reaction diffusion problems residual type a posteriori error estimates in the maximum norm are given for singularly perturbed semilinear reaction diffusion equations posed in polyhedral domains standard finite element approximations are considered the error constants are independent of the diameters of mesh elements and the small perturbation parameter in our analysis we employ sharp bounds on the green s function of the linearized differential operator numerical results are presented that support our theoretical findings