the primal dual active set strategy as a semismooth newton method this paper addresses complementarity problems motivated by constrained optimal control problems it is shown that the primal dual active set strategy which is known to be extremely efficient for this class of problems and a specific semismooth newton method lead to identical algorithms the notion of slant differentiability is recalled and it is argued that the max function is slantly differentiable in lp spaces when appropriately combined with a two norm concept this leads to new local convergence results of the primal dual active set strategy global unconditional convergence results are obtained by means of appropriate merit functions

analysis of two active set type methods to solve unilateral contact problems in this work two active set type methods are considered in order to solve a mathematical problem which describes the frictionless contact between a deformable body and a perfectly rigid obstacle the so called signorini problem these methods are the primal dual active set method and the projection iterative method our aim here is to analyze these two active set type methods and to carry out a comparison with the well known augmented lagrangian method by considering two representative contact problems in the case of large and small deformation after presenting the mechanical formulation in the hyperelasticity framework we establish weak formulations of the problem and the existence result of the weak solution is recalled then we give the finite element approximation of the problem and a description of the numerical methods is presented the main result of this work is to provide a convergence result for the projection iterative method finally we present numerical simulations which illustrate the behavior of the solution and allow the comparison of the numerical methods

twisted groups and locally toroidal regular polytopes in recent years much work has been done on the classification of abstract regular polytopes by their local and global topological type abstract regular polytopes are combinatorial structures which generalize the well known classical geometric regular polytopes and tessellations in this context the classical theory is concerned with those which are of globally or locally spherical type in a sequence of papers the authors have studied the corresponding classification of abstract regular polytopes which are globally or locally toroidal here this investigation of locally toroidal regular polytopes is continued with a particular emphasis on polytopes of ranks 5 and 6 for large classes of such polytopes their groups are explicitly identified using twisting operations on quotients of coxeter groups in particular this leads to new classification results which complement those obtained elsewhere the method is also applied to describe certain regular polytopes with small facets and vertex figures

on h p adaptive bem for frictional contact problems in linear elasticity a mixed formulation for a tresca frictional contact problem in linear elasticity is considered in the context of boundary integral equations which is later extended to coulomb friction the discrete lagrange multiplier an approximation of the surface traction on the contact boundary part is a linear combination of biorthogonal basis functions in case of curved isoparametric elements these are the solutions of local problems in particular the biorthogonality allows to rewrite the variational inequality constraints as a simple set of complementarity problems thus enabling an efficient application of a semi smooth newton solver for the discrete mixed problem converging locally super linearly in the frictional case and quadratically in the frictionless case typically the solution of frictional contact problems is of reduced regularity at the interface between contact to non contact and from stick to slip to identify the a priori unknown locations of these interfaces two a posteriori error estimations are introduced in a first step the error is split into specific error contributions resulting from the contact and friction conditions and from the discretization error of a variational equation for the latter a residual and a bubble error estimation are considered explicitly the numerical experiments show the applicability of the derived error estimations and the superiority of h p adaptivity compared to low order uniform and adaptive approaches

fluid structure interaction for non conforming interfaces based on a dual mortar formulation abstract in the present work the problem of fluid structure interaction fsi with independently space discretized fluid and structure fields is addressed in the context of finite elements to be able to deal with non conforming meshes at the fluid structure interface we propose the integration of a dual mortar method into the general fsi framework this method has lately been used successfully to impose interface constraints in other contexts such as finite deformation contact the main focus is set on monolithic coupling algorithms for fsi here in these cases the dual mortar approach allows for the elimination of the additional lagrange multiplier degrees of freedom from the global system by condensation the resulting system matrices have the same block structure as their counterparts for the conforming case and permit the same numerical treatment partitioned dirichlet neumann coupling is also considered briefly and it is shown that the dual mortar approach permits a numerically efficient mapping between fluid and structure quantities at the interface numerical examples demonstrate the efficiency and robustness of the proposed method we present results for a variety of different element formulations for the fluid and the structure field indicating that the proposed method is not limited to any specific formulation furthermore the applicability of state of the art iterative solvers is considered and the convergence behavior is shown to be comparable to standard simulations with conforming discretizations at the interface

modelling unilateral frictionless contact using the null space method and cubic b spline interpolation the analysis of unilateral sliding contact in elasticity is equivalent to a minimisation problem subjected to a set of inequality constraints however the presence of boundary discontinuities such as those stemming from the spatial discretisation appears as a major problem to determine the set of active constraints this work introduces a smoothing technique of the master surface resorting to cubic b spline interpolation which is c 1 continuous in contact situations between elastic and rigid bodies and g 1 continuous in elastic elastic contact problems the technique is applied in conjunction with the null space method where the solution is searched in an unconstrained manifold the resulting formulation eases the contact transition along the master surface and recovers the quadratic convergence of the iterative newton raphson process the robustness of the method is demonstrated using 2d and 3d examples

semismooth newton method for frictional contact between pseudo rigid bodies recently developed semismooth newton approach is adopted in the context of the frictional contact between three dimensional pseudo rigid bodies the signorini coulomb problem is formulated according to the formalism of the contact dynamics method hybrid linearisation penalty scaling and line search techniques are combined as the global convergence enhancements of the newton algorithm quasi static simulations of dry masonry assemblies exemplify performance of the presented framework

a primal dual active set algorithm for three dimensional contact problems with coulomb friction in this paper efficient algorithms for contact problems with tresca and coulomb friction in three dimensions are presented and analyzed the numerical approximation is based on mortar methods for nonconforming meshes with dual lagrange multipliers using a nonsmooth complementarity function for the three dimensional friction conditions a primal dual active set algorithm is derived the method determines active contact and friction nodes and at the same time resolves the additional nonlinearity originating from sliding nodes no regularization and no penalization are applied and superlinear convergence can be observed locally in combination with a multigrid method it defines a robust and fast strategy for contact problems with tresca or coulomb friction the efficiency and flexibility of the method is illustrated by several numerical examples

improved robustness and consistency of 3d contact algorithms based on a dual mortar approach abstract mortar finite element methods have been successfully applied as discretization scheme to a wide range of contact and impact problems in recent years the 3d finite deformation contact formulation taken up and enhanced in this paper is based on a mortar approach using so called dual lagrange multipliers which substantially facilitate the treatment of interface constraints as compared with standard mortar techniques despite being quite well established in the meantime dual mortar methods may lack robustness or even consistency in certain situations e g when large curvatures occur in the contact zone or when one body slides off another at a sharp edge however since such scenarios are regularly appearing in engineering practice the present contribution provides several new extensions that completely resolve these issues and thus significantly improve the applicability of dual mortar formulations for challenging contact problems the proposed extensions include a consistent biorthogonalization procedure to obtain feasible dual lagrange multiplier shape functions close to boundaries of the contact surfaces an improved conditioning of the global linear system of equations by nodal scaling and a novel approach to unify the advantages of standard and dual mortar methods via a petrov galerkin type of lagrange multiplier interpolation several numerical examples demonstrate the achievable improvements in terms of consistency and robustness for 3d contact analysis including finite deformations

consistent treatment of boundaries with mortar contact formulations using dual lagrange multipliers abstract computational modelling of contact problems raises two basic questions which method should be used to enforce the contact conditions and how should this method be discretised the most popular enforcement methods are the lagrange multiplier method the penalty method and combinations of these two a frequently used discretisation method is the so called node to segment approach however this approach might lead to problems like jumps in contact forces loss of convergence or failure to pass the patch test thus in the last few years several segment to segment contact algorithms based on the mortar method were proposed combination of a mortar discretisation with a penalty based enforcement of the contact conditions leads to unphysical penetrations on the other hand a lagrange multiplier mortar method requires additional unknowns hence condensation of the lagrange multipliers is desirable to preserve the initial size of the system of equations this can be achieved by interpolating the lagrange multipliers with so called dual shape functions discretising two contacting bodies leads to opposed contact surface representations of finite element edges called slave and master elements respectively in current versions of dual lagrange multiplier mortar formulations an inconsistency at the boundary appears when only a part of a slave element instead of the entire element belongs to the contact area we present a modified definition of the dual shape functions in such slave elements the basic idea is to construct dual shape functions that fulfill the so called biorthogonality condition within the contact area this leads to consistent mortar matrices also in the boundary region to avoid ill conditioning of the stiffness matrix the modified mortar matrices are weighted with appropriate weighting factors in doing so the corresponding modified lagrange multiplier nodal values are of the same order as the unmodified ones various examples demonstrate the performance of the modified mortar contact algorithm

finite deformation frictional mortar contact using a semi smooth newton method with consistent linearization a two dimensional finite deformation frictional contact formulation with coulomb s law is presented the approach considers multibody contact and is based on a mortar formulation the enforcement of contact constraints is realized with dual lagrange multipliers these alternative multiplier spaces are constructed in a way that the multipliers can easily be eliminated from the global system of equations by static condensation such that the system size does not increase friction kinematic variables are formulated in an objective way and enter non smooth complementarity functions for expressing the contact constraints an active set strategy is derived by applying a semismooth newton method which treats contact nonlinearities material and geometrical nonlinearities in one single iterative scheme by further carrying out a consistent linearization for both normal and frictional contact forces and constraints a robust and highly efficient algorithm for linear and higher order quadratic interpolation is achieved efficiency of the proposed method and quality of results are demonstrated in serveral exemples

a cal v cycle multigrid approach for mortar finite elements mortar methods based on dual lagrange multipliers provide a flexible tool for the numerical approximation of partial differential equations the associated finite element spaces are in general nonconforming and nonnested optimal multigrid results have previously been established for cal w cycle and the variable cal v cycle multigrid methods in this paper we introduce a new multigrid method based on a nested sequence of modified mortar spaces for which we can establish that the cal v cycle with one smoothing step has contraction numbers uniformly bounded away from one to obtain nested mortar spaces we apply a product form of certain corrections at the interfaces numerical results demonstrate the efficiency of the resulting multigrid solver

a dual lagrange method for contact problems with regularized contact conditions this paper presents an algorithm for solving quasi static non linear elasticity contact problems without friction in the context of rough surfaces here we want to model the transition from soft to hard contact in case of rough surfaces on the micro scale the popular dual mortar method is used to enforce the contact constraints in a variationally consistent way without increasing the algebraic system size the algorithm is deduced from a perturbed lagrange formulation and combined with mass lumping techniques to exploit the full advantages of the duality pairing this leads to a regularized saddle point problem for which a nonlinear complementary function and thus a semi smooth newton method can be derived numerical examples demonstrate the applicability to industrial problems and show a good agreement to experimentally obtained results copyright 2014 john wiley sons ltd

generalized newton s methods for the approximation and resolution of frictional contact problems in elasticity abstract in this paper some new generalized newton s methods for the resolution of elastostatic frictional contact problem approximated by finite elements are presented and compared to existing ones a numerical experimentation is performed to compare the different methods especially with respect to the sensitivity to the method parameter two different strategies to approximate the contact and friction condition are considered a nodal and an integral one existence and uniqueness results of the solution to the discretized problem are also discussed

an augmented lagrangian algorithm for contact mechanics based on linear regression summary the penalty method for the solution of contact problems yields an approximate satisfaction of the contact constraints augmentation schemes which can be adopted to improve the solution either include the contact forces as additional unknowns or are strongly affected by the penalty parameter and display a poor convergence rate in a previous investigation an unconventional augmentation scheme was proposed on the basis of estimating the exact values of the contact forces through linear interpolation of a database extracted by the previous converged states an enhanced version of this method is proposed herein with respect to the original method the enhanced one eliminates some numerical problems and improves the regularity of the convergence path by carrying out the estimate through linear regression methods the resulting convergence rate is superlinear and the method is quite insensitive to the penalty parameter the main underlying concept is that within the iterative solution of a non linear problem linear regression techniques may be used as a tool to shoot faster to the final solution on the basis of a set of intermediate data the generality of this concept makes it potentially applicable to contact problems in more general settings as well as to other categories of non linear problems copyright 2012 john wiley sons ltd

asymptotic analysis of rzf in large scale mu mimo systems over rician channels in this paper we focus on the downlink ergodic sum rate of a single cell large scale multiuser mimo system in which the base station employs n antennas to communicate with k single antenna user equipments ues a regularized zero forcing rzf scheme is used for precoding under the assumption that each ue uses a specific power and each link forms a spatially correlated mimo rician fading channel the analysis is conducted assuming that n and k grow large with a given ratio and perfect channel state information is available at the base station new results from random matrix theory and large system analysis are used to compute an asymptotic expression of the signal to interference plus noise ratio as a function of system parameters spatial correlation matrix and rician factor numerical results are used to validate the accuracy of asymptotic approximations in the finite system regime and to evaluate the performance under different operating conditions it turns out that the asymptotic expressions provide accurate approximations even for relatively small values of n and k

the local average contact lac method abstract the local average contact lac method allows the handling of nonmatching meshes in an easy way by averaging locally the interpenetration between the contacting bodies in this paper we consider the frictionless unilateral contact problem and several numerical experiments involving two and three dimensional bodies discretized with various linear and quadratic finite elements we also present a convergence analysis of the method in the geometrical nonconforming case in which the boundary points of the candidate contact areas do not coincide

efficient simulation of multi body contact problems on complex geometries a flexible decomposition approach using constrained minimization we consider the numerical simulation of non linear multi body contact problems in elasticity on complex three dimensional geometries in the case of warped contact boundaries and non matching finite element meshes particular emphasis has to be put on the discretization of the transmission of forces and the non penetration conditions at the contact interface we enforce the discrete contact constraints by means of a non conforming domain decomposition method which allows for optimal error estimates here we develop an efficient method to assemble the discrete coupling operator by computing the triangulated intersection of opposite element faces in a locally adjusted projection plane but carrying out the required quadrature on the faces directly our new element based algorithm does not use any boundary parameterizations and is also suitable for isoparametric elements the emerging non linear system is solved by a monotone multigrid method of optimal complexity several numerical examples in 3d illustrate the effectiveness of our approach copyright 2008 john wiley sons ltd

a dual lagrange method for contact problems with regularized frictional contact conditions modelling micro slip abstract this paper presents an algorithm for solving quasi static non linear elasticity contact problems with friction in the context of rough surfaces here we want to model the transition from sticking to slipping also called micro slip in a physically correct way in order to reproduce measured frictional damping the popular dual mortar method is used to enforce the contact constraints in a variationally consistent way without increasing the algebraic system size the algorithm is deduced from a perturbed lagrange formulation and combined with a serial parallel iwan model this leads to a regularized saddle point problem for which a non linear complementary function and thus a semi smooth newton method can be derived numerical examples demonstrate the applicability to industrial problems and show good agreement to experimentally obtained results

combined interior point method and semismooth newton method for frictionless contact problems in the present paper a solution scheme is proposed for frictionless contact problems of linear elastic bodies which are discretized using the finite element method with lower order elements an approach combining the interior point method and the semismooth newton method is proposed in this method an initial active set for the semismooth newton method is obtained from the approximate optimal solution by the interior point method the simplest node to node contact model is considered in the present paper that is pairs of matching nodes exist on the contact surfaces however the discussions can be easily extended to a node to segment or segment to segment contact model in order to evaluate the proposed method a number of illustrative examples of the frictionless contact problem are shown the proposed combined method is compared with the interior point method and the semismooth newton method two numerical examples that are difficult to solve using the semismooth newton method are solved effectively using the proposed combined method it is shown that the proposed method converges within far fewer iterations than the semismooth newton methods or the interior point method

a point to segment contact formulation for isogeometric nurbs based finite elements formulation of isogeometric finite elements has received a great deal of attention in the recent past the present study deals with the treatment of problems in structural mechanics including large elastic deformations and contact one decisive difference between isogeometric finite elements based on nurbs functions and standard finite elements fe based on lagrange polynomials is higher inter element continuity this is a promising characteristics to model contact problems as all issues associated with kinks between elements are naturally avoided particularly in cases with large sliding this has the potential to be an attractive feature we present a bilateral isogeometric collocation contact formulation for geometrically non linear two dimensional problems using greville and botella points to collocate the contact integrals different methods to obtain accurate and physically meaningful stress distributions are investigated and compared the results show that the higher inter element continuity in context of non smooth problems may imply undesired effects in the numerical solution such as unphysical stress oscillations similar effects have been observed in standard p fem numerical experiments indicate that these oscillations may be avoided if the basis functions of contact and non contact zones are separated by knot relocation and knot repetition

hybrid finite element methods for the signorini problem we study three mixed linear finite element methods for the numerical simulation of the two dimensional signorini problem applying falk s lemma and saddle point theory to the resulting discrete mixed variational inequality allows us to state the convergence rate of each of them two of these finite elements provide optimal results under reasonable regularity assumptions on the signorini solution and the numerical investigation shows that the third method also provides optimal accuracy

a posteriori error estimates of discontinuous galerkin methods for the signorini problem a reliable and efficient a posteriori error estimator is derived for a class of discontinuous galerkin dg methods for the signorini problem a common property shared by many dg methods leads to a unified error analysis with the help of a constraint preserving enriching map the error estimator of dg methods is comparable with the error estimator of the conforming methods numerical experiments illustrate the performance of the error estimator

unilateral non linear dynamic contact of thin walled structures using a primal dual active set strategy the efficient modelling of three dimensional contact problems is still a challenge in non linear implicit structural analysis we use a primal dual active set strategy siam j optim 2003 13 865 888 based on dual lagrange multipliers siam j numer anal 2000 38 989 1012 to handle the non linearity of the contact conditions this allows us to enforce the contact constraints in a weak integral sense without any additional parameter due to the biorthogonality condition of the basis functions the lagrange multipliers can be locally eliminated we perform a static condensation to achieve a reduced system for the displacements the lagrange multipliers representing the contact pressure can be easily recovered from the displacements in a variationally consistent way for the application to thin walled structures we adapt a three dimensional non linear shell formulation including the thickness stretch of the shell to contact problems a reparametrization of the geometric description of the shell body gives us a surface oriented shell element which allows the application of contact conditions directly to nodes lying on the contact surface shell typical locking phenomena are treated with the enhanced assumed strain method and the assumed natural strain method the discretization in time is done with the implicit generalized method j appl mech 1993 60 371 375 and the generalized energy momentum method comp methods appl mech eng 1999 178 343 366 to compare the development of energies within a frictionless contact description in order to conserve the total energy within the discretized frictionless contact framework we follow an approach from laursen and love int j numer methods eng 2002 53 245 274 who introduced a discrete contact velocity to update the velocity field in a post processing step various examples show the good performance of the primal dual active set strategy applied to the implicit dynamic analysis of thin walled structures

an interior point method for isogeometric contact abstract the interior point method is applied to frictionless contact mechanics problems and is shown to be a viable alternative to the augmented lagrangian approach the method is derived from a mixed formulation which induces a contact discretization scheme in the spirit of the mortar method and naturally delivers slack variables that help constrain the solution to the feasible region the derivation of the algorithm as well as its robustness benefits from the contact interface description that is induced by nurbs based isogeometric volume discretizations various interior point algorithms are discussed including a primal dual approach that satisfies the unilateral contact constraints exactly in addition to two primal approaches that retain an arbitrary barrier parameter the developed algorithms can easily be pursued starting from an augmented lagrangian implementation numerical investigations on benchmark problems demonstrate the efficiency and the robustness of the framework but also highlight current limitations that suggest paths for future research overall the results indicate that the interior point method can challenge the augmented lagrangian method in contact mechanics even displaying potential for higher efficiency and robustness

a posteriori error estimator and error control for contact problems in this paper we consider two error estimators for one body contact problems the first error estimator is defined in terms of h div conforming stress approximations and equilibrated fluxes while the second is a standard edge based residual error estimator without any modification with respect to the contact we show reliability and efficiency for both estimators more over the error is bounded by the first estimator with a constant one plus a higher order data oscillation term plus a term arising from the contact that is shown numerically to be of higher order the second estimator is used in a control based afem refinement strategy and the decay of the error in the energy is shown several numerical tests demonstrate the performance of both estimators

path following and augmented lagrangian methods for contact problems in linear elasticity a certain regularization technique for contact problems leads to a family of problems that can be solved efficiently using infinite dimensional semismooth newton methods or in this case equivalently primal dual active set strategies we present two procedures that use a sequence of regularized problems to obtain the solution of the original contact problem first order augmented lagrangian and path following methods the first strategy is based on a multiplier update while path following with respect to the regularization parameter uses theoretical results about the path value function to increase the regularization parameter appropriately comprehensive numerical tests investigate the performance of the proposed strategies for both a 2d as well as a 3d contact problem

a stable energy conserving approach for frictional contact problems based on quadrature formulas a common approach for the numerical simulation of non linear multi body contact problems is the use of lagrange multipliers to model the contact conditions the stability of standard algorithms is improved by introducing a modified mass matrix which assigns no mass to the potential contact nodes by this the spurious algorithmic oscillations in the multiplier do not occur any more which facilitates the application of the primal dual active set strategy to dynamical contact problems the new mass matrix is calculated via a modified quadrature formula that needs no extra computational cost in addition the conservation properties of the underlying algorithm are transferred to the modified mass version different numerical examples for frictional two body contact problems illustrate the improvement in the results for the contact stresses copyright 2007 john wiley sons ltd

a discontinuous stabilized mortar method for general 3d elastic problems first the present paper is concerned with the extension to linearized elastodynamics of the optimal results known in statics for the mortar method it also analyzes and tests a new couple of displacements lagrange multipliers for the method as proposed independently by ben belgacem f ben belgacem a stabilized domain decomposition method with non matching grids for the stokes problem in three dimensions siam j numer anal 42 2 2004 667 685 and the authors p hauret methodes numeriques pour la dynamique des structures non lineaires incompressibles a deux echelles numerical methods for the dynamic analysis of two scale incompressible nonlinear structures ph d thesis ecole polytechnique 2004 finally questions of practical implementation in the presence of curved interfaces are addressed and validated from the numerical point of view

semi smooth newton methods for mixed fem discretizations of higher order for frictional elasto plastic two body contact problems in this article a semi smooth newton method for frictional two body contact problems and a solution algorithm for the resulting sequence of linear systems are presented it is based on a mixed variational formulation of the problem and a discretization by finite elements of higher order general friction laws depending on the normal stresses and elasto plastic material behavior with linear isotropic hardening are considered numerical results show the efficiency of the presented algorithm

numerical implementation of two nonconforming finite element methods for unilateral contact we consider the finite element approximation of the unilateral contact problem between elastic bodies we are interested in a practical problem which often occurs in finite element computations concerning two independently discretized bodies in unilateral contact it follows that the nodes of both bodies located on the contact surface do not fit together we present two different approaches in order to define unilateral contact on nonmatching meshes the first is an extension of the mortar finite element method to variational inequalities that defines the contact in a global way on the contrary the second one expresses local node on segment contact conditions in both cases the theoretical approximation properties are given then we implement and compare the two methods

an accurate local average contact method for nonmatching meshes the present paper deals with linear and quadratic finite element approximations of the two and three dimensional unilateral contact problems between two elastic bodies with nonmatching meshes we propose a simple noninterpenetration condition on the displacements which is local as the well known node to segment and node to face conditions and accurate like the mortar approach this condition consists of averaging locally on a few elements the noninterpenetration we prove optimal convergence rates in 2d and 3d using various linear and quadratic elements the taylor patch test and the hertzian contact test illustrate the theoretical results and show the capabilities of the method

a large deformation frictional contact formulation using nurbs based isogeometric analysis this paper focuses on the application of nurbs based isogeometric analysis to coulomb frictional contact problems between deformable bodies in the context of large deformations a mortar based approach is presented to treat the contact constraints whereby the discretization of the continuum is performed with arbitrary order nurbs as well as c0 continuous lagrange polynomial elements for comparison purposes the numerical examples show that the proposed contact formulation in conjunction with the nurbs discretization delivers accurate and robust predictions results of lower quality are obtained from the lagrange discretization as well as from a different contact formulation based on the enforcement of the contact constraints at every integration point on the contact surface copyright 2011 john wiley sons ltd

stable lagrange multipliers for quadrilateral meshes of curved interfaces in 3d for the discretization of coupled problems by means of mortar finite elements the choice of the discrete lagrange multiplier space is of crucial importance for the performance of the resulting numerical scheme dual lagrange multipliers offer several advantages since they can be locally eliminated in this paper the lowest order dual lagrange multipliers are extended to arbitrary quadrilateral or triangular surface grids in a first step two alternatives are provided for the case of planar interfaces for the approximation of vector fields two more modifications are given for use on curvilinear interfaces a priori results are obtained and several numerical examples are shown

incorporation of contact for high order finite elements in covariant form the covariant contact description is applied to incorporate the treatment of contact problems into a high order finite element technique a hierarchical enrichment of the shape functions space allows to construct a contact layer finite element combining both exact geometry representation for contact surfaces and a mesh with linear shape functions for the interior of the contacting bodies the developed contact approach can be viewed as a smoothing technique for linear meshes as well as a general application of high order fem the good approximation property for the developed contact layer elements is shown for the classical hertz problem even within a few contact elements discretizing the contact zone

solution of contact problems by feti domain decomposition with natural coarse space projections abstract an efficient non overlapping domain decomposition algorithm of the neumann neumann type for solving both coercive and semicoercive contact problems is presented the discretized problem is first turned by the duality theory of convex programming to the quadratic programming problem with bound and equality constraints and the latter is further modified by means of orthogonal projectors to the natural coarse space introduced by farhat and roux in the framework of their feti method the resulting problem is then solved by an augmented lagrangian type algorithm with an outer loop for the lagrange multipliers for the equality constraints and an inner loop for the solution of the bound constrained quadratic programming problems the projectors are shown to guarantee fast convergence of iterative solution of auxiliary linear problems and to comply with efficient quadratic programming algorithms proposed earlier reported theoretical results and numerical experiments indicate high numerical scalability of the algorithm which preserves the parallelism of the feti methods

hp adaptive ipdg tdg fem for parabolic obstacle problems for a parabolic obstacle problem two equivalent hp fem discretization methods based on interior penalty discontinuous galerkin in space and discontinuous galerkin in time are presented the first approach is based on a variational inequality vi formulation and the second approach on a mixed method in which the non penetration condition is resolved by a lagrange multiplier the discrete lagrange multiplier is a linear combination of biorthogonal basis functions allowing to write the discrete vi constraints as a set of complementarity problems employing a penalized fischer burmeister non linear complementarity function the discrete mixed problem can be solved by a locally q quadratic converging semi smooth newton ssn method the hierarchical a posteriori error estimator for the vi formulation which under the saturation assumption is both efficient and reliable allows hp adaptivity the numerical experiments show improved convergence compared to uniform and h adaptive meshes furthermore an a priori error estimate is given for the vi formulation

a segment to segment mortar contact method for quadratic elements and large deformations abstract this paper describes a new extension of mortar based segment to segment algorithms for large deformation contact analysis permitting their use when quadratic interpolations are used in the underlying finite element method the work builds upon results previously described by puso and laursen m a puso t a laursen a mortar segment to segment contact method for large deformation solid mechanics comput methods appl mech engrg 193 2004 601 629 m a puso t a laursen a mortar segment to segment frictional contact method for large deformations comput methods appl mech engrg 193 2004 601 629 where methods were developed for quasistatic and implicit dynamic analysis in the context of linear element interpolations here we see that many algorithmic ideas from the linear case can be adopted for use with quadratic elements along with different alternative interpolations of the mortar contact multipliers the result is a class of methods appropriate for use for either mesh tying true large sliding simulation frictionless or frictional and readily admits mixing of element types several numerical examples are given to demonstrate the effectiveness of the approach

a mortar method for energy momentum conserving schemes in frictionless dynamic contact problems summary in the present work the mortar method is applied to planar large deformation contact problems without friction in particular the proposed form of the mortar contact constraints is invariant under translations and rotations these invariance properties lay the foundation for the design of energy momentum timestepping schemes for contact impact problems the iterative solution procedure is embedded into an active set algorithm lagrange multipliers are used to enforce the mortar contact constraints the solution of generalized saddle point systems is circumvented by applying the discrete null space method numerical examples demonstrate the robustness and enhanced numerical stability of the newly developed energymomentum scheme copyright q 2008 john wiley sons ltd

monotone multigrid methods on nonmatching grids for nonlinear multibody contact problems nonconforming domain decomposition techniques provide a powerful tool for the numerical approximation of partial differential equations we use a generalized mortar method based on dual lagrange multipliers for the discretization of a nonlinear contact problem between linear elastic bodies in the case of unilateral contact problems pointwise constraints occur and monotone multigrid methods yield efficient iterative solvers here we generalize these techniques to nonmatching triangulations where the constraints are realized in terms of weak integral conditions the basic new idea is the construction of a nested sequence of nonconforming constrained spaces we use suitable basis transformations and a multiplicative correction in contrast to other approaches no outer iteration scheme is required the resulting monotone method is of optimal complexity and can be implemented as a multigrid method numerical results illustrate the performance of our approach in two and three dimensions

a mortar finite element method using dual spaces for the lagrange multiplier the mortar finite element method allows the coupling of different discretization schemes and triangulations across subregion boundaries in the original mortar approach the matching at the interface is realized by enforcing an orthogonality relation between the jump and a modified trace space which serves as a space of lagrange multipliers in this paper this lagrange multiplier space is replaced by a dual space without losing the optimality of the method the advantage of this new approach is that the matching condition is much easier to realize in particular all the basis functions of the new method are supported in a few elements the mortar map can be represented by a diagonal matrix in the standard mortar method a linear system of equations must be solved the problem is considered in a positive definite nonconforming variational as well as an equivalent saddle point formulation

adaptive finite elements for elastic bodies in contact to avoid interpenetration of matter under the small strain assumption the linear contact condition is frequently applied where the distance of bodies is controlled only along a certain direction the standard direction is the normal on the surface where interpenetration might occur in this paper we allow other directions as well we address questions such as the correct mathematical model existence of solutions the penalty method for regularization of the variational inequality finite element discretization and a priori and a posteriori error estimates but exclude the error of penalization the computable upper error bound leads to a criterion for automatic mesh refinements within a finite element method numerical simulations of the hertzian contact problem and a supported cantilever beam are included

a dual mortar approach for 3d finite deformation contact with consistent linearization in this paper an approach for three dimensional frictionless contact based on a dual mortar formulation and using a primal dual active set strategy for direct constraint enforcement is presented we focus on linear shape functions but briefly address higher order interpolation as well the study builds on previous work by the authors for two dimensional problems first and foremost the ideas of a consistently linearized dual mortar scheme and of an interpretation of the active set search as a semi smooth newton method are extended to the 3d case this allows for solving all types of nonlinearities i e geometrical material and contact within one single newton scheme owing to the dual lagrange multiplier approach employed this advantage is not accompanied by an undesirable increase in system size as the lagrange multipliers can be condensed from the global system of equations moreover it is pointed out that the presented method does not make use of any regularization of contact constraints numerical examples illustrate the efficiency of our method and the high quality of results in 3d finite deformation contact analysis

efficient algorithms for problems with friction in this paper a nonconforming discretization method for the frictional contact between two bodies subjected to antiplane shear deformation is considered the method is based on a mixed variational formulation where dual basis functions are used for the discretization of the lagrange multiplier under some regularity assumptions on the solution an optimal a priori error estimate is obtained to solve the discrete nonlinear problem an inexact primal dual active set strategy is introduced finally numerical examples confirming the theoretical result for the a priori error estimate and illustrating the performance of the algorithm are presented the results can easily be generalized to the case of coulomb friction for this case a numerical example is given

a primal dual active set strategy for non linear multibody contact problems abstract non conforming domain decomposition methods provide a powerful tool for the numerical approximation of partial differential equations for the discretization of a non linear multibody contact problem we use the mortar approach with a dual lagrange multiplier space to handle the non linearity of the contact conditions we apply a primal dual active set strategy to find the actual contact zone the algorithm can be easily generalized to multibody contact problems a suitable basis transformation guarantees the same algebraic structure in the multibody situation as in the one body case using an inexact primal dual active set strategy in combination with a multigrid method yields an efficient iterative solver different numerical examples for one and multibody contact problems illustrate the performance of the method