eichler shimura theory for mock modular forms we use mock modular forms to compute generating functions for the critical values of modular l functions and we answer a generalized form of a question of kohnen and zagier by deriving the extra relation that is satisfied by even periods of weakly holomorphic cusp forms to obtain these results we derive an eichler shimura theory for weakly holomorphic modular forms and mock modular forms this includes two eichler shimura isomorphisms a multiplicity two hecke theory a correspondence between mock modular periods and classical periods and a haberland type formula which expresses petersson s inner product and a related antisymmetric inner product on m k in terms of periods

dynamics and control of viscoelastic solids with contact and friction effects the dynamicsand conlrol of distributed systems subjected to frictional resistive forces has been an open problem in partial differential equations for many years largely owing to the absence of a meaningful existence theory for dynamical systems with friction the mathcmatical difficulties associated with coulomb friction in problems of elastodynamics were pointed out in 1972 by duvaut and lions see i and has been the subject of much study in the intervening years the finite dimensional case has proved to be somewhat more tractable and conditions for the existence of solutions of discrete dynamical problems with friction have been recently reported by lbtstedt 2 and jean and pratt 3 the complexity of the problem can be appreciated by reviewing the work on dynamics of systems with frictionless contacts by for example schatzman 4 carrero and pascali 5 lotstedt 6 amerio and prouse 7 amerio 8 and others in recent papers oden and martins 9 10 pointed out that one of the principal sources of mathematical difficulty was the definition of frictional stresses on the contact surfaces charac terized by coulomb s law however an overwhelming volume of experimental data accumu lated over a half century suggests that this law is inadequate for modeling actual contacts and resisiive forces on deformable bodies by characterizing the actual normal compliance of elastic interfaces a constitutive equation for an interface can be developed which yields results in agreement with a sizable collection of experimental results on static and dynamic friction 9 moreover the use of such interface constitutive laws in mathematical models of elastostatics elastodynamics and viscoelastodynamics problems with friction produces a tractable theory results on the existence and local uniqueness of solutions to static problems in elaslicity with the oden martins 9 nonlinear friction laws were recently established by rabier el of ii and of dynamic problems on elasticity and viscoelasticity by martins and oden 10 these new theories and results set the stage for the study of the optimal control of such systems taken up in the present paper in the present study we establish the existence of a class of optimal controls for a broad class of problems in the dynamics of linear viscoelastic bodies with contact and friction laws of the type introduced by oden and martins 9 we are able to show under very mild restrictions

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

error estimation and adaptive meshing in strongly nonlinear dynamic problems we present work aimed at developing a general framework for mesh adaption in strongly nonlinear possibly dynamic problems we begin by showing that the solutions of the incremental boundary value problem for a wide class of materials including nonlinear elastic materials compressible newtonian fluids and viscoplastic solids obey a minimum principle provided that the constitutive updates are formulated appropriately this minimum principle can be taken as a basis for asymptotic error estimation in particular we chose to monitor the error of a lower order projection of the finite element solution the optimal mesh size distribution then follows from a posteriori error indicators which are purely local i e can be computed element by element we demonstrate the robustness and versatility of the computational framework with the aid of convergence studies and selected examples of application

finite deformation post buckling analysis involving inelasticity and contact constraints this paper is concerned with the numerical solution of large deflection structural problems involving finite strains subject to contact constraints and unilateral boundary conditions and exhibiting inelastic constitutive response first a three dimensional finite strain beam model is summarized and its numerical implementation in the two dimensional case is discussed next a penalty formulation for the solution of contact problems is presented and the correct expression for consistent tangent matrix is developed finally basic strategies for tracing limit points are reviewed and a modification of the arc length method is proposed the good performance of the procedures discussed is illustrated by means of numerical examples

calculation of impact contact problems of thin elastic shells taking into account geometrical nonlinearities within the contact region abstract during impact of elastic bodies contact stresses are transmitted in time depending contact surfaces in many impact contact problems large displacements and rotations appear only in the contact surface and in a certain neighbourhood therefore it is efficient to consider geometrical nonlinearities only in this region and to describe the remainder of the body within the geometrical linear theory this leads to substructure techniques where only properties of the nonlinear elements need be modified during the impact contact process the principle of virtual work for nonlinear thin shells is expressed using the total lagrangian formulation and the geometrical nonlinearity of thin shells is described in the frame of moderate rotation theory the contact conditions lead to inequalities for the normal stresses and displacements in the contact interfaces therefore the numerical algorithm involves two superposed iterations for the computation of contact forces and contact areas and for the geometrical nonlinearity the iteration procedure has to be carried out in each time step the spatial discretization using finite element techniques leads to a system of ordinary differential equations which is integrated over the time using the newmark algorithm numerical results were obtained for the impact contact problem of spherical shells for these examples the impact forces and the contact pressure distribution are presented for several parameter combinations results are controlled by conservation laws in integral form and compared with results from geometrical linear theory

a posteriori error estimation of h p finite element approximations of frictional contact problems abstract dynamic and static frictional contact problems are described using the normal compliance law on the contact boundary dynamic problems are recast into quasistatic problems by time discretization an a posteriori error estimator is developed for a nonlinear elliptic equation of corresponding static or quasistatic problems the a posteriori error estimator is applied to a frictionless case and extended to frictional contact problems also an adaptive strategy is introduced and h p finite element meshes are obtained through a procedure based on a priori and a posteriori error estimations numerical examples are given to support the theoretical results

regularity of srb entropy for geometric lorenz attractors we consider the classical geometric lorenz attractors showing that the srb entropy admits gamma holder continuity for any 0 gamma 1

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

variational integrators and the newmark algorithm for conservative and dissipative mechanical systems the purpose of this work is twofold first we demonstrate analytically that the classical newmark family as well as related integration algorithms are variational in the sense of the veselov formulation of discrete mechanics such variational algorithms are well known to be symplectic and momentum preserving and to often have excellent global energy behavior this analytical result is veried through numerical examples and is believed to be one of the primary reasons that this class of algorithms performs so well second we develop algorithms for mechanical systems with forcing and in particular for dissipative systems in this case we develop integrators that are based on a discretization of the lagrange d alembert principle as well as on a variational formulation of dissipation it is demonstrated that these types of structured integrators have good numerical behavior in terms of obtaining the correct amounts by which the energy changes over the integration run

discontinuous variational time integrators for complex multibody collisions the objective of the present work is to formulate a new class of discontinuous variational time integrators that allow the system to adopt two possibly different configurations at each sampling time tk representing predictor and corrector configurations of the system the resulting sequence of configuration pairs then represents a discontinuous or non classical trajectory continuous or classical trajectories are recovered simply by enforcing a continuity constraint at all times in particular in systems subject to one sided contact constraints simulated via discontinuous variational time integrators the predictor configuration is not required to satisfy the one sided constraints whereas the corrector configuration is obtained by a closest point projection cpp onto the admissible set the resulting trajectories are generally discontinuous or non classical but are expected to converge to classical or continuous solutions for decreasing time steps we account for dissipation including friction by means of a discrete lagrange d alembert principle and make extensive use of the spacetime formalism in order to ensure exact energy conservation in conservative systems and the right rate of energy decay in dissipative systems the structure range and scope of the discontinuous variational time integrators and their accuracy characteristics are illustrated by means of examples of application concerned with rigid multibody dynamics

time discretized variational formulation of non smooth frictional contact the present work extends the non smooth contact class of algorithms introduced by kane et al to the case of friction the formulation specifically addresses contact geometries e g involving multiple collisions between tightly packed non smooth bodies for which neither normals nor gap functions can be properly defined a key aspect of the approach is that the incremental displacements follow from a minimum principle the objective function comprises terms which account for inertia strain energy contact friction and external forcing the euler lagrange equations corresponding to this extended variational principle are shown to be consistent with the equations of motion of solids in frictional contact in addition to its value as a basis for formulating numerical algorithms the variational framework offers theoretical advantages as regards the selection of trajectories in cases of non uniqueness we present numerical and analytical examples which demonstrate the good momentum and energy conservation characteristics of the numerical algorithms as well as the ability of the approach to account for stick and slip conditions

geometric integrators and nonholonomic mechanics a geometric derivation of nonholonomic integrators is developed it is based in the classical technique of generating functions adapted to the special features of nonholonomic systems the theoretical methodology and the integrators obtained are different from those obtained in cortes and martinez nonholonomic integrators nonlinearity 14 1365 1392 2001 in the case of mechanical systems with linear constraints a family of geometric integrators preserving the nonholonomic constraints is given

operator split methods for the numerical solution of the elastoplastic dynamic problem the elastoplastic dynamic problem is first formulated in a form that facilitates the application of product formula techniques the additive decomposition of the dynamic equations into elastic and plastic parts is taken as a basis for the definition of product algorithms that exploit such decomposition in the context of a finite element discretization these product algorithms entail for every time step the solution of an elastic problem followed by the application of plastic algorithms that operate on the stresses and internal variables at the integration points and bring in the plastic constitutive relations suitable plastic algorithms are discussed for the cases of perfect and hardening plasticity and viscoplasticity the proposed formalism does not depend on any notion of smoothness of the yield surface and is applicable to arbitrary convex elastic regions with or without corners the stability properties of the product algorithm are identical to those of the elastic algorithm used whereas the computational expense is practically equal to that of an elastic problem

parallel asynchronous variational integrators this paper presents a scalable parallel variational time integration algorithm for nonlinear elastodynamics with the distinguishing feature of allowing each element in the mesh to have a possibly different time step furthermore the algorithm is obtained from a discrete variational principle and hence it is termed parallel asynchronous variational integrator pavi the underlying variational structure grants it outstanding conservation properties based on a domain decomposition strategy pavi combines a careful scheduling of computations with fully asynchronous communications to provide a very efficient methodology for finite element models with even mild distributions of time step sizes numerical tests are shown to illustrate pavi s performance on both slow and fast networks showing scalability properties similar to the best parallel explicit synchronous algorithms with lower execution time finally a numerical example in which pavi needs 100 times less computing than an explicit synchronous algorithm is shown copyright 2006 john wiley sons ltd

formulatin of spatial contact situations in rigid multibody systems the paper gives an overview of possible formulations of spatial contact situations as they occur in dynamic systems composed of rigid bodies we will discuss formulations of the contact problem of sliding and rolling as linear and nonlinear complementarity problems in standard form where general approximations of convex sets by polytopes are also presented for completeness a short treatise on spatial contact kinematics and the structure of the equations of motion is given

computational geometry an introduction from the reviews this book offers a coherent treatment at the graduate textbook level of the field that has come to be known in the last decade or so as computational geometry the book is well organized and lucidly written a timely contribution by two founders of the field it clearly demonstrates that computational geometry in the plane is now a fairly well understood branch of computer science and mathematics it also points the way to the solution of the more challenging problems in dimensions higher than two

formulation and comparison of algorithms for frictional contact problems this paper presents two algorithms for solving the discrete quasi static small displacement linear elastic contact problem with coulomb friction the algorithms are adoptions of a newton method for solving b differentiable equations and an interior point method for solving smooth constrained equations for the application of the former method the contact problem is formulated as a system of b differentiable equations involving the projection operator onto sets with simple structure for the application of the latter method the contact problem is formulated as a system of smooth equations involving complementarity conditions and with the non negativity of variables treated as constraints the two algorithms are numerically tested for two dimensional problems containing up to 100 contact nodes and up to 100 time increments results show that at the present stage of development the newton method is superior both in robustness and speed additional comparison is made with a commercial finite element code

a numerically stable dual method for solving strictly convex quadratic programs an efficient and numerically stable dual algorithm for positive definite quadratic programming is described which takes advantage of the fact that the unconstrained minimum of the objective function can be used as a starting point its implementation utilizes the cholesky and qr factorizations and procedures for updating them the performance of the dual algorithm is compared against that of primal algorithms when used to solve randomly generated test problems and quadratic programs generated in the course of solving nonlinear programming problems by a successive quadratic programming code the principal motivation for the development of the algorithm these computational results indicate that the dual algorithm is superior to primal algorithms when a primal feasible point is not readily available the algorithm is also compared theoretically to the modified simplex type dual methods of lemke and van de panne and whinston and it is illustrated by a numerical example

computational model for 3 d contact problems with friction based on the penalty method the friction forces are assumed to follow the coulomb law with a slip criterion treated in the context of a standard return mapping algorithm consistent linearization of the field equations is performed which leads to a fully implicit scheme with non symmetric tangent stiffness which preserves asymptotic quadratic convergence of the newton raphson method

finite element analysis of nonsmooth contact this work develops robust contact algorithms capable of dealing with complex contact situations involving several bodies with corners amongst the mathematical tools we bring to bear on the problem is nonsmooth analysis following clarke f h clarke optimization and nonsmooth analysis john wiley and sons new york 1983 we specifically address contact geometries for which both the use of normals and gap functions have difficulties and therefore precludes the application of most contact algorithms proposed to date such situations arise in applications such as fragmentation where angular fragments undergo complex collision sequences before they scatter we demonstrate the robustness and versatility of the nonsmooth contact algorithms developed in this paper with the aid of selected two and three dimensional applications

a perturbed lagrangian formulation for the finite element solution of contact problems making use of a perturbed lagrangian formulation a finite element procedure for contact problems is developed for the general case in which node to node contact no longer holds the proposed procedure leads naturally to a discretization of the contact interface into contact segments within the context of a bilinear interpolation for the displacement field a mixed finite element approximation is introduced by assuming discont naous contact pressure constant on the contact segment because of this piece wise constant approximation the gap function enters into the formulation in an average sense instead of through a point wise definition numerical examples are presented that illustrate the performance of the proposed procedure current finite element formulations for contact problems based on either the classical lagrange parameter procedure l 3 12 203 or the penalty function method 4 6 11 are characterized by a point wise enforcement of the contact constraint condition in the sense that penetration of the bodies is established on a nodal basis moreover in this methods the recovery of the contact pressure over the element from the contact nodal forces generally requires an additional procedure within the framework of classical lagrange multiplier methods the contact condition is exactly satisfied by transforming the constrained problem into an unconstrained one with the introduction of additional variables lagrange multipliers these extra variables add computational effort to the solution process which often requires special procedures to handle the presence of zero diagonal terms penalty methods on the other hand enable one to transform the constrained problem into an unconstrained one without introducing additional variables the constraint condition is now satisfied only approximately for finite values of the penalty parameter the main difhculty associated with these methods however lies in the poor conditioning of the problem as the penalty is increased to more accurately enforce the constraint condition this is a well understood phenomenon particularly in the context of the incompressible and nearly incompressible problem in solid and fluid mechanics e g see l 22 25 for a review recently augmented lagrangian procedures have been proposed as a promising way to partially overcome these difficulties and regularize the penalty formulation e g see the survey in 7 8 within the framework of zinearited e u c it is possible to restrict the finite element formulation of contact problems by assuming that node to node contact occurs this is in fact the case often considered in the literature l 2 4 6 12 20 211 in the general context of fully

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

time stepping for three dimensional rigid body dynamics traditional methods for simulating rigid body dynamics involves determining the current contact arrangement e g each contact is either a rolling or sliding contact the development of this approach is most clearly seen in the work of haug et al mech machine theory 21 1986 401 425 and pfeiffer and glocker multibody dynamics with unilateral contacts wiley 1996 however there has been a controversy about the status of rigid body dynamics as a theory due to simple problems in the area which do not appear to have solutions the most famous if not the earliest is due to paul painleve c r acad sci paris 121 1895 112 115 recently a number of time stepping methods have been developed to overcome these difficulties these time stepping methods use integrals of the forces over time steps rather than the actual forces this allows impulsive forces without the need for a separate formulation or special procedures to cover this case the newest of these methods are developed in terms of complementarity problems the complementarity problems that define the time stepping procedure are solvable unlike previous methods for simulating rigid body dynamics with friction proof of the existence of solutions to the continuous problem can be shown in the sense of measure differential inclusions in terms of these methods in this paper a number of these variants will be discussed and their essential properties proven

lagrange constraints for transient finite element surface contact a new approach to enforce surface contact conditions in transient non linear finite element problems is developed in this paper the method is based on the lagrange multiplier concept and is compatible with explicit time integration operators compatibility with explicit operators is established by referencing lagrange multipliers one time increment ahead of associated surface contact displacement constraints however the method is not purely explicit because a coupled system of equations must be solved to obtain the lagrange multipliers an important development herein is the formulation of a highly efficient method to solve the lagrange multiplier equations the equation solving strategy is a modified gauss seidel method in which non linear surface contact force conditions are enforced during iteration the new surface contact method presented has two significant advantages over the widely accepted penalty function method surface contact conditions are satisfied more precisely and the method does not adversely affect the numerical stability of explicit integration transient finite element analysis results are presented for problems involving impact and sliding with friction a brief review of the classical lagrange multiplier method with implicit integration is also included

a newton method for three dimensional fretting problems the present paper concerns the numerical treatment of fretting problems using a finite element analysis the governing equations resulting from a formal finite element discretization of an elastic body with a potential contact surface are considered in a quasi static setting the constitutive equations of the potential contact surface are signorini s contact conditions coulomb s law of friction and archard s law of wear using a backward euler time discretization and an approach based on projections the governing equations are written as an augmented lagrangian formulation which is implemented and solved using a newton algorithm for three dimensional fretting problems of didactic nature details concerning the implementation are provided