Study of variational inequality and equality formulations for elastostatic frictional contact problems

Summary An overview ofvariational inequality andvariational equality formulations for frictionless contact and frictional contact problems is provided. The aim is to discuss the state-of-the-art in these two formulations and clearly point out their advantages and disadvantages in terms of mathematical completeness and practicality. Various terms required to describe the contact configuration are defined.Unilateral contact law and classical Coulomb’s friction law are given.Elastostatic frictional contact boundary value problem is defined. General two-dimensional frictionless and frictional contact formulations for elastostatic problems are investigated. An example problem of a two bar truss-rigid wall frictionless contact system is formulated as an optimization problem based on the variational inequality approach. The problem is solved in a closed form using the Karush-Kuhn-Tucker (KKT) optimality conditions. The example problem is also formulated as a frictional contact system. It is solved in the closed form using a new two-phase analytical procedure. The procedure avoids use of the incremental/iterative techniques and user defined parameters required in a typical implementation based on the variational equality formulation. Numerical solutions for the frictionless and frictional contact problems are compared with the results obtained by using a general-purpose finite element program ANSYS (that uses variational equality formulation). ANSYS results match reasonably well with the solutions of KKT optimality conditions for the frictionless contact problem and the two-phase procedure for the frictional contact problem. The validity of the analytical formulation for frictional contact problems (with one contacting node) is verified. Thevariational equality formulation for frictionless and frictional, contact problems is also studied in detail. The incremental/iterative Newton-Raphson scheme incorporating the penalty approach is utilized. Studies are conducted to provide insights for the numerical solution techniques. Based on the present study it is concluded that alternate formulations and computational procedures need to be developed for analysis of frictional contact problems.     

Optimization of a hyper-elastic structure with multibody contact using continuum-based shape design sensitivity analysis

In this paper, a continuum-based shape design sensitivity formulation is presented for a hyper-elastic structure with multibody frictional contact. A nearly incompressible constraint is treated using the pressure projection method that projects a hydrostatic pressure into a lower order space to avoid a volumetric locking. The variational formulation for multibody frictional contact is developed using a penalty method that regularizes the solution of the variational inequality. The material derivative of continuum mechanics is utilized to develop the continuum-based shape design sensitivity analysis for the hyper-elastic constitutive relation and penalized contact formulation. The sensitivity equation is solved at each converged load step using the same tangent stiffness of response analysis due to the path dependency of the sensitivity of the frictional contact problem. A very accurate and efficient sensitivity results are shown through shape optimization of a windshield wiper. Received August 8, 1999     

Formulation and Preparation for Numerical Evaluation of Linear Complementarity Systems in Dynamics

In this paper, we provide a full instruction on how to formulate and evaluate planar frictional contact problems in the spirit of non-smooth dynamics. By stating the equations of motion as an equality of measures, frictional contact reactions are taken into account by Lagrangian multipliers. Contact kinematics is formulated in terms of gap functions, and normal and tangential relative velocities. Associated frictional contact laws are stated as inclusions, incorporating impact behavior in form of Newtonian kinematic impacts. Based on this inequality formulation, a linear complementarity problem in standard form is presented, combined with Moreau’s time stepping method for numerical integration. This approach has been applied to the woodpecker toy, of which a complete parameter list and numerical results are given in the paper.     

Surface Smoothing Procedures in Computational Contact Mechanics

This work addresses the problems arising in the finite element simulation of contact problems undergoing large deformation. The frictional contact problem is formulated in the continuum framework, introducing the interface laws for the normal and tangential stress components in the contact area. The variational formulation is presented, considering different methods to enforce the contact constraints. The spatial discretization within the finite element method is applied, as well as the temporal discretization required to solve the three sources of nonlinearities: geometric, material and frictional contact. The discretization of contact surfaces is discussed in detail, including different surface smoothing procedures. This numerical strategy allows to solve the difficulties associated with the discontinuities in the contact surface geometry introduced by finite element discretization, which leads to nonphysical oscillations of the contact force for large sliding problems. The geometrical accuracy of different interpolation methods is evaluated, paying particular attention to the Nagata patch interpolation recently proposed. In this framework, the Node-to-Nagata contact elements are developed using the augmented Lagrangian method to regularize the variational frictional contact problem. The techniques used to search for contact in case of large deformations are discussed, including self-contact phenomena. Several numerical examples are presented, comprising both the contact between deformable and rigid obstacles and the contact between deformable bodies. The results show that the accuracy and robustness of the numerical simulations is improved when the contact surface is smoothed with Nagata patches.     

Nonsmooth computational mechanics algorithms, quasidifferentiability and related topics

Several extensions of smooth computational mechanics algorithms for the treatment of nonsmooth and possible nonconvex problems are briefly discussed in this paper. A potential or dissipation energy minimization problem approach is used for the structural analysis problem, so as to make the link with mathematical optimization techniques straightforward. Variational inequality problems, hemivariational inequality problems and systems of variational inequalities can be treated by the methods reviewed in this paper. The use of quasidifferentiable and codifferentiable optimization techniques is proposed for the solution of the more general class of nonconvex, possibly nonsmooth problems. Established and new directions in path-following techniques are discussed and are linked with nonsmooth mechanics algorithms.     

An inexact continuation accelerated proximal gradient algorithm for low n-rank tensor recovery

The low n-rank tensor recovery problem is an interesting extension of the compressed sensing. This problem consists of finding a tensor of minimum n-rank subject to linear equality constraints and has been proposed in many areas such as data mining, machine learning and computer vision. In this paper, operator splitting technique and convex relaxation technique are adapted to transform the low n-rank tensor recovery problem into a convex, unconstrained optimization problem, in which the objective function is the sum of a convex smooth function with Lipschitz continuous gradient and a convex function on a set of matrices. Furthermore, in order to solve the unconstrained nonsmooth convex optimization problem, an accelerated proximal gradient algorithm is proposed. Then, some computational techniques are used to improve the algorithm. At the end of this paper, some preliminary numerical results demonstrate the potential value and application of the tensor as well as the efficiency of the proposed algorithm.     

A variational inequality approach to optimal plastic design of structures via the Prager-Rozvany theory

The theory of optimal plastic design of structures via optimality criteria (W. Prager approach) transforms the optimal design problem into a certain nonlinear elastic structural analysis problem with appropriate stress-strain laws, which are derived by the adopted specific cost function for the members of the structure and which generally have complete vertical branches. Moreover, the concept of structural universe (introduced by G.I.N. Rozvany) permits us to tackle complicated optimal layout problems.On the other hand, a significant effort in the field of nonsmooth mechanics has recently been devoted to the solution of structural analysis problems with complete material and boundary laws, e.g. stress-strain laws or reaction-displacement laws with vertical branches.In this paper, the problem of optimal plastic design and layout of structures following the approach of Prager-Rozvany is revised within the framework of recent progress in the area of nonsmooth structural analysis and it is treated by means of techniques primarily developed for the solution of inequality mechanics problems. The problem of the optimal layout of trusses is used here as a model problem. The introduction of general convex, continuous and piecewise linear specific cost functions for the structural members leads to the formulation of linear variational inequalities or equivalent piecewise linear, convex but nonsmooth optimization problems. An algorithm exploiting the particular structure of the minimization problem is then described for the numerical solution. Thus, practical structural optimization problems of large size can be treated. Finally, numerical examples illustrate the applicability and the advantages of the method.On leave from the Institute of Applied Mechanics, Department of Engineering Sciences, Technical University of Crete, GR-73100 Chania, Greece     

Numerical solution of large deformation problems involving stability and unilateral constraints

The present study is concerned with the numerical treatment of large deformation beam problems where stability as well as post-buckling behaviour is coupled with frictional contact constraints. The flexible beams are described according to a nonlinear rod-type theory which accounts for both finite rotations and large deformations. The contact conditions are introduced via a penalty function method. From these conditions we obtain a linear complementary problem (LCP) resulting from the variational inequality formulation. For the examination of the post-buckling behaviour the displacement control method is applied. Particular attention is paid to the development of the linear complementary problem combining with the computational strategy for tracing limit points. Finally, the modification algorithms of the linear complementary problem, in which the penalty factors have been eliminated, are proposed. The numerical techniques not only allow some limit points to be passed, but also guarantee the computational stability characteristics during the Newton-Raphson's iterative process. Numerical examples are presented that illustrate the performances of the proposed algorithms.     

Finite element algorithms for contact problems

Summary The numerical treatment of contact problems involves the formulation of the geometry, the statement of interface laws, the variational formulation and the development of algorithms. In this paper we give an overview with regard to the different topics which are involved when contact problems have to be simulated. To be most general we will derive a geometrical model for contact which is valid for large deformations. Furthermore interface laws will be discussed for the normal and tangential stress components in the contact area. Different variational formulations can be applied to treat the variational inequalities due to contact. Several of these different techniques will be presented. Furthermore the discretization of a contact problem in time and space is of great importance and has to be chosen with regard to the nature of the contact problem. Thus the standard discretization schemes will be discussed as well as techiques to search for contact in case of large deformations.     

A Relaxation Approach for Estimating Origin–Destination Trip Tables

The problem of estimating origin-destination travel demands from partial observations of traffic conditions has often been formulated as a network design problem (NDP) with a bi-level structure. The upper level problem in such a formulation minimizes a distance metric between measured and estimated traffic conditions, and the lower level enforces user-equilibrium traffic conditions in the network. Since bi-level problems are usually challenging to solve numerically, especially for large-scale networks, we proposed, in an earlier effort (Nie et al., Transp Res, 39B:497–518, 2005), a decoupling scheme that transforms the O–D estimation problem into a single-level optimization problem. In this paper, a novel formulation is proposed to relax the user equilibrium conditions while taking users’ route choice behavior into account. This relaxation approach allows the development of efficient solution procedures that can handle large-scale problems, and makes the integration of other inputs, such as path travel times and historical O–Ds rather straightforward. An algorithm based on column generation is devised to solve the relaxed formulation and its convergence is proved. Using a benchmark example, we compare the estimation results obtained from bi-level, decoupled and relaxed formulations, and conduct various sensitivity analysis. A large example is also provided to illustrate the efficiency of the relaxation method.     

Nonlinear equation approach for inequality elastostatics: a two-dimensional BEM implementation

The numerical solution of variational inequality problems in elastostatics is investigated by means of recently proposed equivalent nonlinear equations. Symmetric and nonsymmetric variational inequalities and linear or nonlinear, but monotone, complementarity problems can be solved this way without explicit use of nonsmooth (nondifferentiable) solvers. As a model application, two-dimentional unilateral contact problems with and without friction effects approximated by the boundary element method are formulated as nonsymmetric variational inequalities, or, for the two-dimensional case as linear complementarity problems, and are numerically solved. Performance comparisons using two standard, smooth, general purpose nonlinear equation solvers are included.     

An augmented Lagrangian optimization method for contact analysis problems, 1: formulation and algorithm

A review of existing augmented Lagrangian methods (ALM) for contact analysis problems reveals that they have not been implemented with automatic penalty updates as intended in their original development. Therefore, although the methods are an improvement over the penalty methods, solution with them still depends on the user-specified penalty values for the contact constraints. To overcome this drawback, an ALM is developed and discussed for contact analysis problems that automatically update the user-specified penalty values to obtain the final appropriate values. Further, to solve the frictional contact analysis problem accurately, a two-phase formulation is proposed. Solution of the Phase 1 problem removes penetration of the contacting nodes and brings them exactly to their initial contact points. In addition, a new contact constraint is introduced which allows determination of the precise friction force at the contacting nodes. Phase 2 of the formulation checks the friction conditions and solves the friction problem to bring the structure to an equilibrium state. Phases 1 and 2 are then combined to provide a general algorithm for multi-node frictional contact problems. The two-phase procedure also removes dependence of the contact solution on the number of load steps for the elastostatic problem. Numerical evaluation of the formulation and the algorithm is presented in Part 2 of the paper.     

On numerical approximation of a variational–hemivariational inequality modeling contact problems for locking materials

This paper is devoted to numerical analysis of a new class of elliptic variational–hemivariational inequalities in the study of a family of contact problems for elastic ideally locking materials. The contact is described by the Signorini unilateral contact condition and the friction is modeled by a nonmonotone multivalued subdifferential relation allowing slip dependence. The problem involves a nonlinear elasticity operator, the subdifferential of the indicator function of a convex set for the locking constraints and a nonconvex locally Lipschitz friction potential. Solution existence and uniqueness result on the inequality can be found in Migórski and Ogorzaly (2017) . In this paper, we introduce and analyze a finite element method to solve the variational–hemivariational inequality. We derive a Céa type inequality that serves as a starting point of error estimation. Numerical results are reported, showing the performance of the numerical method.     

A computational method for parameter optimization problems arising in control†

A parameter optimization procedure is presented for large-scale problems arising in linear control system design that include equality and inequality constraints. The procedure is based on a novel min—max algorithm for locating a constrained relative minimum without the use of penalty functions or slack variables. This algorithm is constructed from an auxiliary minimization problem with equality constraints. Inequality constraints then are introduced using the notion of an effective constraint. Typical problem formulations are discussed, and an extensive design example is presented.     

New second-order cone linear complementarity formulation and semi-smooth Newton algorithm for finite element analysis of 3D frictional contact problem

We propose a new second-order cone linear complementarity problem (SOCLCP) formulation for the numerical finite element analysis of three-dimensional (3D) frictional contact problems by the parametric variational principle. Specifically, we develop a regularization technique to resolve the multi-valued difficulty involved in the frictional contact law, and use a second-order cone complementarity condition to handle the regularized Coulomb friction law in contact analysis. The governing equations of the 3D frictional contact problem is represented by an SOCLCP via the parametric variational principle and the finite element method, which avoids the polyhedral approximation to the Coulomb friction cone so that the problem to be solved has much smaller size and the solution has better accuracy. In this paper, we reformulate the SOCLCP as a semi-smooth system of equations via a one-parametric class of second-order cone complementarity functions, and then apply the non-smooth Newton method for solving this system. Numerical results confirm the effectiveness and robustness of the SOCLCP approach developed.     

A multigrid optimization algorithm for the numerical solution of quasilinear variational inequalities involving the p-Laplacian

In this paper we propose a multigrid optimization algorithm (MG/OPT) for the numerical solution of a class of quasilinear variational inequalities of the second kind. This approach is enabled by the fact that the solution of the variational inequality is given by the minimizer of a nonsmooth energy functional, involving the $p$-Laplace operator. We propose a Huber regularization of the functional and a finite element discretization for the problem. Further, we analyze the regularity of the discretized energy functional, and we are able to prove that its Jacobian is slantly differentiable. This regularity property is useful to analyze the convergence of the MG/OPT algorithm. In fact, we demonstrate that the algorithm is globally convergent by using a mean value theorem for semismooth functions. Finally, we apply the MG/OPT algorithm to the numerical simulation of the viscoplastic flow of Bingham, Casson and Herschel–Bulkley fluids in a pipe. Several experiments are carried out to show the efficiency of the proposed algorithm when solving this kind of fluid mechanics problems.     

Primal/Dual Descent Methods for Dynamics

We examine the relationship between primal, or force-based, and dual, or constraint-based formulations of dynamics. Variational frameworks such as Projective Dynamics have proved popular for deformable simulation, however they have not been adopted for contact-rich scenarios such as rigid body simulation. We propose a new preconditioned frictional contact solver that is compatible with existing primal optimization methods, and competitive with complementarity-based approaches. Our relaxed primal model generates improved contact force distributions when compared to dual methods, and has the advantage of being differentiable, making it well-suited for trajectory optimization. We derive both primal and dual methods from a common variational point of view, and present a comprehensive numerical analysis of both methods with respect to conditioning. We demonstrate our method on scenarios including rigid body contact, deformable simulation, and robotic manipulation.     

A New Projection-Based Neural Network for Constrained Variational Inequalities

This paper presents a new neural network model for solving constrained variational inequality problems by converting the necessary and sufficient conditions for the solution into a system of nonlinear projection equations. Five sufficient conditions are provided to ensure that the proposed neural network is stable in the sense of Lyapunov and converges to an exact solution of the original problem by defining a proper convex energy function. The proposed neural network includes an existing model, and can be applied to solve some nonmonotone and nonsmooth problems. The validity and transient behavior of the proposed neural network are demonstrated by some numerical examples.      

A contact force solution for non-colliding contact dynamics simulation

Rigid-body impact modeling remains an intensive area of research spurred on by new applications in robotics, biomechanics, and more generally multibody systems. By contrast, the modeling of non-colliding contact dynamics has attracted significantly less attention. The existing approaches to solve non-colliding contact problems include compliant approaches in which the contact force between objects is defined explicitly as a function of local deformation, and complementarity formulations in which unilateral constraints are employed to compute contact interactions (impulses or forces) to enforce the impenetrability of the contacting objects. In this article, the authors develop an alternative approach to solve the non-colliding contact problem for objects of arbitrary geometry in contact at multiple points. Similarly to the complementarity formulation, the solution is based on rigid-body dynamics and enforces contact kinematics constraints at the acceleration level. Differently, it leads to an explicit closed-form solution for the normal forces at the contact points. Integral to the proposed formulation is the treatment of tangential contact forces, in particular the static friction. These friction forces must be calculated as a function of microslip velocity or displacement at the contact point. Numerical results are presented for four test cases: (1) a thin rod sliding down a stationary wedge; (2) a cube pushed off a wedge by an applied force; (3) a cube rotating off the wedge under application of an external moment; and (4) the cube and the wedge both moving under application of a moment. To ascertain validity and correctness, the solutions to frictionless and frictional scenarios obtained with the new formulation are compared to those generated by using a commercial simulation tool MSC ADAMS.     

An augmented Lagrangian optimization method for contact analysis problems, 2: numerical evaluation

The ALM2 solution procedure is evaluated by solving two simple contact analysis problems for different friction conditions. These example problems are devised to have closed form solutions. This way there is no uncertainty about the target solution for evaluation of the proposed algorithm as well as existing algorithms. The numerical results with ALM2 are compared with the analytical solutions as well as with the penalty, Lagrange multiplier and existing augmented Lagrangian methods. All the algorithms are analysed for stick and slip friction conditions. The example problems are used to show clearly the dependence of the existing solution methods on the number of load steps and penalty values. It is concluded that convergence of incremental solution schemes employed in these methods does not guarantee accuracy of the contact solution even with the use of solution enhancement schemes such as automatic load stepping and contact load prediction. The example problems are also used to demonstrate solution independence of the proposed ALM2 procedure from penalty values, and from the number of load steps. The proposed formulation for calculation of frictional forces and the ALM solution algorithm have worked quite well for the example problems. However, the algorithm needs to be developed and evaluated for more complex contact analysis problems.