Dynamic contact analysis technique for rapidly sliding elastic bodies with node-to-segment contact and differentiated constraints

For a stabilized Newmark time integration of dynamic contact problems of the rapidly sliding bodies, considering the equality and inequality contact constraints and a high-speed contact point motion sliding on the deforming contact surface, the velocity and acceleration contact constraints are derived. Also, to suppress the numerical oscillations accompanied by the node-to-segment contact of the finite element models, a pseudo-node-to-node contact technique is suggested with the linear shape function elements having the almost equal segment lengths on the contact surface. The numerical simulations are performed with a high-speed punch moving on the beam and the high-speed rotating disks to check the stability and accuracy of the solution.     

Numerical analysis for dynamic contact between high‐speed wheel and elastic beam with Coulomb friction

For numerical analysis of the dynamic contact between a high‐speed wheel and an elastic beam, the equation of motion of each body is time integrated by a simple ODE solution technique and frictional contact conditions are imposed by the augmented Lagrange multiplier method using the contact errors defined in this work. For the stability of the numerical solution, the velocity and acceleration contact conditions as well as the displacement contact condition are imposed with special consideration for the high‐velocity contact point moving on the deformed beam. Especially, it is shown that the Coriolis and centripetal accelerations of the contact point moving rapidly on the deformed beam play crucial roles for the stability of the solution. It is also shown that, for a wheel rolling on a beam with friction, the acceleration constraint in the tangential direction is important for the stability of the solution. Copyright © 2008 John Wiley & Sons, Ltd.     

A numerical solution for dynamic contact problems satisfying the velocity and acceleration compatibilities on the contact surface

For the numerical solution of dynamic contact problems, correct contact points and displacements are determined by iteratively reducing the displacement error vector monotonically toward zero. And spurious oscillations are prevented from the solution by enforcing the velocity and acceleration compatibilities of the contact points with the corresponding error vectors. Economic computation is possible because the accelerated iterative schemes are used and because decomposition of large matrix is not required in the iterations for the contact analysis of elastic bodies. Numerical simulations are conducted to demonstrate the accuracy of the solution and the necessity of the velocity and acceleration compatibilities on the contact surface.This research was supported in part by Korea Research Foundation, 1993.     

A NUMERICAL METHOD FOR DYNAMIC ANALYSIS OF VEHICLES MOVING ON FLEXIBLE STRUCTURES HAVING GAPS

A numerical method is presented for the dynamic analysis of vehicles moving on flexible structures which contain gaps. The Lagrange multipliers associated with the kinematic constraints of the vehicle components and the contact forces between the rigid wheels of the vehicle and the flexible structures are simultaneously computed with the solutions of the equations of motion by using the iterative schemes. On the kinematic joints and on the possible contact points the velocity and acceleration constraints as well as the displacement constraints are satisfied by the monotone reductions of the corresponding error vectors. And a well-developed simple one-step time integration of ordinary differential equation is employed for the solution of the equations of motion. Convergences of the iterative schemes are analysed and numerical simulations are conducted. © 1997 by John Wiley & Sons, Ltd.     

A time‐stepping method for stiff multibody dynamics with contact and friction

We define a time‐stepping procedure to integrate the equations of motion of stiff multibody dynamics with contact and friction. The friction and non‐interpenetration constraints are modelled by complementarity equations. Stiffness is accommodated by a technique motivated by a linearly implicit Euler method. We show that the main subproblem, a linear complementarity problem, is consistent for a sufficiently small time step h. In addition, we prove that for the most common type of stiff forces encountered in rigid body dynamics, where a damping or elastic force is applied between two points of the system, the method is well defined for any time step h. We show that the method is stable in the stiff limit, unconditionally with respect to the damping parameters, near the equilibrium points of the springs. The integration step approaches, in the stiff limit, the integration step for a system where the stiff forces have been replaced by corresponding joint constraints. Simulations for one‐ and two‐dimensional examples demonstrate the stable behaviour of the method. Published in 2002 by John Wiley & Sons, Ltd.     

A nonsmooth frictional contact formulation for multibody system dynamics

We present a new node-to-face frictional contact element for the simulation of the nonsmooth dynamics of systems composed of rigid and flexible bodies connected by kinematic joints. The equations of motion are integrated using a nonsmooth generalized-α time integration scheme and the frictional contact problem is formulated using a mixed approach, based on an augmented Lagrangian technique and a Coulomb friction law. The numerical results are independent of any user-defined penalty parameter for the normal or tangential component of the forces and, the bilateral and the unilateral constraints are exactly fulfilled both at position and velocity levels. Finally, the robustness and the performance of the proposed algorithm are demonstrated by solving several numerical examples of nonsmooth mechanical systems involving frictional contact.     

Asynchronous collision integrators: Explicit treatment of unilateral contact with friction and nodal restraints

This article presents asynchronous collision integrators and a simple asynchronous method treating nodal restraints. Asynchronous discretizations allow individual time step sizes for each spatial region, improving the efficiency of explicit time stepping for finite element meshes with heterogeneous element sizes. The article first introduces asynchronous variational integration being expressed by drift and kick operators. Linear nodal restraint conditions are solved by a simple projection of the forces that is shown to be equivalent to RATTLE. Unilateral contact is solved by an asynchronous variant of decomposition contact response. Therein, velocities are modified avoiding penetrations. Although decomposition contact response is solving a large system of linear equations (being critical for the numerical efficiency of explicit time stepping schemes) and is needing special treatment regarding overconstraint and linear dependency of the contact constraints (for example from double‐sided node‐to‐surface contact or self‐contact), the asynchronous strategy handles these situations efficiently and robust. Only a single constraint involving a very small number of degrees of freedom is considered at once leading to a very efficient solution. The treatment of friction is exemplified for the Coulomb model. Special care needs the contact of nodes that are subject to restraints. Together with the aforementioned projection for restraints, a novel efficient solution scheme can be presented. The collision integrator does not influence the critical time step. Hence, the time step can be chosen independently from the underlying time‐stepping scheme. The time step may be fixed or time‐adaptive. New demands on global collision detection are discussed exemplified by position codes and node‐to‐segment integration. Numerical examples illustrate convergence and efficiency of the new contact algorithm. Copyright © 2013 The Authors. International Journal for Numerical Methods in Engineering published by John Wiley & Sons, Ltd.     

A linearly implicit trapezoidal method for integrating stiff multibody dynamics with contact,joints, and friction

We present a hard constraint, linear complementarity based, method for the simulation of stiff multibody dynamics with contact, joints and friction. The approach uses a linearization of the modified trapezoidal method, incorporates a Poisson restitution model at collision, and solves only one linear complementarity problem per time step when no collisions are encountered. We prove that, under certain assumptions, the method has order two, a fact that is also demonstrated by our numerical simulations. For the unconstrained (ODE) case, the method achieves second‐order convergence and absolute stability while solving only one linear system per step. When we use a special approximation of the Jacobian matrix for the case where the stiff forces originate in springs and dampers attached to two points in the system, the linear complementarity problem can be solved for any value of the time step and numerical simulation demonstrate that the method is stiffly stable. The method was implemented in UMBRA, an industrial‐grade virtual prototyping software. Copyright © 2005 John Wiley & Sons, Ltd.     

A contact‐stabilized Newmark method for dynamical contact problems

The numerical integration of dynamical contact problems often leads to instabilities at contact boundaries caused by the non‐penetration condition between bodies in contact. Even an energy dissipative modification (see, e.g. (Comp. Meth. Appl. Mech. Eng. 1999; 180 :1–26)), which discretizes the non‐penetration constraints implicitly, is not able to circumvent artificial oscillations. For this reason, the present paper suggests a contact stabilization in function space, which avoids artificial oscillations at contact interfaces and is also energy dissipative. The key idea of this contact stabilization is an additional L²‐projection at contact interfaces, which can be easily added to any existing time integration scheme. In case of a lumped mass matrix, this projection can be carried out completely locally, thus creating only negligible additional numerical cost. For the new scheme, an elementary analysis is given, which is confirmed by numerical findings in an illustrative test example (Hertzian two‐body contact). Copyright © 2007 John Wiley & Sons, Ltd.     

An accelerated iterative method for the dynamics of constrained multibody systems

An accelerated iterative method is suggested for the dynamic analysis of multibody systems consisting of interconnected rigid bodies. The Lagrange multipliers associated with the kinematic constraints are iteratively computed by the monotone reduction of the constraint error vector, and the resulting equations of motion are easily time-integrated by a well established ODE technique. The velocity and acceleration constraints as well as the position constraints are made to be satisfied at the joints at each time step. Exact solution is obtained without the time demanding procedures such as selection of the independent coordinates, decomposition of the constraint Jacobian matrix, and Newton Raphson iterations. An acceleration technique is employed for the faster convergence of the iterative scheme and the convergence analysis of the proposed iterative method is presented. Numerical solutions for the verification problems are presented to demonstrate the efficiency and accuracy of the suggested technique.     

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.     

Rate‐dependent projection operators for frictional contact constraints

We develop rate‐dependent regularization approaches for three‐dimensional frictional contact constraints based on the Kelvin and Maxwell viscoelastic constitutive models. With the present regularization schemes, we aim to provide a basis to better model friction and to stabilize the contact analysis while keeping the contact model as simple as possible. The key feature of the regularization approaches, implemented using an implicit time integrator, is that one can recover in the limit the widely used rate‐independent elastoplastic regularization framework without encountering numerical difficulties. Intermediate contact tractions are defined in terms of the relative displacement, the relative velocity, and the regularization parameters. The projection operators operate on the intermediate tractions and yield contact tractions that satisfy all the discretized contact constraints. The use of projection operators allows a systematic implementation of the present regularization schemes. Through numerical simulations, we observed that the Maxwell‐type regularization effectively avoids convergence problems, even for relatively large time step sizes, while the Kelvin‐type regularization does not. Copyright © 2003 John Wiley & Sons, Ltd.     

Decomposition contact response (DCR) for explicit finite element dynamics

We propose a new explicit contact algorithm for finite element discretized solids and shells with smooth and non‐smooth geometries. The equations of motion are integrated in time with a predictor‐corrector‐type algorithm. After each predictor step, the impenetrability constraints and the exchange of momenta between the impacting bodies are considered and enforced independently. The geometrically inadmissible penetrations are removed using closest point projections or similar updates. Penetration is measured using the signed volume of intersection described by the contacting surface elements, which is well‐defined for both smooth and non‐smooth geometries. For computing the instantaneous velocity changes that occur during the impact event, we introduce the decomposition contact response method. This enables the closed‐form solution of the jump equations at impact, and applies to non‐frictional as well as frictional contact, as exemplified by the Coulomb frictional model. The overall algorithm has excellent momentum and energy conservation characteristics, as several numerical examples demonstrate. Copyright © 2005 John Wiley & Sons, Ltd.     

An implicit finite wear contact formulation based on dual mortar methods

Finite deformation contact problems with frictional effects and finite shape changes due to wear are investigated. To capture the finite shape changes, a third configuration besides the well‐known reference and spatial configurations is introduced, which represents the time‐dependent worn state. Consistent interconnections between these states are realized by an arbitrary Lagrangean–Eulerian formulation. The newly developed partitioned and fully implicit algorithm is based on a Lagrangean step and a shape evolution step. Within the Lagrangean step, contact constraints as well as the wear equations are weakly enforced following the well‐established mortar framework. Additional unknowns due to the employed Lagrange multiplier method for contact constraint enforcement and due to wear itself are eliminated by condensation procedures based on the concept of biorthogonality and the so‐called dual shape functions. Several numerical examples in both 2D and 3D are provided to demonstrate the performance and accuracy of the proposed numerical algorithm. Copyright © 2016 John Wiley & Sons, Ltd.     

Transient three-dimensional contact problems: mortar method. Mixed methods and conserving integration

The present work deals with the development of an energy-momentum conserving method to unilateral contact constraints and is a direct continuation of a previous work (Hesch and Betsch in Comput Mech 2011, doi:) dealing with the NTS method. In this work, we introduce the mortar method and a newly developed segmentation process for the consistent integration of the contact interface. For the application of the energy-momentum approach to mortar constraints, we extend an approach based on a mixed formulation to the segment definition of the mortar constraints. The enhanced numerical stability of the newly proposed discretization method will be shown in several examples.     

A finite deformation mortar contact formulation using a primal–dual active set strategy

In recent years, nonconforming domain decomposition techniques and, in particular, the mortar method have become popular in developing new contact algorithms. Here, we present an approach for 2D frictionless multibody contact based on a mortar formulation and using a primal–dual active set strategy for contact constraint enforcement. We consider linear and higher‐order (quadratic) interpolations throughout this work. So‐called dual Lagrange multipliers are introduced for the contact pressure but can be eliminated from the global system of equations by static condensation, thus avoiding an increase in system size. For this type of contact formulation, we provide a full linearization of both contact forces and normal (non‐penetration) and tangential (frictionless sliding) contact constraints in the finite deformation frame. The necessity of such a linearization in order to obtain a consistent Newton scheme is demonstrated. By further interpreting the active set search as a semi‐smooth Newton method, contact nonlinearity and geometrical and material nonlinearity can be resolved within one single iterative scheme. This yields a robust and highly efficient algorithm for frictionless finite deformation contact problems. Numerical examples illustrate the efficiency of our method and the high quality of results. Copyright © 2009 John Wiley & Sons, Ltd.     

A consistent penalty method for contact between a deforming viscoplastic workpiece and a rigid tool

A methodology is presented for enforcing a contact constraint between a viscoplastic workpiece and the tooling which deforms it. The algorithm is based on a consistent penalty approach wherein an elemental discretization permits elimination of contact tractions prior to solution for the velocity field. When written in rate form, the new algorithm allows contact constraints to be embedded in a viscoplastic formulation without altering the structure of the global matrix equation. Examples are presented to illustrate its use in metal forming simulations.     

On the elastodynamic solution of frictional contact problems using variational inequalities

This article is concerned with the development, implementation and application of variational inequalities to treat the general elastodynamic contact problem. The solution strategy is based upon the iterative use of two subproblems. Quadratic programming and Lagrange multipliers are used to solve the respective first and second subproblems and to identify the candidate contact surface and contact stresses. This approach guarantees the imposition of the active kinematic contact constraints, avoids the use of special contact elements and the interference of the user in dictating the accuracy of the solution. A modified Newmark formulation is developed to integrate the resulting time‐dependent variational inequality. This newly devised implicit time integration scheme is unconditionally stable, second‐order accurate, avoids numerical oscillations present in the traditional Newmark method, and does not cause numerical dissipation. To demonstrate the versatility and accuracy of the newly proposed algorithm, several examples are examined and compared with existing solutions where the penalty method has been employed. Copyright © 2001 John Wiley & Sons, Ltd.     

Dynamic and contact analysis of a bimodal ultrasonic motor

A bimodal ultrasonic motor, which operates with only one power amplifier, uses two simultaneously excited modes to drive the rotor; a longitudinal mode and a flexural mode. The equations of motion describing the vibrations and contact behavior are derived by Hamilton's principle and the geometry constraint. The Lagrange multiplier method is used to treat the frictional contact problem. The finite element method and numerical integration scheme are used to simulate the dynamic responses of this system with and without contact. Some important factors are studied for the bimodal ultrasonic motor design. The factors include structure design, amplitude of input voltage, phase displacement, exciting frequency, and contact behavior.