## 1 INTRODUCTION

### Asynchronous integration

The main disadvantage of explicit time integration is its conditional stability, that is, it becomes unstable if the time step exceeds a certain threshold. In many applications only a small number of finite elements of very small size and/or with stiff material properties are responsible for the small size of this critical time step.

A popular approach to overcome this problem are multiple time stepping integrators. One line of development starts with mixed methods by using implicit and explicit time stepping for different domains [1], another line uses different time step sizes known as subcycling [2]. Multiple time stepping creates configurations at discrete times *t*_{k} = *k* ⋅ Δ*t* where all nodal displacements and velocities are synchronous in time. Some parts of the structure are substepped using a smaller time step at configurations in-between where not the whole system is synchronous. These smaller time steps are obtained by bisection, integer ratios or non-integer ratios [3-5].

A generalization of the symplectic-momentum multiple-time stepping scheme r-RESPA [6] are asynchronous variational integrators (AVIs) [7-12]. Therein, time step sizes are individually assigned to each finite element at arbitrary ratios. It is, therefore, generally not possible to obtain configurations in time where all finite elements are evaluated at the same ‘synchronous’ time. The time stepping scheme can be derived as a variational integrator [13] and, thus, is symplectic [14] and momentum preserving [15]. Convergence can be proved for linear elasticity [11]. Reliable stability criteria are difficult to find. A stability analysis was exemplified for a single degree of freedom system with two asynchronous potential functions in [12].

The conditional stability of explicit integrators is the reason why implicit time stepping schemes are often preferred. For linear systems implicit schemes may be unconditionally stable allowing arbitrarily large time steps. In nonlinear dynamics, symplectic implicit schemes are also only conditionally stable. Even energy-conserving time stepping schemes may become unstable when applied to nonlinear systems [16]. Benes and Matous [17, 18] have shown computationally that the critical time step for their implicit asynchronous integrators drops sharply but saturates at a level within the range of reasonable engineering accuracy. They also show that the critical time step for explicit asynchronous integrators improves over the synchronous case. This result agrees with [19] on improving stability by mollified impluses in r-RESPA and the references therein. As noted in [12], however, there may be time step sizes below the numerically assessed stability limit where asynchronous schemes are unstable.

The motivation of this article is to extend the idea behind explicit asynchronous integration to the context of explicit contact/impact dynamics and to explore the potential computational savings by this approach.

### Contact dynamics

Unilateral constraints often arise in contact/impact problems where an impenetrability condition [20] must be satisfied that is represented by an inequality condition. There are generally two approaches to dynamic discretization of the constraint: enforcing impenetrability at discrete points in time or enforcing the time derivative of impenetrability (persistency condition) being zero.

The enforcement of the persistency condition is known as Laursen–Chawla algorithm [21]. Therein, an analysis of the generalized-alpha and Newmark methods leads to the observation that energy conservation is tied to the persistency condition. An application to the augmented Lagrangian and penalty method is presented that are either conservative or dissipative and allow small penetrations. The algorithm was later combined with the impenetrability condition whereby energy conservation is restored by an additional velocity update [22, 23]. A variational treatment based on a discrete version of the classical principle of Hamilton is given by [24] who introduce the collision time as additional degree of freedom in explicit integrators and perform a velocity jump that is equivalent to enforcing the persistency condition. This leads to a nonsmooth trajectory. The approach was further simplified in [25] introducing decomposition contact response (DCR) that is a non-iterative treatment at discrete points in time extending to inelastic and frictional contact problems.

### Asynchronous collisions

This article presents contact algorithms to explicit asynchronous simulation of structural dynamics. When handling contact problems in explicit dynamics, there generally exist two approaches: penalty based and Lagrange multiplier methods. Penalty methods are simple to implement, and penalty forces can be computed efficiently. But they are inaccurate allowing penetrations and may affect the critical time step. A penalty approach to asynchronous contact was presented in [26]. Lagrange multiplier methods, on the other hand, often lead to iterative procedures. The possible large number of highly nonlinear constraints reduces the efficiency. Furthermore, redundant constraints may appear leading to singular systems of equations.

The application of an asynchronous collision integrator may eliminate some problems arising in Lagrange multiplier methods. Let the individual contact constraints be enforced at asynchronous times. If each spatial constraint is considered individually, the system of equations is simplified by two factors: (1) there is only a single constraint to be enforced at one time; and (2) furthermore, only a limited number of degrees of freedom is affected. The size of the equation system is, therefore, very small. By application of DCR, the equations are linear and the constraints can be enforced non-iteratively. The operation only modifies the momentum and can, thus, be interpreted in terms of a kick operator of an asynchronous variational integration algorithm. Because each constraint is considered individually without affecting the critical time step, one may chose the time step size between two contact corrections according to local accuracy conditions, such as relative velocities and finite element sizes. The formulation of the adaptive time step is much easier than for the potential energy where either symplecticity must be restored by additive terms that may lead to iterative schemes even for explicit methods [27] or where instabilities occur because of unsolvable equations [28].

### Objectives and outline

One objective of this article is to develop an efficient implementation of nodal restraint conditions for AVI. The main focus is, however, to derive a suitable explicit collision integrator within asynchronous variational integration. In particular, its efficient implementation and its coupling with nodal restraints is the main contribution of this article.

The outline is as follows: Section 2 recalls the basic ideas of AVI. Section 3 explains the notation used in this article by interpreting AVI as a sequence of drifts with constant motion and a set of events that modify the velocity asynchronously. The novel implementation of nodal restraints will be derived in Section 4 on the basis of the variational RATTLE method. Implications of the asynchronous approach to global contact detection algorithms are briefly discussed in Section 5 by the example of position codes and node-to-surface integration. The asynchronous collision integrator is finally presented in Section 6. For completeness, the decomposition contact response is briefly shown. Subsequently, three novel and efficient solution algorithms of DCR in the context of AVI are formulated: normal contact, normal contact with nodes being subject to restraint conditions and normal contact with friction being exemplified by the Coulomb model. Three configurations of AVI are presented in Section 7 introducing a synchronous setting and asynchronous settings with fixed and variable step sizes. Numerical examples illustrate convergence and efficiency of the new contact algorithm in Section 8.

### Symbols and typography

The notation in Sections 2 to 4 follows the lines of articles on variational integrators, For example [8, 9, 12, 24, 28, 29]. Matrices, vectors, and scalars are characterized by non-bold letters, that is, all generalized coordinates are collected in a vector *q*, whereas its conjugate momenta are defined as *j*. Sections 5 to 7 are related to the spatial and temporal contact formulation and introduce the notation that is more familiar to most engineers in mechanics, that is, small bold letters for vectors, large bold letters for matrices. The generalized coordinates then specialize to the vectors of nodal displacements **u** whereas one often uses the vector of velocities **v** instead of momenta.