## Abstract

Dynamic contact and impact problems are widely applicable. An accurate solution method for these kinds of problems can be used in many fields of mechanical engineering (e.g., cutting metalwork, cogwheel drives, etc.). However, the proper handling of the contact is problematic, as there emerges a substantial amount of nonlinearity in the displacement field. Therefore, a spurious high frequency oscillation is present in the solution. These oscillations must be avoided, as divergence can easily occur in the contact algorithm due to them. In order to eliminate this effect, the applied numerical method must be chosen and set properly. In this study, a comprehensive guide is provided for the appropriate selection of the proper numerical method and its parameters.

## 1 Introduction

By solving contact problems, one of the main challenges is the proper handling of the emerging nonlinearities. Historically, it was Hertz [1], who firstly dealt with these kinds of problems in 1881. In the field of contact problems, there are only a few simple examples, which can be treated analytically. In most cases, only an approximate solution can be achieved, which is usually provided by the Finite Element Method (FEM) [2, 3]. The Finite Element (FE) analysis of contact problems has a very wide literature [4, 5] in which two main approaches can be distinguished. The first technique is the penalty method [6], which is a simpler, faster, but not so accurate way for solving mechanical contact problems using FEM. The second approach is the Lagrange multiplier method [7] in which the contact pressure emerges as an additional unknown variable. This technique fulfills the contact conditions accurately, but its application is computationally more expensive. A third possibility is the augmented Lagrangian method [8], which is a certain kind of combination of the penalty and the Lagrange multiplier method. In this paper the contact boundary conditions are treated with the Lagrange multiplier method.

In most cases, the solution of a dynamic contact problem requires the time integration of the semi-discretized equation of motion, which is discretized in space, but continuous in time. By an equation of this kind, for the integration in time both forward and backward increment, single- and multi-stage numerical methods [9] can be used. This paper is going to analyze the applicability of four single-stage discrete time integration methods and will demonstrate their specific effects on the solution. The adaptation of the very commonly used Central Difference Method (CDM) [10] to the Lagrange multiplier method was presented by Carpenter et al. [11] in 1991. The main drawback of this method is that it possesses only conditional stability. However, the application of it is computationally cheaper and CDM can be applied effectively when a lumped mass matrix [12] is used instead of a consistent one. Furthermore, the application of the Backward Euler (BE) [13], the NewMark (NM) [14, 15] and the Hilber-Hughes-Taylor-*α* (HHT-α) method [16] will be studied, which are widely used backward increment schemes. The main advantage of these methods is that they are unconditionally stable. This is a major aspect; hence these methods are often built in commercial FE software. The Newmark method is the most common implicit method in this kind of software. The formulation of this method contains a parameter through which the numerical damping of the method can be affected directly. The HHT-α method is the improved version of the Newmark method, where the dissipation level can be controlled with two parameters. Nevertheless, the solution produced with a backward increment scheme is more time-consuming to achieve, as a system of equations must be solved in every time step. There are many more time integration schemes, like the Houbolt, the Wilson-

The right choice of the applied numerical method is crucial, as it has a significant impact on the solution characteristics, the computation time and the achievable accuracy. There is not any universal recipe to reach an ideal solution, many different schemes must be taken into consideration to examine, which method gives the best solution as of the user's preferences. It is a further problem that due to the spatial discretization a spurious high frequency oscillation occurs in the resulting fields of displacement and contact pressure, which distorts the correct solution. This difficulty can be at least partially eliminated by applying a certain amount of numerical damping by the solution of the equation of motion. Nevertheless, it is also substantial to filter only the disturbance out by leaving the useful component of the signal intact. Thus, the adjustability of the numerical damping is an important aspect to consider, when choosing the right method. The main difficulty is to find an optimal region for the dissipation parameter to achieve a solution with acceptable accuracy, while significantly reducing the spurious oscillations.

In this article, the effectiveness of the aforementioned kinds of methods is going to be analyzed. The accuracy and the practical applicability of the examined methods will be studied from several aspects. Besides, parameter identifications are going to be made in order to determine the right amount of numerical damping for methods with adjustable dissipation level. The main goal of this paper is to provide a comprehensive application analysis about how to choose the right method with the correct parameters for a dynamic contact problem.

## 2 Time stepping methods

*t*, respectively. Moreover the solution of this differential system of equations requires both initial and boundary conditions from which the former can be written as

### 2.1 Central difference method

Equations (4) and (6) can be substituted into the equation of motion (1). By ordering it the same way, as it was evaluated for Eqs (5) and (7) can be obtained.

### 2.2 Backward Euler method

The general formulation of this method according to [13] can be written as Eqs (8) and (9). Eq. (1) of motion for time step

### 2.3 Newmark method

Its basic formulation is determined by Eqs (11) and (12). Based on [14], the optimal parameter setting can be given as Eq. (13). By setting the

### 2.4 HHT-
$\alpha $
method

If

## 3 Modeling of the impact problem

### 3.1 FEM model

The methods that have been described above should be tested using an example by which it can be judged, how well they can handle the sudden changes in characteristic indicators like the velocity or the contact pressure. For this purpose, a 1D impact problem is an ideal choice as it is a widely used test example in the literature of contact problems [11, 17, 18]. The main benefits of this example are that its solution can be made easily and fast and it has an exact analytical solution, which provides a good reference to assess the results. Besides, it challenges the achievable accuracy of the numerical method at a sufficient level. Thus, if an accurate solution can be produced for this test example, then the applied method can presumably treat more complex geometries too. Therefore, a linearly elastic rod is considered which is moving towards a rigid wall (Fig. 1).

In order to solve this test example, a FEM model must be constructed based on the mechanical model shown in Fig. 1. This FEM model is built using 1D truss elements, which possess 2 nodes with 1 Degree of Freedom (DoF) per node, which is the longitudinal displacement. The FEM model is shown in Fig. 2, where

### 3.2 Treatment of the contact

The proper handling of the contact is an important aspect by the modeling.

**X**means the initial configuration (vector of length

*n*+ 1 filled with the initial coordinates of nodes),

**u**is the displacement vector, and

**G**is the contact constraint matrix. This equation is a certain kind of manifestation of the Hertz-Signorini-Moreau (HSM) conditions [4] in contact mechanics. The detailed description of here applied formulation can be found in [11]. Thus, Eq. (1) must be modified with regard to the contact force acting between the elastic rod and the rigid wall

So, the equation of motion can be written in the form of Eq. (17) where

In this case, that quantity is equivalent with the surface contact pressure. Thus, the Lagrange multiplier method proceeds at the here presented example by treating *λ* as unknown and solving Eqs (16) and (17) simultaneously. The numerical methods examined in Section 2 can easily be deducted for the contacting case by replacing the equation of motion (1) with (17).

## 4 Parameter identification

Thus, it is practical to define certain indicators by which it can be quantified, how accurate the 1D example may be solved and how efficient the elimination of the spurious oscillations is. Firstly, relative errors are considered, which can be calculated by Eq. (18) where

### 4.1 Newmark method

Based on these two indicators the practical range of

a) Relative error of the pressure for varying values of

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a) Relative error of the pressure for varying values of

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a) Relative error of the pressure for varying values of

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### 4.2 HHT-
$\alpha $
method

a) Relative error of the pressure for varying values of

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a) Relative error of the pressure for varying values of

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a) Relative error of the pressure for varying values of

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Based on this examination, Eq. (19) may be concluded to reach a parameter setting that results minimum for the period error. Considering Fig. 4, this connection proves to be a practical compromise between the accuracy and the elimination of the spurious oscillations. Hence, it is beneficial to choose the parameter values according to Eq. (19), especially as it makes the parameter setting much easier. Based on Fig. 4 and in accordance with the concrete numerical tests with the FEM model, the “practical stage” can be ascertained as

## 5 Application in a 1D example

In order to compare the applied methods and check the results of the parameter identification, a concrete function must be plotted that characterizes the 1D impact. As the spurious oscillations are mainly problematic in the contact pressure and the velocity functions, the methods shall be compared using the time evolution of these quantities (Figs 5–6). The magnified area on the left side shows the oscillations emerging directly after the moment of impact. The enlarged region on the right side of Fig. 5 indicates how steep the pressure decreases to zero which illustrates the accuracy of the solutions. The prominence in the zoomed plot on the right side of Fig. 6 is caused by the wave propagation in the rod and must be as narrow as possible. Besides, the indicators that have been used in section 4 are illustrated by identical dissipation for each method in Figs 7 and 8 for the pressure and the velocity, respectively. By studying them, it can be assessed that there cannot be found an ideal method which to use is beneficial in all aspects.

Time evolution of the contact pressure

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Time evolution of the contact pressure

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Time evolution of the contact pressure

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Time evolution of the velocity at the contacting node

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Time evolution of the velocity at the contacting node

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Time evolution of the velocity at the contacting node

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Comparison of the applied methods (pressure function)

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Comparison of the applied methods (pressure function)

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Comparison of the applied methods (pressure function)

Citation: Pollack Periodica 17, 3; 10.1556/606.2022.00599

Comparison of the applied methods (velocity function at the contacting node)

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Comparison of the applied methods (velocity function at the contacting node)

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Comparison of the applied methods (velocity function at the contacting node)

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The central difference method offers a fast and accurate result, but the large amount of oscillations makes it problematic to apply. In contrast, the backward Euler method offers the best performance in the elimination of oscillations, but this feature is coupled with the highest relative error. Based on this study, the Newmark and the HHT-

## 6 Conclusion

In this study a thorough analysis was made about the application of numerical methods in contact problems. Four different, widely used numerical schemes were deducted and applied for a 1D example. In the parameter identification, different aspects were presented, through which a correct parameter setting may be determined for numerical methods with one or more control parameters. The minimization of the period error proved to be a beneficial compromise by the HHT-

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