Authors:
Dániel Serfőző Department of Applied Mechanics, Faculty of Information Technology and Electrical Engineering, Széchenyi István University, Győr, Hungary

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Balázs Pere Department of Applied Mechanics, Faculty of Information Technology and Electrical Engineering, Széchenyi István University, Győr, Hungary

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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.

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- θ , the generalized- α , etc. that can be found in [9]. Besides, there exist several multi stage schemes, like Kim's method [17] and the Kolay-Ricles- α (KR- α ) method [18]. In this paper, only the above mentioned four methods are considered, as they proved to be the most useful for contact problems, based on our experiences.

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

Let us consider a structural dynamic problem, where the interacting bodies are linearly elastic. The semi-discretized equation of motion after spatial discretization by finite element method can be written as a system of linear differential equations of second order according to
M u ¨ t + C u ˙ t + K u t = f t ,
where M , C and K denote the mass, damping and stiffness matrices, respectively. On the right side f t means the applied load vector, while u t , u ˙ t and u ¨ t are the unknown displacement, velocity and acceleration vectors at time instant 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
u t = 0 = u 0 and u ˙ t = 0 = v 0 ,
where u 0 and v 0 denote the initial displacement and velocity vectors, respectively. In the following, four different numerical methods will be introduced for solving numerically the equation of motion (1), hence the deduction will be presented for each method in the next four chapters.

2.1 Central difference method

The CDM is a commonly used forward increment method for various mechanical applications. The main benefits of using this scheme are that it is second order accurate, and computations can be executed very fast due to its explicit formulation. However, the central difference method is only conditionally stable, which reduces the applicable time step region. In this method, the velocity and the acceleration are given using the displacement at the current, previous and next time step the following way.
u ˙ t = u t + Δ t u t Δ t 2 Δ t ,
u ¨ t = u t + Δ t 2 u t + u t Δ t Δ t 2 ,
( Δ t C + 2 M ) u t + Δ t = ( 4 M 2 Δ t 2 K ) u t + ( Δ t C 2 M ) u t Δ t + 2 Δ t 2 f t .
By substituting Eqs (3) and (4) into the Eq. of motion (1) and reordering it, Eq. (5) can be obtained. The main problem with this formula is that it requires the inversion of the term Δ t C + 2 M , which is a computationally expensive procedure,
u ˙ t = u t u t Δ t Δ t .
Hence, it is worthwhile to combine Eq. (4) with Eq. (6), as it results a formulation that only requires the inversion of M . As the inversion of a lumped mass matrix is computationally cheap, it is rewarding to diagonalize the mass matrix by the method described in [12]. It is important to note that the formula applied for the u ˙ t velocities originates from the backward Euler method. Thus, the applied explicit technique is a certain kind of combination of the central difference and the backward Euler method,
M u t + Δ t = ( 2 M Δ t C Δ t 2 K ) u t + ( Δ t C M ) u t Δ t + Δ t 2 f t .

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 backward Euler scheme has an implicit character, so, a system of equations (which is often nonlinear) needs to be solved for every time step, resulting high computational costs. This method is unconditionally stable, first order accurate and has a certain amount of numerical damping. The main problem is, that the dissipation level is not controllable; hence it often leads to energy dissipation of an undesirable rate,
u t + Δ t = u t + Δ t u ˙ t + Δ t ,
u ˙ t + Δ t = u ˙ t + Δ t u ¨ t + Δ t ,
M u ¨ t + Δ t + C u ˙ t + Δ t + K u t + Δ t = f t + Δ t .

The general formulation of this method according to [13] can be written as Eqs (8) and (9). Eq. (1) of motion for time step t + Δ t leads to formulation (10).

2.3 Newmark method

The Newmark method [14, 15] is a commonly used implicit integration scheme, which is built in many commercial FEM software (Ansys, Abaqus, etc.),
u t + Δ t = u t + Δ t u ˙ t + Δ t 2 2 ( ( 1 2 β ) u ¨ t + 2 β u ¨ t + Δ t ) ,
u ˙ t + Δ t = u ˙ t + Δ t ( ( 1 γ ) u ¨ t + γ u ¨ t + Δ t ) ,
β = 1 4 ( 1 + δ ) 2 , and γ = 1 2 + δ , where δ 0.

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 δ parameter according to Eq. (13), the integration accuracy and stability can be obtained by only one parameter. By changing this parameter, the amount of numerical damping also alters; hence the problematic oscillations are fully or partially extinguishable. This method is unconditionally stable and first order accurate (except by γ  = 0.5 which results second order accuracy). The main disadvantage is the lack of high-frequency damping which limits the reduction of spurious noises.

2.4 HHT- α method

The HHT-α method [16] is another scheme, which is widely employed for solving the equation of motion of mechanical systems. This method is the improved version of the Newmark method with controllable numerical dissipation through the α parameter. The main benefit in relation to the Newmark scheme is that by this method numerical damping can be exerted without degrading the order of accuracy,
M u ¨ t + Δ t + ( 1 + α ) C u ˙ t + Δ t α C u ˙ t + ( 1 + α ) K u t + Δ t α K u t = ( 1 + α ) f t + Δ t α f t .
The basic Equations (11) and (12) of the Newmark method remain valid for this scheme. However, the equation of motion (10) must be modified using the α parameter according to Eq. (14). In the interval of α ∈ [−0.5, 0], the amount of numerical damping can be controlled where the smaller value of α means the smaller amount of dissipation,
β = 1 4 ( 1 α ) 2 and γ = 1 2 ( 1 2 α ) .

If α is treated independently, then Eq. (13) parameters from the Newmark method remain unchanged, so the result of the numerical computation can be influenced by two separate parameters, δ and α . However, if β and γ are chosen according to Eq. (15), the scheme becomes second order accurate, although this does not cause the improvement of the numerical results necessarily. In this case the practical range α [ 1 / 3 , 0 ] . In practice, it is usually worthwhile to try both Eq. (15) and the independent choice of δ and α .

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).

Fig. 1.
Fig. 1.

The mechanical model of the examined 1D impact problem

Citation: Pollack Periodica 17, 3; 10.1556/606.2022.00599

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 L = 20 [ mm ] ; h = 0.1 [ mm ] ; n = 200 [ ] ; and v 0 = 1000 [ mm / s ] . The model contains uniform elements with E = 90 [ MPa ] and ρ = 7.85 · 10 9 [ kg / mm 3 ] material parameters. Based on the performed calculations, above the applied element number the results do not improve substantially.

Fig. 2.
Fig. 2.

FEM model

Citation: Pollack Periodica 17, 3; 10.1556/606.2022.00599

3.2 Treatment of the contact

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

As it has been mentioned earlier, by the Lagrange multiplier method, the penetration between the contacted bodies is completely prohibited. Because of this condition, certain displacement constraints are prescribed the following way:
G ( u + X ) = 0.
In the equation above, 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
M u ¨ t + C u ˙ t + K u t + G t T λ t = f t .

So, the equation of motion can be written in the form of Eq. (17) where λ t denotes the Lagrange multiplier at time instant t .

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

In the case of the Newmark and the HHT- α method, the quality of the numerical solution highly depends on the proper choice of the alterable parameter(s). Hence, a parameter identification is needed in order to test the applicability for a contact problem,
ε p = i = 1 n | p i e x a c t p i F E M | i = 1 n | p i e x a c t | .

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 p i e x a c t denotes the variable (pressure or velocity) determined by the exact solution, while p i F E M means the variable calculated from the FEM model at the i th time step. On the other hand, the amount of the spurious oscillations is reviewed by the number of wave peaks in the time evolution of the considered variable denoted as N o P p (Number of Peaks).

4.1 Newmark method

Based on these two indicators the practical range of δ parameter for a contact problem may be ascertained. After determining them for the contact pressure for multiple values of δ inside the interval δ [ 0 , 1 ] , these indicators can be plotted in the function of δ according to Fig. 3a and b, respectively. As it can be seen in these figures, N o P p is decreasing strictly monotonic, while ε p is starting to grow after a certain value of δ is stepped over. Hence, an ideal parameter value cannot be ascertained, by which both of the applied indicators are reaching their minimum, so an acceptable compromise must be made. Based on Fig. 3, a so called “practical stage” may be assigned, inside, which both measures remain at sufficiently low values. In accordance with the numerical tests using the FEM model, this practical stage can be determined as α [ 0.3 , 1 ] .

Fig. 3.
Fig. 3.

a) Relative error of the pressure for varying values of δ ; b) number of wave peaks in the pressure for varying values of δ

Citation: Pollack Periodica 17, 3; 10.1556/606.2022.00599

4.2 HHT- α method

The parameter identification for this method is more complicated, as there are two independent parameters affecting the numerical solution. The applied indicators are the same as by the Newmark method, but by reason of the one more ruling parameter, a surface plot must be applied (Fig. 4a and b). In order to establish a beneficial correlation between α and δ , the Period Error (PE) [19] function must be analyzed,
δ minimal P E = 1.5 α .
Fig. 4.
Fig. 4.

a) Relative error of the pressure for varying values of δ and α ; b) number of wave peaks in the pressure for varying values of δ  and  α

Citation: Pollack Periodica 17, 3; 10.1556/606.2022.00599

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 α [ 0.5 , 0.3 ] .

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 56). 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.

Fig. 5.
Fig. 5.

Time evolution of the contact pressure

Citation: Pollack Periodica 17, 3; 10.1556/606.2022.00599

Fig. 6.
Fig. 6.

Time evolution of the velocity at the contacting node

Citation: Pollack Periodica 17, 3; 10.1556/606.2022.00599

Fig. 7.
Fig. 7.

Comparison of the applied methods (pressure function)

Citation: Pollack Periodica 17, 3; 10.1556/606.2022.00599

Fig. 8.
Fig. 8.

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

Citation: Pollack Periodica 17, 3; 10.1556/606.2022.00599

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- α prove to have the highest potential at this kind of application. Both of them are controllable with one or more parameters, by which an acceptable compromise may be made. Even from these two methods, the HHT- α seems to possess more beneficial damping characteristics, as by identical dissipation it results a better solution. As the Newmark method is built in the ANSYS mechanical APDL software, the performed calculations were verified by that comparison. Applying the same FEM model with identical parameters in ANSYS, the results show a very precise match, proving that the calculations made in Julia language are accurate.

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- α method regarding the examined measures. The results of the application in a concrete 1D example also confirmed that this approach is correct. Nevertheless, a numerical method that is ideal to solve a contact problem in all aspects cannot be ascertained, as the right choice highly depends on the user's actual priorities. The main function of this paper is to provide a comprehensive guide for the correct selection of the numerical method for a concrete contact problem.

References

  • [1]

    H. Hertz , “On the contact of elastic solids(in German), J. Reine Angew. Math., vol. 92, pp. 156171, 1881.

  • [2]

    K. J. Bathe , Finite Element Procedures for Solids and Structures. MIT Open Courseware, MIT, 1982–1986.

  • [3]

    M. Fahad , R. Nagy , and P. Fuleki , “Creep model to determine rut development by autonomous truck axles on pavement,” Pollack Period., vol. 17, no. 1, pp. 6677, 2022.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • [4]

    P. Wriggers , Computational Contact Mechanics. John Wiley & Sons Ltd, 2002.

  • [5]

    T. A Laursen , Computational Contact and Impact Mechanics, Fundamentals of Modeling Interfacial Phenomena in Nonlinear Finite Element Analysis. Springer Science & Business Media, 2013.

    • Search Google Scholar
    • Export Citation
  • [6]

    K Willner and L Gaul , “A penalty approach for contact description by fem based on interface physics,” Trans. Eng. Sci., vol. 7, pp. 257264, 1970.

    • Search Google Scholar
    • Export Citation
  • [7]

    P. Papadopoulos and J. M. Solberg , “A Lagrange multiplier method for the finite element solution of frictionless contact problems,” Math. Comput. Model., vol. 28, nos 4–8, pp. 373384, 1998.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • [8]

    J. C. Simo and T. A. Laursen . “An augmented Lagrangian treatment of contact problems involving friction,” Comput. Struct., vol. 42, no. 1, pp. 97116, 1992.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • [9]

    T. Belytschko and T. J. R. Hughes , Eds, Computational Methods for Transient Analysis. North-Holand, 1983.

  • [10]

    B. Wu , H. Bao , J. Ou , and S. Tian , “Stability and accuracy analysis of the central difference method for real-time substructure testing,” Earthquake Eng. Struct. Dyn., vol. 34, no. 7, pp. 705718, 2005.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • [11]

    N. J. Carpenter , R. L. Taylor , and M. G. Katona , “Lagrange constraints for transient finite element surface contact,” Int. J. Numer. Methods Eng., vol. 32, no. 1, pp. 103128, 1991.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • [12]

    S. R. Wu , “Lumped mass matrix in explicit finite element method for transient dynamics of elasticity,” Comput. Methods Appl. Mech. Eng., vol. 195, nos 44–47, pp. 59835994, 2006.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • [13]

    B. Sumithra and A. Tamilselvan , “Numerical solution of stiff system by backward Euler method,” Appl. Math. Sci., vol. 9, no. 67, pp. 33033311, 2015.

    • Search Google Scholar
    • Export Citation
  • [14]

    N. M. Newmark , “A method of computation for structural dynamics,” J. Eng. Mech. Division, vol. 85, no. 3, pp. 6794, 1959.

  • [15]

    H. Maourane and T. Szabó , “Linear and nonlinear dynamical analysis of a Crane model,” Pollack Period., vol. 15, no. 2, pp. 8293, 2020.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • [16]

    H. M. Hilber , T. J. R. Hughes , and R. L. Taylor , “Improved numerical dissipation for time integration algorithms in structural dynamics,” Earthquake Eng. Struct. Dyn., vol. 5, no. 3, pp. 283292, 1977.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • [17]

    W. Kim , “A new family of two-stage explicit time integration methods with dissipation control capability for structural dynamics,” Eng. Struct., vol. 195, pp. 358372, 2019.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • [18]

    C. Kolay and J. M. Ricles , “Development of a family of unconditionally stable explicit direct integration algorithms with controllable numerical energy dissipation,” Earthquake Eng. Struct. Dyn., vol. 43, no. 9, pp. 13611380, 2014.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • [19]

    G. Noh , S. Ham , and K. J. Bathe , “Performance of an implicit time integration scheme in the analysis of wave propagations,” Comput. Struct., vol. 123, pp. 93105, 2013.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • [1]

    H. Hertz , “On the contact of elastic solids(in German), J. Reine Angew. Math., vol. 92, pp. 156171, 1881.

  • [2]

    K. J. Bathe , Finite Element Procedures for Solids and Structures. MIT Open Courseware, MIT, 1982–1986.

  • [3]

    M. Fahad , R. Nagy , and P. Fuleki , “Creep model to determine rut development by autonomous truck axles on pavement,” Pollack Period., vol. 17, no. 1, pp. 6677, 2022.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • [4]

    P. Wriggers , Computational Contact Mechanics. John Wiley & Sons Ltd, 2002.

  • [5]

    T. A Laursen , Computational Contact and Impact Mechanics, Fundamentals of Modeling Interfacial Phenomena in Nonlinear Finite Element Analysis. Springer Science & Business Media, 2013.

    • Search Google Scholar
    • Export Citation
  • [6]

    K Willner and L Gaul , “A penalty approach for contact description by fem based on interface physics,” Trans. Eng. Sci., vol. 7, pp. 257264, 1970.

    • Search Google Scholar
    • Export Citation
  • [7]

    P. Papadopoulos and J. M. Solberg , “A Lagrange multiplier method for the finite element solution of frictionless contact problems,” Math. Comput. Model., vol. 28, nos 4–8, pp. 373384, 1998.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • [8]

    J. C. Simo and T. A. Laursen . “An augmented Lagrangian treatment of contact problems involving friction,” Comput. Struct., vol. 42, no. 1, pp. 97116, 1992.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • [9]

    T. Belytschko and T. J. R. Hughes , Eds, Computational Methods for Transient Analysis. North-Holand, 1983.

  • [10]

    B. Wu , H. Bao , J. Ou , and S. Tian , “Stability and accuracy analysis of the central difference method for real-time substructure testing,” Earthquake Eng. Struct. Dyn., vol. 34, no. 7, pp. 705718, 2005.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • [11]

    N. J. Carpenter , R. L. Taylor , and M. G. Katona , “Lagrange constraints for transient finite element surface contact,” Int. J. Numer. Methods Eng., vol. 32, no. 1, pp. 103128, 1991.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • [12]

    S. R. Wu , “Lumped mass matrix in explicit finite element method for transient dynamics of elasticity,” Comput. Methods Appl. Mech. Eng., vol. 195, nos 44–47, pp. 59835994, 2006.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • [13]

    B. Sumithra and A. Tamilselvan , “Numerical solution of stiff system by backward Euler method,” Appl. Math. Sci., vol. 9, no. 67, pp. 33033311, 2015.

    • Search Google Scholar
    • Export Citation
  • [14]

    N. M. Newmark , “A method of computation for structural dynamics,” J. Eng. Mech. Division, vol. 85, no. 3, pp. 6794, 1959.

  • [15]

    H. Maourane and T. Szabó , “Linear and nonlinear dynamical analysis of a Crane model,” Pollack Period., vol. 15, no. 2, pp. 8293, 2020.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • [16]

    H. M. Hilber , T. J. R. Hughes , and R. L. Taylor , “Improved numerical dissipation for time integration algorithms in structural dynamics,” Earthquake Eng. Struct. Dyn., vol. 5, no. 3, pp. 283292, 1977.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • [17]

    W. Kim , “A new family of two-stage explicit time integration methods with dissipation control capability for structural dynamics,” Eng. Struct., vol. 195, pp. 358372, 2019.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • [18]

    C. Kolay and J. M. Ricles , “Development of a family of unconditionally stable explicit direct integration algorithms with controllable numerical energy dissipation,” Earthquake Eng. Struct. Dyn., vol. 43, no. 9, pp. 13611380, 2014.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • [19]

    G. Noh , S. Ham , and K. J. Bathe , “Performance of an implicit time integration scheme in the analysis of wave propagations,” Comput. Struct., vol. 123, pp. 93105, 2013.

    • Crossref
    • Search Google Scholar
    • Export Citation
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Editorial Board

  • Bálint Bachmann (Institute of Architecture, Faculty of Engineering and Information Technology, University of Pécs, Hungary)
  • Jeno Balogh (Department of Civil Engineering Technology, Metropolitan State University of Denver, Denver, Colorado, USA)
  • Radu Bancila (Department of Geotechnical Engineering and Terrestrial Communications Ways, Faculty of Civil Engineering and Architecture, “Politehnica” University Timisoara, Romania)
  • Charalambos C. Baniotopolous (Department of Civil Engineering, Chair of Sustainable Energy Systems, Director of Resilience Centre, School of Engineering, University of Birmingham, U.K.)
  • Oszkar Biro (Graz University of Technology, Institute of Fundamentals and Theory in Electrical Engineering, Austria)
  • Ágnes Borsos (Institute of Architecture, Department of Interior, Applied and Creative Design, Faculty of Engineering and Information Technology, University of Pécs, Hungary)
  • Matteo Bruggi (Dipartimento di Ingegneria Civile e Ambientale, Politecnico di Milano, Italy)
  • Petra Bujňáková (Department of Structures and Bridges, Faculty of Civil Engineering, University of Žilina, Slovakia)
  • Anikó Borbála Csébfalvi (Department of Civil Engineering, Institute of Smart Technology and Engineering, Faculty of Engineering and Information Technology, University of Pécs, Hungary)
  • Mirjana S. Devetaković (Faculty of Architecture, University of Belgrade, Serbia)
  • Szabolcs Fischer (Department of Transport Infrastructure and Water Resources Engineering, Faculty of Architerture, Civil Engineering and Transport Sciences Széchenyi István University, Győr, Hungary)
  • Radomir Folic (Department of Civil Engineering, Faculty of Technical Sciences, University of Novi Sad Serbia)
  • Jana Frankovská (Department of Geotechnics, Faculty of Civil Engineering, Slovak University of Technology in Bratislava, Slovakia)
  • János Gyergyák (Department of Architecture and Urban Planning, Institute of Architecture, Faculty of Engineering and Information Technology, University of Pécs, Hungary)
  • Kay Hameyer (Chair in Electromagnetic Energy Conversion, Institute of Electrical Machines, Faculty of Electrical Engineering and Information Technology, RWTH Aachen University, Germany)
  • Elena Helerea (Dept. of Electrical Engineering and Applied Physics, Faculty of Electrical Engineering and Computer Science, Transilvania University of Brasov, Romania)
  • Ákos Hutter (Department of Architecture and Urban Planning, Institute of Architecture, Faculty of Engineering and Information Technolgy, University of Pécs, Hungary)
  • Károly Jármai (Institute of Energy and Chemical Machinery, Faculty of Mechanical Engineering and Informatics, University of Miskolc, Hungary)
  • Teuta Jashari-Kajtazi (Department of Architecture, Faculty of Civil Engineering and Architecture, University of Prishtina, Kosovo)
  • Róbert Kersner (Department of Technical Informatics, Institute of Information and Electrical Technology, Faculty of Engineering and Information Technology, University of Pécs, Hungary)
  • Rita Kiss  (Biomechanical Cooperation Center, Faculty of Mechanical Engineering, Budapest University of Technology and Economics, Budapest, Hungary)
  • István Kistelegdi  (Department of Building Structures and Energy Design, Institute of Architecture, Faculty of Engineering and Information Technology, University of Pécs, Hungary)
  • Stanislav Kmeť (President of University Science Park TECHNICOM, Technical University of Kosice, Slovakia)
  • Imre Kocsis  (Department of Basic Engineering Research, Faculty of Engineering, University of Debrecen, Hungary)
  • László T. Kóczy (Department of Information Sciences, Faculty of Mechanical Engineering, Informatics and Electrical Engineering, University of Győr, Hungary)
  • Dražan Kozak (Faculty of Mechanical Engineering, Josip Juraj Strossmayer University of Osijek, Croatia)
  • György L. Kovács (Department of Technical Informatics, Institute of Information and Electrical Technology, Faculty of Engineering and Information Technology, University of Pécs, Hungary)
  • Balázs Géza Kövesdi (Department of Structural Engineering, Faculty of Civil Engineering, Budapest University of Engineering and Economics, Budapest, Hungary)
  • Tomáš Krejčí (Department of Mechanics, Faculty of Civil Engineering, Czech Technical University in Prague, Czech Republic)
  • Jaroslav Kruis (Department of Mechanics, Faculty of Civil Engineering, Czech Technical University in Prague, Czech Republic)
  • Miklós Kuczmann (Department of Automations, Faculty of Mechanical Engineering, Informatics and Electrical Engineering, Széchenyi István University, Győr, Hungary)
  • Tibor Kukai (Department of Engineering Studies, Institute of Smart Technology and Engineering, Faculty of Engineering and Information Technology, University of Pécs, Hungary)
  • Maria Jesus Lamela-Rey (Departamento de Construcción e Ingeniería de Fabricación, University of Oviedo, Spain)
  • János Lógó  (Department of Structural Mechanics, Faculty of Civil Engineering, Budapest University of Technology and Economics, Hungary)
  • Carmen Mihaela Lungoci (Faculty of Electrical Engineering and Computer Science, Universitatea Transilvania Brasov, Romania)
  • Frédéric Magoulés (Department of Mathematics and Informatics for Complex Systems, Centrale Supélec, Université Paris Saclay, France)
  • Gabriella Medvegy (Department of Interior, Applied and Creative Design, Institute of Architecture, Faculty of Engineering and Information Technology, University of Pécs, Hungary)
  • Tamás Molnár (Department of Visual Studies, Institute of Architecture, Faculty of Engineering and Information Technology, University of Pécs, Hungary)
  • Ferenc Orbán (Department of Mechanical Engineering, Institute of Smart Technology and Engineering, Faculty of Engineering and Information Technology, University of Pécs, Hungary)
  • Zoltán Orbán (Department of Civil Engineering, Institute of Smart Technology and Engineering, Faculty of Engineering and Information Technology, University of Pécs, Hungary)
  • Dmitrii Rachinskii (Department of Mathematical Sciences, The University of Texas at Dallas, Texas, USA)
  • Chro Radha (Chro Ali Hamaradha) (Sulaimani Polytechnic University, Technical College of Engineering, Department of City Planning, Kurdistan Region, Iraq)
  • Maurizio Repetto (Department of Energy “Galileo Ferraris”, Politecnico di Torino, Italy)
  • Zoltán Sári (Department of Technical Informatics, Institute of Information and Electrical Technology, Faculty of Engineering and Information Technology, University of Pécs, Hungary)
  • Grzegorz Sierpiński (Department of Transport Systems and Traffic Engineering, Faculty of Transport, Silesian University of Technology, Katowice, Poland)
  • Zoltán Siménfalvi (Institute of Energy and Chemical Machinery, Faculty of Mechanical Engineering and Informatics, University of Miskolc, Hungary)
  • Andrej Šoltész (Department of Hydrology, Faculty of Civil Engineering, Slovak University of Technology in Bratislava, Slovakia)
  • Zsolt Szabó (Faculty of Information Technology and Bionics, Pázmány Péter Catholic University, Hungary)
  • Mykola Sysyn (Chair of Planning and Design of Railway Infrastructure, Institute of Railway Systems and Public Transport, Technical University of Dresden, Germany)
  • András Timár (Faculty of Engineering and Information Technology, University of Pécs, Hungary)
  • Barry H. V. Topping (Heriot-Watt University, UK, Faculty of Engineering and Information Technology, University of Pécs, Hungary)

POLLACK PERIODICA
Pollack Mihály Faculty of Engineering
Institute: University of Pécs
Address: Boszorkány utca 2. H–7624 Pécs, Hungary
Phone/Fax: (36 72) 503 650

E-mail: peter.ivanyi@mik.pte.hu 

or amalia.ivanyi@mik.pte.hu

Indexing and Abstracting Services:

  • SCOPUS

 

2021  
Web of Science  
Total Cites
WoS
not indexed
Journal Impact Factor not indexed
Rank by Impact Factor

not indexed

Impact Factor
without
Journal Self Cites
not indexed
5 Year
Impact Factor
not indexed
Journal Citation Indicator not indexed
Rank by Journal Citation Indicator

not indexed

Scimago  
Scimago
H-index
12
Scimago
Journal Rank
0,26
Scimago Quartile Score Civil and Structural Engineering (Q3)
Materials Science (miscellaneous) (Q3)
Computer Science Applications (Q4)
Modeling and Simulation (Q4)
Software (Q4)
Scopus  
Scopus
Cite Score
1,5
Scopus
CIte Score Rank
Civil and Structural Engineering 232/326 (Q3)
Computer Science Applications 536/747 (Q3)
General Materials Science 329/455 (Q3)
Modeling and Simulation 228/303 (Q4)
Software 326/398 (Q4)
Scopus
SNIP
0,613

2020  
Scimago
H-index
11
Scimago
Journal Rank
0,257
Scimago
Quartile Score
Civil and Structural Engineering Q3
Computer Science Applications Q3
Materials Science (miscellaneous) Q3
Modeling and Simulation Q3
Software Q3
Scopus
Cite Score
340/243=1,4
Scopus
Cite Score Rank
Civil and Structural Engineering 219/318 (Q3)
Computer Science Applications 487/693 (Q3)
General Materials Science 316/455 (Q3)
Modeling and Simulation 217/290 (Q4)
Software 307/389 (Q4)
Scopus
SNIP
1,09
Scopus
Cites
321
Scopus
Documents
67
Days from submission to acceptance 136
Days from acceptance to publication 239
Acceptance
Rate
48%

 

2019  
Scimago
H-index
10
Scimago
Journal Rank
0,262
Scimago
Quartile Score
Civil and Structural Engineering Q3
Computer Science Applications Q3
Materials Science (miscellaneous) Q3
Modeling and Simulation Q3
Software Q3
Scopus
Cite Score
269/220=1,2
Scopus
Cite Score Rank
Civil and Structural Engineering 206/310 (Q3)
Computer Science Applications 445/636 (Q3)
General Materials Science 295/460 (Q3)
Modeling and Simulation 212/274 (Q4)
Software 304/373 (Q4)
Scopus
SNIP
0,933
Scopus
Cites
290
Scopus
Documents
68
Acceptance
Rate
67%

 

Pollack Periodica
Publication Model Hybrid
Submission Fee none
Article Processing Charge 900 EUR/article
Printed Color Illustrations 40 EUR (or 10 000 HUF) + VAT / piece
Regional discounts on country of the funding agency World Bank Lower-middle-income economies: 50%
World Bank Low-income economies: 100%
Further Discounts Editorial Board / Advisory Board members: 50%
Corresponding authors, affiliated to an EISZ member institution subscribing to the journal package of Akadémiai Kiadó: 100%
Subscription fee 2022 Online subsscription: 327 EUR / 411 USD 321
Print + online subscription: 393 EUR / 492 USD
Subscription fee 2023 Online subsscription: 336 EUR / 411 USD
Print + online subscription: 405 EUR / 492 USD
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Pollack Periodica
Language English
Size A4
Year of
Foundation
2006
Volumes
per Year
1
Issues
per Year
3
Founder Akadémiai Kiadó
Founder's
Address
H-1117 Budapest, Hungary 1516 Budapest, PO Box 245.
Publisher Akadémiai Kiadó
Publisher's
Address
H-1117 Budapest, Hungary 1516 Budapest, PO Box 245.
Responsible
Publisher
Chief Executive Officer, Akadémiai Kiadó
ISSN 1788-1994 (Print)
ISSN 1788-3911 (Online)

Monthly Content Usage

Abstract Views Full Text Views PDF Downloads
Apr 2022 0 0 0
May 2022 0 0 0
Jun 2022 0 0 0
Jul 2022 0 0 0
Aug 2022 0 88 50
Sep 2022 0 42 21
Oct 2022 0 0 0