 Research article
 Open Access
Explicit Verlet timeintegration for a Nitschebased approximation of elastodynamic contact problems
 Franz Chouly^{1} and
 Yves Renard^{2}Email authorView ORCID ID profile
https://doi.org/10.1186/s4032301801245
© The Author(s) 2018
 Received: 15 June 2018
 Accepted: 11 December 2018
 Published: 21 December 2018
Abstract
The aim of the present paper is to study theoretically and numerically the Verlet scheme for the explicit timeintegration of elastodynamic problems with a contact condition approximated by Nitsche’s method. This is a continuation of papers (Chouly et al. ESAIM Math Model Numer Anal 49(2), 481–502, 2015; Chouly et al. ESAIM Math Model Numer Anal 49(2), 503–528, 2015) where some implicit schemes (thetascheme, Newmark and a new hybrid scheme) were proposed and proved to be wellposed and stable under appropriate conditions. A theoretical study of stability is carried out and then illustrated with both numerical experiments and numerical comparison to other existing discretizations of contact problems.
Keywords
 Unilateral contact
 Elastodynamics
 Nitsche’s method
 Explicit timemarching schemes
 Stability
 Explicit dynamics
Introduction and problem setting
Explicit timemarching schemes for the dynamics of deformable solids with impact has already been the subject of an abundant literature (see, e.g., [1–3] for some recent contributions). They are appealing since they can be easy to implement, fast and adapted to parallel architectures. Nevertheless, there still remains important difficulties to design robust explicit methods and to obtain reliable numerical simulations in this context (see, e.g., [4]). Among these difficulties, numerical stability and energy conservation remains one of the most important ones. Another one is to preserve the quality and the accuracy of the numerical solution, which can present spurious oscillations in the displacement, the velocity or the contact stress. A last one is to enforce properly the contact condition, particularly the nonpenetration condition.
A precursory method is the one developed by Taylor and Flanagan [5] in the framework of PRONTO3D software (see also the description in [6]). Nevertheless, the method is not fully explicit, except in a nodetonode contact approximation, in the sense that the contact pressure is computed in an iterative process on the whole contact surface. To mention some other of the most important contributions, we can say that a widely resumed theoretical work in dynamic impact problems is due to Moreau [7, 8] for the impact of rigid body systems. The (implicit) schemes proposed by Moreau have been extended quite naturally to the elasticity case through finite element semidiscretization in space (for instance in [9]) which transforms the continuous impact problem into a discrete one very close to a rigid body system. These discrete impact problems, governed by a socalled measure differential inclusion are notoriously illposed and of very low regularity.
The illposedness can be fixed (for the most part) by the addition of an impact law with a restitution coefficient. As a matter of fact, standard schemes, such as the commonly used ones of Newmark’s family [10], have an erratic behavior when they are applied to dynamic contact problems. This is mainly because they select a solution corresponding to an arbitrary (and potentially very large) restitution coefficient (see [11]). Alternatively, a valuable scheme in this context is that of Paoli and Schatzman [12, 13] who implicitly takes into account this restitution coefficient. However, the addition of a restitution coefficient can be considered as artificial in the context of deformable solids. This does not diminish the interest for the Paoli–Schatzman scheme which will be a point of comparison with our proposed approach. The implicit inclusion of a restitution coefficient has also been considered in [14] to develop a wide range of schemes based on a time discontinuous Galerkin framework.
As noticed in [11], even in the case where the continuous problem is wellposed (see, e.g., [15, 16] for wellposedness results), the illposed measure differential inclusion that results from finite element semidiscretization in space has an infinite number of solutions, depending on the choice of a restitution coefficient on each node of the contact boundary. Moreover, it is not possible to decide which solution is more suitable than other. Indeed, the two most remarkable solutions are, on the one hand, the one for a unitary restitution coefficient which ensures conserving energy but which causes very important spurious oscillations of the contact nodes and unexploitable contact stress, and, on the other hand, the solution for a vanishing restitution coefficient which ensures stability and a better approximation of the contact stress but is energy dissipative, while the continuous problem is not. This resulted in [11] to design the mass redistribution method (generalized in [17, 18]) which allows a compromise in this context, i.e. wellposedness of the space semidiscretized problem, conservation of the energy and an improved quality of the contact stress. However, and this is also the case for the Paoli–Schatzman scheme, it introduces a global problem to be solved (at least on the contact nodes) when an explicit timemarching scheme is used. In the same spirit, a timemarching scheme has been designed in [19] for dynamic fracture problems, in which the cohesive forces are treated implicitly, while an explicit scheme is used for the dynamics of interior nodes.
For explicit timeintegration, primal formulations of contact conditions are better suited. Indeed, since no additional unknown such as a Lagrange multiplier are introduced, they allow to enforce the contact conditions at the previous timestep, instead of the current one, so that the contact term appears at the righthand side and does not require global (and nonlinear) solving. A first possibility is to penalize / regularize the contact conditions (see, e.g., [20, 21]): the resulting penalty method is simple to implement and only an inversion of the massmatrix is needed at each timestep to solve the resulting fully discretized problem (and the scheme becomes fully explicit when the mass matrix is lumped). Nevertheless, the penalty method is not consistent and the choice of the penalty parameter remains a difficulty (see, e.g., [22]). The alternative we explore in this paper is a Nitsche treatment of contact conditions, which is still a primal method, with the same advantages as penalty, but that remains consistent with the original problem, and more robust with respect to the Nitsche parameter. Nitsche’s method, originally designed to enforce weakly Dirichlet boundary conditions [23, 24], was adapted to unilateral contact in [25, 26] (see also [27] for an overview of recent results on this topic).
We studied previously in [28, 29] the behavior of Nitsche’s method for contact in elastodynamics, when combined to various implicit timemarching schemes. Particularly, when applied to contactimpact in elastodynamics, Nitsche’s method has the good property of leading to a wellposed semidiscrete problem in time (i.e., a system of Lipschitz differential equations) as it is shown in [28]. This feature is shared also by the penalty method and modified mass methods. Moreover the symmetric variant of Nitsche’s space semidiscretization conserves an augmented energy [28], as does the penalty method [30]. We studied as well theoretically the wellposedness, the stability and energy conservation properties of fully discrete schemes based on space semidiscretization with Nitsche’s method combined with the thetascheme, the Newmark scheme and a new Hybrid scheme. This study was illustrated with some numerical experiments.
The aim of this paper is to study mathematically and numerically the approximation of contact problems in elastodynamics by Nitsche’s method combined with the explicit Verlet timemarching scheme. The choice of the Verlet scheme is motivated both by its simplicity and its attractive theoretical properties (symplecticity) [31]. We will also make comparisons with some of the existing methods mentioned above and with the approximation by penalized contact. The numerical comparison will be mainly performed on the onedimensional problem introduced in [15] whose advantage is to present a known periodic solution and to make clear the occurrence of parasitic oscillations, the convergence and energy conservation properties. Comparisons for 2D and 3D problems will also be presented.
Let us introduce some useful notations. In what follows, bold letters like \(\mathbf{u },\mathbf{v }\), indicate vector or tensor valued quantities, while the capital ones (e.g., \(\mathbf{V },\mathbf{K }\ldots \)) represent functional sets involving vector fields. As usual, we denote by \((H^{s}(.))^d\), \(s\in \mathbb {R}, d=1,2,3\) the Sobolev spaces in one, two or three space dimensions (see [32]). The usual scalar product of \((H^{s}(D))^d\) is denoted by \((\cdot ,\cdot )_{s,D}\) and the corresponding norm is denoted by \(\Vert \cdot \Vert _{s,D}\)—we keep the same notation when \(d=1\) or \(d>1\). The letter C stands for a generic constant, independent of the discretization parameters.
We consider an elastic body \(\Omega \) in \(\mathbb {R}^d\) with \(d=1,2,3\). Small strain assumptions are made (as well as plane strain when \(d=2\)). The boundary \(\partial \Omega \) of \(\Omega \) is polygonal (\(d=2\)) or polyhedral (\(d=3\)). The normal unit outward vector on \(\partial \Omega \) is denoted \(\mathbf{n }\). We suppose that \(\partial \Omega \) consists in three nonoverlapping parts \(\Gamma _D\), \(\Gamma _N\) and the contact boundary \(\Gamma _C\), with meas\((\Gamma _D) > 0\) and meas\((\Gamma _C) > 0\). In its initial stage, the body is in contact on \(\Gamma _C\) with a rigid foundation and we suppose that the unknown contact zone during deformation is included into \(\Gamma _C\). The body is clamped on \(\Gamma _D\) for the sake of simplicity. It is subjected to volume forces \(\mathbf{f }\) in \(\Omega \) and to surface loads \(\mathbf{g }\) on \(\Gamma _N\).
To our knowledge, the wellposedness of Problems (1), (2) is still an open issue. The few available existence results concern simplified model problems involving the (scalar) wave equation with Signorini’s conditions (see, e.g., [16, 33–36]) or thin structures like membranes, beams (see [37]) or plates (see [38]). Even in these simplified cases, obtaining uniqueness and energy conservation still involves difficulties in 2D or 3D. For a review on some of these results, one can refer to the book [39].
Note that, even if it is expected that solutions to Problems (1), (2) satisfy the persistency Condition \(\sigma _n(\mathbf{u }(t)) \dot{u}_n (t) = 0\) in order to respect the nondissipative character of the frictionless contact condition, it has only been rigorously proved in a one dimensional framework (elastic bar) for instance in [36, Lemma 2.5].
The rest of our paper is outlined as follows. The first section is dedicated to the description of the fully discrete formulation for dynamic contact with Nitsche and Verlet explicit timeintegration. Then, a stability analysis is carried out, and finally, some numerical comparisons with other classical methods are investigated and analysed.
Discrete setting: Nitsche’s method with Verlet scheme
We begin this section with preliminary notations and results. Then, we introduce our Nitschebased finite element semidiscretization in space, and we recall its main properties of wellposedness and energy conservation. Finally we describe the fully discretized problem based on the Verlet explicit timemarching scheme.
Preliminary notations and results

Regular, i.e., there exists \(\sigma >0\) such that \(\forall K \in \mathcal{T}^h, h_K / \rho _K \le \sigma \) where \(\rho _K\) denotes the radius of the inscribed ball in K,

Conformal to the subdivision of the boundary into \(\Gamma _D\), \(\Gamma _N\) and \(\Gamma _C\), which means that a face of an element \(K \in \mathcal{T}^h\) is not allowed to have simultaneous nonempty intersection with more than one part of the subdivision,

Quasiuniform, i.e., there exists \(c>0\), such that, \(\forall h>0,~\forall K\in \mathcal{T}^h, ~h_K \ge c h\).
We next define convenient meshdependent norms, in fact weighted \(L^2(\Gamma _C)\)norm (since \((\gamma _0 / {\gamma _h})_K = h_K\)).
Definition 1
Additionally, it will be convenient to endow \(\mathbf{V }^h\) with the following mesh and parameterdependent scalar product:
Definition 2
We end this section with the following statement: a discrete trace inequality (see, e.g., [43]), that is a key ingredient for the whole mathematical analysis of Nitsche’s based methods.
Lemma 3
Semidiscrete problem in space
Our Nitschebased discretization of the contact condition comes from the following result (see [44] and as well [25] for a detailed formal proof).
Proposition 4
Remark 5
Note that, as in [27], we adopted in this presentation a different convention for notations compared to previous works [28, 29]. This is in order to get closer to the formulations provided in most of the papers on Nitsche’s method and on the augmented Lagrangian method.
Theorem 6
Remark 7
Note that, conversely to the static case (see [25, 26, 45]) and the fullydiscrete case there is no condition on \(\gamma _0\) for the space (semi)discretization, which remains wellposed even if \(\gamma _0\) is arbitrarily small. However, this does not imply that the solution remains consistent when \(\gamma _0\) becomes small (see Remark 19 and Fig. 4 in the sequel).
We recall that the standard (mixed) finite element semidiscretization for elastodynamics with unilateral contact leads to illposed problems (see, e.g., [11, 22]), which is not the case of Nitsche’s formulation that leads to a wellposed (Lipschitz) system of differential equations. This feature is shared with the standard penalty method, the difference being that Nitsche’s method remains consistent in a strong sense (see [28]). Note that the standard (mixed) finite element semidiscretization is consistent as well as the singular dynamic method introduced in [18]. The mass redistribution method introduced in [11] is asymptotically consistent when h vanishes.
Proposition 8
Proof
This result is obtained using the coercivity of \(a(\cdot ,\cdot )\) and applying Lemma 3. \(\square \)
Remark 9
Proposition 8 states that the energy \(E^h_{\Theta }(t)\) remains always positive (if \(E^h(0)\) is) for \(\Theta \ge 0\) and \(\gamma _0\) large enough. For \(\gamma _0\) small, the existence of zero energy spurious modes cannot be excluded.
Remark 10
For \(\Theta < 0\), such a result with a constant independent of the mesh parameter h cannot be obtained. As a consequence, for \(\Theta < 0\), the quantity \(E^{h}_{\Theta } (t)\) cannot be used for optimal energy evolution estimates and might become even negative for h small.
Still in [28], the following evolution of \(E^h_{\Theta }\) is obtained:
Theorem 11
This result links the nonsatisfaction of the energy conservation to the nonsatisfaction of the socalled persistency condition. However, it appears in the present study that it would be preferable to use \(E^h_{1} (t)\) even for the variants \(\Theta \ne 1\) (see Remark 10), for which the following result can be established:
Theorem 12
Proof
Remark 13
The above result still states that \(E_1^h(t)\) is conserved for the symmetric variant \(\Theta =1\), and, for \(\Theta \ne 1\) the variations of \(E_1^h(t)\) come from the nonfulfillment of the contact Condition (7) by \(\mathbf{u }^h\).
Verlet scheme
Stability properties of Verlet scheme
First, we present different energies associated to the solution to Problem (13), and make explicit their relationships. Then, we derive energy estimates associated to the fully discrete Problem (13), and a (nonoptimal) stability result is deduced. We conclude with some comments and arguments that a better result may be expected.
Discrete energies
Proposition 14
Proposition 15
Proof
Energy evolution estimates
First, the straightforward adaptation of [29, Proposition 3.4], taking \(\gamma =\frac{1}{2}\) and \(\beta =0\) for Verlet scheme gives the following energy identity:
Proposition 16
This result can be easily adapted as follows when the energy \(E^{h,n}_{1}\) is considered instead, even for \(\Theta \ne 1\):
Proposition 17
Proof
As an interesting consequence, we obtain the following result for the discrete energy \(E^{h,\tau , n}\) by simplifying the previous one:
Proposition 18
Proof
Remark 19
For \(\gamma _0\) small, the property of Proposition 15 can be lost and the energy \(E^{h,n}_{1}\) may become negative. In that case, some deformation corresponding to a negative energy may exist, which is of course a nonphysical situation. This highlights that, even thought the semidiscrete problem (9) has a unique solution for \(\gamma _0\) small, the reliability of the discretization is guaranteed only for \(\gamma _0\) large enough.
Remark 20
Still referring to [29, Proposition 3.4], and instead of Verlet scheme, if we consider the explicit Newmark scheme \(\gamma =1\) and \(\beta =0\) and \(\Theta = 1\) as Nitsche’s variant, the pending energy evolution corresponding to Proposition 18 in that case involves the sole term \(\left[ \mathrm {P}^{n+1} \right] _{_{\mathbb {R}^}} \left[ \mathrm {P}^n \right] _{_{\mathbb {R}^+}}\) (instead of \(\left[ \mathrm {P}^{n+1} \right] _{_{\mathbb {R}^}} \left[ \mathrm {P}^n \right] _{_{\mathbb {R}^+}} \left[ \mathrm {P}^n \right] _{_{\mathbb {R}^}} \left[ \mathrm {P}^{n+1} \right] _{_{\mathbb {R}^+}}\) for Verlet scheme). This term being nonpositive, the stability of this explicit scheme can be established thanks to Proposition 15 for \(\frac{\tau }{h}\) small enough.
Stability analysis in the case \(\Theta = 1\)
The main result of this section is the following (nonoptimal) stability result for the Scheme (13) in the case \(\Theta =1\):
Proposition 21
Proof
Comments on the stability analysis
Lemma 22
Proof
Proposition 23
Proof
Remark 24
For a linear problem, we would conclude that the scheme is stable, under the Condition \(\frac{\tau }{h}\) small enough. However, in a nonlinear framework, the conclusion cannot be drawn since the matrix \(\mathbf{C }^{h,n}\) changes from an iteration to another. Moreover, it seems difficult to pursue the reasoning made on two iterations to an arbitrary number of iterations.
Numerical experiments
We first carry out numerical experiments in 1D, where we can compare our results with an exact solution. Then, numerical experiments in 2D/3D will be described. These numerical tests are performed with the help of our freely available finite element library GetFEM++ (see http://getfem.org).
1D numerical experiments: multiple impacts of an elastic bar
We first present the setting, and then the results obtained by combination of Nitsche’s contact discretization and Verlet scheme. These results are also compared with computations using other methods: the Paoli–Schatzman scheme, the Taylor–Flanagan scheme, the mass redistribution method and the penalty method. This section is ended with numerical convergence tests.
Setting
 1.
The displacement u at the contact point \(\Gamma _C\), given at time \(t^n\) by \(u^{h,n} (0) (= U^n_0)\).
 2.The contact pressure \(\sigma _C\), which, in the discrete case, is different from \(\sigma (u)\). If a standard (mixed) method is used for the treatment of contact, it is directly given by the Lagrange multiplier, i.e., \( \sigma _C^n := \lambda ^{h,n}\) at time \(t^n\). In the case of the Nitschebased formulation, it can be computed as follows at time \(t^n\):which comes from the contact Condition (7).$$\begin{aligned} \sigma _C^n := \left[ \sigma _n (u^{h,n}) (0)  {\gamma _h}(u^{h,n} (0)) \right] _{_{\mathbb {R}^}} = \left[ \frac{E}{h} (U_1^n  U_0^n) + \frac{\gamma _0}{h} U^n_0 \right] _{_{\mathbb {R}^}}, \end{aligned}$$
 3.The discrete energy \(E^h\) which is at time \(t^n\)and the discrete augmented energy \(E^h_{1}\):$$\begin{aligned} E^{h,n} = \frac{1}{2} \left( (\dot{\mathbf{U }}^n)^{\mathrm {T}} \mathbf{M }{\dot{\mathbf{U }}^n} + (\mathbf{U }^n)^{\mathrm {T}} \mathbf{K }\mathbf{U }^n \right) , \end{aligned}$$$$\begin{aligned} E^{h,n}_{1} = E^{h,n}  R^{h,n}, ~~R^{h,n} = \frac{h}{2 \gamma _0} \left( (\sigma _n (u^{h,n}) (0))^2  (\sigma ^n_C)^2 \right) . \end{aligned}$$
Numerical results for Nitsche’s method
We discretize the bar with \(M=20\) linear finite elements (\(k=1\), \(h=0.05\)) and take \(\tau =0.01\). The resulting Courant number is \(\nu _C = 0.2\). Note that almost all the parameters have been taken identical as in [29] for comparison purposes. The number of element is smaller (\(M=20\) instead of 100 in [29]) and the timestep \(\tau \) is 0.01 for stability reasons. We first investigate the variant \(\Theta = 0\) with a parameter \(\gamma _0 = 1\). The mass matrix is computed in a standard fashion. The choice \(\gamma _0\) equal to 1 is guided by the concern to obtain a stiffness associated with the degree of freedom on the contact boundary comparable to the stiffnesses obtained by the finite element discretization inside the bar.
The calculation for the variant \(\Theta = 1\) and for Nitsche’s parameter \(\gamma _0\) still equal to 1 is presented on Fig. 3. It can be seen that the nonpenetration condition is slightly better respected, which indicates that the additional terms compared to the variant \(\Theta = 0\) reinforce the consistency of the method. However, this is at the price of stronger oscillations on the velocity at the contact point. The approximation of the contact stress remains comparable to the \(\Theta = 0\) variant, as well as the energy evolution.
Finally, Figs. 7 and 8 show the evolution of the solution for decreasing discretization parameters, and for the variant \(\Theta =1\), \(\gamma _0=2\), and the standard mass matrix. We note in Fig. 7 a rapid decrease of the oscillations on the displacement with the refinement of the discretization. Conversely, the convergence of the contact stress as depicted in Fig. 8 is rather slow, as it could be expected from the very low regularity of the solution. Indeed we observe a very gradual decrease in the amplitude of the oscillations.
Comparison with Paoli–Schatzman scheme
The numerical tests for \(h=0.05\) and \(\tau =0.01\) are presented in Figs. 9, 10 and 11 for a restitution coefficient equal to 0, 1 / 2 and 1, respectively. The results for \(e=0\) and \(e=1/2\) are very similar to each other despite the difference between the restitution coefficients, and we observe a very similar loss of energy for each impact. The approximation of the displacement and of the nonpenetration condition are quite good. The results for \(e=1\) show an excessive bounce of the contact point which leads to very noisy contact point velocity and contact stress.
Comparison with Taylor–Flanagan scheme
Since the Taylor–Flanagan scheme prescribes the contact condition with an implicited Lagrange multiplier and enforces the persistency condition, it is very close to the Paoli–Schatzman scheme with a restitution coefficient \(e=0\) even if the contact condition is prescribed in a slightly different way. The consequence is that the results of the simulations shown on Fig. 12 for the Taylor–Flanagan scheme are almost identical to the results shown on Fig. 9 for the Paoli–Schatzman scheme with \(e=0\). Particularly, a loss of energy occurs at each impact.
Comparison with the mass redistribution method
Since the mass matrix admits a kernel containing the vectors being only nonzero on the contact boundary, the system (32), (33) consists in an algebraic variational inequality when reduced on this kernel. Due to the Lipschitz continuity, with respect to the data, of the solution to this variational inequality, Problems (32), (33) reduces to a system of ordinary differential equations on the orthogonal of the kernel. This property, detailed in [11] allows to use quite arbitrary timemarching schemes to approximate (32), (33), among others the Verlet scheme. Of course, the method is not strictly an explicit one since a global solving has to be done on the kernel of the modified mass matrix. However, in the onedimensional test case, this kernel is onedimensional which allows an explicit solving.
The corresponding simulations can be seen on Fig. 13. One characteristic of the mass redistribution method is to produce low oscillating velocity and contact stress compared to other discretizations. One can see that the energy is conserved, but slightly modified compared to the standard energy.
Comparison with the penalty method
It is worth comparing Fig. 15 to Figs. 2, 3 and 4 for Nitsche’s method and the same value of the parameter \(\gamma _0\). The nonpenetration condition is better satisfied with Nitsche’s method, which highlights its consistency. However, energy conservation is better preserved by the penalty method except when the variant \(\Theta =1\) of Nitsche’s method is used.
Numerical convergence
A comparison of Figs. 17 and 18 leads to the conclusion that despite the very low regularity of the exact solution (velocity and stress are discontinuous), there is a substantial gain in using quadratic elements. It even improves the convergence rate for the \(L^2(0,T,L^2(\Omega ))\)norm of the error of the displacement. Globally, the mass redistribution with quadratic elements provides the best compromise. However, as the Paoli–Schatzman scheme, it necessitates to solve a global problem on the contact surface.
2D/3D numerical experiments: multiple impacts of a disc / a sphere
Numerical experiments are then carried out in 2D and 3D, to assess the behavior of Nitsche’s method in a more realistic situation. We study the impact of a disc and a sphere on a rigid support. The physical parameters are the following: the diameter of the disc is \(D=40\), the Lamé coefficients are \(\lambda =30\) and \(\mu =30\), and the material density is \(\rho =1\). The total simulation time is \(T=120\).
For space semidiscretization, Lagrange isoparametric finite elements of order \(k=2\) have been used. The mesh size is \(h=4\) for the ball and \(h=8\) for the sphere (see Fig. 19). Integrals of the nonlinear term on \(\Gamma _C\) are computed with standard quadrature formulas of order 4. A snapshot of the evolution of the disc and the sphere during the first impact can be seen on Figs. 20 and 21.
The comparison of the simulations for the different methods is depicted Fig. 22 for the twodimensional case and Fig. 23 for the threedimensional case. For the sake of shortness, only the penalty, the singular mass and Nitsche methods are compared. First of all, a conclusion that can be drawn from these numerical experiments is that the tested methods are all capable of reliably approximating two and threedimensional dynamic contact problems. An important difference between simulations in dimension 2 and 3 is a much smaller oscillation of the contact stress in dimension 3, except for the mass redistribution method which is not subjected to spurious oscillations. The energy is conserved more strictly with the penalty method, the mass redistribution method and the variant \(\Theta = 1\) of Nitsche’s method, the other two variants presenting significant disturbances in the energy evolution. The mass redistribution method appears to give the best compromise between energy conservation and the low level of oscillation on the contact boundary. Note however that it produces a weakening of the rebound, mainly in dimension 3, which we do not explain. The lake of consistency of the penalty method is illustrated on the normal displacement graph where we can note a larger interpenetration compared to the other methods. Finally, among the variants of Nitsche’s method, the symmetric variant \(\Theta = 1\) is the one that achieves the best compromise between energy conservation and the level of oscillations of the contact stress, which remains moderate.
Concluding remarks
In this paper, we studied the application of an explicit Verlet scheme for the approximation of elastodynamic contact problems with Nitsche’s method. The explicit method being commonly used in elastodynamic contact problems, it seemed important to complete the study that had been performed in [28, 29] for implicit schemes. We tried to characterize the stability properties of the different variants of Nitsche’s method for the Verlet scheme and we introduced a number of necessary tools for this analysis. Of course, we are aware that the stability result we establish is very partial (only for \(\Theta = 1\)) and certainly suboptimal: a stability condition such as \(\tau = \mathcal {O}(h)\) would be more satisfactory and would correspond to what we noted in numerical tests. This result remains to be refined. Moreover, it would certainly be possible to prove a convergence result in dimension one, as in [15], because in this context the existence and uniqueness of the solution is theoretically proven.
We numerically compared the Nitsche method to the main existing methods that can support an explicit scheme: the Paoli–Schatzman scheme, the Taylor–Flanagan scheme, the mass redistribution method and the penalty method.
We first performed this comparison on a onedimensional test case whose exact solution consists of a shock wave indefinitely travelling between the two ends of a bar. We can see globally, by comparing the Figs. 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 and 16 that Nitsche’s method, especially the variant corresponding to \(\Theta = 1\), yields an approximation of a comparable quality as the one obtained with the schemes using an implicitation of the contact force (Paoli–Schatzman scheme, Taylor–Flanagan scheme, mass redistribution). Only the mass redistribution method results in lower oscillation levels. Compared to the penalty method, the oscillation levels are of the same magnitude, but contact penetration is more limited. This is reflected in Figs. 17 and 18 by quite similar convergence rates for all the methods, except for penalty. For this latter, a compromise remains difficult to find between a large penalization coefficient, which corresponds to a good approximation of the displacement but a poor approximation of the contact stress, and a small penalization coefficient, for which the interpenetration becomes too large. The decisive advantage of Nitsche’s method over the other methods, with the exception of the penalty method, is to be a primal method for which there is no need for an implicit resolution of the contact force. This allows a really explicit resolution in case of lumped mass matrix. The 2D and 3D test cases we performed also confirm the good behavior of Nitsche’s method. We can see in Figs. 22 and 23 the advantage in comparison to the penalty method in term of interpenetration, which is less. Still some better approximation results are obtained for the variant \(\Theta = 1\).
We can thus conclude that, among the variants of Nitsche’s method, the symmetric variant \(\Theta =1\) seems to be the most suitable for solving dynamic contact problems mainly because of its energy conservation properties. For the other variants a gain of energy can be observed, especially for low values of Nitsche’s parameter \(\gamma _0\). Some perspectives of this work could be to gain further insight into the properties of energy conservation, for instance using other definitions of the discrete energies, such as in [57, Theorem 4.1, Remark 4.1] in which an energy that remains positive irrespectively of the value of numerical parameters is introduced. Also some new explicit timemarching schemes endowed with appealing properties of energy conservation could be considered (see, e.g., [58]). Moreover, further study of the effect of the mass matrix lumping, particularly on the stability of the method, and of the proper choice of the Nitsche’s parameter \(\gamma _0\) are other perspectives of this work.
Declarations
Authors' contributions
The two authors collaborated at each stage of this study: obtaining mathematical results, developing test codes, analyzing numerical results, drafting the manuscript and finalizing it. Both authors read and approved the final manuscript.
Acknowledgements
We thank both referees for their constructive comments that helped to improve the presentation of the results. We thank Patrick Letallec for the rewarding discussions on existing explicit strategies in contact dynamics.
Competing interests
The authors declare that they have no competing interests.
Availability of data and materials
The datasets used and analysed during the current study are available from the corresponding author on reasonable request.
Funding
Not applicable.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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