The inequality levelset approach to handle contact: membrane case
 Matthieu Graveleau^{1},
 Nicolas Chevaugeon^{1}Email author and
 Nicolas Moës^{1}
DOI: 10.1186/s4032301500348
© Graveleau et al. 2015
Received: 28 November 2014
Accepted: 2 June 2015
Published: 15 July 2015
Abstract
Background
Contact mechanics involves models governed by inequality constraints. Even for the simplest contact problem, inequalities arise from the lack of information on the contact zone position. In addition to increasing the difficulty to solve such problems, an unknown contact zone makes it difficult to use an appropriate mesh and to represent efficiently phenomena on the contact zone boundary. Nevertheless these phenomena are often crucial to have an accurate representation of the problem such as weak discontinuity of the displacement.
Methods
In this paper, we propose a method specifically designed to solve inequality constraint problems linked to an unknown domain without remeshing. In order to do so, level sets coupled with XFEM is used to define the unknown domain and take into account the specific behavior at the contact zone boundary. The key idea of the method is to split the problem involving inequality constraints into two problems. In the first problem, the unknown domain is set and therefore it only involves equalities. Nevertheless, the constraints might be violated, meaning the set domain has to be changed. Then, the other problem is a shape optimization of this domain and leads to an updated set domain. These two problems are iterated up to convergence of the algorithm. Moreover, the addition of adhesion to the problem will be considered.
Results
The studied case in this paper is a membrane in the context of small deformations. First, a 1D example will be given to illustrate the method with and without adhesion. Then 2D cases will be studied. Finally an example with an evolving load will be given. Comparison will be made with a classical activeset method.
Conclusions
The ILS is proved to be an efficient method giving a convincing accuracy for the contact boundary without need of remeshing. It is also able to naturally handle adhesion.
Keywords
XFEM Levelset Contact ILS Adhesion MembraneBackground
Being able to predict how bodies in contact are going to behave is important. Indeed, such situations are omnipresent in daily life. Thus, contact mechanics has been intensively studied. Nowadays several methods are available to tackle contact problems. In this paper we will focus on the methods based on the finite element method [1] even if other options are available such as isogeometric methods, for which a review can be found in [2]. Furthermore, only contact with a rigid body will be studied, therefore avoiding, for now, the problem of nonmatching meshes. Methods have been designed to overcome this challenge. Among them, the mortar methods [3, 4] have shown great effectiveness. For our focus of study, the most common methods are the Lagrange multipliers method [5, 6], the penalty method [5, 6] and the augmented Lagrangian method [6, 7]. One of the main difficulty of contact is that the contact area is a priori unknown. This leads to the introduction of an inequality and consequently, computational challenges even in the simplest contact problem [5]. They usually require iterative algorithms and can be conditionally convergent. These difficulties are common to any inequality problems. Another issue to be addressed is the representation of the contact boundary area and the phenomena on it. In fact, the finite element method is strongly dependent on a predefined mesh. However, as the contact area is unknown, it is impossible to design a suitable mesh at first. In addition, low regularity of the solution are often present at the boundary of the contact area. The XFEM [8] is a definite asset in representing these discontinuities, leading to a higher order convergence rate with respect to element size. We propose in this paper a new approach for contact, the ILS, which is using XFEM and is designed to handle these problems.
In the first section of this paper we will state a general framework for the class of problem we are going to study. Then we will give a brief review of the most used methods. In the next section, we will set up the ILS framework applied to contact. We will explain how the variational inequality problem is transformed into an equality problem coupled with a shape optimization. We will also emphasize the particular tools required such as the notion of levelset and the XFEM. The addition of adhesion behavior to the method will be discussed. Finally we will give examples of our method applied to membrane problems. First, a 1D case will be presented to allow an easier understanding of the method. Then the 2D case will be studied in order to use the full extent of the method.
Description of the problem and classical methods
The problem
Our problem being parametrized, we can build the mathematical framework needed to find the equilibrium solution of this problem. Of course, this framework has already been settled and the reader can refer to [5] for more details. Indeed, we will only outline the main results needed here.
Problem 1
In Problem 1, all the equations have been generalized to the whole domain. However, if we split the domain into \(\Omega ^+\) and \(\Omega ^\) (being the actual contact zone), as described previously, Problem 1 is equivalent to:
Problem 2
Find u, \(\tilde{p}\) and \(\Omega ^\) such that:
In \(\Omega ^+\)  In \(\Omega ^\)  On \(\Gamma\)  

Equality  \(\begin{aligned}T\Delta u +f_d =0 \\ u=0 \;{on} \;\partial \Omega \end{aligned}\)  \(\begin{aligned} &T\Delta u +f_d+\tilde{p} =0\\ &u =d \end{aligned}\)  \(u^+=u^\) 
Inequality  \(u\le d\)  \(\tilde{p}\ge 0\) 
The separation of the two domains will be a key point for the method exposed in "Method: the ILS applied to contact problems".
Another problem, which is closer to what is solved in practice, is to find the equilibrium solution assuming a contact zone.
Problem 3
In \(\Omega ^+\)  In \(\Omega ^\)  On \(\Gamma\)  

Equality  \(\begin{aligned} T\Delta u +f_d =0 \\ u=0 \;on\; \partial \Omega \end{aligned}\)  \(\begin{aligned} &T\Delta u +f_d+\tilde{p} =0 \\ & u =d \end{aligned}\)  \(\begin{aligned} &T[\![\nabla u]\!]\cdot \underline{n}=\tilde{\lambda }\\& u^+=u^ \end{aligned}\) 
Let us show that if the contact conditions are fulfilled, i.e. the contact zone is the right one, we must satisfy \([\![\nabla u]\!]\cdot \underline{n}=0\). First, let us look at the case \([\![\nabla u]\!]\cdot \underline{n}> 0\), thus \(\tilde{\lambda }>0\). This means that the membrane is attracted by the rigid body. Without adhesion this is not a physical solution. Now, if \([\![\nabla u]\!]\cdot \underline{n}< 0\) and therefore \(\nabla u^+(s)\cdot \underline{n}<\nabla u^(s)\cdot \underline{n}\). As by definition \(\nabla d(s)\) is continuous and \(\nabla u^(s)=\nabla d(s)\) we must have \(\nabla u^+(s)\cdot \underline{n}<\nabla d(s)\cdot \underline{n}\) and therefore there is penetration of the rigid surface.
Problem 4
Problem 5
Some existing approaches
As we said, this problem is numerically difficult due to the inequality constraint and iterative algorithms are needed. The most common approach is to iterate on the set of degrees of freedom for which the constraint is active. Such methods are called activeset methods. Nonetheless, the contribution of the contact constraint have to be taken into account. The most spread methods in the industry are the penalty method and the Lagrange multipliers methods. An overview of these methods can be found in [5, 6]. Some methods combine both of them. This is the case of the perturbed Lagrange formulation [9] or the augmented Lagrange formulation [7]. Another approach is to use tools from mathematical programming like in [10].
The penalty method being the easier to implement is widely used. It also does not need to compute extra degrees of freedom as in the Lagrange multipliers method. Nevertheless, it relies on the choice of a penalty parameter. If the latter is too small, penetration is allowed whereas if it is too big, the problem might become illconditioned. Another spread method is the Lagrange multipliers method which leads to an exact fulfillment of the contact condition. However it does require additional degrees of freedom and a special care is to be taken for rigid body motion between two contacting bodies. To overcome these difficulties, an augmented Lagrange formulation is often coupled to an Uzawa algorithm. Combining both advantages of the penalty and Lagrange multiplier methods is more reliable but also more difficult to implement. Again, a comprehensive survey of these methods can be found in [6] or [11] for instance.
Even with the latter one, other difficulties arise. One of the most obvious one is that the representation of the contact zone is strongly linked with the discretization. As the contact zone is a priori unknown it is impossible to use an appropriate mesh at first. A solution to this problem is to move the nodes of the mesh to the approximated boundary of the contact zone [12]. A comparison of this method against more classical ones can be found in [13]. Obviously this method needs to update the mesh at each iteration. In the next section, we propose a new method to tackle contact especially designed to solve inequality constrained problems on a fixed mesh.
Method: the ILS applied to contact problems
The ILS method has been designed to be able to treat problems involving inequality constraints arising from an unknown constrained region. This means that if we can predict the constrained region there is no inequality anymore. Several problems fall into this category. Between them, a differentiation can be made. In the case of 3D solids the constrained domain can be a volume or a boundary of the domain (thus a surface). An example of the first case was treated in [14] where the ILS was first applied. It dealt with volumetric kinematic constraints inside a 3D domain. Contact problems are clearly part of the other category. Indeed if we know the contact area on the boundary we can prescribe the distance of the body to the rigid surface to be strictly zero and no more greater than zero. Nevertheless, even if the problem we are dealing with is also a contact problem we neglected the thickness of the membrane. Therefore the dimension of the contact zone is the same as the dimension of the membrane. This does not change the main idea of the ILS method but only local details. The Dirichlet boundary condition on \(\Omega ^\) can be enforced using Lagrange multipliers for instance. Therefore the challenge is to find this contact zone. Starting from an initial contact zone, a shape optimization is set to make the contact zone evolve toward one which allows all the contact conditions to be fulfilled. In short, the main idea of the method is to transform the inequality problem to a sequence of equality problems involving a shape optimization at each step.
The equilibrium problem
A straightforward choice for F would be a ridge function allowing a weak discontinuity of the displacement field. Then Lagrange multipliers have to be defined on \(\Gamma\) in order to represent the linear force needed to physically allow a slope discontinuity in the membrane. Nonetheless, an equivalent strategy is to set F to be an Heaviside function and to use the above mentioned Lagrange multipliers to cancel the jump of the displacement \([\![u]\!]\). A better numerical behavior of the Heaviside function, especially for high order finite element, motivates this choice.
Problem 6
The size of the matrix \(\mathbf {A}^{\!\!ee}\) is usually very small compare to the size of \(\mathbf {A}^{\!\!cc}\). This last matrix is not changing through the iteration, which is numerically convenient. Furthermore, from a broader point of view, when it will come to deal with plain solid both the size of \(\mathbf {B}\) and \(\mathbf {C}\) will be modest compare to \(\mathbf {A}^{\!\!cc}\). Therefore a great deal of computational effort can be spared by working on \(\mathbf {A}^{\!\!cc}\) only once.
Shape optimization
Here, the obvious choice for \(\varrho\) is \([\![\nabla u ]\!]\cdot \underline{n}\), see Eq. (4).
Then the zero of \(\varrho\) can be found by solving:
Problem 7
We will denote the normal directional derivative of a quantity a: \(D(a)[\underline{w}_{\!\Gamma }]=\mathring{a}\). In this particular case, \(\mathring{\varrho }= \mathring{\overline{[\![\nabla u ]\!]\cdot \underline{n}}}=[\![\nabla \mathring{u} ]\!]\cdot \underline{n}[\![\nabla u ]\!]\cdot \nabla \underline{w}_{\!\Gamma }\cdot \underline{n} +[\![\nabla u ]\!]\cdot \nabla \underline{n}\cdot \underline{w}_{\!\Gamma }\). To compute it, we are using the same trick as in [18] or [19]. We assimilate the change of shape to a body motion which is a classical problem with plenty of tools at our disposal.
Problem 8
It is important to note that Problem 8 has the same lefthand side as Problem 5. Using the discretization given in (11, 12, 13) Problem 8 allows the following discrete form:
Problem 9
Problem 7 can thereby be approximated using a Galerkin approximation:
Problem 10
Again, Problem 10 can be written as:
Problem 11
An easy adaptation to adhesion problems
It is interesting to note that the introduction of adhesive energy changes the solution from an inflection point to a minimum point.
Results
In this section we are going to study the case of a membrane with a homogeneous loading. Then, the full 2D problem will be dealt with.
The axisymmetric problem
With axisymmetric hypothesis
Let us illustrate what was described in "Method: the ILS applied to contact problems" with this simplified problem. In order to impose \(u = d\) on [0, rc] we use Lagrange multipliers p. Even if p is going to be defined on [0, R] we will be able to take into account their influence only on the contact zone. Indeed, taking advantage of the levelset framework we integrate only on the contact zone.
Problem 12
As we are dealing with a 1D problem we will use only one mode for \(w\) which correspond to a unit displacement of \(r_c\) in \(\underline{e_r}\) direction If we take the normal directional derivative of these equations we obtain the following sensitivity problem.
Problem 13
We compared this method to an active set method with nodal Lagrange multipliers (which will be referred as “classical”) in the case without adhesion. The set where the constraint is active (namely where the contact is imposed) is a set of \(N_c\) nodes. The coordinates of the \(k^{\text {th}}\) node in this set is noted \(\underline{x}_c^k\). In order to make this contact zone evolve, the penetration and the reaction of the support are checked as explained in the last section. Then, every node which violates the penetration is added in the active set whereas the ones violating the support reaction are removed. The same discretization of u is used without the enrichment. The problem for a given \(\Omega ^\) is then:
Problem 14
Without axisymmetric hypothesis
Time comparison between ILS and active set method. First on fixed meshes then for a targeted error
Mesh  ILS  Classical  

1/h  \(t^{\text {ILS}}_{\text {tot}}\) (s)  \(\overline{t^{\text {ILS}}_{\text {iter}}}\) (s)  \(t^{\text {class}}_{\text {tot}}\) (s)  \(\overline{t^{\text {class}}_{\text {iter}}}\) (s) 
2  0.4  0.08  0.06  0.02 
4  1.06  0.212  0.39  0.08 
8  2.53  0.506  1.06  0.151 
16  7.59  1.52  5.24  0.476 
32  36.4  7.28  48.7  2.43 
64  243.9  48.78  617.8  19.92 
128  2,062  412.4  20,971  349.5 
Error on \(\Gamma\)  ILS  Classical 

\(t^{\text {ILS}}_{\text {tot}}\) (s)  \(t^{\text {class}}_{\text {tot}}\) (s)  
5 × 10^{−2}  0.4  1.06 
5 × 10^{−3}  2.53  617.8 
5 × 10^{−4}  36.4  \(\times\) 
10^{−2}  2,062  \(\times\) 
Nonaxisymmetric problems
The method seems to be consistent as the results are similar for the loading and the unloading of the membrane. The differences come from the different initial levelset between the loading and the unloading. Indeed, using Fourier modes, their number is limited by the number of elements cut by \(\Gamma\) and therefore local errors (mainly arising when penetration or release checks are done) are difficult to overcome. In a future work localized modes will be developed to deal with this issue. Nevertheless, it is notable that few iterations are needed between each steps.
Conclusion

precisely the contact zone without mesh constraints,

the low regularity of the solution on the boundary of the contact zone.
This leads to a good quality solution without use of an excessively refined mesh. On top of it, the ILS takes advantage of configurational mechanics to make the contact zone evolve. It has been shown in this paper to be an efficient way to find the contact zone with and without adhesion. Numerical experiments did show that the convergence is fast.
In a future work, we will investigate the case of plain solids, particularly punch problems. Also the ILS framework offers a natural way to numerically analyze the bifurcation of systems involving contact and adhesion. An example of such system is given in [31].
Abbreviations
 ILS:

inequality LevelSet
 XFEM:

eXtended Finite Element Method
 BB condition:

Babuška–Brezzi condition
Declarations
Authors’ contribution
MG worked on the algorithms, performed the computations and drafted the manuscript. NC worked on the algorithms, performed the computations and carried out detailed revision. NM worked on the algorithms and carried out detailed revision. All authors read and approved the final manuscript.
Acknowledgements
The support of the ERC Advanced Grant XLS no 291102 is greatfully acknowledged. We would also like to thank Peter Wriggers for his advices and Anthony Nouy for his help on the mathematical formulation.
Compliance with ethical guidelines
Competing interests The authors declare that they have no competing interests.
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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