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In contact mechanics, a unilateral contact, also called unilateral constraint, denotes a mechanical constraint which prevents penetration between two rigid/flexible bodies. This kind of constraints is omnipresent in non-smooth multibody dynamics applications, such as

• granular flows;
• Legged robot;
• vehicle dynamics;
• particle damping;
• imperfect joints;
• landing rokcet;
• ...

In these applications, the unilateral contstraints result in impacts happening, so it is important to give a proper method to deal with the unilateral constraints.

Modeling of the unilateral constraints

There are mainly two kinds of method to model the unilateral constraints. The first kind of method is intherited from continuum mechanics, and the second kind of method is based on the non-smooth theory.

Continuum mechanics based method

Fig. 1. Hertz contact model

In this method, normal forces generated from the unilateral constraints are obtained according to the local material properties of bodies. In detail, contact force models are derived from the continuum mechanics, and expressed as functions of gap and the impact velocity between bodies.The animation of classic Hertz contact law is shown in Fig. 1, where the contact is explained by the local defomation of bodies. And more contact models can refer to the introduction in contact mechanics, and some reviews.

Non-smooth theory based method

In the non-smooth method, unilateral interactions between bodies are fundamentally modelled by the Signorini condition for the non-penetration condition, and impact laws to define the impact process. The Signorini condition can be concluded as the complementarity problem: $g\geq 0,\lambda \geq 0,\lambda \perp g$, where $g$ means the distance between two bodies and $\lambda$ denotes the contact force generated by the unilateral constraints, as shown in Fig. 2b. Also, using the proximal point of convex theory, the Signorini condition can be equivalently expressed as: $\lambda ={\rm {proj}}_{\mathbb {R} ^{+}}(\lambda -\rho g)$, where $\rho >0$ denotes an auxiliary parameter, and ${\rm {proj_{\bf {C}}()}}$ represents the proximal point of a set ${\bf {C}}$ to the variable. Both two expressions above represent the dynamic behaviors of unilateral constraints: when the normal distance $g$ is above zero, the contact is open, which means that there is no contact force between bodies, $\lambda =0$; while when the normal distance $g$ is equal to zero, the contact is closed, resulting in $\lambda \geq 0$.

Figure 2: a) unilateral contact, b) the Signorini graph, c) continuum mechanics based model

During the implementation of non-smooth theory based method, velocity Signorini condition or the acceleration Signorini condition are actually employed in most cases. The velocity Signorini condition is expressed as: $U^{+}\geq 0,\lambda \geq 0,U^{+}\lambda =0$, where $U^{+}$means the relative normal velocity after impact. This velocity Signorini condition should be understood together with the Signorini $g\geq 0,\lambda \geq 0,\lambda \perp g$. The acceleration Signorini condition is considered under the close contact ($g=0,U^{+}=0$), as: ${\ddot {g}}\geq 0,\lambda \geq 0,{\ddot {g}}\lambda =0$, where the over dot means the derivative with repect to time.

When using this method for unilateral constraints between two rigid bodies, the Signorini condition alone is not enough for the impact process, so impact laws, which give the information for states before the impact and after the impact, are needed to handle the impact process. For example, when the Newton restitution law is employed, a coefficient of the restitution will be defined as: $e=-{U^{+}}/{U^{-}}$, where $U^{-}$denotes the relative normal velocity before impact, to give information for the impact process. A detailed framework for unilateral constraints between two rigid bodies can refer to some published papers.

Solution techniques

If the unilateral constraints are modelled by the contact models, the contact forces generated from the unilateral constraints can be calculated directly from the explicit mathematical formular, wich can refer to the reviews listed in Section Continuum mechanics based method. If the non-smooth theory based method is employed, there are mainly two kinds of formulations to solve the Signorini conditions: the nonlinear/linear complementarity problem (N/LCP) formulation and the augmented Lagrangian formulation. Compared to the solution processusing the contact models, the non-smooth method is more tedious, but can be simulated faster. More comparisons between solutions using contact models and non-smooth theory can refer to the paper.

N/LCP formulations

In this way, solution of dynamics equations with unilateral constraints is transformed into the solution of N/LCPs. In detail, for the frictionless unilateral constraints or the unilateral constraints with planar friction, the problem is transformed into LCPs, and for frcitional unilateral constraints, the problem is transformed into NCPs. For the LCPs, the pivoting algorithm, originating from the algotithm of Lemek and Dantzig, can be employed, but for a large number of unilateral constraints, this alogorithm may lead to .

Further reading

Books and articles

• Acary V., Brogliato B. Numerical Methods for Nonsmooth Dynamical Systems. Applications in Mechanics and Electronics. Springer Verlag, LNACM 35, Heidelberg, 2008.
• Brogliato B. Nonsmooth Mechanics. Communications and Control Engineering Series Springer-Verlag, London, 1999 (2dn Ed.)
• Glocker, Ch. Dynamik von Starrkoerpersystemen mit Reibung und Stoessen, volume 18/182 of VDI Fortschrittsberichte Mechanik/Bruchmechanik. VDI Verlag, Düsseldorf, 1995
• Glocker Ch. and Studer C. Formulation and preparation for Numerical Evaluation of Linear Complementarity Systems. Multibody System Dynamics 13(4):447-463, 2005
• Jean M. The non-smooth contact dynamics method. Computer Methods in Applied mechanics and Engineering 177(3-4):235-257, 1999
• Moreau J.J. Unilateral Contact and Dry Friction in Finite Freedom Dynamics, volume 302 of Non-smooth Mechanics and Applications, CISM Courses and Lectures. Springer, Wien, 1988
• Pfeiffer F., Foerg M. and Ulbrich H. Numerical aspects of non-smooth multibody dynamics. Comput. Methods Appl. Mech. Engrg 195(50-51):6891-6908, 2006
• Potra F.A., Anitescu M., Gavrea B. and Trinkle J. A linearly implicit trapezoidal method for integrating stiff multibody dynamics with contacts, joints and friction. Int. J. Numer. Meth. Engng 66(7):1079-1124, 2006
• Stewart D.E. and Trinkle J.C. An Implicit Time-Stepping Scheme for Rigid Body Dynamics with Inelastic Collisions and Coulomb Friction. Int. J. Numer. Methods Engineering 39(15):2673-2691, 1996
• Studer C. Augmented time-stepping integration of non-smooth dynamical systems, PhD Thesis ETH Zurich, ETH E-Collection, to appear 2008
• Studer C. Numerics of Unilateral Contacts and Friction -- Modeling and Numerical Time Integration in Non-Smooth Dynamics, Lecture Notes in Applied and Computational Mechanics, Volume 47, Springer, Berlin, Heidelberg, 2009

Open-source softwares

In most cases, friction among bodies should also be considered, and the A unilateral contact is usually associated with a gap function g which measures the distance between the two bodies and a contact force. The behaviour of a unilateral contact is modeled by a force law which states a relation between the gap function and the contact force. Set-valued force laws of type Upr assume a hard contact and clearly distinguish between open contact (contact force equal to zero, gap g is positive) and closed contact (contact force is positive, gap g is zero), see figure 1b. Regularized force laws are associated to compliance models. These laws write the contact force as function of the gap, i.e. the models have the same underlying mathematical structure for closed and open contacts. Unilateral contacts are used in contact dynamics and/or contact mechanics.

See also

• contact dynamics – Motion of multibody systems
• contact mechanics – Study of the deformation of solids that touch each other
• Siconos – Open source scientific software for modeling non-smooth dynamical systems