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

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

Continuum mechanics based method
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 the left figure, where the contact is explained by the local deformation 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 the below figure. Also, using the proximal point of convex theory, the Signorini condition can be equivalently expressed as:

$$\lambda ={\rm{proj}}_{\R^+}(\lambda -\rho g )$$,

where $$\rho>0$$ denotes an auxiliary parameter, and $${\rm proj}_{\bf C}(x)$$ represents the proximal point in the set $$C$$ to the variable $$x$$, expressed as:

$${\rm proj}_{\bf C}(x)={\rm argmin}_{y\in C}\|y-x\|$$.

Both two expressions above represent the dynamic behaviours of unilateral constraints: when the normal distance $$g_{\rm N} $$ is above zero, the contact is open, which means that there is no contact force between bodies, $$\lambda =0 $$; while when the normal distance $$g_{\rm N} $$ is equal to zero, the contact is closed, resulting in $$\lambda \geq0$$.

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 :

$$U_{\rm N}^{+}\geq 0, \lambda \geq0, U^{+}\lambda =0$$,

where $$U_{\rm N}^{+}$$ 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 ($$g =0, U_{\rm N}^{+}=0$$), as :

$$\ddot g \geq 0, \lambda \geq0, \ddot g \lambda =0$$,

where the over dot means the derivative with respect 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: $$e=-{U_{\rm N}^{+}}/{U_{\rm N}^{-}}$$, where $$U_{\rm N}^{-}$$denotes the relative normal velocity before impact, to give information for the impact process.

Frictional unilateral constraints
For frictional unilateral constraints, the normal contact forces are described by the methods above, while the friction forces are commonly described by the Coulomb's friction law. The Coulomb's friction law can be concluded as: when the tangential velocity $$U_{\rm T}$$ is not equal to zero, namely the sliding state, the friction force $$\lambda_{\rm T}$$ is proportional to the normal contact force $$\lambda$$; when the tangential velocity $$U_{\rm T}$$ is equal to zero, namely the stick state, the friction force $$\lambda_{\rm T}$$ is nor more than the maximum of the static friction force. This relationship can be summarised using the maximum dissipation principle, as

$$\lambda_{\rm T} \in D(\mu \lambda)\forall S\in D(\mu \lambda)(S-\lambda_{\rm T})U_{\rm T}\geq 0,$$

where

$$D(\mu \lambda)=\{\forall x|-\mu \lambda\leq\|x\|\leq \mu \lambda\}$$

represents the friction cone, and $$\mu$$ denotes the kinematic friction coefficient. Similar as the normal contact force, using the proximal point of convex theory, the formulation above can be equivalently expressed as :

$$\lambda_{\rm T}={\rm{proj}}_{D(\mu\lambda)}(\lambda_T-\rho U_{\rm T})$$,

where $$\rho>0$$ denotes an auxiliary parameter.

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 formula, which 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 preprocessing the contact models, the non-smooth method is more tedious, but requires less computation cost. 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, while for frictional unilateral constraints, the problem is transformed into NCPs. To solve LCPs, the pivoting algorithm, originating from the algorithm of Lemek and Dantzig, is the most popular method. But unfortunately, numerical experiences show that the pivoting algorithm may fail when handling systems with a large number of unilateral contacts, even using the deepest optimizations. For the NCPs, using a polyhedral approximation can transform the NCPs into a set of LCPs, and then be solved by the LCP solver. Besides, other methods, like NCP-functions, cone complementarity problems (CCP) based method  , are also employed to solve NCPs.

Augmented Lagrangian formulation
Different from the N/LCP formulations, the augmented Lagrangian formulation uses the proximal functions listed above, $$\lambda={\rm{proj}}_{\R^+}(\lambda-\rho g)$$. Together with dynamics equations, this formulation is solved by root finding algorithms. A comparative study between LCP formulations and the augmented Lagrangian formulation can be found in the reference.

Open-source software
Open-source codes and non-commercial packages using the non-smooth based method:
 * Chrono, an open source multi-physics simulation engine, see also project website
 * Chrono, an open source multi-physics simulation engine, see also project website

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

Category:Mechanics