Unilateral contact

2021-01-20T02:32:21Z

210.6.125.151: /* Frictional unilateral constraints */ stedy -> steady

{{short description|A mechanical constraint which prevents penetration between two bodies;}}
In [[contact mechanics]], the term '''unilateral contact''', also called '''unilateral constraint''', denotes a mechanical [[constraint (classical mechanics)|constraint]] which prevents penetration between two rigid/flexible bodies.
Constraints of this kind are omnipresent in [[non-smooth mechanics|non-smooth multibody dynamics]] applications, such as granular flows,<ref name="Anitescu-Tasora-2010">{{cite journal |last1=Anitescu |first1=Mihai |last2=Tasora |first2=Alessandro |title=An iterative approach for cone complementarity problems for nonsmooth dynamics |journal=Computational Optimization and Applications |date=26 November 2008 |volume=47 |issue=2 |pages=207–235 |doi=10.1007/s10589-008-9223-4|s2cid=1107494 |url=https://hal.archives-ouvertes.fr/hal-01631636/file/MAATA.pdf }}</ref> [[legged robot]], [[vehicle dynamics]], [[particle damping]], imperfect joints,<ref name="Flores-2010">{{cite journal |last1=Flores |first1=Paulo |title=A parametric study on the dynamic response of planar multibody systems with multiple clearance joints |journal=Nonlinear Dynamics |date=7 March 2010 |volume=61 |issue=4 |pages=633–653 |doi=10.1007/s11071-010-9676-8|hdl=1822/23520 |s2cid=92980088 |hdl-access=free }}</ref> or rocket landings. In these applications, the unilateral constraints result in impacts happening, therefore requiring suitable methods to deal with such constraints.

== Modelling of the unilateral constraints ==
There are mainly two kinds of methods to model the unilateral constraints. The first kind is based on [[continuum mechanics|smooth contact dynamics]], including methods using Hertz's models, penalty methods, and some regularization force models, while the second kind is based on the [[contact dynamics|non-smooth contact dynamics]], which models the system with unilateral contacts as [[variational inequalities]].

=== Smooth contact dynamics ===
{{See also|Contact mechanics}}[[File:Hertz_contact_animated.gif|thumb|upright|Hertz contact model|alt=]]
In this method, normal forces generated by the unilateral constraints are modelled according to the local material properties of bodies. In particular, contact force models are derived from continuum mechanics, and expressed as functions of the gap and the impact velocity of bodies. As an example, an illustraction of the classic [[contact mechanics|Hertz contact model]] is shown in the figure on the right. In such model, the contact is explained by the local deformation of bodies. More contact models can be found in some review scientific works<ref name="Machado-Moreira-Flores-Lankarani-2012">{{cite journal |last1=Machado |first1=Margarida |last2=Moreira |first2=Pedro |last3=Flores |first3=Paulo |last4=Lankarani |first4=Hamid M. |title=Compliant contact force models in multibody dynamics: Evolution of the Hertz contact theory |journal=Mechanism and Machine Theory |date=July 2012 |volume=53 |pages=99–121 |doi=10.1016/j.mechmachtheory.2012.02.010|hdl=1822/19623 |hdl-access=free }}</ref><ref name="Gilardi-Sharf-2002">{{cite journal |last1=Gilardi |first1=G. |last2=Sharf |first2=I. |title=Literature survey of contact dynamics modelling |journal=Mechanism and Machine Theory |date=October 2002 |volume=37 |issue=10 |pages=1213–1239 |doi=10.1016/S0094-114X(02)00045-9}}</ref><ref name="Alves-Peixinho-Silva-Flores-Lankarani-2015">{{cite journal |last1=Alves |first1=Janete |last2=Peixinho |first2=Nuno |last3=da Silva |first3=Miguel Tavares |last4=Flores |first4=Paulo |last5=Lankarani |first5=Hamid M. |title=A comparative study of the viscoelastic constitutive models for frictionless contact interfaces in solids |journal=Mechanism and Machine Theory |date=March 2015 |volume=85 |pages=172–188 |doi=10.1016/j.mechmachtheory.2014.11.020|hdl=1822/31823 |hdl-access=free }}</ref> or in the article dedicated to [[contact mechanics]].

=== Non-smooth contact dynamics ===
{{See also|Contact dynamics#Non-smooth approach}}
In non-smooth method, unilateral interactions between bodies are fundamentally modelled by the [[Signorini problem|Signorini condition]]<ref name="Jean-1999">{{cite journal |last1=Jean |first1=M. |title=The non-smooth contact dynamics method |journal=Computer Methods in Applied Mechanics and Engineering |date=July 1999 |volume=177 |issue=3–4 |pages=235–257 |doi=10.1016/S0045-7825(98)00383-1|url=https://hal.archives-ouvertes.fr/hal-01390459/file/MJ.pdf }}</ref> for non-penetration, and impact laws are used to define the impact process.<ref name="Pfeiffer-2012">{{cite journal |last1=Pfeiffer |first1=Friedrich |title=On non-smooth multibody dynamics |journal=Proceedings of the Institution of Mechanical Engineers, Part K: Journal of Multi-body Dynamics |date=14 March 2012 |volume=226 |issue=2 |pages=147–177 |doi=10.1177/1464419312438487|s2cid=123605632 }}</ref> The Signorini condition can be expressed as the complementarity problem:

<math>g \geq 0, \quad \lambda \geq 0, \quad \lambda \perp g </math>,

where <math>g </math> denotes the distance between two bodies and <math>\lambda </math> denotes the contact force generated by the unilateral constraints, as shown in the figure below. Moreover, in terms of the concept of proximal point of convex theory, the Signorini condition can be equivalently expressed<ref name="Jean-1999" /><ref name=":0">{{Cite journal|last1=Pfeiffer|first1=Friedrich|last2=Foerg|first2=Martin|last3=Ulbrich|first3=Heinz|date=October 2006|title=Numerical aspects of non-smooth multibody dynamics|url=https://linkinghub.elsevier.com/retrieve/pii/S0045782505003646|journal=Computer Methods in Applied Mechanics and Engineering|language=en|volume=195|issue=50–51|pages=6891–6908|doi=10.1016/j.cma.2005.08.012}}</ref> as:

<math>\lambda ={\rm{proj}}_{\R^+}(\lambda -\rho g )</math>,

where <math>\rho>0</math> denotes an auxiliary parameter, and <math>{\rm proj}_{\bf C}(x)</math> represents the proximal point in the set <math>C</math> to the variable <math>x</math>,<ref name="link.springer.com">{{Cite journal|last1=Jalali Mashayekhi|first1=Mohammad|last2=Kövecses|first2=József|date=August 2017|title=A comparative study between the augmented Lagrangian method and the complementarity approach for modeling the contact problem|url=http://link.springer.com/10.1007/s11044-016-9510-2|journal=Multibody System Dynamics|language=en|volume=40|issue=4|pages=327–345|doi=10.1007/s11044-016-9510-2|s2cid=123789094|issn=1384-5640}}</ref> defined as:

<math>{\rm proj}_{\bf C}(x)={\rm argmin}_{y\in C}\|y-x\|</math>.

Both the expressions above represent the dynamic behaviour of unilateral constraints: on the one hand, when the normal distance <math>g_{\rm N} </math> is above zero, the contact is open, which means that there is no contact force between bodies, <math>\lambda =0 </math>; on the other hand, when the normal distance <math>g_{\rm N} </math> is equal to zero, the contact is closed, resulting in <math>\lambda \geq0</math>.

[[Image:contact dynamics unilateral.jpg|frame|center|Figure 2: a) unilateral contact, b) the Signorini graph, c) continuum mechanics based model|alt=]]When implementing non-smooth theory based methods, the velocity Signorini condition or the acceleration Signorini condition are actually employed in most cases. The velocity Signorini condition is expressed as:<ref name="Jean-1999" /><ref>{{Cite journal|last1=Tasora|first1=A.|last2=Anitescu|first2=M.|date=January 2011|title=A matrix-free cone complementarity approach for solving large-scale, nonsmooth, rigid body dynamics|url=https://linkinghub.elsevier.com/retrieve/pii/S0045782510001970|journal=Computer Methods in Applied Mechanics and Engineering|language=en|volume=200|issue=5–8|pages=439–453|doi=10.1016/j.cma.2010.06.030}}</ref>

<math>U_{\rm N}^{+}\geq 0,\quad \lambda \geq0,\quad U^{+}\lambda =0</math>,

where <math>U_{\rm N}^{+}</math> denotes the relative normal velocity after impact. The velocity Signorini condition should be understood together with the previous conditions <math>g \geq 0,\;\lambda \geq 0,\;\lambda \perp g </math>. The acceleration Signorini condition is considered under closed contact (<math>g =0, U_{\rm N}^{+}=0</math>), as:<ref name=":0" />

<math>\ddot g \geq 0,\quad \lambda \geq0,\quad \ddot g \lambda =0</math>,

where the overdots denote the second-order derivative with respect to time.

When using this method for unilateral constraints between two rigid bodies, the Signorini condition alone is not enough to model the impact process, so impact laws, which give the information about the states before and after the impact,<ref name="Jean-1999" /> are also required. For example, when the Newton restitution law is employed, a [[coefficient of restitution]] will be defined as: <math>e=-{U_{\rm N}^{+}}/{U_{\rm N}^{-}}</math>, where <math>U_{\rm N}^{-}</math>denotes the relative normal velocity before impact.

=== Frictional unilateral constraints ===
For frictional unilateral constraints, the normal contact forces are modelled by one of the methods above, while the friction forces are commonly described by means of [[Friction|Coulomb's friction law]]. Coulomb's friction law can be expressed as follows: when the tangential velocity <math>U_{\rm T}</math> is not equal to zero, namely when the two bodies are sliding, the friction force <math>\lambda_{\rm T}</math> is proportional to the normal contact force <math>\lambda</math>; when instead the tangential velocity <math>U_{\rm T}</math> is equal to zero, namely when the two bodies are relatively steady, the friction force <math>\lambda_{\rm T}</math> is no more than the maximum of the static friction force. This relationship can be summarised using the maximum dissipation principle,<ref name="Jean-1999" /> as

<math>\lambda_{\rm T} \in D(\mu \lambda)~~~~~~\forall S\in D(\mu \lambda)~~~~~~(S-\lambda_{\rm T})U_{\rm T}\geq 0,</math>

where

<math>D(\mu \lambda)=\{\forall x|-\mu \lambda\leq\|x\|\leq \mu \lambda\}</math>

represents the friction cone, and <math>\mu</math> denotes the kinematic friction coefficient. Similarly to the normal contact force, the formulation above can be equivalently expressed in terms of the notion of proximal point as:<ref name="Jean-1999" />

<math>\lambda_{\rm T}={\rm{proj}}_{D(\mu\lambda)}(\lambda_T-\rho U_{\rm T})</math>,

where <math>\rho>0</math> denotes an auxiliary parameter.

== Solution techniques ==
If the unilateral constraints are modelled by the continuum mechanics based contact models, the contact forces can be computed directly through an explicit mathematical formula, that depends on the contact model of choice. If instead the non-smooth theory based method is employed, there are two main formulations for the solution of the Signorini conditions: the [[Nonlinear complementarity problem|nonlinear]]/[[linear complementarity problem]] (N/LCP) formulation and the augmented Lagrangian formulation. With respect to the solution of contact models, the non-smooth method is more tedious, but less costly from the computational viewpoint. A more detailed comparison of solution methods using contact models and non-smooth theory was carried out by Pazouki et al.<ref>{{Cite journal|last1=Pazouki|first1=Arman|last2=Kwarta|first2=Michał|last3=Williams|first3=Kyle|last4=Likos|first4=William|last5=Serban|first5=Radu|last6=Jayakumar|first6=Paramsothy|last7=Negrut|first7=Dan|date=2017-10-13|title=Compliant contact versus rigid contact: A comparison in the context of granular dynamics|url=https://link.aps.org/doi/10.1103/PhysRevE.96.042905|journal=Physical Review E|language=en|volume=96|issue=4|pages=042905|doi=10.1103/PhysRevE.96.042905|pmid=29347540|issn=2470-0045|doi-access=free}}</ref>

=== N/LCP formulations ===
Following this approach, the solution of dynamics equations with unilateral constraints is transformed into the solution of N/LCPs. In particular, for frictionless unilateral constraints or 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 [[Simplex algorithm|pivoting algorithm]], originating from the algorithm of Lemek and Dantzig, is the most popular method.<ref name=":0" /> Unfortunately, however, numerical experiments show that the pivoting algorithm may fail when handling systems with a large number of unilateral contacts, even using the best optimizations.<ref name="Anitescu-Tasora-2008">{{cite journal|last1=Anitescu|first1=Mihai|last2=Tasora|first2=Alessandro|date=26 November 2008|title=An iterative approach for cone complementarity problems for nonsmooth dynamics|journal=Computational Optimization and Applications|volume=47|issue=2|pages=207–235|doi=10.1007/s10589-008-9223-4|s2cid=1107494|url=https://hal.archives-ouvertes.fr/hal-01631636/file/MAATA.pdf}}</ref> For NCPs, using a polyhedral approximation can transform the NCPs into a set of LCPs, which can then be solved by the LCP solver.<ref>{{Cite journal|last1=Xu|first1=Ziyao|last2=Wang|first2=Qi|last3=Wang|first3=Qingyun|date=December 2017|title=Numerical method for dynamics of multi-body systems with two-dimensional Coulomb dry friction and nonholonomic constraints|url=http://link.springer.com/10.1007/s10483-017-2285-8|journal=Applied Mathematics and Mechanics|language=en|volume=38|issue=12|pages=1733–1752|doi=10.1007/s10483-017-2285-8|s2cid=125402414|issn=0253-4827}}</ref> Other approaches beyond these methods, such NCP-functions<ref>{{Cite journal|last=Mangasarian|first=O. L.|date=July 1976|title=Equivalence of the Complementarity Problem to a System of Nonlinear Equations|url=http://epubs.siam.org/doi/10.1137/0131009|journal=SIAM Journal on Applied Mathematics|language=en|volume=31|issue=1|pages=89–92|doi=10.1137/0131009|issn=0036-1399}}</ref><ref>{{Cite journal|last=Fischer|first=A.|date=January 1992|title=A special newton-type optimization method|url=http://www.tandfonline.com/doi/abs/10.1080/02331939208843795|journal=Optimization|language=en|volume=24|issue=3–4|pages=269–284|doi=10.1080/02331939208843795|issn=0233-1934}}</ref> or cone complementarity problems (CCP) based methods<ref>{{Cite journal|last1=Melanz|first1=Daniel|last2=Fang|first2=Luning|last3=Jayakumar|first3=Paramsothy|last4=Negrut|first4=Dan|date=June 2017|title=A comparison of numerical methods for solving multibody dynamics problems with frictional contact modeled via differential variational inequalities|journal=Computer Methods in Applied Mechanics and Engineering|language=en|volume=320|pages=668–693|doi=10.1016/j.cma.2017.03.010|doi-access=free}}</ref><ref>{{Cite journal|last1=Negrut|first1=Dan|last2=Serban|first2=Radu|last3=Tasora|first3=Alessandro|date=2018-01-01|title=Posing Multibody Dynamics With Friction and Contact as a Differential Complementarity Problem|url=https://asmedigitalcollection.asme.org/computationalnonlinear/article/doi/10.1115/1.4037415/384577/Posing-Multibody-Dynamics-With-Friction-and|journal=Journal of Computational and Nonlinear Dynamics|language=en|volume=13|issue=1|pages=014503|doi=10.1115/1.4037415|issn=1555-1415|doi-access=free}}</ref> are also employed to solve NCPs.

=== Augmented Lagrangian formulation ===
Different from the N/LCP formulations, the augmented Lagrangian formulation uses the proximal functions described above, <math>\lambda={\rm{proj}}_{\R^+}(\lambda-\rho g)</math>. Together with dynamics equations, this formulation is solved by means of [[root-finding algorithms]]. A comparative study between LCP formulations and the augmented Lagrangian formulation was carried out by Mashayekhi et al.<ref name="link.springer.com"/>

==See also==
*{{annotated link|Multibody dynamics}}
*{{annotated link|contact dynamics}}
*{{annotated link|contact mechanics}}
*{{annotated link|discrete element method}}
*{{annotated link|Non-smooth mechanics}}
*[[Collision response]]
*[[Variational inequalities]]

== References ==
{{reflist}}

== Further reading ==

=== Open-source software ===
Open-source codes and non-commercial packages using the non-smooth based method:
* {{annotated link|Siconos}}
* [https://github.com/projectchrono/chrono/ Chrono], an open source multi-physics simulation engine, see also project [http://projectchrono.org 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

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