Unilateral contact
In contact mechanics, the term unilateral contact, also called unilateral constraint, denotes a mechanical constraint which prevents penetration between two rigid/flexible bodies.
Constraints of this kind are omnipresent in non-smooth multibody dynamics applications, such as granular flows,[1] legged robot, vehicle dynamics, particle damping, imperfect joints,[2] or rocket landings.
There are mainly two kinds of methods to model the unilateral constraints.
The first kind is based on smooth contact dynamics, including methods using Hertz's models, penalty methods, and some regularization force models, while the second kind is based on the non-smooth contact dynamics, which models the system with unilateral contacts as variational inequalities.
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 illustration of the classic Hertz contact model is shown in the figure on the right.
In such model, the contact is explained by the local deformation of bodies.
In non-smooth method, unilateral interactions between bodies are fundamentally modelled by the Signorini condition[6] for non-penetration, and impact laws are used to define the impact process.
[7] The Signorini condition can be expressed as the complementarity problem:
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[6][8] as:
Both the expressions above represent the dynamic behaviour of unilateral constraints: on the one hand, when the normal distance
When implementing non-smooth theory based methods, the velocity Signorini condition or the acceleration Signorini condition are actually employed in most cases.
denotes the relative normal velocity after impact.
The acceleration Signorini condition is considered under closed contact (
, where the overdots denote the second-order derivative with respect to time.
denotes the relative normal velocity before impact.
Coulomb's friction law can be expressed as follows: when the tangential velocity
is not equal to zero, namely when the two bodies are sliding, the friction force
is equal to zero, namely when the two bodies are relatively steady, the friction force
This relationship can be summarised using the maximum dissipation principle,[6] as
Similarly to the normal contact force, the formulation above can be equivalently expressed in terms of the notion of proximal point as:[6]
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.
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.[11] Following this approach, the solution of dynamics equations with unilateral constraints is transformed into the solution of N/LCPs.
[8] 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.
[13] Other approaches beyond these methods, such NCP-functions[14][15][16] or cone complementarity problems (CCP) based methods[17][18] are also employed to solve NCPs.
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.[9] Open-source codes and non-commercial packages using the non-smooth based method:
