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Nonsmooth dynamics

Nonsmooth Mechanics is that part of Mechanics that deals with mechanical systems subject to various types of nonsmooth interaction laws, such as impacts, friction, or any kind of piecewise linear contact laws (there are many!). One typical such interface law is represented by so-called complementarity relations, which are a set of relationships between a function h(q) of the generalized position q (that one may think of as being a "distance" between the system and some obstacle), and a Lagrange multiplier lambda (that represents the contact force between the system and the obstacle). The complementarity between h(q) and lambda says that they are both non-negative (componentwise) and their scalar product is always zero (said another way, they are orthogonal vectors). The physical meaning of such a law is as follows: if there is no contact between the system and its obstacle (equivalently h(q) > 0), then the contact force has to be zero. Magnetic forces are forbidden in such a model. Second, if the contact force is positive (equivalently lambda > 0) then the system has to be in contact with its obstacle, i.e. h(q)=0. Moreover one sees that negative contact forces are not encompassed by this model, which therefore cannot represent adhesive effects (but, more complex, similar models can). Finally, the inequality constraint (or unilateral constraint) on the position f(q) > 0, simply means that the two systems (the mechanical system and its environment) cannot interpenetrate one each other. There is a hard boundary in the configuration space, within which the system has to evolve. Surprizingly enough (at least for beginners in the field), this does not necessarily implies that the bodies are modelled as perfectly rigid bodies: contact flexibilities are permitted (think of a mass colliding a spring: they do not interpenetrate, however the spring is deformable, and the distance between the mass and the contact point on the spring is always non-negative). But, the modelling as rigid bodies has an important consequence: velocities have sometimes to be discontinuous. This gives rise to instantaneous impacts, or collisions, which are mathematically represented by impulsive contact forces, or, more rigorously, by Dirac measures.

This complementarity relation is one example of a nonsmooth contact law, that is to be coupled to the usual (smooth) dynamical equations (which most often are the Lagrange equations). Coulomb's friction is another typical and most encountered example. These two, associated to some impact laws, are widely used in the field of multibody dynamics, for control, stability analysis, modelling, numerical simulation (see for example this article, mathematical analysis, bifurcation analysis, etc.

Historically, this field of Mechanics has been settled by Jean Jacques Moreau, from the university of Montpellier, in a seminal paper published in 1963. Moreau developped Nonsmooth Mechanics in parallel with Convex Analysis, that is the field of analysis dealing with convex, possibly non-differentiable functions, and convex sets (see here and there). Indeed, nonsmooth systems need nonsmooth analysis! In parallel to Moreau's works on Mechanics, Rockafellar also developped convex analysis in order to solve problems of nonsmooth Optimization. The reader who is aware of optimization will have notice that indeed, complementarity relations look like very much the so-called Kuhn-Tucker conditions of constrained optimization. It is a fact that nonsmooth mechanics and nonsmooth optimization, share a common mathematical language and both rely upon similar mathematical tools (convex analysis, nonsmooth analysis, variational inequalities). Nonsmooth mechanics may even be a rich source of nonsmooth optimization problems! It is therefore natural and fruitful to encourage the cross-fertilizing between researchers of both fields: this is what the BIPOP project intends to achieve, among other things.

Let us get back to the nonsmooth models briefly described above. As we said such systems are made of three basic ingredients:

The mathematical formalisms which can be used to write down these systems are: complementarity dynamical systems, standard and unbounded differential inclusions, evolution variational inequalities. This means in turn that their study is closely linked to complementarity theory (essentially developed by people in the field of economics), variational inequalities and all their variants and extensions (like Panagiotopoulos' hemivariational inequalities), and the theory of differential inclusions. All the systems (physical or abstract) which are written down with such tools, are consequently of the same "family": the family of nonsmooth dynamical systems, to which nonsmooth mechanical systems belong. From the application point of view, not only mechanical systems are encompassed by these models. Electrical systems (circuits with ideal diodes), are another important class. Other applications exist in transportation science, macro-economics, etc. Other mathematical formalisms exist, like ordinary differential equations with impulsive excitations, piecewise linear systems, that model nonsmooth effects in various fields like predator-prey and biological systems, etc.

The BIPOP project develops works related to both nonsmooth dynamical systems (with an emphasis on mechanics and electrical systems) and nonsmooth optimization. Nonsmooth optimization activities are described elsewhere on the project's web page. The studies developed by BIPOP on nonsmooth dynamics concern:

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