# Nonsmooth problems formulation and solve¶

When dynamical systems and their interactions have been properly defined inside a model and its nonsmooth dynamical system, a proper formulation for the latter must be chosen, associated to a nonsmooth solver.

## Linear nonsmooth problems¶

$w = q + M z, M \in R^{n \times n }, q \in R^{n}$

where $$w \in R^{n}, z \in R^{n}$$ are the unknowns.

• Linear Complementarity Problems (LCP)
$\begin{split}w = q + M z, M \in R^{n \times n }, q \in R^{n} \\ w \geq 0, z \geq 0, z^{T} w =0\end{split}$
$\begin{split}0 = Au + Cv + a\\ z = Du + Bv + b\\ v \geq 0, z \geq 0, z^{T} v =0\end{split}$

where

$\begin{split}u \in R^{n}, v \in R^{m}, z \in R^{m} \ the \ unknowns, \\ a \in R^{n}, b \in R^{m} \\ A \in R^{n \times n }, B \in R^{m \times m }\\ C \in R^{n \times m }, D \in R^{m \times n }\end{split}$
• Relay
• Equality
• AVI
• 2D or 3D friction contact problem FrictionContact
$\begin{split}velocity = q + M reaction \\ velocity \geq 0, reaction \geq 0, reaction^{T} velocity =0\end{split}$

and a Coulomb friction law. With $$velocity \in R^{n}, reaction \in R^{n}$$ the unknowns, and $$M \in R^{n \times n }, q \in R^{n}$$

$\begin{split}M velocity = q + reaction \\ localVelocity = H^T velocity + tildeLocalVelocity\\ reaction = H localReaction \\\end{split}$

and $$localVelocity,localReaction$$ belongs to the Coulomb friction law with unilateral contact.

With $$velocity \in R^{n}, reaction \in R^{n}, localVelocity \in R^{m}, localReaction \in R^{m}$$ the unknowns, $$M \in R^{n \times n }, q \in R^{n}$$. $$tildeLocalVelocity \in R^{m}$$ is the modified local velocity ($$e U_{N,k}$$), $$M \in R^{n \times n }, q \in R^{n}, H \in R^{n \times m }$$.

• Generic mechanical problem (GenericMechanical)

Complete problem with bilateral equality, complementarity, impact and friction.

## The Simulation process¶

As for Event-Driven, we introduce level index sets, with level = 0 for first order systems and level=1 for second order systems (this is related to the relative degrees but we won’t get into details about that here).

$$I_0$$ is the set of all the potential UnitaryRelations (UR). For second order systems: $$I_1 = \{ ur_\alpha\in I_{0} , y^p_{\alpha} = 0 \}$$. Thus, the LCP is built only for unitary relations that belongs to $$I_level$$, level=0 for first order and level=1 for second order systems.

Then, the steps of a Moreau’s Time-Stepping simulation will be:

Knowing all values at the beginning of the time step $$[t_i,t_{i+1}]$$,

-# compute the free solutions -# for $$ur \in I_level$$ formalize and solve a LCP -# update the state (according to the possibly LCP results) -# go to next time step

SP::TimeStepping s(new TimeStepping(myModel));
SP::TimeDiscretisation t(new TimeDiscretisation(timeStep,s));

s->initialize();

int N = t->getNSteps(); // Number of time steps

// --- Time loop ---
while(k < N)// for each time step ...
{
// compute xFree, or qFree,vFree
s->computeFreeStep();
// Formalize and solve a LCP
computeOneStepNSProblem("timeStepping");
// Update state, using last computed values
s->update(level); //
// transfer of state i+1 into state i and time incrementation
s->nextStep();
}


Note that all time-independent operators are computed during simulation initialisation.

## Customize simulation behavior¶

Each time ComputeOneStepNS() function, i.e. the numerics solver, is called, it returns an int, giving some information about the convergence of the solver:

By default, when the convergence is not achieved, an exception is throwed and the process stops. Change this behavior is possible by defining a specific function of the form:

//
//
void myF(int info, SP::Simulation s)
{
// do what you need ...
}

int main(int argc, char* argv[])
{
//
// ...
SP::TimeStepping your_simulation = ...
your_simulation->setCheckSolverFunction(&myF);


Then after each call to your_simulation->computeOneStepNS(…), the function myF will be called. That may be usefull to change the solver type, the tolerance or whatever is needed.