Function finiteelement::solve_fes

pub fn solve_fes<'a, F: Float + 'a, B: FiniteElement<F>>(
    system: &[B], 
    positions: &mut [Point<F>], 
    nb_iter: usize, 
    epsilon_stop: F, 
    gradient_switch: F, 
    nb_gradient_steps: usize, 
    snapshot_steps: usize
) -> Vec<Vec<Point<F>>>

Solve a system of finite element.

Arguments:

• system is a slice of finite elements.
• posistions is the initial vector of positions of the points of the system, it is updated at each step of the optimization
• nb_iter maximum number of optimization step (see below)
• epsilon_stop threshold at witch the optimization is considered finished (see below)
• gradient_swith threshold for switching to gradient descent (see below)
• nb_gradient_steps number of gradient descent steps before switching back to Newton's method (see below).
• snapshot_step number of optimization between each snapshot (see below).

Optimization steps are performed untill one of the following happens

• nb_iter steps have been performed
• There is no point in the system on which an acceleration of norm greater than epsilon_stop is applied

The method used here is an hybrid between Newton's method and gradient descent. In the first iterations, Newton's method will be used. More precesily, it will perform the following opperation: for i in 0..posistions.len() { positions[i] += delta[i]} where delta is a solution to the equation J*delta = F where F and J are respectively the acceleration vector and its Jacobian.

If during one step of Newton's method the norm of delta is smaller than gradient_switch, nb_gradient_steps steps of gradient descent are performed. Each gradient step do the following opperation: for i in 0..positions.len() { positions[i] -= rate * force[i] } where rate is a parameter that is updated using the rms prop heuristic. Once these gradient descent steps have been done, the next steps are Newton's method steps.

Every snapshot_step steps, the current value of positions is pushed in a Vec that will be returned by this function. If snapshot_step is set to 0, snapshots are never made and an empty vector will be returned.

A value of about 5 is recommended for nb_gradient_steps.