Coupling convergence can be improved by applying a relaxation parameter. Ideally, it has to be small enough to keep the iteration from diverging, but as large as possible to minimize the amount of iteration.
$$ u_1^{i + 1} = \hat u_2^{i + 1} = \alpha u_2^{i + 1} + (1 - \alpha )u_2^i$$
To maximize the efficiency, Aitken’s formulation dynamically calculates a suitable relaxation parameter for each iteration by taking into account the result from the previous iterations.
$$ {\alpha ^{i + 1}} = {\alpha ^i}\frac{(r^i)^T(r^{i+1}-r^i)}{||r^{i+1}-r^i||}$$
$$ r^{i+1}=u_2^{i+1}-u_1^{i+1}$$
The default parameters specific to this solver are:
"init_alpha" : 0.1,
"init_alpha_max" : 0.45,
"alpha_max" : 2.0,
"alpha_min" : -2.0
- init_alpha: Relaxation factor in the first time step.
- init_alpha_max: Maximum relaxation factor for the first iteration in each time step.
- alpha_max: Upper bound for the dynamic relaxation factor.
- alpha_min: Lower bound for the dynamic relaxation factor.
Reference: Ulrich Küttler et al., “Fixed-point fluid–structure interaction solvers with dynamic relaxation”