Relaxation Scheme

`FMM(; p::Int=4, ncrit::Int=50, theta::Real=0.4, phi::Real=0.3)`

Parameters for FMM solver.

Arguments

• p : Order of multipole expansion (number of terms).
• ncrit : Maximum number of particles per leaf.
• theta : Neighborhood criterion. This criterion defines the distance where the far field starts. The criterion is that if θ*r < R1+R2 the interaction between two cells is resolved through P2P, where r is the distance between cell centers, and R1 and R2 are each cell radius. This means that at θ=1, P2P is done only on cells that have overlap; at θ=0.5, P2P is done on cells that their distance is less than double R1+R2; at θ=0.25, P2P is done on cells that their distance is less than four times R1+R2; at θ=0, P2P is done on cells all cells.
• phi : Regularizing neighborhood criterion. This criterion avoid approximating interactions with the singular-FMM between regularized particles that are sufficiently close to each other across cell boundaries. Used together with the θ-criterion, P2P is performed between two cells if φ < σ/dx, where σ is the average smoothing radius in between all particles in both cells, and dx is the distance between cell boundaries ( dx = r-(R1+R2) ). This means that at φ = 1, P2P is done on cells with boundaries closer than the average smoothing radius; at φ = 0.5, P2P is done on cells closer than two times the smoothing radius; at φ = 0.25, P2P is done on cells closer than four times the smoothing radius.

`Relaxation(relax, nsteps_relax, rlxf)`

Defines a relaxation method implemented in the function relax(rlxf::Real, p::Particle) where rlxf is the relaxation factor between 0 and 1, with 0 == no relaxation, and 1 == full relaxation. The simulation is relaxed every nsteps_relax steps.

`relax_Pedrizzetti(rlxf::Real, p::Particle)`

Relaxation scheme where the vortex strength is aligned with the local vorticity.

`relax_correctedPedrizzetti(rlxf::Real, p::Particle)`

Relaxation scheme where the vortex strength is aligned with the local vorticity. This version fixes the error in Pedrizzetti's relaxation that made the strength to continually decrease over time. See notebook 20200921 for derivation.