Marey.Sim 0.7.0

Marey.Sim

Simulation for Marey, a dimensionally-typed visualization system for F#. A simulation is a start and a step, and its trajectory is an animation: it seeks, it films, and a seek to any time is exactly the state a straight run reaches.

#r "nuget: Marey.Sim"
#r "nuget: Marey.Backend.Svg"

open Marey
open Marey.UnitsOfMeasure

// A driven, damped pendulum: an angle in rad and a rate in rad/s, stepped by RK4 at 1 ms.
let pendulum: Flow2T<rad, rad/s, s> =
    fun t th om -> struct (om, -9.81<rad/s^2> * sin (float th) - 0.2</s> * om + 0.5<rad/s^2> * cos (1.3 * float t))

let sim = Sim.ofStep 1e-3<s> (Ode.rk4 pendulum) (struct (0.5<rad>, 0.0<rad/s>))
let trajectory = Sim.trajectory 1.0<s> 60.0<s> sim

let path = Array.init 600 (fun i -> trajectory.At(float i * 0.1<s>))

Chart.line Derived.radian (U.div Derived.radian U.second) (path |> Array.map (fun (struct (th, _)) -> th)) (path |> Array.map (fun (struct (_, om)) -> om))
|> Chart.toSvg SvgBackend.render "pendulum.svg" (Chart.size 640.0 480.0)

• Integrators typed by the shape of the state. Ode.verlet and Ode.leapfrog over positions and velocities in one unit, with the acceleration checked to be in that unit per second squared; Ode.euler, Ode.rk4 and Ode.dopri5 (Dormand–Prince 4(5), dense output) over a plane flow with time; and eulerWith, rk4With, dopri5With over any record given a Tangent. A rate in the wrong unit does not compile.
• Trajectories. Sim.trajectory every horizon checkpoints one straight run and seeks by replay, bit for bit equal to the run, on any thread. Invariants (Sim.withInvariant) and their drift (Sim.drift, Drift.chart).
• Events. A function of the state crossing zero, located in time, with a response; a minimum separation ends a Zeno sequence at rest instead of looping. Collide.step resolves hard-disc contacts exactly within a step, with a grid broad phase.
• Sections and particles. Poincare.stroboscopic and Poincare.section. Unsteady fields, analytic or from grids (ofSnapshots, ofFunction), carried by Particles.step stage by stage; streaklines and pathlines.
• Grid solvers. Fdm.heat (explicit, with its stability limit refused in seconds), Fdm.heatImplicit (Crank–Nicolson by ADI), Fdm.wave, Fdm.poisson, on the compact five-point stencil with Dirichlet, Neumann and periodic sides.
• Fluids. Stam.step, stable fluids on a staggered grid with the pressure by multigrid, walls or periodic edges, viscosity, a dye in its own unit and a buoyancy per unit of it; Lbm.step, lattice Boltzmann D2Q9 with obstacles, walls, an inflow and an outflow, refusing a relaxation time at or below ½ with the numbers. Marey.Backend.Gpu steps both on a GPU (GpuSim).

Compiles under Fable. A chaotic system run in a browser and the same one under .NET are different trajectories after tens of seconds, because JavaScript's sin and exp differ in the last bit; both are correct to the integrator's order.

• The guide's chapter
• Examples: integrators.fsx, double-pendulum.fsx, double-gyre.fsx, heat.fsx, fluid.fsx, cylinder.fsx