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M = 10
m = 1 # mass of particles
function calculate_force(
left_pos,
middle_pos,
right_pos,
K,
alpha = 0.0,
beta = 1000.0,
)
linear_force = K * (left_pos + right_pos - 2 * middle_pos)
quadratic_force = alpha * (left_pos - middle_pos)^2 + alpha * (right_pos - middle_pos)^2
cubic_force = beta * (left_pos - middle_pos)^3 + beta * (right_pos - middle_pos)^3
return linear_force + quadratic_force + cubic_force
end
function tendency!(du, u, p, t)
# unpack the params
N, K, m = p
# get the positions and momenta
qs = u[1:2:end]
ps = u[2:2:end]
# go over the points in the lattice and update the state
for i in 1:N-1
mass = m
if i == Int(N / 2)
mass = M
end
left_index = max(1, i - 1)
right_index = min(N, i + 1)
du[i*2-1] = ps[i] / mass
force = calculate_force(qs[left_index], qs[i], qs[right_index], K)
du[i*2] = force / mass
end
# make last point same as first
du[N*2-1] = du[1] = 0 # set to 0
du[N*2] = du[2] = 0
if t % 100000 == 0
println("TIME UPDATE: ", t)
end
# if ps[Int(N / 2)] / M >= 1
# println("(in sim!) Time: ", t, " Vel: ", ps[Int(N / 2)] / M)
# # println("Other Positions: ", qs)
# println("Other Velocities: ", ps, "\n")
# end
end
function get_initial_state(
N,
initial_displacement = 2,
initial_velocity = 0,
)
state = zeros(2 * N)
middle_index = 2 * Int(N / 2) - 1 # middle mass
state[middle_index] = initial_displacement
state[middle_index+1] = initial_velocity * M
return state
end
using DifferentialEquations
function run_simulation(
N,
K,
m,
final_time,
initial_displacement = 2,
initial_velocity = 0,
)
println("Running simulation with N = $N, K = $K, m = $m, final_time = $final_time, initial_displacement = $initial_displacement, initial_velocity = $initial_velocity\n")
s_0 = get_initial_state(N, initial_displacement, initial_velocity)
calculate_energy(s_0)
# pack the params
p = N, K, m
t_span = (0.0, final_time)
prob = ODEProblem(tendency!, s_0, t_span, p)
sol = solve(prob, Tsit5(), reltol = 1e-5, abstol = 1e-5, maxiters = 1e10) # control simulation
calculate_energy(sol.u[end])
println("Done Running Sim!\n\n")
return sol
end
using Plots
function animate_positions(
states,
time_steps,
time_min = 0,
time_max = 30,
red_threshold = 2,
shift = true,
)
println("Animating positions")
anim = @animate for i in 1:length(time_steps)
t = time_steps[i]
if t < time_min
continue
end
if t > time_max
break
end
positions = states[i][1:2:end]
v_middle = states[i][Int(length(states[1]) / 2)] / M
p_middle = states[i][Int(length(states[1]) / 2)-1]
y_lims = shift ? (-3 + p_middle, 3 + p_middle) : (-3, 3)
# plot(positions, label = "t = $(round(t, digits = 3)), v_middle=$(round(v_middle, digits=3))", marker = :circle, xlabel = "Mass Number", ylabel = "Displacement", ylim = (-3, 3))
if v_middle >= red_threshold
plot(positions, label = "t = $(round(t, digits = 7)), v_middle=$(round(v_middle, digits=7))", marker = :circle, xlabel = "Mass Number", ylabel = "Displacement", ylim = y_lims,
color = :red, legend = :topright,
)
else
plot(positions, label = "t = $(round(t, digits = 7)), v_middle=$(round(v_middle, digits=7))", marker = :circle, xlabel = "Mass Number", ylabel = "Displacement", ylim = y_lims,
color = :blue, legend = :topright,
)
end
end
mp4(anim, "t/animate-positions.mp4", fps = 30)
println("Done animating positions")
end
function plot_starting_and_final_positions(
states,
time_steps,
)
# plot the positions
middle_index = Int(length(states[1]) / 2) - 1
pos_init = [x - states[1][middle_index] for x in states[1][1:2:end]]
pos_final = [x - states[end][middle_index] for x in states[end][1:2:end]]
p1 = plot(pos_init, label = "Initial", marker = :circle, xlabel = "Mass Number", ylabel = "Displacement", title = "Positions Over Time")
plot!(p1, pos_final, label = "Final", marker = :circle)
# plot the vels
vels_init = [x / M for x in states[1][2:2:end]]
vels_final = [x / M for x in states[end][2:2:end]]
p2 = plot(states[1][2:2:end], label = "Initial", marker = :circle, xlabel = "Mass Number", ylabel = "Velocity", title = "Velocities Over Time")
plot!(p2, states[end][2:2:end], label = "Final t = $(time_steps[end])", marker = :circle)
# save the plots
savefig(p1, "t/initial-final-positions.png")
savefig(p2, "t/initial-final-velocities.png")
end
function analyize_vels(
states,
time_steps,
threshold = 1.975,
)
println("Analyzing velocities:\n")
output = []
for i in 1:length(states)
v = states[i][Int(length(states[i]) / 2)] / M
if v >= threshold - 10e-6
push!(output, i)
println("Time: ", time_steps[i], " Vel: ", v)
end
end
data = []
for i in 1:length(states)
push!(data, states[i][Int(length(states[i]) / 2)])
end
p = plot(data, label = "Velocity Over Time", xlabel = "Time", ylabel = "Velocity")
savefig(p, "t/velocity-over-time.png")
println("\nDone!\n\n")
return output
end
function analyize_pos(
states,
time_steps,
threshold = 1.975,
)
println("Analyzing positions:\n")
output = []
for i in 1:length(states)
if states[i][Int(length(states[i]) / 2)-1] >= threshold
push!(output, i)
println("Time: ", time_steps[i], " Position: ", states[i][Int(length(states[i]) / 2)])
end
end
# plot the first 10 seconds of Velocity
data = []
for i in 1:length(states)
if time_steps[i] > 10
break
end
push!(data, states[i][Int(length(states[i]) / 2)] - 1)
end
p = plot(data, label = "Position Over Time", xlabel = "Time", ylabel = "Position")
savefig(p, "t/pos-over-time.png")
println("\nDone!\n\n")
return output
end
function calculate_energy(state)
# calculate the kinetic energy
kinetic_energy = 0
vels = state[2:2:end]
for i in 1:N
mass = i == Int(N / 2) - 1 ? M : m
# calculate the kinetic energy
kinetic_energy += 0.5 * vels[i] * vels[i] / mass
end
# calcaute the potential energy
potential_energy = 0
pos = state[1:2:end]
for i in 1:N-1
left_index = max(1, i - 1)
right_index = min(N, i + 1)
potential_energy += 0.5 * K * (pos[left_index] - pos[i])^2
potential_energy += 0.5 * K * (pos[right_index] - pos[i])^2
end
# print the energy
println("Kinetic Energy: ", kinetic_energy)
println("Potential Energy: ", potential_energy)
println("Total Energy: ", kinetic_energy + potential_energy, "\n")
end
function plot_middle_mass_phase_space(states)
# get the index of the middle mass
middle_index = Int(length(states[1]) / 2) - 1
# build an array of the pos and vel over time
pos = []
vel = []
for i in 1:length(states)
push!(pos, states[i][middle_index])
push!(vel, states[i][middle_index+1])
end
# plot the phase space
p = plot(pos, vel, xlabel = "Position", ylabel = "Momentum", title = "Phase Space of Middle Mass in FPU", label = "Beta = 10, K = 1, N = 64, M = $M")
# save the plot
savefig(p, "t/phase-space.png")
end
# Run the simulation
N = 64 # number of masses
K = 1 # spring constant
final_time = 1000 # seconds
plot_data = []
my_vel = 100
sol = run_simulation(N, K, m, final_time, 0, my_vel)
println("final time: ", sol.t[end])
# s = sol.u[1:2:end]
# analyize_vels(sol.u, sol.t, my_vel)
# analyize_pos(sol.u, sol.t, 1.4)
plot_starting_and_final_positions(sol.u, sol.t)
# animate_positions(sol.u, sol.t, 0, 100, my_vel) # expect 80913.35854226245 for k=10?? rip
plot_middle_mass_phase_space(sol.u)
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