remoll/computational_astrophysics_nbody_solver
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cleanup and presentable

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Remy Moll
2025-02-03 20:14:33 +01:00
parent da8a7d4574
commit facb52b33e
12 changed files with 734 additions and 443 deletions
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nbody/presentation/helpers.typ
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@@ -2,7 +2,7 @@
#import "@preview/based:0.2.0": base64
#let code_font_scale = 0.6em
#let code_font_scale = 0.5em
#let cell_matcher(cell, cell_tag) = {
// Matching function to check if a cell has a specific tag

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nbody/presentation/main.pdf
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nbody/presentation/main.typ
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@@ -1,8 +1,5 @@
#import "@preview/diatypst:0.2.0": *
// #set text(font: "Cantarell")
// #set heading(numbering: (..nums)=>"")
#show: slides.with(
title: "N-Body project ",
subtitle: "Computational Astrophysics, HS24",
@@ -13,15 +10,13 @@
// ratio: 16/9,
)
#show footnote.entry: set text(size: 0.6em)
#set footnote.entry(gap: 3pt)
#set align(horizon)
#import "helpers.typ"
// KINDA COOL:
// _diatypst_ defines some default styling for elements, e.g Terms created with ```typc / Term: Definition``` will look like this
// / *Term*: Definition
// Setup of code location
#let t1 = json("../task1.ipynb")
@@ -41,10 +36,7 @@
== Overview - the system
Get a feel for the particles and their distribution. [#link(<task1:plot_particle_distribution>)[code]]
Get a feel for the particles and their distribution
#columns(2)[
#helpers.image_cell(t1, "plot_particle_distribution")
// Note: for visibility the outer particles are not shown.
@@ -54,33 +46,39 @@ Get a feel for the particles and their distribution. [#link(<task1:plot_particle
- a _spherical_ distribution
$==>$ treat the system as a *globular cluster*
#footnote[Unit handling [#link(<task1:function_apply_units>)[code]]]
]
// It is a small globular cluster with
// - 5*10^4 stars => m in terms of msol
// - radius - 10 pc
// Densities are now expressed in M_sol / pc^3
// Forces are now expressed
== Density
We compare the computed density with the analytical model provided by the _Hernquist_ model:
Compare the computed density
#footnote[Density sampling [#link(<task1:function_density_distribution>)[code]]]
with the analytical model provided by the _Hernquist_ model:
#grid(
columns: (1fr, 2fr),
columns: (3fr, 4fr),
inset: 0.5em,
block[
$
rho(r) = M/(2 pi) a / (r dot (r + a)^3)
rho(r) = M/(2 pi) a / (r dot (r + a)^3)
$
where we infer $a$ from the half-mass radius:
$
r_"hm" = (1 + sqrt(2)) dot a
r_"hm" = (1 + sqrt(2)) dot a
$
#text(size: 0.6em)[
Density sampling [#link(<task1:function_density_distribution>)[code]];
]
],
block[
#helpers.image_cell(t1, "plot_density_distribution")
]
)
// Note that by construction, the first shell contains no particles
// => the numerical density is zero there
// Having more bins means to have shells that are nearly empty
@@ -89,45 +87,41 @@ We compare the computed density with the analytical model provided by the _Hernq
== Force computation
// N Body and variations
#grid(
columns: (2fr, 1fr),
columns: (3fr, 2fr),
inset: 0.5em,
block[
#helpers.image_cell(t1, "plot_force_radial")
// The radial force is computed as the sum of the forces of all particles in the system.
#text(size: 0.6em)[
Analytical force [#link(<task1:function_analytical_forces>)[code]];
$N^2$ force [#link(<task1:function_n2_forces>)[code]];
$epsilon$ computation [#link(<task1:function_interparticle_distance>)[code]];
]
#helpers.image_cell(t1, "plot_force_radial")
],
block[
Discussion:
- the analytical method replicates the behavior accurately
- at small softenings the $N^2$ method has noisy artifacts
- a $1 dot epsilon$ softening is a good compromise between accuracy and stability
- the analytical
#footnote[Analytical force [#link(<task1:function_analytical_forces>)[code]]]
method replicates the behavior accurately
- at small softenings the $N^2$
#footnote[$N^2$ force [#link(<task1:function_n2_forces>)[code]]]
method has noisy artifacts
- a $1 dot epsilon$
#footnote[$epsilon$ computation [#link(<task1:function_interparticle_distance>)[code]]]
softening is a good compromise between accuracy and stability
]
)
// basic $N^2$ matches analytical solution without dropoff. but: noisy data from "bad" samples
// $N^2$ with softening matches analytical solution but has a dropoff. No noisy data.
// => softening $\approx 1 \varepsilon$ is a sweet spot since the dropoff is "late"
== Relaxation
We express system relaxation in terms of the dynamical time of the system.
$
t_"relax" = overbrace(N / (8 log N), n_"relax") dot t_"crossing"
$
where the crossing time of the system can be estimated through the half-mass velocity $t_"crossing" = v(r_"hm")/r_"hm"$.
We find a relaxation of [#link(<task1:compute_relaxation_time>)[code]].
We find a relaxation of $approx 30 "Myr"$ ([#link(<task1:compute_relaxation_time>)[code]])
// === Discussion
#grid(
columns: (1fr, 1fr),
inset: 0.5em,
@@ -140,6 +134,7 @@ We find a relaxation of [#link(<task1:compute_relaxation_time>)[code]].
- $=>$ relaxation time increases
]
)
// The estimate for $n_{relax}$ comes from the contribution of each star-star encounter to the velocity dispersion. This depends on the perpendicular force
// $\implies$ a bigger softening length leads to a smaller $\delta v$.
@@ -164,7 +159,7 @@ We find a relaxation of [#link(<task1:compute_relaxation_time>)[code]].
)[
#helpers.image_cell(t2, "plot_particle_distribution")
$=>$ use $M_"sys" approx 10^4 M_"sol" + M_"BH"$
$==>$ use $M_"sys" approx 10^4 M_"sol" + M_"BH"$
]
@@ -180,55 +175,83 @@ We find a relaxation of [#link(<task1:compute_relaxation_time>)[code]].
inset: 0.5em,
block[
#helpers.image_cell(t2, "plot_force_radial_single")
// The radial force is computed as the sum of the forces of all particles in the system.
#text(size: 0.6em)[
$N^2$ force [#link(<task1:function_n2_forces>)[code]];
$epsilon$ computation [#link(<task1:function_interparticle_distance>)[code]];
Mesh force [#link(<task2:function_mesh_force>)[code]];
]
],
block[
Discussion:
- using the (established) baseline of $N^2$ with $1 dot epsilon$ softening
- small grids are stable but inaccurate at the center
- using the (established) baseline of $N^2$
#footnote[$N^2$ force [#link(<task1:function_n2_forces>)[code]]]
with $1 dot epsilon$
#footnote[$epsilon$ computation [#link(<task1:function_interparticle_distance>)[code]]]
softening
- small grids
#footnote[Mesh force [#link(<task2:function_mesh_force>)[code]]]
are stable but inaccurate at the center
- very large grids have issues with overdiscretization
$==> 75 times 75 times 75$ as a good compromise
// Some other comments:
// - see the artifacts because of the even grid numbers (hence the switch to 75)
// overdiscretization for large grids -> vertical spread even though r is constant
// this becomes even more apparent when looking at the data without noise - the artifacts remain
]
)
// Some other comments:
// - see the artifacts because of the even grid numbers (hence the switch to 75)
// overdiscretization for large grids -> vertical spread even though r is constant
// this becomes even more apparent when looking at the data without noise - the artifacts remain
//
// We can not rely on the interparticle distance computation for a disk!
// Given softening length 0.037 does not match the mean interparticle distance 0.0262396757880128
//
// Discussion of the discrepancies
// TODO
#helpers.image_cell(t2, "plot_force_computation_time")
// Computed for 10^4 particles => mesh will scale better for larger systems
== Time integration
=== Runge-Kutta
*Integration step*
#helpers.code_reference_cell(t2, "function_runge_kutta")
*Timesteps*
Chosen such that displacement is small (compared to the inter-particle distance) [#link(<task2:integration_timestep>)[code]]:
$
op(d)t = 10^(-4) dot S / v_"part"
$
// too large timesteps lead to instable systems <=> integration not accurate enough
*Full integration*
[#link(<task2:function_time_integration>)[code]]
#pagebreak()
=== Results
#align(center, block(
height: 1fr,
== First results
#helpers.image_cell(t2, "plot_system_evolution")
== Varying the softening
#helpers.image_cell(t2, "plot_second_system_evolution")
== Stability [#link("../task2_nsquare_integration.gif")[1 epsilon]]
#page(
columns: 2
)[
#helpers.image_cell(t2, "plot_system_evolution")
])
#helpers.image_cell(t2, "plot_integration_stability")
]
== Particle mesh solver
sdlsd
#helpers.image_cell(t2, "plot_pm_solver_integration")
#helpers.image_cell(t2, "plot_pm_solver_stability")
= Appendix - Code <appendix>
== Code
#helpers.code_cell(t1, "plot_particle_distribution")
<task1:plot_particle_distribution>
#helpers.code_reference_cell(t1, "function_apply_units")
<task1:function_apply_units>
#pagebreak(weak: true)
@@ -260,6 +283,15 @@ sdlsd
#helpers.code_reference_cell(t2, "function_mesh_force")
<task2:function_mesh_force>
#pagebreak(weak: true)
#helpers.code_cell(t2, "integration_timestep")
<task2:integration_timestep>
#pagebreak(weak: true)
#helpers.code_cell(t2, "function_time_integration")
<task2:function_time_integration>