Particle packing

[LasNikas]

based on #439 and #436

[svchb]

@LasNikas Please fix the tests first than I will review this PR.

[efaulhaber]

Review of the first part. Remaining: Examples, NHS, signed distance and following.

[efaulhaber]

Why not add the whole particle packing directory?

[LasNikas]

How?

[efaulhaber]

See e.g. the setups:

```@autodocs
Modules = [TrixiParticles]
Pages = map(file -> joinpath("setups", file), readdir(joinpath("..", "src", "setups")))
```

[efaulhaber]

Why did this change?

[LasNikas]

This version is more robust. consider a deleteat! on geometry.

[svchb]

Suggested change

	function triangle_plane_intersection(point, plane_normal, min_corner, max_corner, cell_size)
	dirx = SVector(cell_size[1], zero(eltype(point)), zero(eltype(point)))
	diry = SVector(zero(eltype(point)), cell_size[2], zero(eltype(point)))
	dirz = SVector(zero(eltype(point)), zero(eltype(point)), cell_size[3])
	cell_vertex_1 = min_corner
	cell_vertex_2 = min_corner + dirx
	pos_diff_1 = cell_vertex_1 - point
	pos_diff_2 = cell_vertex_2 - point
	# Corners: bottom north-west and bottom north-east
	dot1 = sign(dot(pos_diff_1, plane_normal))
	dot2 = sign(dot(pos_diff_2, plane_normal))
	!(dot1 == dot2) && return true
	cell_vertex_3 = min_corner + diry
	pos_diff_3 = cell_vertex_3 - point
	# Corners: bottom north-east and top north-west
	dot3 = sign(dot(pos_diff_3, plane_normal))
	!(dot2 == dot3) && return true
	cell_vertex_4 = min_corner + dirz
	pos_diff_4 = cell_vertex_4 - point
	# Corners: top norht-west and bottom south-west
	dot4 = sign(dot(pos_diff_4, plane_normal))
	!(dot3 == dot4) && return true
	cell_vertex_5 = max_corner
	pos_diff_5 = cell_vertex_5 - point
	# Corners: bottom south-west and top south-east
	dot5 = sign(dot(pos_diff_5, plane_normal))
	!(dot4 == dot5) && return true
	cell_vertex_6 = max_corner - dirx
	pos_diff_6 = cell_vertex_6 - point
	# Corners: top south-east and top south-west
	dot6 = sign(dot(pos_diff_6, plane_normal))
	!(dot5 == dot6) && return true
	cell_vertex_7 = max_corner - diry
	pos_diff_7 = cell_vertex_7 - point
	# Corners: top south-west and bottom south-east
	dot7 = sign(dot(pos_diff_7, plane_normal))
	!(dot6 == dot7) && return true
	cell_vertex_8 = max_corner - dirz
	pos_diff_8 = cell_vertex_8 - point
	# Corners: bottom south-east and top north-east
	dot8 = sign(dot(pos_diff_8, plane_normal))
	!(dot7 == dot8) && return true
	# All edge vertices are on one side of the plane
	return false
	end
	function triangle_plane_intersection(point, plane_normal, min_corner, max_corner, cell_size)
	# Generate all eight corners of the cell (axis-aligned bounding box).
	# Each corner is calculated by adding a combination of cell_size components
	# to the min_corner, based on the indices i, j, k, which take values 0 or 1.
	# This creates all possible combinations, resulting in the eight corners of the cell.
	corners = [
	min_corner + SVector(i * cell_size[1], j * cell_size[2], k * cell_size[3])
	for i in (0, 1), j in (0, 1), k in (0, 1)
	]
	# For each corner, compute the vector from a known point on the plane to the corner.
	# Then, calculate the dot product of this vector with the plane's normal vector.
	# The dot product indicates the relative position of the corner to the plane:
	# - Positive value: corner is on the same side as the normal vector points.
	# - Zero: corner lies exactly on the plane.
	# - Negative value: corner is on the opposite side from where the normal vector points.
	# The sign() function converts the dot product into -1, 0, or 1 accordingly.
	signs = [sign(dot(corner - point, plane_normal)) for corner in corners]
	# Determine if the plane intersects the cell by checking the signs.
	# - If all signs are the same (all corners on one side of the plane), there's no intersection.
	# - If signs vary (some corners on different sides), the plane intersects the cell.
	# If maximum(signs) > minimum(signs), it means there are both positive and negative signs.
	return maximum(signs) > minimum(signs)
	end

[LasNikas]

Thanks for this. Your version is definitely more readable. I'm wondering if my version is slightly faster as we don't have to check each combination in best case.

However, I prefer your readable version. What's @efaulhaber opinion on this?

[svchb]

Please just test what the performance is like compared to the current version.

[LasNikas]

That's not that easy. I mean it's case dependent. In best case (first condition is true), it should be faster. But in worst case it should be same performance, no?

[svchb]

Yes but I would just compare it for a large example and check if the impact is actually measurable.

[LasNikas]

I think the performance difference is negligible, that's why I also prefer your version.

[svchb]

Please align

[LasNikas]

it is aligned?

[svchb]

we have usually also aligned the ":"

[LasNikas]

ah didn't know that. But it seems that we haven't done that consistently.

[LasNikas]

like so?