## simulating binary relation using unary relation and 2-arguments function

Let $$Sigma={c,f(cdot),R(cdot,cdot)}$$, $$Sigma_2={c,g(cdot,cdot),R(cdot)}$$.

Show an algorithm, that given $$varphi$$ above $$Sigma$$, returns $$varphi’$$ such that:
$$varphi$$ is satisfiable $$iff$$ $$varphi’$$ is satisfiable

I had some ideas, non of them seems to work out.
Would appreciate help, or hints. Thanks!

## unity – Simulating Road Traffic/Overtaking – Overtake works mostly, but starts clipping other vehicle

I am trying to improve on my Car class, which is used to make a small road scene with 3 lanes each direction with the ability to overtake by changing lanes.

I had a very basic version working with all box colliders and a predictive movement script in my code (so basically I performed the movement of the object, then immediately check a collision and move it to previous_position in same frame). It worked OK but i found i was a bit clunky and I had to have oversized collision boxes on the car which caused me other problems elsewhere in the game.

Now I am using Raycast. At first I used just one Raycast from the centre of the car. But of course once the centre of the car had past the vehicle in front it would think road was clear. I tried to fix this by having 3 rays (centre, left, right in local space). This improved it a lot but the wingmirrors of the car (and often times a slice of the body) would clip into each-other.

I’ve ended up adding around 10 Raycasts trying to make sure it always sees the car in front. I even tried making the edge-most ones slightly past the width of the car.

The only thing I can think to mention is that the cars are rotated 90 degree. So transform.forward is actually facing either +/- X-Axis in world space depending on if isTopLane is true/false.

Here is the Car class in full. For the life of me I cannot figure out why the cars overtake but stop overtaking too early (ie. they continue straight through the edge of the car in front):

public class Car : MonoBehaviour
{
bool isTopLane;
float speed;

float vehicle_width;
float vehicle_length;
Material material;

AudioSource audioSource;
float start_pitch;
float max_pitch;

float ray_length_forwards;
float ray_length_sideways;

public void InitCar(int spawn_z)
{
Color color = new Color(Random.Range(0f, 1f), Random.Range(0f, 1f), Random.Range(0f, 1f), 1);

if (spawn_z <= 0)
isTopLane = false;
else
isTopLane = true;

Vector3 rot = Vector3.zero;
rot.y = 90;
if (isTopLane)
{
x *= -1;
rot.y *= -1;
}

transform.position = new Vector3(x, 0, spawn_z);
transform.eulerAngles = rot;

material = GetComponentInChildren<MeshRenderer>().material;
material.color = color;

audioSource = GetComponent<AudioSource>();
start_pitch = audioSource.pitch + (speed * 0.01f);
max_pitch = start_pitch * 2.6f;

vehicle_width = GetComponent<Collider>().bounds.size.x;
vehicle_length = GetComponent<Collider>().bounds.size.z;
ray_length_forwards = GetComponent<Collider>().bounds.size.z * 1.4f;
ray_length_sideways = GetComponent<Collider>().bounds.size.x * 1.4f;

if ((isTopLane && transform.position.z < 1.1f) || (!isTopLane && transform.position.z > -1.1f))

}

private void FixedUpdate()
{
if (GameManager.instance.Game_Paused)
return;

audioSource.enabled = GameManager.instance.Options_SoundOn;

audioSource.pitch += (speed * 0.04f) * Time.fixedDeltaTime;
if (audioSource.pitch > max_pitch)
audioSource.pitch = start_pitch;

Drive();
}

void Drive()
{
Vector3 centre = transform.position;
Vector3 right1 = centre + Vector3.up * (vehicle_width * 0.1f);
Vector3 right2 = centre + Vector3.up * (vehicle_width * 0.2f);
Vector3 right3 = centre + Vector3.up * (vehicle_width * 0.3f);
Vector3 right4 = centre + Vector3.up * (vehicle_width * 0.4f);
Vector3 right5 = centre + Vector3.up * (vehicle_width * 0.5f);
Vector3 right6 = centre + Vector3.up * (vehicle_width * 0.65f);
Vector3 left1 = centre + Vector3.down * (vehicle_width * 0.1f);
Vector3 left2 = centre + Vector3.down * (vehicle_width * 0.2f);
Vector3 left3 = centre + Vector3.down * (vehicle_width * 0.3f);
Vector3 left4 = centre + Vector3.down * (vehicle_width * 0.4f);
Vector3 left5 = centre + Vector3.down * (vehicle_width * 0.5f);
Vector3 left6 = centre + Vector3.down * (vehicle_width * 0.65f);
Vector3() ray_origins = { centre, right1, right2, right3, right4, right5, right6, left1, left2, left3, left4, left5, left6 };
for (int i = 0; i < ray_origins.Length; i++)
{
if (Physics.Raycast(new Ray(ray_origins(i), transform.forward), out RaycastHit hit, ray_length_forwards))
{
if (hit.transform.gameObject.GetComponent<Car>())
{
Overtake();
return;
}

}
}

DriveForwards();

}

void DriveForwards()
{
Vector3 motion = transform.forward * speed * Time.fixedDeltaTime;
transform.position += motion;
}

void Overtake()
{
if (isTopLane && transform.position.z <= 1)
return;

if (!isTopLane && transform.position.z >= -1)
return;

if (Physics.Raycast(new Ray(transform.position, -transform.right), out RaycastHit hit, ray_length_sideways))
{
if (hit.transform.gameObject.GetComponent<Car>())
return;
}

Vector3 motion = -transform.right * speed * Time.fixedDeltaTime;
transform.position += motion;
}

}

Thank you for your time and help.

## java – Simulating a dice throw with threadlocalrandom does not converge to the expected value

I was trying to verify the expected value formula in java.
I did something very trivial, I simulated the roll of a dice, but I mapped each side to a specific value. So each value has probability of 1/6.
The expected value was determined as: 232.65
I run the dice simulation a very long number of runs, 10000000 to be precise but the number did not converge to 232.65 but was 232.74. Smaller runs also were fluxuating up to the value 233.97. Runs up to 2100000000gave 232.63.
So I am wondering if I am doing something wrong. My main premise here is that I shouldn’t need to simulate 2 billion throws of a dice to eventually the expected value calculated via the formula.
So I am wondering if I am using the wrong random API. I can not get the seed so I can’t verify if that changes per iteration. My code is the following:

Map<Integer, Float> map = new HashMap<>();
map.put(1, 17.2f);
map.put(2, 11f);
map.put(3, 128f);
map.put(4, 1f);
map.put(5, 1200f);
map.put(6, 38.7f);

double sum = 0;
int count = 0;

for(Map.Entry<Integer, Float> entry: map.entrySet()) {
sum += (entry.getValue() * (1/6f));
}

System.out.println("EV " + sum);
sum = 0;

for(int j = 0; j < 2100000000; j++) {
sum += map.get(dice);
++count;
}
System.out.println(sum / count);

Output:

EV 232.65000109374523
232.63201358102154

I was expected after a much earlier number of runs I would consistently get something ~232.650…

Also tried using BigDecimal but there was no difference:

//double sum = 0;
BigDecimal sum = BigDecimal.valueOf(0);
int count = 0;

for(Map.Entry<Integer, Float> entry: map.entrySet()) {
}

System.out.println("EV " + sum);

sum = BigDecimal.valueOf(0);

for(int j = 0; j < 10000000; j++) {
++count;
}

System.out.println(sum.divide(BigDecimal.valueOf(count)));

Output:

EV 232.6500010937452306
232.3528356843933100044325

## differential equations – simulating many Ito processes

I need to simulate a large number of correlated Ito processes and I am hoping to use Table to define the set of equations. Here is what I tried with just 2 processes which does not work:

k1[t_] := Sum[Subscript[x, i]
eqns[Subscript[a, 1] _, Subscript[a, 2] _, Subscript[s, 1] _,
Subscript[s, 2] _] =
Table[ItoProcess[{[DifferentialD]Subscript[x, i][
t] == -k1
Subscript[s, i] [DifferentialD]Subscript[W, i]
Subscript[x, i]
Subscript[W, i] [Distributed] WienerProcess[]}], {i, 2}]

Subscript[a, 1] = 0.1;
Subscript[a, 2] = 0.6;
Subscript[s, 1] = 0.3;
Subscript[s, 2] = 0.4;
sample = RandomFunction[
eqns[Subscript[a, 1], Subscript[a, 2], Subscript[s, 1], Subscript[s,
2]], {0, 10}]
ListPlot[sample]

## simulation – Simulating a multi-network environment in one device using Python

I am deploying a blockchain on a localhost where all nodes can communicate with each other using ports and APIs; however, a node/group of nodes can leave the main blockchain group.

Some questions came up as I was working on it:

• When a group splits, can we reassign new ports for the split such that the nodes can communicate among themselves and prevent the split from communicating with the main blockchain group?
• Are there any Python tools that can facilitate the simulation of the network environment?

## python – From Hummingbirds to Ornithopters: Simulating the Aerodynamics of Flapping Wings

A Flapping Wing Aerodynamics Simulator Written in Python

## Motivation

About a year ago, I became fascinated by how animals fly. As an aerospace engineering student, it surprised me that natural flight was barely covered in my courses. I soon realized that the reason why flapping wing aerodynamics isn’t taught to undergraduate engineers is that it’s insanely complicated.

This only made me more interested. I had some experience with UAV design, so I wanted to find a tool that would help me create a flapping wing micro-air vehicle (MAV), such as the TU Delft’s DelFly. Unfortunately, there were no open-source, easy to use, and fast programs for doing so. Therefore, in a moment of temporary insanity, I decided to write my own!

## Initial Goals

My initial goal for Ptera Software was to create a tool that other researchers and I can use to design the aerodynamics of ornithopters. This is goal zero. I also wanted my software to be:

1. Easy to use
3. Easy to maintain
4. Validated
5. Attractive to contributors in the open-source community
6. Fast enough to analyze at least 1000 configurations in 24 hours
Based on these requirements, I decided that the solver would be a Python implementation of the Unsteady Vortex-Lattice Method (UVLM). I also decided to host the repository on GitHub, and distribute it via PyPI.

## Current State

Like many ignorant non-CS engineers before me, I was confident that creating, debugging, testing, and maintaining a massive project would be as easy as writing a 100-line MATLAB script. Months of coding and therapy later, I had released my ‘stable’ 1.0.0 package.

I’ve copied the source code of my unsteady solver below. I did this to abide by the rule that we keep things as native to this site as possible. If you can, I recommend going to the vectorization branch on GitHub and navigate to unsteady_ring_vortex_lattice_method.py. If the vectorization branch is no longer active, go to the latest release’s master branch.

"""This is an aerodynamics solver that uses an unsteady ring vortex lattice method."""

"""This is the initialization method."""

# Initialize this solution's attributes.

# Initialize attributes to hold aerodynamic data that pertains to this problem.
self.current_step = None
self.current_airplane = None
self.current_operating_point = None
self.current_freestream_velocity_geometry_axes = None
self.current_wing_wing_influences = None
self.vectorized_current_wing_wing_influences = None
self.current_freestream_wing_influences = None
self.vectorized_current_freestream_wing_influences = None
self.current_wake_wing_influences = None
self.vectorized_current_wake_wing_influences = None
self.current_vortex_strengths = None
self.vectorized_current_vortex_strengths = None
self.streamline_points = None

# Initialize attributes to hold geometric data that pertains to this problem.
self.panels = None
self.panel_normal_directions = None
self.panel_areas = None
self.panel_centers = None
self.panel_collocation_points = None
self.panel_back_right_vortex_vertices = None
self.panel_front_right_vortex_vertices = None
self.panel_front_left_vortex_vertices = None
self.panel_back_left_vortex_vertices = None
self.panel_right_vortex_centers = None
self.panel_right_vortex_vectors = None
self.panel_front_vortex_centers = None
self.panel_front_vortex_vectors = None
self.panel_left_vortex_centers = None
self.panel_left_vortex_vectors = None
self.panel_back_vortex_centers = None
self.panel_back_vortex_vectors = None
self.seed_points = None

# Initialize variables to hold aerodynamic data that pertains details about
# this panel's location on its wing.
self.panel_is_trailing_edge = None
self.panel_is_left_edge = None
self.panel_is_right_edge = None

# Initialize variables to hold aerodynamic data that pertains to this
# problem's last time step.
self.last_panel_collocation_points = None
self.last_panel_vortex_strengths = None
self.last_panel_back_right_vortex_vertices = None
self.last_panel_front_right_vortex_vertices = None
self.last_panel_front_left_vortex_vertices = None
self.last_panel_back_left_vortex_vertices = None
self.last_panel_right_vortex_centers = None
self.last_panel_front_vortex_centers = None
self.last_panel_left_vortex_centers = None
self.last_panel_back_vortex_centers = None

# Initialize variables to hold aerodynamic data that pertains to this
# problem's wake.
self.wake_ring_vortex_strengths = None
self.wake_ring_vortex_front_right_vertices = None
self.wake_ring_vortex_front_left_vertices = None
self.wake_ring_vortex_back_left_vertices = None
self.wake_ring_vortex_back_right_vertices = None

def run(
self,
verbose=True,
prescribed_wake=True,
calculate_streamlines=True,
):
"""This method runs the solver on the unsteady problem."""

# Initialize all the airplanes' panels' vortices.
if verbose:
print("Initializing all airplanes' panel vortices.")
self.initialize_panel_vortices()

# Iterate through the time steps.
for step in range(self.num_steps):

# Save attributes to hold the current step, airplane, and operating point.
self.current_step = step
self.current_step
).operating_point
self.current_freestream_velocity_geometry_axes = (
self.current_operating_point.calculate_freestream_velocity_geometry_axes()
)
if verbose:
print(
"nBeginning time step "
+ str(self.current_step)
+ " out of "
+ str(self.num_steps - 1)
+ "."
)

# Initialize attributes to hold aerodynamic data that pertains to this
# problem.
self.current_wing_wing_influences = np.zeros(
(self.current_airplane.num_panels, self.current_airplane.num_panels)
)
self.vectorized_current_wing_wing_influences = np.zeros(
(self.current_airplane.num_panels, self.current_airplane.num_panels)
)
self.current_freestream_velocity_geometry_axes = (
self.current_operating_point.calculate_freestream_velocity_geometry_axes()
)
self.current_freestream_wing_influences = np.zeros(
self.current_airplane.num_panels
)
self.vectorized_current_freestream_wing_influences = np.zeros(
self.current_airplane.num_panels
)
self.current_wake_wing_influences = np.zeros(
self.current_airplane.num_panels
)
self.vectorized_current_wake_wing_influences = np.zeros(
self.current_airplane.num_panels
)
self.current_vortex_strengths = np.ones(self.current_airplane.num_panels)
self.vectorized_current_vortex_strengths = np.ones(
self.current_airplane.num_panels
)

# Initialize attributes to hold geometric data that pertains to this
# problem.
self.panels = np.empty(self.current_airplane.num_panels, dtype=object)
self.panel_normal_directions = np.zeros(
(self.current_airplane.num_panels, 3)
)
self.panel_areas = np.zeros(self.current_airplane.num_panels)
self.panel_centers = np.zeros((self.current_airplane.num_panels, 3))
self.panel_collocation_points = np.zeros(
(self.current_airplane.num_panels, 3)
)
self.panel_back_right_vortex_vertices = np.zeros(
(self.current_airplane.num_panels, 3)
)
self.panel_front_right_vortex_vertices = np.zeros(
(self.current_airplane.num_panels, 3)
)
self.panel_front_left_vortex_vertices = np.zeros(
(self.current_airplane.num_panels, 3)
)
self.panel_back_left_vortex_vertices = np.zeros(
(self.current_airplane.num_panels, 3)
)
self.panel_right_vortex_centers = np.zeros(
(self.current_airplane.num_panels, 3)
)
self.panel_right_vortex_vectors = np.zeros(
(self.current_airplane.num_panels, 3)
)
self.panel_front_vortex_centers = np.zeros(
(self.current_airplane.num_panels, 3)
)
self.panel_front_vortex_vectors = np.zeros(
(self.current_airplane.num_panels, 3)
)
self.panel_left_vortex_centers = np.zeros(
(self.current_airplane.num_panels, 3)
)
self.panel_left_vortex_vectors = np.zeros(
(self.current_airplane.num_panels, 3)
)
self.panel_back_vortex_centers = np.zeros(
(self.current_airplane.num_panels, 3)
)
self.panel_back_vortex_vectors = np.zeros(
(self.current_airplane.num_panels, 3)
)
self.seed_points = np.empty((0, 3))

# wing.
self.panel_is_trailing_edge = np.zeros(
self.current_airplane.num_panels, dtype=bool
)
self.current_airplane.num_panels, dtype=bool
)
self.panel_is_left_edge = np.zeros(
self.current_airplane.num_panels, dtype=bool
)
self.panel_is_right_edge = np.zeros(
self.current_airplane.num_panels, dtype=bool
)

# Initialize variables to hold details about the last airplane's panels.
self.last_panel_collocation_points = np.zeros(
(self.current_airplane.num_panels, 3)
)
self.last_panel_vortex_strengths = np.zeros(
self.current_airplane.num_panels
)
self.last_panel_back_right_vortex_vertices = np.zeros(
(self.current_airplane.num_panels, 3)
)
self.last_panel_front_right_vortex_vertices = np.zeros(
(self.current_airplane.num_panels, 3)
)
self.last_panel_front_left_vortex_vertices = np.zeros(
(self.current_airplane.num_panels, 3)
)
self.last_panel_back_left_vortex_vertices = np.zeros(
(self.current_airplane.num_panels, 3)
)
self.last_panel_right_vortex_centers = np.zeros(
(self.current_airplane.num_panels, 3)
)
self.last_panel_front_vortex_centers = np.zeros(
(self.current_airplane.num_panels, 3)
)
self.last_panel_left_vortex_centers = np.zeros(
(self.current_airplane.num_panels, 3)
)
self.last_panel_back_vortex_centers = np.zeros(
(self.current_airplane.num_panels, 3)
)

self.wake_ring_vortex_strengths = np.empty(0)
self.wake_ring_vortex_front_right_vertices = np.empty((0, 3))
self.wake_ring_vortex_front_left_vertices = np.empty((0, 3))
self.wake_ring_vortex_back_left_vertices = np.empty((0, 3))
self.wake_ring_vortex_back_right_vertices = np.empty((0, 3))

# Collapse this problem's geometry matrices into 1D ndarrays of attributes.
if verbose:
print("Collapsing geometry.")
self.collapse_geometry()

# Find the matrix of wing-wing influence coefficients associated with
# this current_airplane's geometry.
if verbose:
print("Calculating the wing-wing influences.")
self.calculate_wing_wing_influences()

# Find the vector of freestream-wing influence coefficients associated
# with this problem.
if verbose:
print("Calculating the freestream-wing influences.")
self.calculate_freestream_wing_influences()

# Find the vector of wake-wing influence coefficients associated with
# this problem.
if verbose:
print("Calculating the wake-wing influences.")
self.calculate_wake_wing_influences()

# Solve for each panel's vortex strength.
if verbose:
print("Calculating vortex strengths.")
self.calculate_vortex_strengths()

# Solve for the near field forces and moments on each panel.
if self.current_step >= self.first_results_step:
if verbose:
print("Calculating near field forces.")
self.calculate_near_field_forces_and_moments()

# Solve for the near field forces and moments on each panel.
if verbose:
print("Shedding wake vortices.")
self.populate_next_airplanes_wake(prescribed_wake=prescribed_wake)

# Solve for the location of the streamlines if requested.
if calculate_streamlines:
if verbose:
print("nCalculating streamlines.")
self.calculate_streamlines()

def initialize_panel_vortices(self):
"""This method calculates the locations every problem's airplane's bound
vortex vertices, and then initializes
its panels' bound vortices.

Every panel has a ring vortex, which is a quadrangle whose front vortex leg
is at the panel's quarter chord.
The left and right vortex legs run along the panel's left and right legs. If
the panel is not along the
trailing edge, they extend backwards and meet the back vortex leg at a length
of one quarter of the rear
panel's chord back from the rear panel's front leg. Otherwise, they extend
back backwards and meet the back
vortex leg at a length of one quarter of the current panel's chord back from
the current panel's back leg.
"""

# Iterate through all the steady problem objects.

this_freestream_velocity_geometry_axes = (
)

# Iterate through this problem's airplane's wings.

# Iterate through the wing's chordwise and spanwise positions.
for chordwise_position in range(wing.num_chordwise_panels):
for spanwise_position in range(wing.num_spanwise_panels):

panel = wing.panels(chordwise_position, spanwise_position)

front_left_vortex_vertex = panel.front_left_vortex_vertex
front_right_vortex_vertex = panel.front_right_vortex_vertex

# Define the back left and right vortex vertices based on
# whether the panel is along the trailing edge or not.
if not panel.is_trailing_edge:
next_chordwise_panel = wing.panels(
chordwise_position + 1, spanwise_position
)
back_left_vortex_vertex = (
next_chordwise_panel.front_left_vortex_vertex
)
back_right_vortex_vertex = (
next_chordwise_panel.front_right_vortex_vertex
)
else:
# As these vertices are directly behind the trailing
# edge, they are spaced back from their
# panel's vertex by one quarter the distance traveled
# during a time step. This is to more
# accurately predict drag.
back_left_vortex_vertex = (
front_left_vortex_vertex
+ (panel.back_left_vertex - panel.front_left_vertex)
+ this_freestream_velocity_geometry_axes
* self.delta_time
* 0.25
)
back_right_vortex_vertex = (
front_right_vortex_vertex
+ (panel.back_right_vertex - panel.front_right_vertex)
+ this_freestream_velocity_geometry_axes
* self.delta_time
* 0.25
)

# Initialize the panel's ring vortex.
panel.ring_vortex = ps.aerodynamics.RingVortex(
front_right_vertex=front_right_vortex_vertex,
front_left_vertex=front_left_vortex_vertex,
back_left_vertex=back_left_vortex_vertex,
back_right_vertex=back_right_vortex_vertex,
strength=None,
)

def collapse_geometry(self):
"""This method converts attributes of the problem's geometry into 1D
ndarrays. This facilitates vectorization, which speeds up the solver."""

global_panel_position = 0

# Iterate through the current airplane's wings.
for wing in self.current_airplane.wings:

# Convert this wing's 2D ndarray of panels into a 1D ndarray.
panels = np.ravel(wing.panels)
wake_ring_vortices = np.ravel(wing.wake_ring_vortices)

# Iterate through the 1D ndarray of this wing's panels.
for panel in panels:

# Update the solver's list of attributes with this panel's attributes.
self.panels(global_panel_position) = panel
self.panel_normal_directions(
global_panel_position, :
) = panel.normal_direction
self.panel_areas(global_panel_position) = panel.area
self.panel_centers(global_panel_position) = panel.center
self.panel_collocation_points(
global_panel_position, :
) = panel.collocation_point
self.panel_back_right_vortex_vertices(
global_panel_position, :
) = panel.ring_vortex.right_leg.origin
self.panel_front_right_vortex_vertices(
global_panel_position, :
) = panel.ring_vortex.right_leg.termination
self.panel_front_left_vortex_vertices(
global_panel_position, :
) = panel.ring_vortex.left_leg.origin
self.panel_back_left_vortex_vertices(
global_panel_position, :
) = panel.ring_vortex.left_leg.termination
self.panel_right_vortex_centers(
global_panel_position, :
) = panel.ring_vortex.right_leg.center
self.panel_right_vortex_vectors(
global_panel_position, :
) = panel.ring_vortex.right_leg.vector
self.panel_front_vortex_centers(
global_panel_position, :
) = panel.ring_vortex.front_leg.center
self.panel_front_vortex_vectors(
global_panel_position, :
) = panel.ring_vortex.front_leg.vector
self.panel_left_vortex_centers(
global_panel_position, :
) = panel.ring_vortex.left_leg.center
self.panel_left_vortex_vectors(
global_panel_position, :
) = panel.ring_vortex.left_leg.vector
self.panel_back_vortex_centers(
global_panel_position, :
) = panel.ring_vortex.back_leg.center
self.panel_back_vortex_vectors(
global_panel_position, :
) = panel.ring_vortex.back_leg.vector
self.panel_is_trailing_edge(
global_panel_position
) = panel.is_trailing_edge
global_panel_position
self.panel_is_right_edge(global_panel_position) = panel.is_right_edge
self.panel_is_left_edge(global_panel_position) = panel.is_left_edge

# Check if this panel is on the trailing edge.
if panel.is_trailing_edge:
# If it is, calculate it's streamline seed point and add it to
# the solver's ndarray of seed points.
self.seed_points = np.vstack(
(
self.seed_points,
panel.back_left_vertex
+ 0.5 * (panel.back_right_vertex - panel.back_left_vertex),
)
)

# Increment the global panel position.
global_panel_position += 1

for wake_ring_vortex in wake_ring_vortices:
self.wake_ring_vortex_strengths = np.hstack(
(self.wake_ring_vortex_strengths, wake_ring_vortex.strength)
)
self.wake_ring_vortex_front_right_vertices = np.vstack(
(
self.wake_ring_vortex_front_right_vertices,
wake_ring_vortex.front_right_vertex,
)
)
self.wake_ring_vortex_front_left_vertices = np.vstack(
(
self.wake_ring_vortex_front_left_vertices,
wake_ring_vortex.front_left_vertex,
)
)
self.wake_ring_vortex_back_left_vertices = np.vstack(
(
self.wake_ring_vortex_back_left_vertices,
wake_ring_vortex.back_left_vertex,
)
)
self.wake_ring_vortex_back_right_vertices = np.vstack(
(
self.wake_ring_vortex_back_right_vertices,
wake_ring_vortex.back_right_vertex,
)
)

# Initialize a variable to hold the global position of the panel as we
# iterate through them.
global_panel_position = 0

if self.current_step > 0:

# Iterate through the current airplane's wings.
for wing in last_airplane.wings:

# Convert this wing's 2D ndarray of panels into a 1D ndarray.
panels = np.ravel(wing.panels)

# Iterate through the 1D ndarray of this wing's panels.
for panel in panels:
# Update the solver's list of attributes with this panel's
# attributes.
self.last_panel_collocation_points(
global_panel_position, :
) = panel.collocation_point

self.last_panel_vortex_strengths(
global_panel_position
) = panel.ring_vortex.strength

self.last_panel_back_right_vortex_vertices(
global_panel_position, :
) = panel.ring_vortex.right_leg.origin

self.last_panel_front_right_vortex_vertices(
global_panel_position, :
) = panel.ring_vortex.right_leg.termination

self.last_panel_front_left_vortex_vertices(
global_panel_position, :
) = panel.ring_vortex.left_leg.origin

self.last_panel_back_left_vortex_vertices(
global_panel_position, :
) = panel.ring_vortex.left_leg.termination

self.last_panel_right_vortex_centers(
global_panel_position, :
) = panel.ring_vortex.right_leg.center

self.last_panel_front_vortex_centers(
global_panel_position, :
) = panel.ring_vortex.front_leg.center

self.last_panel_left_vortex_centers(
global_panel_position, :
) = panel.ring_vortex.left_leg.center

self.last_panel_back_vortex_centers(
global_panel_position, :
) = panel.ring_vortex.back_leg.center

# Increment the global panel position.
global_panel_position += 1

def calculate_wing_wing_influences(self):
"""This method finds the matrix of wing-wing influence coefficients
associated with this airplane's geometry."""

# Find the matrix of normalized velocities induced at every panel's
# collocation point by every panel's ring
# vortex. The answer is normalized because the solver's vortex strength list
# was initialized to all ones. This
# will be updated once the correct vortex strength's are calculated.
total_influences = ps.aerodynamics.calculate_velocity_induced_by_ring_vortices(
points=self.panel_collocation_points,
back_right_vortex_vertices=self.panel_back_right_vortex_vertices,
front_right_vortex_vertices=self.panel_front_right_vortex_vertices,
front_left_vortex_vertices=self.panel_front_left_vortex_vertices,
back_left_vortex_vertices=self.panel_back_left_vortex_vertices,
strengths=self.current_vortex_strengths,
collapse=False,
)

# Take the batch dot product of the normalized velocities with each panel's
# normal direction. This is now the
# problem's matrix of wing-wing influence coefficients.
self.current_wing_wing_influences = np.einsum(
"...k,...k->...",
total_influences,
np.expand_dims(self.panel_normal_directions, axis=1),
)

def calculate_freestream_wing_influences(self):
"""This method finds the vector of freestream-wing influence coefficients
associated with this problem."""

# Find the normal components of the freestream velocity on every panel by
# taking a batch dot product.
freestream_influences = np.einsum(
"ij,j->i",
self.panel_normal_directions,
self.current_freestream_velocity_geometry_axes,
)

# Get the current flapping velocities at every collocation point.
current_flapping_velocities_at_collocation_points = (
self.calculate_current_flapping_velocities_at_collocation_points()
)

# Find the normal components of every panel's flapping velocities at their
# collocation points by taking a batch
# dot product.
flapping_influences = np.einsum(
"ij,ij->i",
self.panel_normal_directions,
current_flapping_velocities_at_collocation_points,
)

# Calculate the total current freestream-wing influences by summing the
# freestream influences and the
# flapping influences.
self.current_freestream_wing_influences = (
freestream_influences + flapping_influences
)

def calculate_wake_wing_influences(self):
"""This method finds the vector of the wake-wing influences associated with
the problem at this time step."""

# Check if this time step is not the first time step.
if self.current_step > 0:

# Get the wake induced velocities. This is a (M x 3) ndarray with the x,
# y, and z components of the velocity
# induced by the entire wake at each of the M panels.
wake_induced_velocities = ps.aerodynamics.calculate_velocity_induced_by_ring_vortices(
points=self.panel_collocation_points,
back_right_vortex_vertices=self.wake_ring_vortex_back_right_vertices,
front_right_vortex_vertices=self.wake_ring_vortex_front_right_vertices,
front_left_vortex_vertices=self.wake_ring_vortex_front_left_vertices,
back_left_vortex_vertices=self.wake_ring_vortex_back_left_vertices,
strengths=self.wake_ring_vortex_strengths,
collapse=True,
)

# Set the current wake-wing influences to the normal component of the
# wake induced velocities at each panel.
self.current_wake_wing_influences = np.einsum(
"ij,ij->i", wake_induced_velocities, self.panel_normal_directions
)

else:

# If this is the first time step, set the current wake-wing influences to
# zero everywhere, as there is no
# wake yet.
self.current_wake_wing_influences = np.zeros(
self.current_airplane.num_panels
)

def calculate_vortex_strengths(self):
"""This method solves for each panel's vortex strength."""

# Solve for the strength of each panel's vortex.
self.current_vortex_strengths = np.linalg.solve(
self.current_wing_wing_influences,
-self.current_wake_wing_influences
- self.current_freestream_wing_influences,
)

# Iterate through the panels and update their vortex strengths.
for panel_num in range(self.panels.size):
# Get the panel at this location.
panel = self.panels(panel_num)

# Update this panel's ring vortex strength.
panel.ring_vortex.update_strength(self.current_vortex_strengths(panel_num))

def calculate_solution_velocity(self, points):
"""This function takes in a group of points. At every point, it finds the
induced velocity due to every vortex
and the freestream velocity.

:param points: 2D ndarray of floats
This variable is an ndarray of shape (N x 3), where N is the number of
points. Each row contains the x, y,
and z float coordinates of that point's position in meters.
:return solution_velocities: 2D ndarray of floats
The output is the summed effects from every vortex, and from the
freestream on a given point. The result
will be of shape (N x 3), where each row identifies the velocity at a
point. The results units are meters
per second.
"""

# Find the vector of velocities induced at every point by every panel's ring
# vortex. The effect of every ring
# vortex on each point will be summed.
ring_vortex_velocities = (
ps.aerodynamics.calculate_velocity_induced_by_ring_vortices(
points=points,
back_right_vortex_vertices=self.panel_back_right_vortex_vertices,
front_right_vortex_vertices=self.panel_front_right_vortex_vertices,
front_left_vortex_vertices=self.panel_front_left_vortex_vertices,
back_left_vortex_vertices=self.panel_back_left_vortex_vertices,
strengths=self.current_vortex_strengths,
collapse=True,
)
)

# Find the vector of velocities induced at every point by every wake ring
# vortex. The effect of every wake ring
# vortex on each point will be summed.
wake_ring_vortex_velocities = (
ps.aerodynamics.calculate_velocity_induced_by_ring_vortices(
points=points,
back_right_vortex_vertices=self.wake_ring_vortex_back_right_vertices,
front_right_vortex_vertices=self.wake_ring_vortex_front_right_vertices,
front_left_vortex_vertices=self.wake_ring_vortex_front_left_vertices,
back_left_vortex_vertices=self.wake_ring_vortex_back_left_vertices,
strengths=self.wake_ring_vortex_strengths,
collapse=True,
)
)

# Find the total influence of the vortices, which is the sum of the influence
# due to the bound ring vortices and
# the wake ring vortices.
total_vortex_velocities = ring_vortex_velocities + wake_ring_vortex_velocities

# Calculate and return the solution velocities, which is the sum of the
# velocities induced by the vortices and
# freestream at every point.
solution_velocities = (
total_vortex_velocities + self.current_freestream_velocity_geometry_axes
)
return solution_velocities

def calculate_near_field_forces_and_moments(self):
"""This method finds the the forces and moments calculated from the near field.

Citation:
This method uses logic described on pages 9-11 of "Modeling of
aerodynamic forces in flapping flight with the Unsteady Vortex Lattice
Method" by Thomas Lambert.

Note: The forces and moments calculated are in geometry axes. The moment is
about the airplane's reference point, which should be at the center of
gravity. The units are Newtons and Newton-meters.

:return: None
"""

# Initialize a variable to hold the global panel position as the panel's are
# iterate through.
global_panel_position = 0

# Initialize three lists of variables, which will hold the effective strength
# of the line vortices comprising
# each panel's ring vortex.
effective_right_vortex_line_strengths = np.zeros(
self.current_airplane.num_panels
)
effective_front_vortex_line_strengths = np.zeros(
self.current_airplane.num_panels
)
effective_left_vortex_line_strengths = np.zeros(
self.current_airplane.num_panels
)

# Iterate through the current_airplane's wings.
for wing in self.current_airplane.wings:

# Convert this wing's 2D ndarray of panels into a 1D ndarray.
panels = np.ravel(wing.panels)

# Iterate through this wing's 1D ndarray panels.
for panel in panels:

# Check if this panel is on its wing's right edge.
if panel.is_right_edge:

# Change the effective right vortex line strength from zero to
# this panel's ring vortex's strength.
effective_right_vortex_line_strengths(
global_panel_position
) = self.current_vortex_strengths(global_panel_position)

else:

# Get the panel directly to the right of this panel.
panel_to_right = wing.panels(
panel.local_chordwise_position,
panel.local_spanwise_position + 1,
)

# Change the effective right vortex line strength from zero to
# the difference between this panel's
# ring vortex's strength, and the ring vortex strength of the
# panel to the right of it.
effective_right_vortex_line_strengths(global_panel_position) = (
self.current_vortex_strengths(global_panel_position)
- panel_to_right.ring_vortex.strength
)

# Check if this panel is on its wing's leading edge.

# Change the effective front vortex line strength from zero to
# this panel's ring vortex's strength.
effective_front_vortex_line_strengths(
global_panel_position
) = self.current_vortex_strengths(global_panel_position)
else:

# Get the panel directly in front of this panel.
panel_to_front = wing.panels(
panel.local_chordwise_position - 1,
panel.local_spanwise_position,
)

# Change the effective front vortex line strength from zero to
# the difference between this panel's
# ring vortex's strength, and the ring vortex strength of the
# panel in front of it.
effective_front_vortex_line_strengths(global_panel_position) = (
self.current_vortex_strengths(global_panel_position)
- panel_to_front.ring_vortex.strength
)

# Check if this panel is on its wing's left edge.
if panel.is_left_edge:

# Change the effective left vortex line strength from zero to
# this panel's ring vortex's strength.
effective_left_vortex_line_strengths(
global_panel_position
) = self.current_vortex_strengths(global_panel_position)
else:

# Get the panel directly to the left of this panel.
panel_to_left = wing.panels(
panel.local_chordwise_position,
panel.local_spanwise_position - 1,
)

# Change the effective left vortex line strength from zero to the
# difference between this panel's
# ring vortex's strength, and the ring vortex strength of the
# panel to the left of it.
effective_left_vortex_line_strengths(global_panel_position) = (
self.current_vortex_strengths(global_panel_position)
- panel_to_left.ring_vortex.strength
)

# Increment the global panel position.
global_panel_position += 1

# Calculate the solution velocities at the centers of the panel's front leg,
# left leg, and right leg.
velocities_at_ring_vortex_front_leg_centers = (
self.calculate_solution_velocity(points=self.panel_front_vortex_centers)
+ self.calculate_current_flapping_velocities_at_front_leg_centers()
)
velocities_at_ring_vortex_left_leg_centers = (
self.calculate_solution_velocity(points=self.panel_left_vortex_centers)
+ self.calculate_current_flapping_velocities_at_left_leg_centers()
)
velocities_at_ring_vortex_right_leg_centers = (
self.calculate_solution_velocity(points=self.panel_right_vortex_centers)
+ self.calculate_current_flapping_velocities_at_right_leg_centers()
)

# Using the effective line vortex strengths, and the Kutta-Joukowski theorem
# to find the near field force in
# geometry axes on the front leg, left leg, and right leg. Also calculate the
# force on each panel, which is derived from the unsteady Bernoulli equation.
near_field_forces_on_ring_vortex_right_legs_geometry_axes = (
self.current_operating_point.density
* np.expand_dims(effective_right_vortex_line_strengths, axis=1)
* nb_explicit_cross(
velocities_at_ring_vortex_right_leg_centers,
self.panel_right_vortex_vectors,
)
)
near_field_forces_on_ring_vortex_front_legs_geometry_axes = (
self.current_operating_point.density
* np.expand_dims(effective_front_vortex_line_strengths, axis=1)
* nb_explicit_cross(
velocities_at_ring_vortex_front_leg_centers,
self.panel_front_vortex_vectors,
)
)
near_field_forces_on_ring_vortex_left_legs_geometry_axes = (
self.current_operating_point.density
* np.expand_dims(effective_left_vortex_line_strengths, axis=1)
* nb_explicit_cross(
velocities_at_ring_vortex_left_leg_centers,
self.panel_left_vortex_vectors,
)
)
self.current_operating_point.density
* np.expand_dims(
(self.current_vortex_strengths - self.last_panel_vortex_strengths),
axis=1,
)
* np.expand_dims(self.panel_areas, axis=1)
* self.panel_normal_directions
)

# Sum the forces on the legs, and the unsteady force, to calculate the total
# near field force, in geometry
# axes, on each panel.
near_field_forces_geometry_axes = (
near_field_forces_on_ring_vortex_front_legs_geometry_axes
+ near_field_forces_on_ring_vortex_left_legs_geometry_axes
+ near_field_forces_on_ring_vortex_right_legs_geometry_axes
)

# Find the near field moment in geometry axes on the front leg, left leg,
# and right leg. Also find the
# moment on each panel due to the unsteady force.
near_field_moments_on_ring_vortex_front_legs_geometry_axes = nb_explicit_cross(
self.panel_front_vortex_centers - self.current_airplane.xyz_ref,
near_field_forces_on_ring_vortex_front_legs_geometry_axes,
)
near_field_moments_on_ring_vortex_left_legs_geometry_axes = nb_explicit_cross(
self.panel_left_vortex_centers - self.current_airplane.xyz_ref,
near_field_forces_on_ring_vortex_left_legs_geometry_axes,
)
near_field_moments_on_ring_vortex_right_legs_geometry_axes = nb_explicit_cross(
self.panel_right_vortex_centers - self.current_airplane.xyz_ref,
near_field_forces_on_ring_vortex_right_legs_geometry_axes,
)
self.panel_collocation_points - self.current_airplane.xyz_ref,
)

# Sum the moments on the legs, and the unsteady moment, to calculate the
# total near field moment, in
# geometry axes, on each panel.
near_field_moments_geometry_axes = (
near_field_moments_on_ring_vortex_front_legs_geometry_axes
+ near_field_moments_on_ring_vortex_left_legs_geometry_axes
+ near_field_moments_on_ring_vortex_right_legs_geometry_axes
)

# Initialize a variable to hold the global panel position.
global_panel_position = 0

# Iterate through this solver's panels.
for panel in self.panels:
# Update the force and moment on this panel.
panel.near_field_force_geometry_axes = near_field_forces_geometry_axes(
global_panel_position, :
)
panel.near_field_moment_geometry_axes = near_field_moments_geometry_axes(
global_panel_position, :
)

# Update the pressure on this panel.
panel.update_pressure()

# Increment the global panel position.
global_panel_position += 1

# Sum up the near field forces and moments on every panel to find the
# total force and moment on the geometry.
total_near_field_force_geometry_axes = np.sum(
near_field_forces_geometry_axes, axis=0
)
total_near_field_moment_geometry_axes = np.sum(
near_field_moments_geometry_axes, axis=0
)

# Find the total near field force in wind axes from the rotation matrix and
# the total near field force in
# geometry axes.
self.current_airplane.total_near_field_force_wind_axes = (
np.transpose(
self.current_operating_point.calculate_rotation_matrix_wind_axes_to_geometry_axes()
)
@ total_near_field_force_geometry_axes
)

# Find the total near field moment in wind axes from the rotation matrix and
# the total near field moment in
# geometry axes.
self.current_airplane.total_near_field_moment_wind_axes = (
np.transpose(
self.current_operating_point.calculate_rotation_matrix_wind_axes_to_geometry_axes()
)
@ total_near_field_moment_geometry_axes
)

# Calculate the current_airplane's induced drag coefficient
induced_drag_coefficient = (
-self.current_airplane.total_near_field_force_wind_axes(0)
/ self.current_operating_point.calculate_dynamic_pressure()
/ self.current_airplane.s_ref
)

# Calculate the current_airplane's side force coefficient.
side_force_coefficient = (
self.current_airplane.total_near_field_force_wind_axes(1)
/ self.current_operating_point.calculate_dynamic_pressure()
/ self.current_airplane.s_ref
)

# Calculate the current_airplane's lift coefficient.
lift_coefficient = (
-self.current_airplane.total_near_field_force_wind_axes(2)
/ self.current_operating_point.calculate_dynamic_pressure()
/ self.current_airplane.s_ref
)

# Calculate the current_airplane's rolling moment coefficient.
rolling_moment_coefficient = (
self.current_airplane.total_near_field_moment_wind_axes(0)
/ self.current_operating_point.calculate_dynamic_pressure()
/ self.current_airplane.s_ref
/ self.current_airplane.b_ref
)

# Calculate the current_airplane's pitching moment coefficient.
pitching_moment_coefficient = (
self.current_airplane.total_near_field_moment_wind_axes(1)
/ self.current_operating_point.calculate_dynamic_pressure()
/ self.current_airplane.s_ref
/ self.current_airplane.c_ref
)

# Calculate the current_airplane's yawing moment coefficient.
yawing_moment_coefficient = (
self.current_airplane.total_near_field_moment_wind_axes(2)
/ self.current_operating_point.calculate_dynamic_pressure()
/ self.current_airplane.s_ref
/ self.current_airplane.b_ref
)

self.current_airplane.total_near_field_force_coefficients_wind_axes = np.array(
(induced_drag_coefficient, side_force_coefficient, lift_coefficient)
)
self.current_airplane.total_near_field_moment_coefficients_wind_axes = np.array(
(
rolling_moment_coefficient,
pitching_moment_coefficient,
yawing_moment_coefficient,
)
)

def calculate_streamlines(self, num_steps=10, delta_time=0.1):
"""Calculates the location of the streamlines coming off the back of the wings."""

# Initialize a ndarray to hold this problem's matrix of streamline points.
self.streamline_points = np.expand_dims(self.seed_points, axis=0)

# Iterate through the streamline steps.
for step in range(num_steps):
# Get the last row of streamline points.
last_row_streamline_points = self.streamline_points(-1, :, :)

# Add the freestream velocity to the induced velocity to get the total
# velocity at each of the last row of
# streamline points.
total_velocities = self.calculate_solution_velocity(
points=last_row_streamline_points
)

# Interpolate the positions on a new row of streamline points.
new_row_streamline_points = (
last_row_streamline_points + total_velocities * delta_time
)

# Stack the new row of streamline points to the bottom of the matrix of
# streamline points.
self.streamline_points = np.vstack(
(
self.streamline_points,
np.expand_dims(new_row_streamline_points, axis=0),
)
)

def populate_next_airplanes_wake(self, prescribed_wake=True):
"""This method updates the next time step's airplane's wake.

:param prescribed_wake: Bool, optional
This parameter determines if the solver uses a prescribed wake model. If
false it will use a free-wake,
which may be more accurate but will make the solver significantly slower.
The default is True.

:return: None
"""

# Populate the locations of the next airplane's wake's vortex vertices:
self.populate_next_airplanes_wake_vortex_vertices(
prescribed_wake=prescribed_wake
)

# Populate the locations of the next airplane's wake vortices.
self.populate_next_airplanes_wake_vortices()

def populate_next_airplanes_wake_vortex_vertices(self, prescribed_wake=True):
"""This method populates the locations of the next airplane's wake vortex
vertices.

:param prescribed_wake: Bool, optional
This parameter determines if the solver uses a prescribed wake model. If
false it will use a free-wake,
which may be more accurate but will make the solver significantly slower.
The default is True.
:return: None
"""

# Check if this is not the last step.
if self.current_step < self.num_steps - 1:

# Get the next airplane object and the current airplane's number of wings.
num_wings = len(self.current_airplane.wings)

# Iterate through the wing positions.
for wing_num in range(num_wings):

# Get the wing objects at this position from the current and the next
# airplane.
this_wing = self.current_airplane.wings(wing_num)
next_wing = next_airplane.wings(wing_num)

# Check if this is the first step.
if self.current_step == 0:

# Get the current wing's number of chordwise and spanwise panels.
num_spanwise_panels = this_wing.num_spanwise_panels
num_chordwise_panels = this_wing.num_chordwise_panels

# Set the chordwise position to be at the trailing edge.
chordwise_position = num_chordwise_panels - 1

# Initialize a matrix to hold the vertices of the new row of wake
# ring vortices.
first_row_of_wake_ring_vortex_vertices = np.zeros(
(1, num_spanwise_panels + 1, 3)
)

# Iterate through the spanwise panel positions.
for spanwise_position in range(num_spanwise_panels):

# Get the next wing's panel object at this location.
next_panel = next_wing.panels(
chordwise_position, spanwise_position
)

# The position of the next front left wake ring vortex vertex
# is the next panel's ring vortex's
# back left vertex.
next_front_left_vertex = next_panel.ring_vortex.back_left_vertex

# Add this to the new row of wake ring vortex vertices.
first_row_of_wake_ring_vortex_vertices(
0, spanwise_position
) = next_front_left_vertex

# Check if this panel is on the right edge of the wing.
if spanwise_position == (num_spanwise_panels - 1):
# The position of the next front right wake ring vortex
# vertex is the next panel's ring
# vortex's back right vertex.
next_front_right_vertex = (
next_panel.ring_vortex.back_right_vertex
)

# Add this to the new row of wake ring vortex vertices.
first_row_of_wake_ring_vortex_vertices(
0, spanwise_position + 1
) = next_front_right_vertex

# Set the next wing's matrix of wake ring vortex vertices to a
# copy of the row of new wake ring
# vortex vertices. This is correct because this is the first time
# step.
next_wing.wake_ring_vortex_vertices = np.copy(
first_row_of_wake_ring_vortex_vertices
)

# Initialize variables to hold the number of spanwise vertices.
num_spanwise_vertices = num_spanwise_panels + 1

# Initialize a new matrix to hold the second row of wake ring
# vortex vertices.
second_row_of_wake_ring_vortex_vertices = np.zeros(
(1, num_spanwise_panels + 1, 3)
)

# Iterate through the spanwise vertex positions.
for spanwise_vertex_position in range(num_spanwise_vertices):

# Get the corresponding vertex from the first row.
wake_ring_vortex_vertex = next_wing.wake_ring_vortex_vertices(
0, spanwise_vertex_position
)

if prescribed_wake:

# If the wake is prescribed, set the velocity at this
# vertex to the freestream velocity.
velocity_at_first_row_wake_ring_vortex_vertex = (
self.current_freestream_velocity_geometry_axes
)
else:

# If the wake is not prescribed, set the velocity at this
# vertex to the solution velocity at
# this point.
velocity_at_first_row_wake_ring_vortex_vertex = (
self.calculate_solution_velocity(
np.expand_dims(wake_ring_vortex_vertex, axis=0)
)
)

# Update the second row with the interpolated position of the
# first vertex.
second_row_of_wake_ring_vortex_vertices(
0, spanwise_vertex_position
) = (
wake_ring_vortex_vertex
+ velocity_at_first_row_wake_ring_vortex_vertex
* self.delta_time
)

# Update the wing's wake ring vortex vertex matrix by vertically
# stacking the second row below it.
next_wing.wake_ring_vortex_vertices = np.vstack(
(
next_wing.wake_ring_vortex_vertices,
second_row_of_wake_ring_vortex_vertices,
)
)

# If this isn't the first step, then do this.
else:

# Set the next wing's wake ring vortex vertex matrix to a copy of
# this wing's wake ring vortex
# vertex matrix.
next_wing.wake_ring_vortex_vertices = np.copy(
this_wing.wake_ring_vortex_vertices
)

# Get the number of chordwise and spanwise vertices.
num_chordwise_vertices = next_wing.wake_ring_vortex_vertices.shape(
0
)
num_spanwise_vertices = next_wing.wake_ring_vortex_vertices.shape(1)

# Iterate through the chordwise and spanwise vertex positions.
for chordwise_vertex_position in range(num_chordwise_vertices):
for spanwise_vertex_position in range(num_spanwise_vertices):

# Get the wake ring vortex vertex at this position.
wake_ring_vortex_vertex = (
next_wing.wake_ring_vortex_vertices(
chordwise_vertex_position, spanwise_vertex_position
)
)

if prescribed_wake:

# If the wake is prescribed, set the velocity at this
# vertex to the freestream velocity.
velocity_at_first_row_wake_vortex_vertex = (
self.current_freestream_velocity_geometry_axes
)
else:

# If the wake is not prescribed, set the velocity at
# this vertex to the solution
# velocity at this point.
velocity_at_first_row_wake_vortex_vertex = np.squeeze(
self.calculate_solution_velocity(
np.expand_dims(wake_ring_vortex_vertex, axis=0)
)
)

# Update the vertex at this point with its interpolated
# position.
next_wing.wake_ring_vortex_vertices(
chordwise_vertex_position, spanwise_vertex_position
) += (
velocity_at_first_row_wake_vortex_vertex
* self.delta_time
)

# Set the chordwise position to the trailing edge.
chordwise_position = this_wing.num_chordwise_panels - 1

# Initialize a new matrix to hold the new first row of wake ring
# vortex vertices.
first_row_of_wake_ring_vortex_vertices = np.empty(
(1, this_wing.num_spanwise_panels + 1, 3)
)

# Iterate spanwise through the trailing edge panels.
for spanwise_position in range(this_wing.num_spanwise_panels):

# Get the panel object at this location on the next
# airplane's wing object.
next_panel = next_wing.panels(
chordwise_position, spanwise_position
)

# Add the panel object's back left ring vortex vertex to the
# matrix of new wake ring vortex
# vertices.
first_row_of_wake_ring_vortex_vertices(
0, spanwise_position
) = next_panel.ring_vortex.back_left_vertex

if spanwise_position == (this_wing.num_spanwise_panels - 1):
# If the panel object is at the right edge of the wing,
# add its back right ring vortex
# vertex to the matrix of new wake ring vortex vertices.
first_row_of_wake_ring_vortex_vertices(
0, spanwise_position + 1
) = next_panel.ring_vortex.back_right_vertex

# Stack the new first row of wake ring vortex vertices above the
# wing's matrix of wake ring vortex
# vertices.
next_wing.wake_ring_vortex_vertices = np.vstack(
(
first_row_of_wake_ring_vortex_vertices,
next_wing.wake_ring_vortex_vertices,
)
)

def populate_next_airplanes_wake_vortices(self):
"""This method populates the locations of the next airplane's wake vortices."""

# Check if the current step is not the last step.
if self.current_step < self.num_steps - 1:

# Get the next airplane object.

# Iterate through the copy of the current airplane's wing positions.
# for wing_num in range(len(current_airplane_copy.wings)):
for wing_num in range(len(self.current_airplane.wings)):

this_wing = self.current_airplane.wings(wing_num)
next_wing = next_airplane.wings(wing_num)

# Get the next wing's matrix of wake ring vortex vertices.
next_wing_wake_ring_vortex_vertices = (
next_wing.wake_ring_vortex_vertices
)

# Get the wake ring vortices from the this wing copy object.
pickle.dumps(
self.current_airplane.wings(wing_num).wake_ring_vortices
)
)

# Find the number of chordwise and spanwise vertices in the next
# wing's matrix of wake ring vortex
# vertices.
num_chordwise_vertices = next_wing_wake_ring_vortex_vertices.shape(0)
num_spanwise_vertices = next_wing_wake_ring_vortex_vertices.shape(1)

# Initialize a new matrix to hold the new row of wake ring vortices.
new_row_of_wake_ring_vortices = np.empty(
(1, num_spanwise_vertices - 1), dtype=object
)

# Stack the new matrix on top of the copy of this wing's matrix and
# assign it to the next wing.
next_wing.wake_ring_vortices = np.vstack(
(new_row_of_wake_ring_vortices, this_wing_wake_ring_vortices_copy)
)

# Iterate through the vertex positions.
for chordwise_vertex_position in range(num_chordwise_vertices):
for spanwise_vertex_position in range(num_spanwise_vertices):

# Set booleans to determine if this vertex is on the right
# and/or trailing edge of the wake.
has_right_vertex = (
spanwise_vertex_position + 1
) < num_spanwise_vertices
has_back_vertex = (
chordwise_vertex_position + 1
) < num_chordwise_vertices

if has_right_vertex and has_back_vertex:

# If this position is not on the right or trailing edge
# of the wake, get the four vertices
# that will be associated with the corresponding ring
# vortex at this position.
front_left_vertex = next_wing_wake_ring_vortex_vertices(
chordwise_vertex_position, spanwise_vertex_position
)
front_right_vertex = next_wing_wake_ring_vortex_vertices(
chordwise_vertex_position, spanwise_vertex_position + 1
)
back_left_vertex = next_wing_wake_ring_vortex_vertices(
chordwise_vertex_position + 1, spanwise_vertex_position
)
back_right_vertex = next_wing_wake_ring_vortex_vertices(
chordwise_vertex_position + 1,
spanwise_vertex_position + 1,
)

if chordwise_vertex_position > 0:
# If this is isn't the front of the wake, update the
# position of the ring vortex at this
# location.
next_wing.wake_ring_vortices(
chordwise_vertex_position, spanwise_vertex_position
).update_position(
front_left_vertex=front_left_vertex,
front_right_vertex=front_right_vertex,
back_left_vertex=back_left_vertex,
back_right_vertex=back_right_vertex,
)

if chordwise_vertex_position == 0:
# If this is the front of the wake, get the vortex
# strength from the wing panel's ring
# vortex direction in front of it.
this_strength_copy = this_wing.panels(
this_wing.num_chordwise_panels - 1,
spanwise_vertex_position,
).ring_vortex.strength

# Then, make a new ring vortex at this location,
# with the panel's ring vortex's
# strength, and add it to the matrix of ring vortices.
next_wing.wake_ring_vortices(
chordwise_vertex_position, spanwise_vertex_position
) = ps.aerodynamics.RingVortex(
front_left_vertex=front_left_vertex,
front_right_vertex=front_right_vertex,
back_left_vertex=back_left_vertex,
back_right_vertex=back_right_vertex,
strength=this_strength_copy,
)

def calculate_current_flapping_velocities_at_collocation_points(self):
"""This method gets the velocity due to flapping at all of the current
airplane's collocation points."""

# Check if the current step is the first step.
if self.current_step < 1:
# Set the flapping velocities to be zero for all points. Then, return the
# flapping velocities.
flapping_velocities = np.zeros((self.current_airplane.num_panels, 3))
return flapping_velocities

# Get the current airplane's collocation points, and the last airplane's
# collocation points.
these_collocations = self.panel_collocation_points
last_collocations = self.last_panel_collocation_points

# Calculate and return the flapping velocities.
flapping_velocities = (these_collocations - last_collocations) / self.delta_time
return flapping_velocities

def calculate_current_flapping_velocities_at_right_leg_centers(self):
"""This method gets the velocity due to flapping at the centers of the
current airplane's bound ring vortices'
right legs."""

# Check if the current step is the first step.
if self.current_step < 1:
# Set the flapping velocities to be zero for all points. Then, return the
# flapping velocities.
flapping_velocities = np.zeros((self.current_airplane.num_panels, 3))
return flapping_velocities

# Get the current airplane's bound vortices' right legs' centers, and the
# last airplane's bound vortices' right
# legs' centers.
these_right_leg_centers = self.panel_right_vortex_centers
last_right_leg_centers = self.last_panel_right_vortex_centers

# Calculate and return the flapping velocities.
flapping_velocities = (
these_right_leg_centers - last_right_leg_centers
) / self.delta_time
return flapping_velocities

def calculate_current_flapping_velocities_at_front_leg_centers(self):
"""This method gets the velocity due to flapping at the centers of the
current airplane's bound ring vortices'
front legs."""

# Check if the current step is the first step.
if self.current_step < 1:
# Set the flapping velocities to be zero for all points. Then, return the
# flapping velocities.
flapping_velocities = np.zeros((self.current_airplane.num_panels, 3))
return flapping_velocities

# Get the current airplane's bound vortices' front legs' centers, and the
# last airplane's bound vortices' front
# legs' centers.
these_front_leg_centers = self.panel_front_vortex_centers
last_front_leg_centers = self.last_panel_front_vortex_centers

# Calculate and return the flapping velocities.
flapping_velocities = (
these_front_leg_centers - last_front_leg_centers
) / self.delta_time
return flapping_velocities

def calculate_current_flapping_velocities_at_left_leg_centers(self):
"""This method gets the velocity due to flapping at the centers of the
current airplane's bound ring vortices'
left legs."""

# Check if the current step is the first step.
if self.current_step < 1:
# Set the flapping velocities to be zero for all points. Then, return the
# flapping velocities.
flapping_velocities = np.zeros((self.current_airplane.num_panels, 3))
return flapping_velocities

# Get the current airplane's bound vortices' left legs' centers, and the last
# airplane's bound vortices' left
# legs' centers.
these_left_leg_centers = self.panel_left_vortex_centers
last_left_leg_centers = self.last_panel_left_vortex_centers

# Calculate and return the flapping velocities.
flapping_velocities = (
these_left_leg_centers - last_left_leg_centers
) / self.delta_time
return flapping_velocities

### Goals 0-4

I think that my tool satisfies design goals zero through four. It is capable of analyzing the forces and moments on any user defined geometry that is flapping with any user defined motion. Setting and running up a simulation can be done with a script such as:

import pterasoftware as ps

example_airplane = ps.geometry.Airplane(
wings=(
ps.geometry.Wing(
symmetric=True,
wing_cross_sections=(
ps.geometry.WingCrossSection(
airfoil=ps.geometry.Airfoil(name="naca2412",),
),
ps.geometry.WingCrossSection(
y_le=5.0, airfoil=ps.geometry.Airfoil(name="naca2412",),
),
),
),
),
)

example_operating_point = ps.operating_point.OperatingPoint()

airplane=example_airplane, operating_point=example_operating_point,
)

)

example_solver.run()

ps.output.draw(
solver=example_solver, show_delta_pressures=True, show_streamlines=True,
)

The source code is heavily commented, uses the Black python formatter, gets an A code quality grade via CodeFactor, has 94% testing coverage, and implements some basic CI techniques. Additionally, I just finished a validation study that compared the solver’s output to experimental data:

Those curves might not be identical, but they are pretty close for a mid-fidelity unsteady fluids solver! A full Navier-Stokes simulation using software like Ansys would be more accurate, but unsteady high-fidelity solvers are the stuff of HPC clusters running one simulation for hours or even days.

While I feel okay with goals zero through four, I’ve learned that my intuition about these things is most often wrong. So, please feel free to rip apart my project on any of these topics!

### Goal 5

This is my first ever open-source project, so maybe my expectations are unrealistic. However, I was hoping for more community engagement and use. I’ve posted about my repository on Reddit, FaceBook, and LinkedIn, and only two people have opened issues on GitHub. Does that sound reasonable given how niche the idea is? Is there anything I can do to increase engagement?

### Goal 6

The latest version can run a typical simulation in around 156 seconds. This means that in 24 hours, I could do about 553 runs. The latest, non-packaged version (the vectorization branch on GitHub) leans heavily on NumPy and Numba to increase speed. Until recently, a single function in aerodynamics.py, calculate_velocity_induced_by_line_vortices was responsible for around 60% of my run time. Thanks to StackOverflow user Jérôme Richard, who answered my optimization question here, this is no longer the case.

However, my code is still two times too slow. How can I optimize the system as a whole? For example, to increase ease-of-use, I went with a heavily object-oriented approach and store many large instances of custom objects. My class hierarchy can be found in geometry, aerodynamics, and movement.py. I wasn’t able to include it because of character limit.

Should I simply focus on parallelizing my calls to the computationally intensive functions? How close to the theoretical speed limit of this simulation using Python and a workstation laptop?

## Future Work

Other features I have planned are:

• Developing a workflow for using Ptera Software to perform system identification for a flapping wing control system
• Modifying the algorithm to analyze hovering flapping wings (right now, the assumptions of the model aren’t valid for situations where the vehicle isn’t moving forward at some velocity)
• Creating either a CLI or GUI
• Implementing aeroelastic (flexible wing) effects
• Tightening the speed requirement even further (i.e., could the solver run in under 10 seconds?)

How important do you think each of these features would be to users?

## Closing

You can’t talk about ornithopters without mentioning Dune…so I hope this project pleases you, and the house Atreides.

## physics – Simulating linear inertia

My question is simple.
I have an aabb (player) which can move dynamically – to get acceleration I am saving the previous previous position (through prevPrevPos, prevPos and pos acceleration will be calculated). Now I have a tiny aabb which could fall on that player aabb (collision friction is a thing in my system).
How would I simulate inertia?
My current approach is not quite right – on collision I pass acceleration into the tiny aabb’s velocity and set it’s position in such a way that it sticks on the top. In the next iteration the tiny aabb would then calculate friction etc. so eventually it will not move and stay in one place and move with the player.

Problem: when it falls off the player it has an acceleration and therefore flys away with high speeds in the opposite direction of the player’s movement, which is not quite the right behaviour if I am understanding inertia correctly.

What would be common approaches for inertia?

## wi fi – Simulating WiFi device in emulator with mac80211_hwsim kernel module

I am looking to simulate an additional WiFi device in a rooted android emulator (to simulate a device my company makes). So far I have found the linux kernel module, mac80211_hwsim, which seems like it would help me create virtual networks like this. I believe this is also the mechanism that the emulator uses to create its fake wifi network already. I have found /vendor/bin/mac80211_create_radios which seems to allow me to access that module (to create more channels / radios which appear in ip show link as wlan*).

If this is in fact the right way forward, once I create some more channels for that kernel module, I want to be able to connect to a program on my local computer after connected to this other WiFi network (similar to how 10.0.2.2 maps to the host computer). (this looks helpful with the redir command)

Does this sound reasonable and does anyone have some tips going forward? It’s been challenging to find information that is relevant to Android in this area.

## physics – Simulating Gas Density and Pressure in a 2D World

I’m building a small spaceship simulation app that looks a lot like a game for an upcoming talk I’m giving where I use this sample app to teach the F# programming language.

This small app is something like FTL meets Oxygen Not Included where you have a top-down 2D grid of tiles (similar to an old RPG) where each tile has its own mixture of gasses – right now oxygen and carbon dioxide, but potentially others.

I’ve got a few things I’m trying to simulate:

1. When new gasses are added to a tile by something like a vent or a life support system, that gas should expand to neighboring tiles if possible
2. When a pressure changes (e.g. opening a door to another area of the ship or a hull breach), air should flow from the high pressure tile to the low pressure tile next to it.

Given this, and given that some gasses naturally sift to the top of others, I’m trying to figure out a small set of simple rules to govern this behavior.

Previously I had all gasses equalizing with their neighbors and no concept of pressure, but that made it very difficult to treat scenarios like hull ruptures, so I’m looking for something a bit more realistic without getting complex or hyper-accurate.

For example, given tile A with 15g oxygen and 6g CO2 and neighboring tile B of 3g oxygen and 1g CO2, some air should clearly flow from A to B. However, what flows? Is it the lightest gasses? The heaviest gasses? A random or representative sampling of gasses in A? Are there any relevant physics principles I should be aware of?

Note: I posted here instead of in physics because I don’t care extremely about nuanced accuracy, just something simple and believable

## Stop problem – help find an argumentation error when simulating large Turing machines with smaller ones

I have an argument that, if it goes through, almost proves that either:

• Programming languages ​​are more powerful than Turing machines
• The busy beaver function ($$BB ()$$) is calculable on Turing machines

Now I understand that my argument is far more likely to have an error that I can't find. But it's more interesting to me How I'm more wrong than if I am wrong.

Meanwhile, I use $$mathbb {S} (x)$$ either means the number of states the Turing machine $$x$$ or the number of states that a Turing machine needs to write $$x$$ on the tape.

Build a Turing machine $$M_1$$ as what takes as arguments (on the tape) $$n, k$$simulates all Turing machines with $$n$$ States to $$k$$ of them stop and then stop. This is easy to write in a programming language, as the following Python snippet shows:

def M1(n, k):
all_machines = generate_turing_machines(n)
is_halted = (0) * len(generate_turing_machines)
while sum(is_halted) < k:
for (i, machine) in enumerate(all_machines):
machine.step()
if machine.is_halted():
is_halted(i) = 1

Now leave $$mathbb {S} (M_1) = m_1$$ be the number of $$M_1$$. fix $$n$$ much larger than $$m_1$$. To let $$k_1$$ be the largest number so that $$M_1 (n, k_1)$$ stops and $$k_0$$ be the smallest number, so if $$M_1 (n, k_0)$$ stops $$k_1$$ Simulated Turing machines were stopped (since all equivalent machines will stop in the same step). Choose $$k$$ With $$k_0 leq k leq k_1$$. It means that $$M_1 (n, k)$$ stops approximately $$BB (n)$$ Steps.

To construct $$M_2$$ that's the same as $$M_1$$ other than the first thing it does is write $$n$$ and $$k$$ on the tape. To let $$mathbb {S} (M_2) = m_2$$. Then $$m_1 + mathbb {S} (n) + mathbb {S} (k) + C = m_2$$ for some little ones $$C$$ (which is probably constant and likely $$0$$).

Now, $$mathbb {S} (n)$$ is at most $$O (log (n))$$. $$k$$ is about $$n ^ n$$, so $$mathbb {S} (k)$$ is at most $$O (n)$$. That sets $$m_2$$ just a little bigger than $$n$$. But here we have a problem: if $$k$$ is then a little easier to write on the tape $$m_2$$ would be a little smaller than $$n$$. That would mean $$BB (m_2)> BB (n)$$ and $$m_2 a clear contradiction.

In my eyes, these are the possible solutions:

• $$M_1$$ It is impossible to create a Turing machine, which means that Python is more powerful than Turing machines.
• There is a transfinite extension for Turing machines that is not much more powerful than Turing machines in general $$M_1$$ can be written in this extension. In other words, $$M_1$$ is the limit of a number of machines $$M_ {1, N}$$that everyone can deal with $$n . This would likely mean that the busy beaver function is predictable.
• There are a large number of numbers that a Turing machine cannot write in much less than $$log (k) = n$$ States (we need $$mathbb {S} (k) ). It seems impossible to me that there is no candidate for $$(n, k)$$ could be compressed sufficiently.

What is the flaw in this logic?