Introduction

When designing hypersonic flight vehicles, heating is a first order concern. The skin temperatures

can exceed $1000$ Kelvin and the internal structures also get very hot. It is important to analyse the

expected temperatures in the vehicle so that appropriate engineering decisions are made early in the

design process. In this assignment, you will build a method $($using Python$)$ to estimate temperatures

on internal structures of a hypersonic vehicle. The particular vehicle for analysis is shown in Figure $1$

with the black outline depicting the cross$-$section of interest. We assume there is an internal structure

at this location to which the aircraft skin attaches. ${?}^1$

Figure $1$: A hypersonic vehicle. Top: view from underside. Bottom: view from top. The black outline

represents the cross$-$section of interest.

You will use a method called "walk$-$on$-$spheres" to calculate the temperature at an internal point of

the cross section. This method uses a simple but powerful physical idea, and a simple but powerful

algorithm efficiency idea. The physical idea for estimating temperature at a point $(x_0)$ inside a domain

$()$ is:

take a number of random walks starting from the point $(x_0)$ and grab the temperature value at

the boundary when the walk hits the boundary;

average the boundary temperature values obtained from all random walks to get the estimate of

internal temperature.

This estimate gets better as more walks are taken. An example of a single walk is shown in Figure $2($a$).$ Now, the random walk can be slow. We can speed this up with an efficiency idea. At any point, let's

find the largest circle that touches the boundary. Instead of a random walk using a small step, we will

take a $($large$)$ random jump on the circle. This moves us across the domain much more quickly. We

repeat the process until we hit $($or get very close to$)$ a boundary. This idea is shown in Figure $2($b$).$

This is why the method is named walk$-$on$-$spheres: we are taking a random walk across spheres. This

idea was demonstrated by Muller $[2]$ in $1956.$ In this assignment, we will work in two$-$dimensions; we

use circles $($in $2$D$)$ instead of spheres. Do not worry about the details now. The assignment tasks will

guide you to build up the method.

The assignment is divided into two parts:

Part $1$

In this part, you will build the walk$-$on$-$spheres method for a square domain using procedural

programming techniques. This will allow you to understand the method and, by using a square

domain, allows us to test our code by comparing to an answer computed by an independent

method.

Part $2$

In the second part, you will build the walk$-$on$-$spheres method using an object$-$oriented design.

This program will be more powerful than the one from Part $1$ because you will be able to compute

temperatures on geometries more complex than a square. You will apply your program to compute

a temperature on the interior of the cross$-$section of a hypersonic vehicle, as shown in Figure $1.$

PART $1$: procedural program for walk$-$on$-$spheres on a square domain

The algorithm for performing a walk$-$on$-$spheres estimate on a square domain is given in Figure $3.$ In

the tasks that follow, we first build pieces of the algorithm and then assemble the whole.Write a function, nearest$\_$edge$\_$square $(x,y,$ length$),$ which returns a tuple containing an integer

or $3$