# Example of real-life problem solved with numerical methods? [closed]

I hope this is a suitable question to ask here.

Well, I'm taking Numerical Methods this semester (also called 'Numerical Analysis' in some places).

The term-assignment is to find a real-life problem which is solvable by numerical methods. And then present two different methods to solve it.

It doesn't have to be something new, simply presenting someone else's solution is acceptable.

This seems like an incredibly easy thing but I'm having a hard time finding something of reasonable difficulty to use.

Any ideas or suggestions?

• The development of the first atomic bomb ? Sep 14 '18 at 21:48
• While I appreciate your interest, these types of questions are really just polls. There is no one answer. This site is set up to have direct questions with direct answers. This is different from your typical forum.
– hazzey
Sep 15 '18 at 1:19
• Problem 1. Consider a ship to be represented by it's midship cross-section. For a fixed girth (this is an isoperimetric optimization problem) and a given ballast density, what is the shape of the cross-section, weight of ballast, and downflooding angle that maximizes the righting moment? Assume hull is thin and weightless, thus CG is that of the ballast which is fixed in the bottom of the hull. Solve for ballast densities of 1, 4 and 10. Problem #2. This takes a bit more creativity. Design a hot air balloon. As a bag in a net with a weight hanging below, explain its flying shape. Sep 15 '18 at 2:14

## 2 Answers

You could develop some saturated steam equations by running a regression on steam table data. Even if it was a giant equation, sure would be nice to not have to do a a table lookup in an otherwise automatic spreadsheet or script. I attempted to use some I found on the internet the other day and only one of them worked (saturated steam equations SE question).

Anyhow, there is one idea. If you do it you should definitely share ;-)

A heat transfer problem might be suitable here. Temperature is an intuitive parameter and a simple scalar, and energy balances are easy to write down but often challenging to solve analytically because of temperature-dependent material properties, for example.

I wrote here about the parabolic and catenary temperature profiles that arise in a simple 1-D geometry with heat generation. One practical application is described here: a microfabricated suspended silicon beam that heats up from an applied current, expands, and deflects—thus, a microscale linear actuator:

Let's consider the temperature profile only and forget about the motion. In the second link, I write about how the time-dependent analysis diverges from the experimental results because the analytical solution doesn't incorporate the temperature dependence of certain material properties.

As you request, there are at least two ways to obtain a more accurate temperature distribution numerically:

(1) One could use a lookup table for the temperature-dependent material properties (for simplicity, maybe just one material property, say, the thermal conductivity of silicon) and perform a 1-D finite-difference heat transfer analysis by discretizing the beam into segments, each with a uniform temperature.

(2) One could fit the temperature-dependent material property of interest by an analytical function and solve the resulting differential heat transfer equation (namely, $k(T)\frac{\partial^2T(x,t)}{\partial x^2}+J^2r=c\rho\frac{\partial T(x,t)}{\partial t}$, where $k$ is the thermal conductivity, $T$ is temperature, $x$ is the position, $t$ is time, $J$ is the current density, $r$ is the resistivity, $c$ is the heat capacity, and $\rho$ is the density) using a preferred numerical scheme. Or for simplicity, one could drop the time dependence and simply solve $k(T)\frac{\partial^2T(x,t)}{\partial x^2}+J^2r=0$. (Here, if $k$ were constant, we'd simply obtain a parabola.)