If you walk down any given hallway in a place where STEM (Science, Technology, Engineering, Math) happens, it is becoming increasingly likely that you will see colorful 3D printed props adorning offices and cubicles. Affordable 3D printing and 3D printers are taking the world by storm, enabling home workshops, public schools, and, yes, even math departments to enjoy this new and exciting technology. If you have ever wanted to be surrounded by your math (in a literal sense), this can easily happen. Mathematical constructs are leaving the dimension of the mind and preparing for a full-scale invasion of the dimension of the desks! Here we provide some highlights from our experience with 3D printing in research and teaching over the past year, specifically in regards to partial differential equations (PDEs) with time-dependent solutions. The article is separated into 3 sections: (1) Printing for teaching PDEs, (2) How to make 3D prints of PDEs, and (3) Printing PDEs for research.

1. 3D Printing for PDE Pedagogy

In math, multivariable calculus is the course that typically comes to mind for incorporating 3D printing into the classroom. Another good fit is Boundary Value Problems, an undergraduate course full of beautiful solutions, both mathematically and physically. The course can be taught from several perspectives, but it is safe to say that all approaches incorporate the study of solutions to PDEs describing diffusion (heat) and/or convection (waves).   

Figure 1: Solutions to (a) a modified 1D wave equation [2] (Eq. 22) with homogeneous Neumann conditions on each side and (b) the standard 1D wave equation with a homogeneous Dirichlet condition on the left side and a homogeneous Neumann condition on the right side. These pieces are part of a set used in a workshop where students match up boundary conditions to 3D prints. The idea is that, early in the course, students see the effect of these boundary conditions on the solution.

Analytical solution methods for PDEs (if given in detail) can be rather lengthy, and the payoff of a beautiful solution can be a welcome sight. For time-varying PDEs, computer animations of a melted or propagated shape are useful for visualizing solutions. Taking this one step further, imagine—as a student—holding an object in your hand that represents a time-history of the solution to a PDE. Consider the 3D print in Figure 1a with homogenous Neumann boundary conditions (i.e., reflective walls). Here, one can make horizontal scans with the index finger from the middle to the edges of the object and feel that the slope at the boundaries approaches zero for all time (shifting the horizontal scan up and down), even though the solution itself (height of the printed object) varies wildly with time. A comparison can then be made for prints of the same (or similar) PDE with different boundary conditions (see Figure 1b). This exercise is part of a 20-minute workshop that we run in class, in which we provide a set of 3D prints to each group of 3-4 students. This type of activity promotes discussion among students as they try to pair boundary conditions with the printed objects. 

An example of the 1D heat equation with non-homogeneous boundary conditions is given in Figure 2, where one may track with their fingers in the direction of time from the initial condition (bottom of figure) to steady state (top of figure) and feel the loss of time dependence but not space dependence on the solution.

Figure 2: The above pieces are used to illustrate the concept of steady-state solutions to the 1D heat equation, when the boundary conditions are non-homogeneous. These are useful for demonstrating that different initial conditions lead to the same steady-state solution in the limit of long time.

For certain topics in a PDE course, a bit of imagination may be needed to bring the problems to life. One workshop focuses on the type of functions that can be represented by a Fourier Series in a uniformly convergent way. The students are given eight 3D printed puzzle pieces that together form a solution to the 1D heat equation (Figure 3). They are asked to assemble the pieces such that the initial condition (the inner face of the objects) is continuous on its domain and on its periodic extension. This leads to eight different possibilities. They are then asked to calculate the Fourier series of the initial condition. If they arrange the pieces such that the initial condition is odd, computing the Fourier coefficient integrals becomes easier. While this exercise can be fun and contains some strategy, appropriate time and guidance is needed to allow students to solve the "puzzle" in their own  way.

Figure 3: Students assemble a solution to the 1D heat equation with an eight-piece piecewise initial condition (photo credit: Ginny Gross). The pieces fit together when the initial condition satisfies the requirements for its Fourier series to converge: continuous, piecewise continuous derivatives, and continuous on its periodic extension. The second picture demonstrates the latter requirement by connecting several sets in a loop (the inner face is the initial condition).

 

Currently, we are using 3D prints in lectures for demos, for workshops (as illustrated above), and also for homework problems. A gallery containing the 3D printed solutions of the homework problems is on display in a window corner of a locked room, allowing students to check their final answer after solving a PDE and plotting it in MATLAB (or equivalent). A snapshot of a problem in the current gallery is given in Figure 4.

Figure 4: A PDE gallery for students to verify their homework solutions.

 

Problems in a PDE course become arguably more interesting near the end of the year, as additional dimensions and coordinate systems become fair game. A modification of the 2D wave equation is shown in Figure 5, were individual time frames have been printed and placed in sequence. Here, plastic jewelry display stands (purchased cheaply online) are used to highlight the progression of a 2D wave in time. 

Figure 5: Solution to a modified 2D wave equation shown in increments of 0.5 time units. Top: homogeneous Neumann boundary conditions on all 4 sides (reflective walls). Bottom: homogeneous Dirichlet conditions on all 4 sides (drum).

 

The examples given thus far illustrate how 3D printing can augment the presentation of PDE solutions. Switching roles, PDEs can assist with 3D printing itself. For example, fractals can be difficult to print in 3D, due to sharp gradients (see Figure 6b). The problem is that these "spikes" may be smaller than the resolution attainable from a modest 3D printer. Fortunately for those of us who use modest 3D printers, the heat equation can be used to diffuse a fractal until it is smooth enough to be printed, as shown in Figures 6b and 6c. This type of example may be useful as a hands-on supplement to a lesson on image noise reduction using the 2D heat equation.

Figure 6: The Julia fractal (a) as it typically appears (bird's-eye view) (b) viewed from an angle (c) diffused by the 2D heat equation and (d) 3D printed.

2. How the 3D Prints Are Made: Getting Started

All of our 3D prints begin as matrices. For each problem, two independent variables are chosen for the dimensions of the matrix and the dependent variable is represented by the values in the matrix. We then use Sven Holcombe's MATLAB functions that transform a matrix into an .stl file [3, 4], making it ready for the 3D printer. The process is illustrated in Figure 7.

Figure 7: Process for making 3D prints of PDEs using MATLAB. (a) Solve the problem and plot the solution. Here, a familiar silhouette is used as an initial condition to the 1D wave equation. (b) Use the "mesh" (or equivalent) command to view the solution as a surface using time as a coordinate. Make sure that the axes have equal spacing, as this is how it will appear when printed. (c) Use the above commands (after downloading functions [3] and [4]) to generate the .stl file. (d) Open the .stl file in a freeware program such as CURA and follow the appropriate steps (associated with your printer) to print the object. The above print took 45 minutes.

If you feel a barrier to getting started, you are not alone. The following is what worked for us:

• If you are thinking of buying a 3D printer, it may be helpful to find a student or colleague who has already purchased a particular brand of printer and knows how to set it up and operate it. For the 3D prints shown and described here, the printed objects are made using various colors of Hatchbox PLA plastic ($20-$35, 1 kg spool) in a Maker Select 3D v2 printer ($300).
• Spread good will in your department and organization by printing objects for your colleagues. If your 3D printer requires ventilation, perhaps a colleague in engineering or chemistry will let you store and operate the printer in a ventilated area when labs are not being run.
• Print, print, print,  for every occasion! This will give you the trial and error you may need before printing objects for class. Print 3D business cards for conferences. If you are giving a talk in a dark room, use glow-in-the-dark material. Make mini-museum exhibits and ask for space to show them off (great for open houses!). Can you think of other ideas?

3. 3D Printing PDEs for Research

In the March SIAM news article, "Telling a Good Story at Conference" [5], an eye-opening suggestion was given to pass around 3D prints to the audience during a research presentation. The author (N. Higham) also points out that "Finding something suitable to print, however, may not be easy!" We couldn't agree more, but if 3D printing is at least on one's radar, ideas will likely come later.

When introducing our research to new group members or talking at conferences, 3D prints have become the perfect tool for showing the difference between a linearly stable, convectively unstable, and absolutely unstable response to a localized initial disturbance. A comparison is given in Figure 8, where wave propagation differences may be characterized by the spreading character of the packet in space and time.