How Do Models Fit into the Processes of Science and Engineering?

Now that we’ve seen a bunch of examples of models, let’s discuss how models fit into scientific and engineering practice. We’ll start with science.

Models in the Scientific Process

The process of science can be represented as a cycle of four undertakings (Figure 2.6.1):

1. Modeling and Theorizing
2. Questioning
3. Investigating
4. Analyzing

This diagram is in fact a process model of the scientific inquiry. As all models are only approximations, the real process is of course quite messier than this, but it is still a useful model. "Modeling and Theorizing" is placed at the top of the cycle because, in some sense, it is the beginning and end of a successful scientific inquiry process. This is reflected in the standard structure of scientific papers:

• Background: explains current theory/models and empirical findings. Then identifies a gap in current understanding
• Research questions: designed to fill a gap in current understanding
• Methods: explains the investigation process
• Analysis: analyzes data that resulted from the investigation to construct findings
• Discussion: links findings back to theory to explain what has been added to our understanding

That said, the real scientific process is rarely this tidy and linear. For example, scientist may have to repeat certain parts of the process many, many times before going to the next part. It also neglects a number crucial elements of the scientific process, such as why scientists get interested in certain questions to begin with.

Example: The Scientific Process and Boyle's Law

Here is an example of a scientific inquiry process that resulted in the development of a mathematical model now known as Boyle's Law.

• Questioning: how are pressure and volume related for a ideal gas at fixed temperature and fixed number of moles of the gas?
• Investigating: Boyle (together with Robert Hooke, who we come back to later) built an apparatus to measure pressure and volume and then used it gather data.
• Analyzing: analyzed the data graphically
• Creating model: created mathematical model that proposed $P=\frac{k}{V}$, where $P$ is the pressure of the gas, $V$ is the volume of the gas, and $k$ is some proportionality constant.

Boyle's Law was later incorporated into the the more complete model known as the ideal gas law, a mathematical model that relates a number of measurable quantities together: (PV=nRT). This more advanced model also included the system temperature and the number of particles in the gas (which in Boyle's approach were held constant and combined into a factor $k$).

Example of (Computational) Model-Based Scientific Investigation

In the case of Boyle's Law, the investigation phase of the process involved physical experiments. However, it is often the case that a cycle of scientific inquiry happens fully within the realm of models. Here is an example using the computational forest fire model from earlier.

• Existing model or theory: We are starting with the simple model we already built.
• Questioning: how do fire dynamics in the model differ when forests have differing densities?
• Investigating: our simple experimental plan will be to run the model with a range of densities and see how that changes the amount of the forest that burns.
• Analyzing: we'll plot percent burned vs tree density and then draw conclusions from this plot.

Now use NetLogo model 2.6.1 below to answer the two questions in Exercise 2.6.1 below it.

These two examples of the scientific process - Boyle's experimental and the forest fire's computational - are complementary. Investigations of the physical world can lead us to create new models or revise existing ones. Investigations of existing models can point us in the direction of new experiments in the physical world to confirm or disconfirm the model.

Models in Engineering Process

In the previous section we showed how to use models in the scientific process to understand and describe physical phenomenon. Models are also used extensively in engineering practice. Models are used frequently in design. Before building anything physical, models can be used to explore the design space and come up with some potential solutions to an engineering problem. Models can be used to predict the performance of a design under various conditions. In modern engineering, the line between engineering and science is also often blurred. If an engineered article fails for an unknown reason, part of the engineering process is figuring out why. This will essentially entail a scientific inquiry process, including constructing models to explain what happened.

You may have encountered engineering modeling in your own studies: these include computer-aided design (3D models), mathematical models in continuum mechanics (e.g., statics or mechanics courses), finite element modeling, optimization modeling in industrial engineering, financial modeling to predict market behavior, etc. In MSE, there are numerous modeling approaches that maximize materials performance. Indeed, efficient implementation of modeling into MSE design workflows has developed it's own sub-discipline, called integrated computational materials engineering. Those who want to learn how this burgeoning technique helps advances critical areas of MSE.