It is often important to develop new theories and models that allow prediction of materials properties or performance without the necessity to measure them all first. There are a whole host of ways to achieve this, each with their own merits, assumptions, simplifications and complexities.
Analytical models attempt to create a set of mathematical equations that link a parameter that can be measured (such as electrical polarisation for example) to a known theory that reveals the physics of the problem. Along the way, mathematical simplifications are often introduced that act to realise an analytical expression. Sometimes however, this isn’t possible to achieve, and then we might explore alternative ways of modelling or predicting our materials response.
Empirical methods are based on a set of experimentally validated equations, that have not simply been derived from the principle physics of interest. These methods are perfectly adequate for many systems but they do not provide any deeper understanding of the origins of the materials performance, or its response to some excitation.
Finally, a set of computer based approaches exist that aims to solve the problem by applying known physics to a cut-and-diced representation of the material sample. These methods are known as finite element or finite difference approaches and use fast computers to iteratively solve and refine the problem until a unique solution becomes apparent.