Multiscale modelling: a new challenge for physicists and mathematicians.
The aim of multiscale modelling is to predict the behaviour of complex materials, including biomaterials such as proteins, across a range of length and time scales.
At the molecular scale, quantum mechanics methods are required to describe the interactions between atoms and electrons. As one moves from the atomic scale towards the macroscale in solids, for example, the performance at the micrometre scale is governed by the behaviour of defects and dislocations. At the next larger scale, the effects of grain boundaries and ensembles of defects become important. Finally, at the macro or continuum scale, the behaviour of materials may be dominated by environmental or loading factors such as applied stresses or temperature gradients.
The key questions for mathematical modellers are:
- How does one develop a range of computational techniques which can tackle this wide range of scales?
- How does one ensure that the results of modelling at one length (or time) scale can be connected to results at the next higher or lower scale?
Ideally one would like to begin at the quantum mechanical level and perform calculations on electrons and nuclei, using only the tools of theoretical physics, that is, knowledge of the particles present and the interactions between them. In the past, such calculations could only be performed for relatively small systems, but modern computing developments have made them tractable for large systems of atoms and molecules, usually with the employment of various approximation methods. Currently researchers are developing new quantum-mechanics-based modelling techniques for simulating large systems containing many thousands of atoms.
Graphical demonstration of the range of problems to which multiscale modelling can make a contribution, from the electronic scale of ab initio methods to the meso- and continuum scale and from the study of DNA and proteins, to crystal structures, and to bulk materials properties.
Such large quantum mechanics systems are still a long way from what is required to understand materials at the micro and mesoscales. The final goal is "seamless modelling from atoms to structures", with quantum mechanics models at one end of the length scale and finite element models at the mesoscale and beyond. On the way to this goal mathematicians and computer engineers will have a vital role to play in ensuring that computational resources are used efficiently and that the results of models can be trusted and transferred from one length scale to the next.
This is an abridged version of an article that first appeared in Counting On IT Issue 13.