Multi-level modelling of elastic filament networks

Many materials are made up of interconnected networks of narrow fibres. This includes everyday items such as paper, felt and nappies, but also sophisticated materials fabricated for specific purposes, such as scaffolds used in tissue engineering, and self-assembled networks of small proteins that have applications including restoring lost tooth enamel. It is often important to control the mechanical properties of these networks for their given application. To help design and fabricate better materials it is advantageous to have a structure-function relation between controllable microscopic properties (e.g. fibre thickness) and the corresponding macroscopic properties. However, no trusted relation exists, necessitating costly and time-consuming experiments to be performed.

Not all fibre networks are synthetic. Mammalian cells contain a network of protein filaments known as the cellular cytoskeleton which plays a load-bearing role in a number of important cellular functions, and for this reason has long been the subject of scrutiny from biophysicists. A key development was the introduction of computer modelling just over a decade ago, which lead to a rapid increase in our understanding of such networks. For example, it became possible to measure to what extent the network deforms uniformly (or 'affinely'), and delineate combinations of network density and fibre thickness for which affinity is, or is not, expected. Since affinity was also shown to be strongly coupled to material stiffness, this issue is of central importance.

Current computer modelling of elastic fibre networks is limited due to the sub-optimal algorithms being employed. Although two-dimensional networks are straightforward, in three-dimensions only small networks, not representative of the real material, can be simulated. Placing the fibres on a lattice increases the system size, but again no longer represents real materials. The simulation methodology follows the original template set down over 10 years ago, where the network response is formulated as a matrix equation that is solved using an iterative algorithm. System sizes can be increased by preconditioning the matrix (roughly, guessing a partial solution in advance), but only basic preconditioners have been employed. This is despite the fact that the computer science community have already devised far more advanced methods, including one known as algebraic multi-grid which has been proven to give enormous increases in speed for standard problems.

The purpose of this project is to design, implement and optimise a computer model for determining the mechanical properties of elastic fibre networks that employs algebraic multigrid preconditioning, making it the most efficient software solution for this class of material. This will be used to construct the first structure-function relation for large systems quantitatively representative of real materials. The linear solver will be implemented first, and this will be used to answer outstanding questions with regards cytoskeleton research. We shall then implement a non-linear solver for when the load applied to the system (or the deformation imposed upon it) is no longer small, and employ this to quantitatively understand recent puzzling experimental measurements on tissue scaffolds and protein networks. Beneficiaries will be companies and other academic groups who will be able to freely use our algorithms, which will be optimal (up to scaling) ad therefore future proof. We will also exploit the range of contacts within Leeds to further develop and apply the model to non-woven fibres, peptide gels and tissue engineering scaffolds.