Multi-level modelling of elastic filament networks

许多材料是由相互连接的窄纤维网络组成的。这包括纸、毛毯和尿布等日常用品，也包括为特定目的制造的复杂材料，如用于组织工程的支架，以及具有修复丢失牙釉质等应用的小蛋白自组装网络。对于给定的应用，控制这些网络的机械性能通常是很重要的。为了帮助设计和制造更好的材料，在可控的微观性能(例如纤维厚度)和相应的宏观性能之间具有结构-功能关系是有利的。然而，不存在可信关系，这就需要进行昂贵且耗时的实验。哺乳动物细胞含有一种被称为细胞细胞骨架的蛋白质细丝网络，它在许多重要的细胞功能中发挥着承重作用，因此长期以来一直是生物物理学家仔细研究的对象。一项关键的发展是十多年前引入了计算机模拟，这导致我们对这种网络的理解迅速增加。例如，可以测量网络均匀(或‘亲和’)变形的程度，并描绘网络密度和纤维厚度的组合，其中亲和力是或不是期望的。由于亲和力也被证明与材料的刚性强耦合，所以这个问题是至关重要的。目前弹性纤维网络的计算机建模由于采用次优算法而受到限制。虽然二维网络是直截了当的，但在三维中，只能模拟不能代表真实材料的小网络。将纤维放置在晶格上会增加系统大小，但也不再表示真实材质。模拟方法遵循10多年前制定的原始模板，其中网络响应被表示为使用迭代算法求解的矩阵方程。系统大小可以通过对矩阵进行预处理(粗略地说，预先猜测部分解)来增加，但只使用了基本的预处理子。尽管计算机科学界已经设计出了更先进的方法，包括一种被称为代数多重网格的方法，它已经被证明可以极大地提高标准问题的速度。这个项目的目的是设计、实现和优化一个计算机模型，用于确定弹性纤维网络的机械性能，它使用代数多重网格预处理，使其成为这类材料的最有效的软件解决方案。这将被用来为真实材料的定量表示的大系统建立第一个结构-功能关系。线性解算器将首先实现，这将用于回答与细胞骨架研究有关的突出问题。然后，我们将实现一个非线性求解器，用于当施加到系统上的负载(或施加在系统上的变形)不再小时，并利用这一点来定量理解最近令人困惑的组织支架和蛋白质网络实验测量。受益者将是公司和其他学术团体，他们将能够自由使用我们的算法，这将是最优的(最大规模)，因此是未来的证明。我们还将利用利兹的接触范围，进一步开发和应用该模型，并将其应用于无纺布纤维、多肽凝胶和组织工程支架。

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.