Applied Mathematical Modelling of Industrial Metal Forming

金属成形是对金属进行成形；制造业的例子包括轧制金属以制造薄板和冲压金属平板以形成汽车车身面板。为了创造新产品，减少废料，弥补更多可变材料（如回收金属），降低成本和减少能源使用，目前工业需要更智能的金属成型。在21世纪，人们可能期望金属成形过程的实时计算机控制，它将在成形过程中监控金属工件，并调整过程以纠正任何问题，并始终如一地获得期望的最终结果。然而，计算机控制器需要一个理论模型来预测如果它要做出改变会发生什么，以便找到正确的改变，这样的理论模型目前是不可用的；计算机模拟（例如使用有限元素）对于实时使用来说太慢了。目前的技术状况是在工艺开发过程中使用计算机有限元模拟或诊断问题，然后使用简单的控制器（如PID控制器）盲目地遵循预先规定的成形程序。显然，新的建模技术有望带来实质性的改进。该项目雄心勃勃的目标是研究连续固体力学和塑性的数学建模技术，其结果可用于为工业金属成形提供预测理论模型。现有的计算机模拟（如有限元）在所有情况下都可以工作，但速度很慢，而这里的目标是利用特定金属成形过程的特性（如对称性，或小参数，如薄板，小变形等），并为每个过程创建特定的简化模型。通过以严格的方式计算这些属性，并使用最佳实践数学技术（例如渐近性和稳定性理论），可以生成具有可保证精度的快速计算模型。这种模型将非常适合用于金属成形过程的在线控制。这个项目的目的是研究如何在具体细节上做到这一点，并产生一些工业相关的例子。有一些相关的研究领域，虽然它们肯定会为这项工作提供信息，但它们不会直接在本项目中进行。这些研究包括对金属和合金的原子尺度建模的研究，以及对塑性连续介质力学（纯数学分析的一个分支）控制方程正确性的研究。在这个项目中，我们将采用在工业金属成形有限元计算中被接受和使用的控制方程，而不是在计算机上求解，我们将研究基于这些方程的建模技术和简化模型的理论发展。希望这个项目的结果将是一些技术，用于创建特定金属成型过程的定制数学模型，以及一些特定于工业相关的特定金属成型过程的模型。选定的具体模式与英国工业相关，特别是与塔塔钢铁（Tata Steel）和Primetals Technologies相关。这几个模型将通过数值结果或实际实验进行验证。该项目还将产生一支熟练的研究团队和一个经过验证的新应用数学知识体系，这将使工业界有信心与PI合作，并投资于将这些知识应用于工业实践的后续项目。

Metal forming is the shaping of metal; examples from manufacturing include rolling metal to create thin sheets and stamping flat sheets of metal to form car body panels. There is currently an industrial need for smarter metal forming in order to create new products, to reduce scrap, to compensate for more variable materials (e.g. recycled metals), to reduce costs, and to reduce energy usage. In the 21st century, one might expect real-time computer control of metal forming processes, which would monitor the metal workpiece during the forming process and adapt the process to correct any problems and consistently obtain the desired end result. However, the computer controller needs a theoretical model to predict what would happen if it were to make a change, in order to find the right changes to make, and such theoretical models are currently unavailable; computer simulations (using finite elements for example) are too slow for use in real-time. The current state of the art is to use computer finite element simulations during process development or to diagnose problems, and then to use simple controllers (such as PID controllers) to blindly follow the pre-prescribed forming procedure. Clearly new modelling techniques could be expected to give a substantial improvement.The ambitious aim of this project is to investigate techniques for mathematical modelling in continuum solid mechanics and plasticity, the outcome of which could be used to provide predictive theoretical models for industrial metal forming. Unlike existing computer simulations (such as finite elements) which work in all situations but which are slow, the aim here is to take advantage of properties of particular metal forming processes (such as symmetry, or small parameters such as thin sheets, small deformations, etc), and create bespoke simplified models specific to each of these processes. By accounting for these properties in a rigorous way, and using best practice mathematical techniques (such as asymptotics and stability theory), quick-to-compute models with a guaranteeable accuracy could be produced. Such models would be eminently suitable for use in online control of the metal forming process. The aim of this project is to work out how to do this in specific detail, and to produce a number of industrially relevant examples.There are a number of related areas of research which, while they will certainly inform this work, they will not be directly worked on in this project. These include research into atomic scale modelling of metals and alloys, and research into the correctness of the governing equations of plastic continuum mechanics (a branch of pure mathematical analysis). In this project, we will take the governing equations that are accepted and used in finite element computations of metal forming used in industry, and instead of solving them on a computer, we will investigate the theoretical development of modelling techniques and simplified models based on these equations.It is hoped the results of this project will be a number of techniques for creating bespoke mathematical models of particular metal forming processes, together with a few such models specific to particular metal forming processes of industrial relevance. The specific models have been chosen to be relevant to UK industry, in particular to Tata Steel and Primetals Technologies. These few models will be validated against either numerical results or practical experiments. The project would also result in a skilled team of researchers and a validated body of new applied mathematical knowledge, which would give industry the confidence to partner with the PI and invest in subsequent projects applying this knowledge in industrial practice.