Applied Nonlinear Mathematics: Making it Real

许多具有实际重要性的问题会引起非常相似的潜在应用数学问题。拟议的研究将解决生命科学和工程领域的四个现实挑战。它们的共同点是由于不同部件之间的耦合而存在延迟，以及由于自由发挥和摩擦而存在不平滑。此外，我们必须解决它们的空间扩展和数学模型的鲁棒性的首要问题。我们要解决的第一个现实挑战是人类大脑的神经生理学。具体来说，我们将开发具有本地和分布式组件的不同复杂性的模型。延迟是由电化学信号引起的。这项工作的一个主要焦点是癫痫发作的预测，在英国每年有超过2000人死于癫痫。更一般地说，问题是：微观神经元参数的控制（例如通过药物传递或电刺激）如何影响整体宏观行为？生物力学系统的性能通常比任何人造设备都要好得多。例子包括哺乳动物和昆虫对声音的探测，以及陆地动物（从猎豹到蟑螂）的运动，它们可以以惊人的速度和效率在崎岖的地形上行走。我们在现实世界面临的第二个挑战是理解这是如何工作的，并帮助构建更好的通信系统或快速的有腿机器人。一个重要的因素是碰撞，例如腿与地面的碰撞，这会导致数学模型中的不平滑。我们的第三个现实挑战是混合测试。这种新方法提供了测试全尺寸单个部件的机会，比如飞机发动机、桥梁电缆或摩天大楼的地板，就好像它是整个工程结构的一部分一样。除了可能比传统测试节省大量成本外，混合测试还允许进行目前不可能进行的测试。为了使混合测试在工程上成为现实，我们需要在测试样品和结构其余部分的计算机模型之间实时关闭涉及传感器和执行器的控制回路。试样和模型一般都具有较大的空间范围，并受周期性强迫等外部影响。我们将研究在控制回路中存在延迟的情况下这些不同影响之间的相互作用。即使在我们日益电子化的世界里，许多工程系统的性能也主要依赖于能量的有效机械传输。我们的第四个现实挑战是如何巧妙地避免机械传动系统中的噪音和振动。自由发挥和冲击是影响整体性能的主要问题。虽然汽车行业的目标是降低噪音，提高驾驶员的舒适度，但过大的振动可能会导致风力涡轮机发生灾难性故障。另一个重要的应用是微尺度上的能量清除，它可以用来驱动起搏器甚至移动电话。机械传动系统的参数空间和相空间非常大，因此模型的鲁棒性问题至关重要。这项资助涉及8名研究人员、4名博士后研究人员、8名研究生与布里斯托尔大学的工程和生命科学同事、以及来自世界各地的工业合作伙伴和访问研究人员的合作。

Many problems of practical importance give rise to very similarunderlying applied mathematics problems. The proposed research willaddress four real-world challenges in the Life Sciences and inEngineering. They have in common the presence of delays due to thecoupling between different components and of nonsmoothness due tofreeplay and friction. Furthermore, we must address their spatial extentand the overarching question of robustness of mathematical models.The first real-world challenge we address is the neurophysiology of thehuman brain. Specifically, we will develop models of differentcomplexity with both local and distributed components. Delays enternaturally due to electro-chemical signalling. A major focus of this workis the prediction of the onset of epilepsy, which causes more than 2000deaths annually in the UK. More generally, the question is: how does thecontrol of microscopic neuronal parameters (e.g. via drug delivery orelectrical stimulation) affect the overall macroscopic behaviour?Biomechanical systems frequently perform far better than any man-madedevice. Examples include the detection of sounds by mammals and insects,as well as the locomotion of land animals (from the cheetah to thecockroach) that can travel over rough terrain at amazing speed andefficiency. Our second real-world challenge is to understand how thisreally works, and to help construct better communication systems or fast,legged robots. An important element is impact, for example of the legswith the ground, which gives rise to nonsmoothness in the mathematicalmodels.Our third real-world challenge is hybrid testing. This new method offersthe opportunity to test a full-scale individual component, such as anaircraft engine, bridge cable or floor of a skyscraper, as if it werepart of the entire engineering structure. Apart from potentially givinghuge cost savings over conventional tests, hybrid testing also allowsone to perform tests that are currently impossible. To make hybridtesting an engineering reality we need to close in real-time the controlloop involving sensors and actuators between the test specimen and acomputer model of the remainder of the structure. Both specimen andmodel are generally of large spatial extent and subject to externalinfluences such as periodic forcing. We will study the interplay betweenthese different effects in the presence of delays in the control loop.Even in our increasingly electronic world the performance of manyengineering systems depends crucially on an efficient mechanicaltransmission of energy. Our fourth real-world challenge concerns cleverways of avoiding noise and vibrations in mechanical transmissionsystems. Freeplay and impacts are the major concerns affecting theoverall performance. While in the car industry the goal is to reducenoise and achieve driver comfort, large vibrations may lead tocatastrophic failure of wind turbines. Another important application ispower scavenging at microscales, which could be used to drive pacemakersor even mobile phones. The parameter and phase spaces of mechanicaltransmission systems are very large, so that the issue of modelrobustness is crucial.The grant involves the collaboration of eight investigators, fourpostdoctoral researchers, and eight postgraduate students withengineering and life sciences colleagues at Bristol, and industrialpartners and visiting researchers from around the world.