Adaptive finite element computations of nonlinear elasticity problems

我们建议用数学来帮助在医院工作的医生为两组病人提供更好的护理。这两组患者分别是乳腺癌患者和心脏病患者。我们首先描述数学如何有助于为这些患者群体提供更好的护理，然后简要描述我们使用的数学。在发达国家，大约10%的妇女会在一生中患上乳腺癌。癌症肿瘤可以通过一种或多种临床成像技术来定位，其中一种技术叫做磁共振成像，患者需要仰卧。另一种技术是x射线乳房x光摄影，女性将站着，乳房被压缩在两个板之间。对于另一种称为超声波的技术，女性将侧躺。对于每一种技术，乳房的形状都是非常不同的。如果女性需要手术切除肿瘤，她将在手术过程中仰卧，乳房将变成另一种形状。这将使外科医生难以确定乳房肿瘤的位置。我们打算用数学方法帮助外科医生确定肿瘤的位置。心脏病是西方世界的主要死亡原因。心跳所需的力量是由心脏细胞内的生化反应产生的。在健康的心脏中，这些细胞几乎同时收缩，这使得血液能够有效地泵入全身。这并不总是发生在不健康的心脏上。我们建议用数学来解释发生了什么。我们使用的数学将乳房或心脏的变形与施加的力联系起来。当身体器官受到外力作用时，或者当一个人改变姿势时，例如站起来而不是躺下时，它们的形状就会改变。这种形状的变化可以用方程来描述，这些方程可以通过计算机编程来使用数学算法来求解。由于人体器官非常复杂，计算机可能需要很长时间才能使用这些算法解决问题。本提案中描述的工作旨在建立最合适的算法，即在确保答案正确的同时在计算机上以最短的时间执行的算法。

We propose to use mathematics to help doctors working in hospitals provide better care for two groups of patinets. these groups of patients are women with breast cancer, and patients with heart disease. We begin this summary by describing how mathematics may help provide better care for these groups of patients, and then brifly describe the mathematics we use.Approximately 10% of women in developed countries will develop breast cancer during the course of their lives. A cancerous tumour may be located using one or more clinical imaging techniques from a range of clinical technique, one of these techniques, called magnetic resonance imaging, the woman will be lying on her front. For another technique, called X-ray mammography, the woman will be standing with her breast compressed between two plates. For another technique, called ultrasound, the woman will be lying on her side. The breast will take a very differnt shape for each of these techniques. Should the woman require surgery to remove a tumour, she will lie on her back during surgery and the breast will take yet another shape. This will make hard for the surgeon to locate the tumour in the breast. We intend to use mathematics to help the surgeon locate the tumour.Heart disease is the major cause of death in the western world. The forces needed for a heartbeat are generated by biochemical reactions inside cells in the heart. In a healthy heart the cells will contract at roughly the same time, which enables the efficient pumping of blood around the body. This doesn't always happen in an unhealthy heart. We propose to use mathematics to explain what happens instead.The mathematics we use relates the dformation of the breast or the heart to the forces that are applied. Body organs change shape when a force is applied to them, or when a person changes prosition, e.g. stands up rather than lies down. This change in shape is described by equations that may be solved by programming a computer to use a mathematical algorithm. As body organs can be very complex, a computer may take a long time to solve the problem using these algorithms. The work described in this proposal intends to establish the most suitable algorithm, i.e. the alorithm that will execute on a computer in the shortest time whilst ensuring that the answer is correct.