Inverse problem in energy beam controlled-depth machining

磨料水射流加工(AWJ)、脉冲激光烧蚀(PLA)和离子束加工(IBM)等技术都是将能量转移到表面并去除材料的能束加工方法，这些技术可以通过可控深度加工来生成自由曲面。虽然每种方法的能量传递方式都有很大的不同(AWJ：空气、砂砾和水的高速混合物机械侵蚀表面；解放军：激光脉冲蒸发表面；IBM：高速带电粒子侵蚀表面)，但它们可以在统一的数学框架下处理，从而用偏微分方程式描述表面的侵蚀速率。这个方程将能量束的足迹(它的瞬时去除率，它可能是受侵蚀表面的几何形状、它与束源的距离以及在束内的位置和束方向的函数)与表面的演变联系起来。这项建议中的研究人员在使用这一数学框架确定给定波束足迹和动态波束路径的最终机械加工表面方面取得了重大成功；这是前向问题。然而，工业上感兴趣的问题是逆问题；给定所需的最终表面，梁应该如何移动才能准确地加工它？目前，在学术研究和行业中，这个问题都是通过试错(工艺)来解决的。这个项目的目的是开发算法解决逆问题的方法，以便这组技术(即能束控制深度加工)的最终用户可以将他们所需的曲面输入到软件包中，并自动生成光束路径。我们将通过解决一系列日益现实的数学问题来实现这一点，这些问题可能与真实的能束过程有关，并得到一个实验计划的支持，我们的模型可以对照该实验计划进行验证。

Techniques such as abrasive water jet machining (AWJ), pulsed laser ablation (PLA) and ion beam machining (IBM) are all methods of energy beam processing, by which energy is transferred to a surface and material is removed; this group of technologies can be employed to generate freeforms surfaces by controlled-depth machining. Although the way in which the energy is transferred in each of these methods is very different (AWJ: a high speed mixture of air, grit and water mechanically erodes the surface; PLA: laser pulses vaporize the surface; IBM: high speed charged particles erode the surface), they can be dealt under a unified mathematical framework whereby the rate of erosion of the surface is described by a partial differential equation. This equation relates the footprint of an energy beam (its instantaneous rate of removal, which may be a function of the geometry of the eroding surface, its distance from the source of the beam as well as position within the beam and beam orientation) to the evolution of the surface. The Investigators in this proposal have had significant success in using this mathematical framework to determine the final, machined surface for a given beam footprint and dynamic beam path; this is the forward problem. However, the problem that is of industrial interest is the inverse problem; given a required final surface, how should the beam be moved in order to accurately machine it? Currently, in both academic research and industry, this problem is solved by trial and error (craftmanship). The aim of this project is to develop methods for solving the inverse problem algorithmically, so that end users of this group of technologies (i.e. energy beam controlled-depth machining) can input their required surface into a software package and automatically generate a beam path. We will do this by tackling a series of increasingly realistic mathematical problems which can be related to real energy beam processes, backed up by an experimental programme against which our models can be verified.