Inverse problem in energy beam controlled-depth machining

诸如磨料水射流加工（AWJ）、脉冲激光烧蚀（PLA）和离子束加工（IBM）等技术都是能量束加工的方法，通过这些方法，能量被转移到表面并去除材料;这组技术可以用于通过控制深度加工来生成自由曲面。虽然在这些方法中的每一种中能量转移的方式非常不同（AWJ：空气、砂砾和水的高速混合物机械地侵蚀表面; PLA：激光脉冲蒸发表面; IBM：高速带电粒子侵蚀表面），但是它们可以在统一的数学框架下处理，其中表面的侵蚀速率由偏微分方程描述。该方程将能量束的足迹（其瞬时去除速率，其可以是侵蚀表面的几何形状、其距束源的距离以及束内的位置和束取向的函数）与表面的演变相关联。本提案中的研究人员在使用该数学框架确定给定光束覆盖区和动态光束路径的最终加工表面方面取得了重大成功;这是向前的问题。然而，工业上感兴趣的问题是逆问题;给定所需的最终表面，应该如何移动梁以精确地加工它？目前，在学术研究和工业中，这个问题都是通过试错（工艺）来解决的。该项目的目的是开发方法来解决反问题的算法，使这组技术的最终用户（即能量束控制深度加工）可以将其所需的表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与真实的能量束工艺相关的日益现实的数学问题来实现这一点，由一个实验计划支持，我们的模型可以根据该计划进行验证。

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.