Data from: Waterjet and laser etching: the nonlinear inverse problem

In waterjet and laser milling, material is removed from a solid surface in a succession of layers to create a new shape, in a depth-controlled manner. The inverse problem consists of defining the control parameters, in particular, the two-dimensional beam path, to arrive at a prescribed freeform surface. Waterjet milling (WJM) and pulsed laser ablation (PLA) are studied in this paper, since a generic nonlinear material removal model is appropriate for both of these processes. The inverse problem is usually solved for this kind of process by simply controlling dwell time in proportion to the required depth of milling at a sequence of pixels on the surface. However, this approach is only valid when shallow surfaces are etched, since it does not take into account either the footprint of the beam or its overlapping on successive passes. A discrete adjoint algorithm is proposed in this paper to improve the solution. Nonlinear effects and non-straight passes are included in the optimization, while the calculation of the Jacobian matrix does not require large computation times. Several tests are performed to validate the proposed method and the results show that tracking error is reduced typically by a factor of two in comparison to the pixel-by-pixel approach and the classical raster path strategy with straight passes. The tracking error can be as low as 2–5% and 1–2% for WJM and PLA, respectively, depending on the complexity of the target surface.在水射流铣削（waterjet milling）与激光铣削（laser milling）工艺中，通过逐层去除固体表面材料以加工出目标形状，整个加工过程采用深度控制模式执行。此类工艺的逆问题（inverse problem）指通过确定控制参数（尤其是二维光束路径），以加工出指定的自由曲面。本文针对水射流铣削（WJM）与脉冲激光烧蚀（PLA）展开研究，因为通用非线性材料去除模型可同时适配这两类加工工艺。针对这类工艺的逆问题，传统求解思路通常是根据工件表面各像素点所需的铣削深度，按比例调控驻留时间（dwell time）。但该方法仅适用于浅表层蚀刻加工，因为其未考虑光束作用覆盖区（beam footprint）与后续加工路径的重叠效应。本文提出一种离散伴随算法（discrete adjoint algorithm）以优化该类问题的求解过程。该优化框架涵盖非线性效应与非直线路径加工，且雅可比矩阵（Jacobian matrix）的计算无需大量计算耗时。本文通过多组实验验证所提方法的有效性，结果表明：与逐像素方法（pixel-by-pixel approach）及经典直线路径光栅加工策略相比，跟踪误差（tracking error）平均可降低至原先的1/2。根据目标曲面的复杂程度差异，WJM与PLA的跟踪误差最低可分别达到2%~5%与1%~2%。