Inverse methods to gradient etch three-dimensional features with prescribed topographies using abrasive jet micro-machining: Part I - Modeling

Abrasive jet micro-machining (AJM) can be used to machine a wide range of features on the micro-scale. Given a set of process parameters, surface evolution models capable of predicting the shape of straight, constant-depth channels in a wide variety of materials are well-established. This paper presents a novel framework for solving the unaddressed more challenging and industrially relevant inverse problem of determining the erosive source traverse velocity and path required to machine a particular user-specified feature topography. It is shown that the problem can be expressed as a Fredholm integral equation of the first kind where, for a given desired topography, the source erosive efficacy is a difference kernel, and the unknown function to be determined is the dependency of the instantaneous source velocity on its location. For Gaussian sources typical of AJM, an analytical solution based on the Gaussian transform is presented that can be used for a wide range of desired topographies. For non-Gaussian sources, an approximate approach based on a point-source solution is suggested, and its accuracy is discussed. Once the velocity function to create the desired shallow topography has been determined, it can be used in existing surface evolution models to predict the shape at any depth. The methodologies are demonstrated to be applicable for the AJM of constant depth micro-channels with user-specified cross-sectional shape, gradient etched micro-channels with specified texture along their length, and pockets with texture in two perpendicular directions. The accompanying paper validates the methodology by comparison with measured three-dimensional profiles arising from the predicted velocity functions. The study is the first to consider the inverse problem for AJM, and to demonstrate the ability to etch and predict user specified three-dimensional topographies. (C) 2016 Elsevier Inc. All rights reserved.