What makes this a challenging simulation problem is the wide range of timescales that characterize the overall cutting process. The slowest dynamics are associated with the evolution of the cut, which is described by a spatially dependent differential equation in time and in which the cutting rate is modeled much in the same manner as the Chemical Mechanical Planarization (CMP) process used in microelectronic device manufacturing. Cutting rate is a direct function of the distance between the wire and bottom of the cut, and so the wire deflection is modeled by approximating the wire as a static circular beam (the wire dynamics are orders of magnitude faster than cut evolution) which is subjected to a force in the axial direction (wire tension) as well as a distributed load in the transverse direction (the upward force of the ingot). We assume a linear stress-strain relationship, that the applied stress is below the elastic limit of the wire, and that wire deﬂection obeys Hooke's law. Solving for the equilibrium equations for the vertical forces and the sum of the moments at point ultimately leads to a fourth-order boundary-value problem.
The numerical solution for this fourth order BVP is found by employing a collocation spectral method to discretize the the diﬀerential equation. This discretization procedure gives a set of algebraic equations which are solved Newton's method. Objected-oriented programming concepts were employed in MATLAB to reduce the complexity of these weighted residual and Newton numerical methods. Thus, a numerical solution for wire deflection can be found for the current applied force and other operating conditions, as well as the current ingot shape, and the solution then is used to advance the evolution of the cut. Representative solutions will be presented as well as the results of sensitivity analysis to show the eﬀects of process parameters on cutting rate.