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Abstract
Recent technological developments require advanced manufacturing technologies for miniaturized and micro-scale components in various types of industrial products, e.g., fuel injection nozzle for automobiles, miniaturized medical tools, microprobes used to measure surface properties, integrated micro-channels used for drug delivery systems, micro-gears, and various aeronautical components. Through-mask electrochemical micromachining (TM-ECMM) is a feasible process for such products. In the present work, the finite elements method (FEM) is used for solving 2-D Laplace equation in the inter electrode gap to determine the potential and flux distribution for the anode shape prediction in TM-ECMM. Algorithm has been developed and implemented in MATLAB to estimate surface erosion of anode for finite time steps. Anode shape and undercut are predicted using FEM and compared with the experimental results. The shape evolution through finite element simulation is approximately complying with the experimental anode profile. This work would later help in tool (cathode) design for TM-ECMM.
NOMENCLATURE
| A | = | Voltage, V |
| A1 | = | Area of an element |
| Ac | = | Constant as defined in text |
| ai, bi, ci | = | Constants for ith element |
| B | = | Interelectrode gap, mm |
| C | = | Electrolyte concentration,% |
| C’ | = | Constant as defined in the text |
| Ce | = | Specific heat of electrolyte, J/g-K |
| d | = | Mask thickness, mm |
| E | = | Electrochemical equivalent, g |
| F | = | Faraday's constant, A-s |
| = | Force vector | |
| f | = | Feed rate, mm/s |
| I | = | Current, A |
| J | = | Current density, A/mm2 |
| K | = | Universal stiffness matrix |
| ke | = | Specific electrical conductivity of electrolyte, Ω−1 mm−1 |
| ko | = | Electrolyte initial electrical conductivity, Ω−1 mm−1 |
| Kij | = | Element stiffness matrix |
| M | = | Amount of material removal, g |
| n′ | = | No. of input parameters |
| n, nx, ny | = | Normal vector |
| Ni | = | ith shape function |
| T | = | Temperature, K |
| t | = | Time, s |
| U | = | Electrolyte flow velocity, mm/s, |
| UCexp. | = | Experimental undercut, mm |
| UCsim | = | Simulated undercut, mm |
| V | = | Applied voltage, V |
| w, wi, | = | Weight function |
| X | = | Axis |
| Xw | = | Machined cavity radius, mm, |
| Y | = | Interelectrode gap, μm, axis |
| Yr | = | Response |
| Yo | = | Initial interelectrode gap, mm |
| Ye | = | Equilibrium gap, mm |
| Ymax | = | Machined depth |
| Δt | = | Time step, s |
| ΔT | = | Change in temperature |
| ∇V | = | Overpotential, V |
| ρw | = | Density of workpiece material, g/mm3 |
| ρe | = | Electrolyte density, g/mm3 |
| Φ, Φi | = | Electric field potential, V |
| ϕe | = | Electric field inside an element |
| = | Flux | |
| α | = | Temperature coefficient of specific conductance |
| αo | = | Constant |
| αi | = | Coefficient |
| βi | = | Input variable |
| ∈ | = | Error |
| η | = | Current efficiency |
| τ | = | Boundary of the domain specified |
Subscript
| a | = | Anode |
| c | = | Cathode |
| x | = | x–axis direction |
| y | = | y–axis direction |
Acronyms
| ANOVA | = | Analysis of variance |
| BEM | = | Boundary element method |
| CAD | = | Computer-aided design |
| DC | = | Direct current |
| ECM | = | Electrochemical machining |
| ECMM | = | Electrochemical micro-machining |
| FEM | = | Finite element method |
| FDM | = | Finite difference method |
| IEG | = | Interelectrode gap, μm |
| MRR | = | Material removal rate, mm3/s |
| MRR1 | = | Linear material removal rate |
| TM | = | Through-mask |
| TM-ECMM | = | Through-mask-ECMM |