1 Introduction
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it uses NURBS to define the geometries to be able to model complex shapes at different resolutions: shape of the craters locally, and shape of the wear globally;
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it mimics the physical \(\mu \)EDM process while performing local surface warping to generate micro-craters;
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it is driven by physical parameters so as to generate an accurate approximation of the wear and roughness;
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it can be inserted in a shape optimization loop so as to find the optimal shape of the initial tool that would produce a targeted workpiece.
2 Literature review
2.1 \(\mu \)EDM modeling
2.1.1 Introduction
2.1.2 Thermo-electrical modelling
2.1.3 Electro-mechanical modeling
2.1.4 Geometric modeling
2.1.5 Conclusion
2.2 Surface deformation techniques
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Algorithmic speed considering the high number of discharges that occur during a \(\mu \)EDM manufacturing process (the exact number depends on the parameters being used but can easily be in the millions), the chosen method must be fast in order to be repeated numerously.
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Local/global control this criterion illustrates the fact that it is needed to be able to deform a very specific location of our geometry without affecting the rest of it. The global deformation will result from a set of local deformations.
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Model preservation in order to avoid added complexity in an iterative process, it is desirable that no change in the initial model be made. For example if the initial number of patches describing the geometry is one, a method preserving the model will keep it that way.
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Shape control this criterion is linked to the variety of deformation that are available.
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Control point repositioning the most common way of modifying NURBS curves or surfaces. The user generally specifies one or multiple curve or surface points to be displaced. Since the problem is often under-constrained, an energy function to be minimized is added to this optimization problem [16, 20, 21].
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Weight or knot modification this method is used in the specific case where the displacement of a point towards or further away from a control point is required. While local control can be increased similarly to control point repositioning, the nature of this method prevents it to be used for shape-specific deformations. Also the modification of knots is not intuitive and the deformations are indirectly performed. The use of non-linear constraints slows down the deformation process [22‐26].
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Surface warping this deformation strategy relies on the notion of control point repositioning. However, instead of specifying a set of constraints linked to points’ positions or derivatives’ values, the control points are moved in respect to the following general formula:where f(i, j) is the warp function and \(\mathbf {N}(i,j)\) is the warp direction. A certain number of common strategies exist such as surface flattening or bending [27‐29].$$\begin{aligned} \mathbf {P'}_{i,j} = \mathbf {P}_{i,j} + f(i,j) \cdot \mathbf {N}(i,j) \end{aligned}$$(10)
Algorithmic speed | Local control | Model preservation | Shape control | |
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Control point repositioning |
\(\oplus \)
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\(\oplus \oplus \)
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\(\oplus \oplus \)
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\(\oplus \)
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Weight, knot sequence modification |
\(\oplus \oplus \)
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\(\oplus \oplus \)
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\(\oplus \oplus \)
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\(\ominus \)
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Surface warping |
\(\oplus \)
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\(\oplus \oplus \)
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\(\oplus \oplus \)
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\(\oplus \oplus \)
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3 The \(\mu \)EDM simulation framework
3.1 Simulation process overview
3.2 Volume enclosed by a NURBS patch
Nb. points | Time (ms) |
\(V_w\) (μm\(^3\)) |
\(D_b\) (%) |
\(D_f\) (%) |
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25 | 24 | 283329 | 15.66 | 100 |
50 | 97 | 360872 | 7.42 | 24.74 |
100 | 379 | 344769 | 2.63 | 6.33 |
200 | 1508 | 334685 | 0.38 | 1.59 |
400 | 3397 | 336873 | 0.27 | 0.70 |
800 | 24652 | 335950 | 0.00 | 0.09 |
Method | Time (ms) |
\(V_w\)(μm\(^3\)) |
\(D_b\) (%) |
\(D_f\) (%) |
---|---|---|---|---|
Sampling | 3397 | 335950 | 100 | 1.79 |
Control points | 61 | 335137 | 0.24 | 100 |
3.3 Volume to be removed for each crater
3.4 Craters insertion
4 Experimental validation
4.1 Measurements
4.2 Vertical wear and surface roughness evaluation
Experiment | 1 | 2 |
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Energy level (index) | 300 | 13 |
Voltage (V) | 60 | 60 |
Current (index) | 20 | 20 |
Time on (ms) | 5 | 5 |
Experiment | 1 | 2 |
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Hole depth (μm) | 50.8 | 50.2 |
Tool vertical wear (μm) | 12.5 | 11.3 |
Roughness Ra (μm) | 1.27 | 0.82 |
Workpiece crater diameter (μm) | 15 | 3 |
Workpiece crater depth (μm) | 3 | 1 |
Tolerance \(T_v\) (%) | 10 | 5 | 1 |
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Volume removed (μm\(^3\)) | 172058 | 169978 | 167586 |
Average volume per crater (μm\(^3\)) | 286.76 | 283.30 | 279.31 |
Roughness Ra (μm) | 1.87 | 1.88 | 1.43 |
Characteristic | Experi. (μm) | Simu. (μm) | Deviation (%) |
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Tool vertical wear | 11.3 | 11.2 | 0.89 |
Roughness Ra | 0.82 | 0.87 | 6.09 |
4.3 Shape difference assessment
Parameter | Value |
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Energy level (index) | 200 |
Voltage (V) | 90 |
Current (index) | 20 |
Time on (ms) | 5 |
Objective depth (μm) | 100 |
Parameter | Value |
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Workpiece crater diameter (μm) | 13.30 |
Workpiece crater depth (μm) | 4.42 |
Tool crater diameter (μm) | 12.40 |
Tool crater depth (μm) | 4.39 |
Tool | Workpiece | |
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\(d_H\) min (μm) | 0.000000 | 0.000107 |
\(d_H\) max (μm) | 8.629291 | 14.886533 |
\(d_H\) mean (μm) | 1.449477 | 3.073571 |
\(d_H\) RMS (μm) | 2.521132 | 3.626015 |