1. Dynamic Programming Approach to Adaptive Slicing for Optimization under a Global Volumetric Error Constraint
Andrew Deng, Yasmine Badr, Puneet Gupta
NanoCAD Lab,
Electrical and Computer Engineering Department,
UCLA

2. Laser Based Additive Manufacturing
Stereolithography (SLA) Printers
Solidify material using laser
Laser moves quickly over build platform plane
Direct Light Projection (DLP) Printers
Flash image of light onto build platform
Also relatively quick to form layers
Both methods involve building objects through flat layers

3. Staircase Effect Results from Construction using Flat Layers
Error inherent in building objects from flat layers
Cannot perfectly match vertical profile
Leads to missing or extra material, affecting print quality
Illustration of Staircase Effect

4. Staircase Effect is Proportional to Thickness of Slices
Staircase Error can be reduced by varying slice widths
Thick slice widths result in exaggerated jagged prints
Using thinner slice widths means print matches model vertical profile more often
Illustration of effect of slice thickness on staircase effect

5. Slice Thickness trades off Print Time
Error reduction from thinner slices comes at cost of speed
Using thinner slices requires more
For laser printers, fixed amount of time to move between layers
Dominated by moving between layers
Print time is proportional to total number of layers
Simple plot showing slice count – time relationship

6. Adaptive Slicing Improves on Uniform Slicing
Model shape varies across entire height
Some regions may require very thin layers
Some regions may be acceptable to print with thick layers
Adaptive Slicing
Vary the print thickness during the print to achieve balance between error and speed
How to select slices in an intelligent way?
Sample Adaptive Slicing Scheme

7. Problem Formulation
Focus on print time reduction from adaptive slicing
Cannot minimize print time and error at the same time
Set total error to be bounded
Provide some control over acceptable error
Minimize the number of slices for a given global volumetric error constraint

8. Dynamic Programming Approach to Optimization
Large set of slice combinations to choose from
Exhaustive search for minimum would be very time consuming
Solved through use of dynamic programming
Recursively traverse through possible subproblems
Keep track of only the best solution at each subproblem
Illustration of dynamic programming1

9. Algorithm Detail
Define subproblems as different heights and error budgets
Choosing a slice at a height leads to a new subproblem with new height and error budget
Choose subproblem with smallest number of slices as optimal at each height
Create large memoization array 𝑆
Follow the recurrence relation:
𝑆 0, 𝐸 =0
𝑆 ℎ, 𝐸 =∞ if error bound is expended
𝑆 ℎ, 𝐸 = min 𝑡 𝑖 <ℎ (1+𝑆 ℎ− 𝑡 𝑖 ,𝐸−𝑒𝑟𝑟( 𝑡 𝑖 ,ℎ) )
Error is continuous rather than discrete
Cannot store cleanly in array as-is, must round
If rounding is kept precise, then solution will approach optimal

10. Algorithm Runtime
Every element of the solutions array is computed only once
Optimal solution is stored, then accessed if needed again
Therefore, in worst case runtime is proportional to size of array
Array has dimensions 𝐻 𝑡 𝑚𝑖𝑛 and 𝐸 𝑟𝑜𝑢𝑛𝑑 , where 𝐻 is height, 𝐸 is starting error budget, 𝑡 𝑚𝑖𝑛 is proportional to smallest slice thickness, and 𝑟𝑜𝑢𝑛𝑑 is error rounding factor
Therefore time complexity is 𝑂 𝐻𝐸 𝑡 𝑚𝑖𝑛 ×𝑟𝑜𝑢𝑛𝑑 , compared to ~𝑂 # 𝑡ℎ𝑖𝑐𝑘𝑛𝑒𝑠𝑠𝑒𝑠 𝐻

11. Error Computation Method
Object was voxelized to approximate error for speed purposes
Error for each slice was computed from voxels using scheme pictured
Error for objects was computed and stored in array for quick access
Illustration of error computation from voxelization

12. Testing Parameters and Methodology
Testing used a Formlabs Form 1+ printer
Slice thicknesses possible: 25, 50, 100, 200 𝜇𝑚
Algorithm was run on 4 different objects with these slice thicknesses
Returned set of slices to use
Estimated error from generated slices using error array
Estimated time by using printer’s software
OpenFL software used to access printer’s print time prediction
Predicted print time tested against actual print time
For comparison against other adaptive slicing methods, was run against an adaptive slicing scheme devised in a recent paper based on octree decomposition2

13. Time-Error Plots on Tested Objects
Uniform Slicing
Octree-based Slicing
Dynamic Programming Approach
1.32x speedup
1.47x speedup
1.34x speedup
1.45x speedup

14. Numerical Comparison Uniform Slicing Dynamic Programming Method
Dynamic Programming Method
Error (mm3)
Time (min)
Slice Count
Error Change
Speed Increase
Slice Count Change
Model Volume Error Change
19.909 79
400
12.714 72
356
-36.1%
1.09x
-11.0%
-0.097%
32.452 37
189
+63.1%
2.14x
-52.8%
+0.170%
40.049 42
200
-18.9%
1.14x
-5.5%
-0.103%
Octree Method 
Dynamic Programming Method
Error (mm3)
Time (min)
Slice Count
Error Change
Speed Increase
Slice Count Change
Model Volume Error Change
12.73
106
556
12.714 72
356
-0.13%
1.47x
-35.9% %
26.948 52
248
26.629   44
215
-1.18%
1.22x
-13.3% %

15. Conclusion
Algorithm provides adaptive slicing scheme optimized for print speed
Minimizes number of layers needed
Dynamic programming solution can provide similar or better results in all cases
Best solution possible under constraint found

16. Questions

17. References
[1]
[2] Siraskar, N., Ratnadeep P., and Sam A., "Adaptive slicing in additive manufacturing process using a modified boundary octree data structure." Journal of Manufacturing Science and Engineering (2015):