Geometric Algorithms for Layered Manufacturing: Part II

1. Geometric Algorithms for Layered Manufacturing: Part II Ravi Janardan Department of Computer Science & Engg. University of Minnesota, Twin Cities

2. Model Acquisition • CAD Software • CT Scans • Laser Scanning • 3D Photography Computer-Aided Process Planning • File repair • Model orientation • Slicing • Support creation Model Building via Layered Manufacturing Postprocessing • Remove supports • Improve finish • Inspect model LAN or Internet Rapid Physical Prototyping • “3D printing” technology that creates physical prototypes of 3D solids from their digital models • Used in the automotive, aerospace, medical industries, etc., to speed up the design cycle

3. Layered Manufacturing • Builds 3D models as a stack of 2D layers Stereolithography

4. volume of supports number of layers surface finish contact-area of supports Geometric Considerations • The choice of build direction affects quality and performance measures

5. Overview of Recent LM Research(http://www.cs.umn.edu/~janardan/layered) • Geometric algorithms for • minimizing surface roughness • minimizing # of layers • protecting critical facets • minimizing support requirements and trapped area in 2D • Exact/approx. geometric algorithms for tool path planning (“polygon hatching”) • Decomposition-based approach to LM • Algorithms to approximate the optimal support requirements

6. Sampling of Related Work • “Plane-sweep slicer for LM”, McMains, Séquin • “Preferred direction of build for RP processes”, Frank, Fadel • “Quantification of errors in RP”, Bablani, Bagchi • “Determination of support structures in LM”, Allen, Dutta • “Accurate slicing for LM”, Kulkarni, Dutta • “Slicing procedures for LM techniques”, Dolenc, Mäkelä • “Double-sided LM”, McMains • “Voxel-based method for LM”, Chandru et al. • Etc... • “Feasibility of design in stereolithography”, Asberg et al • “Approximation algorithms for LM”, Agarwal, Desikan • “Data front-end for LM”, Barequet, Kaplan • Related work in injection-mold design

7. Decomposition-Based Approach • Decompose the model with a plane into a small number of pieces • Build the pieces separately • Glue the pieces back together

8. P+ d H -d P- Polyhedral Decomposition • Decompose a polyhedron P into k pieces with a plane H normal to a given direction d • Goal: Minimize volume of supports or contact area when the pieces are built in directions d and -d

9. d d nf nf nf -d -d CAf = ah2+bh+c CAf = area(f) CAf = 0 Minimizing Contact-Area (CA) for Convex Polyhedra • CA depends on height of H and orientation of facets • e.g. back facet f (nf • d < 0)

10. Overall Algorithm • sweep-based algorithm • initialize (sort vertices, set CA term) • general step at vertex v (update CA term) • minimize new CA term

11. sub: area(f) add: a0h2+b0h+c0 sub: a0h2+b0h+c0 add: a1h2+b1h+c1 sub: a1h2+b1h+c1 • Minimize Ah2 + Bh + C • Run-time: O(n log n), space: O(n). Overall Algorithm (cont’d) • General step details — update CA term

12. Experimental Results • random points on a sphere of radius 100

13. Experimental Results • random points on a rotated “ice-cream” cone

14. Non-convex Polyhedra • the structure of supports is more complex convex non-convex

15. Volume Minimization • partition each facet into two classes of triangles: black tri. — always in contact with supports gray tri. — contact with supports depends on the position of H

16. Computing Black/Gray Triangles • Use cylindrical decomposition

17. Overall Algorithm • compute cylindrical decomposition • apply convex support-volume algorithm on gray triangles • Run-time: O(n2 log n), space: O(n2)

18. Experimental Results (Volume)

19. Controlling Decomp. Size (K ) • Partition the d-direction into intervals Ijs.t. any plane in Ij splits P into same number of pieces kj • Optimize only within intervals where kj[K Two-sweep algorithm • up-sweep: #pieces for P- • dn-sweep: #pieces for P+ Combine results of sweeps Use Union-Find data str.

20. Approximating the Optimal Support Requirements • Given a polyhedral model, compute a build direction for which the support contact-area is close to the minimum (there is no model decomposition here). • Identify heuristics for choosing candidate directions • Design efficient algorithms to compute contact-area for chosen directions • Develop a criterion to evaluate the quality of each heuristic, via easy-to-compute quantities

21. Preliminaries • CA(d) — contact area for build direction d • CA(d) = BFA(d) + FFA(d) + PFA(d) • BFA(d) — back facet area for d • FFA(d) — front facet area for d • PFA(d) — parallel facet area for d d d d

22. CA(d^) CA(d*) CA(d^) BFA(d*) CA(d^) BFA(d’) R = R  R  • Obtain upper bound on • CA(d*)  BFA(d*) therefore • BFA(d*)  BFA(d’) therefore Evaluation Criterion d^ — build direction computed by heuristic d* — optimal build direction d’ — direction which minimizes BFA

23. d d exact algorithm heuristic Compute CA • compute BFA, FFA and PFA for direction d • compute FFA:

24. FFA Results

25. Minimize BFA • Run-time: O(n2 log n), space: O(n) space

26. Heuristics • Min BFA— direction that minimizes the area of back facets • Max PFA— direction that maximizes the area of parallel facets • Max PFC— direction that maximizes the number of parallel facets • PC— direction that corresponds to the principal components of the object • Flat —direction that corresponds to a facet of the convex hull of the object

27. Experimental Results prism pyramid ecc4 triad1 tod21 f0m27 mj 3857438 top_case carcasse oldbasex bot_case

28. CA(d^) BFA(d’) R  Experimental Results (cont’d) • Columns shows upper bound on

29. Future Work • Globally optimal decomposition direction • Multi-way decomposition • Approximating support volume • Exact algorithms for support optimization Conclusions • Efficient algorithms for decomposing polyhedral models • Heuristics and evaluation criterion for approximating optimal build direction so as to minimize contact-area • Applications to Layered Manufacturing

30. Acknowledgements • Research Collaborators: P. Castillo, P. Gupta, M. Hon, I. Ilinkin, E. Johnson, J. Majhi, R. Sriram, M. Smid, and J. Schwerdt • STL models courtesy Stratasys, Inc. • Research supported in part by NSF, NIST, Army HPC Center (U of Minn.), and DAAD (Germany) • Papers at http://www.cs.umn.edu/~janardan/layered