1. Geometric Algorithms for Layered Manufacturing: Part II Ravi Janardan Department of Computer Science & Engg. University of Minnesota, Twin Cities Research Collaborators: P. Castillo, P. Gupta, M. Hon, I. Ilinkin, Research Collaborators: P. Castillo, P. Gupta, M. Hon, I. Ilinkin, E. Johnson, J. Majhi, R. Sriram, M. Smid, and J. Schwerdt

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 LAN or Interne t LAN or Interne t Postprocessing Remove supports Improve finish Inspect model  “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 Rapid Physical Prototyping

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

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

5. Overview of Recent LM Research (http://www.cs.umn.edu/~janardan/layered)  Geometric algorithms for o minimizing surface roughness o minimizing # of layers o protecting critical facets o minimizing support requirements and trapped area in 2D 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. Problem 1 Decomposition-Based Approach  Decompose the model with a plane into a small number of pieces  Build the pieces separately  Glue the pieces back together

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

8. Minimizing Contact-Area (CA) for Convex Polyhedra  CA depends on height of H and orientation of facets e.g. back facet f (n f d < 0) e.g. back facet f (n f d < 0) CA f = area(f) nfnf d CA f = 0 nfnf -d CA f = ah 2 +bh+c nfnf d -d-d

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

10. Overall Algorithm (cont’d)  General step details — update CA term sub: area(f) add: a 0 h 2 +b 0 h+c 0 sub: a 1 h 2 +b 1 h+c 1 sub: a 0 h 2 +b 0 h+c 0 add: a 1 h 2 +b 1 h+c 1  Run-time: O(n log n), space: O(n).  Minimize Ah 2 + Bh + C

11. Experimental Results  random points on a rotated “ice-cream” cone

12. Non-convex Polyhedra  the structure of supports is more complex convexnon-convex

13.  partition each front and each back facet into two classes of triangles: Black/Gray Triangles black tri. — always in contact with supports gray tri. — contact with supports depends on the position of H on the position of H

14. Computing Black/Gray Triangles  Compute supports for undecomposed polyhedron using cylindrical decomposition

15. Overall Algorithm  compute cylindrical decomposition  apply convex algorithm on gray triangles  Run-time: O(n 2 log n), space: O(n 2 )

16. Experimental Results (Volume)

17. Problem 2 Approximating the Optimal Support Requirements Identify heuristics for choosing candidate directions Identify heuristics for choosing candidate directions Design efficient algorithms to compute contact-area for chosen 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 Develop a criterion to evaluate the quality of each heuristic, via easy-to-compute quantities  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). (there is no model decomposition here).

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

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

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

21. FFA Results

22. Minimize BFA  Run-time: O(n 2 log n), space: O(n) space

23. 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

24. Experimental Results prism bot_caseoldbasexcarcassetop_case 3857438f0m27mjtod21 triad1ecc4pyramid

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

26. 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  Globally optimal decomposition direction  Multi-way decomposition  Approximating support volume  Exact algorithms for support optimization Future Work

27. Acknowledgements  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

28. Controlling Decomp. Size (K) Two-sweep algorithm up-sweep: #pieces for P - up-sweep: #pieces for P - dn-sweep: #pieces for P + dn-sweep: #pieces for P + Combine results of sweeps Use Union-Find data str.  Partition the d-direction into intervals I j s.t. any plane in I j splits P into same number of pieces k j  Optimize only within intervals where k j <= K