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Geometric Algorithms for Layered Manufacturing: Optimizing Model Acquisition and Support Requirements

This paper explores advanced geometric algorithms for layered manufacturing, focusing on model acquisition, support creation, and post-processing techniques. It investigates methods to minimize surface roughness, the number of layers, and support requirements while optimizing build direction for quality and performance. Relevant techniques such as slicing, file repair, and orientation are discussed, with applications in industries like automotive and aerospace. The research also reviews decomposition-based approaches and recent advancements in the computation of support structures, enhancing the efficiency of 3D printing technologies.

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Geometric Algorithms for Layered Manufacturing: Optimizing Model Acquisition and Support Requirements

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  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

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