Geometric Algorithms for Layered Manufacturing: Part II

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

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Layered Manufacturing

• Builds 3D models as a stack of 2D layers

Stereolithography
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Geometric Considerations

• The choice of build direction affects quality and
  performance measures

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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

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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

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Decomposition-Based Approach

• Decompose the model with a plane into a small
  number of pieces
• Build the pieces separately
• Glue the pieces back together

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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

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Minimizing Contact-Area (CA) for Convex Polyhedra

• CA depends on height of H and orientation of
  facets
• e.g. back facet f (nf d lt 0)

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Overall Algorithm

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

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Overall Algorithm (contd)

• General step details update CA term

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Experimental Results

• random points on a sphere of radius 100

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Experimental Results

• random points on a rotated ice-cream
  cone

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Non-convex Polyhedra

• the structure of supports is more complex

convex
non-convex
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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
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Computing Black/Gray Triangles

• Use cylindrical decomposition

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Overall Algorithm

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

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Controlling Decomp. Size (K )

• Partition the d-direction into intervals Ij s.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.

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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

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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
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Evaluation Criterion
d build direction computed by heuristic d
optimal build direction d direction which
minimizes BFA
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Compute CA

• compute BFA, FFA and PFA for direction d
• compute FFA

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Minimize BFA

• Run-time O(n2 log n), space O(n) space

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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

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prism
triad1
ecc4
pyramid
3857438
f0m27
mj
tod21
bot_case
oldbasex
carcasse
top_case
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• Columns shows upper bound on

 
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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

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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