1. Computer-Aided Design and Manufacturing Laboratory: Rotational Drainability Sara McMains UC Berkeley

2. University of California, Berkeley Drainability Testing a rotation axis for drainability 2

3. University of California, Berkeley 3 Problem Find an orientation relative to the horizontal rotation axis to drain trapped water Re-orientation is not allowed Can rotate either CW or CCW gravity Does not drain Does drain cross-section rotation axis trapped water http://www.mtm-gmbh.com/ 3

4. University of California, Berkeley 4 Motivation Should run interactively Monitor/check design at any time Feedback to designer if design is not drainable Solve purely from geometric perspective Physics-based method such as CFD is too slow Test a given orientation as a first step [Yasui, McMains ‘11] Assume force applied to water is gravity only Rotation is slow enough 4

5. University of California, Berkeley 5 Geometric Analysis of Manufacturing Process Filling analysis in gravity casting [Bose et al. 98] Rolling a ball out of a polygon [Aloupis et al. 08] Tool accessibility analysis using visibility [Woo et al. 94] Find a rotation axis that minimizes number of setups in planning for 4-axis NC machining [Tang et al. 98, Tang & Liu 03] Related Work 5

6. University of California, Berkeley Outline Motivation and background Testing a rotation axis for drainability Solution in 2D space Solution in 3D space 6

7. University of California, Berkeley 7 All water traps contain a concave vertex Drain all concave vertices! Trapped water gravity 7

8. University of California, Berkeley 8 Consider... One water particle approximates a water trap gravity 8

9. University of California, Berkeley 9 Gravity directions that trap particle at vertex v: Fix geometry, consider gravity rotating relative to geometry Describe gravity as a point on the Gaussian circle Gaussian circle 9 CW CCW

10. University of California, Berkeley 10 Draining Graph A BC D E OUT CW CCW D CB A E Draining graph Particles trapped at concave vertices Capture transitions between concave vertices 10

11. University of California, Berkeley 11 Drainability Checking A BC D E CW CCW CW rotation CCW rotation E D A CB OUT 11

12. University of California, Berkeley Outline Motivation and background Testing a rotation axis for drainability Solution in 2D space Solution in 3D space Input is triangulated boundary representation Results and conclusions 12

13. University of California, Berkeley 13 Construct T v, find, in 3D Describe gravity as a point on the Gaussian Sphere. 13

14. University of California, Berkeley 14 Set rotation axis along z-axis Possible gravity direction where xy-plane intersects sphere 14 Construct T v, find, in 3D

15. University of California, Berkeley 15 Incremental calculation of,

16. University of California, Berkeley 16 Cases for particle tracing in 3D From each concave vertex v Trace along geometric features under / Construct 3D draining graph edges Vertex cases Ridge edge cases Valley edge cases 16

17. University of California, Berkeley 17 Procedure Find concave vertices For each Set as node in draining graph Calculate its,, and Under and, trace paths Add directed edges according to the transitions Check drainability by checking whether there is a path from each node to “out”

18. University of California, Berkeley Outline Motivation and background Testing a rotation axis for drainability Solution in 2D space Solution in 3D space Results and conclusions 18

19. University of California, Berkeley Results outlet Not outlet

20. University of California, Berkeley 20 Results outlet Not outlet 20 gravity

21. University of California, Berkeley 21 # of triangles # of concave vertices Time (sec) Performance: Avg. Testing Time (2.66 GHz CPU, 4GB of RAM) #triangles3,572120,004160,312289,956

22. University of California, Berkeley 22 Future Work Relax simplifying assumptions Pauses required? Multiple rotations required? Consider initial filling state Finding an orientation to drain trapped water Estimating remaining water if not completely drainable 22

23. University of California, Berkeley Conclusions First formulation of solutions to drainability feedback Concave vertex drainability graph Critical gravity directions for transitions Less than 1 second per orientation 23

24. University of California, Berkeley Acknowledgements Yusuke Yasui Peter Cottle Daimler NSF 24