computer-aided design and manufacturing laboratory: rotational drainability sara mcmains uc berkeley
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Computer-Aided Design and Manufacturing Laboratory: Rotational Drainability
Sara McMains
UC Berkeley
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University of California, Berkeley
Drainability Testing a rotation axis for
drainability
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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/
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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
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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
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Outline
Motivation and background Testing a rotation axis for drainability
Solution in 2D space Solution in 3D space
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All water traps contain a concave vertex
Drain all concave vertices!
Trapped water
gravity
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Consider...
One water particle approximates a water trap
gravity
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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
v
2e1e
1H 2H
v v
vTCCWg CWg
}1,0)(|{ vpvpepH ii i
iv HT
1e 2e
CWgCCWg Gaussian circle
vT
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CWCCW
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Draining Graph
A
BC
D
EOUT
CWCCW
D
C B
AE
Draining graph
Particles trapped at concave vertices Capture transitions between concave vertices
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Drainability Checking
A
BC
D
E
CWCCW
CW rotation
CCW rotation
ED A
C B
OUT
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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
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Construct Tv , find , in 3D
}1,0)(|{ vpvpepH ii
i
iv HT
1H 2H vT3H
2e1e
3e1e 2e 3e
v
Describe gravity as a point on the Gaussian Sphere.
CWg CCWg
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Set rotation axis along z-axis Possible gravity direction where xy-plane intersects sphere
)plane()( xyTT vxyv )plane()( xyHH ixyi
ixyixyv HT )()( 14
Construct Tv , find , in 3DCWg CCWg
iCWg
iCCWg
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1H 2H 3H
2e
1e
3e1e
2e 3e
v
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i
xyixyv HT )()( )plane()( xyHH ixyi )plane()( xyTT vxyv
Incremental calculation of , CWg CCWg
2CWg
2CCWg
1CCWg1CWg
1CWg
1CCWg2CCWg
2CWg
3CWg3CCWg
CCWgCWg
3CCWg 3CWg
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Cases for particle tracing in 3D From each concave vertex v
Trace along geometric features under / CWg CCWg
g
Construct 3D draining graph edges
Vertex cases
Ridge edge casesValley edge cases
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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”
vT CWg CCWg
CWg CCWg
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Outline
Motivation and background Testing a rotation axis for drainability
Solution in 2D space Solution in 3D space
Results and conclusions
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Results
outlet
Not outlet
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ResultsoutletNot outlet
202020
gravity
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0.0
0.2
0.4
0.6
0.8
1.0
0 100,000 200,000 300,0000.0
0.2
0.4
0.6
0.8
1.0
0 20000 40000 60000
# of triangles # of concave vertices
Time (sec)Time (sec)
Performance: Avg. Testing Time
(2.66 GHz CPU, 4GB of RAM)
#triangles
3,572 120,004 160,312 289,956
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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
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Conclusions First formulation of solutions to
drainability feedback Concave vertex drainability graph Critical gravity directions for transitions Less than 1 second per orientation
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Acknowledgements Yusuke Yasui Peter Cottle Daimler NSF
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