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

University of California, Berkeley

Drainability Testing a rotation axis for

drainability

2

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/

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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

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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

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University of California, Berkeley

Outline

Motivation and background Testing a rotation axis for drainability

Solution in 2D space Solution in 3D space

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University of California, Berkeley 7

All water traps contain a concave vertex

Drain all concave vertices!

Trapped water

gravity

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University of California, Berkeley 8

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

University of California, Berkeley 10

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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University of California, Berkeley 11

Drainability Checking

A

BC

D

E

CWCCW

CW rotation

CCW rotation

ED A

C B

OUT

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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

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University of California, Berkeley 13

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

University of California, Berkeley

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

University of California, Berkeley

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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

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

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University of California, Berkeley

Results

outlet

Not outlet

University of California, Berkeley 20

ResultsoutletNot outlet

202020

gravity

University of California, Berkeley 21

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

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

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

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Acknowledgements Yusuke Yasui Peter Cottle Daimler NSF

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