Computer-Aided Design and Manufacturing Laboratory: Rotational Drainability

Sara McMains

UC Berkeley

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

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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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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