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Real Time Collaboration and Sharing

01111100101101010100101010National Science Foundation Industry/University Cooperative Research Center fore-Design: IT-Enabled Design and Realization of Engineered Products and Systems

University of Pittsburgh UMassAmherst

Solving Interval Constraints in Computer-Aided Design

Yan WangDepartment of Industrial Engineering

NSF Center for e-DesignUniversity of Pittsburgh

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N National Science Foundation Industry/University Cooperative Research Center for e-Design

University of Pittsburgh UMassAmherst

Outline

Parametric geometric modeling Interval geometric modeling Constraint solving

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Parametric Geometric Modeling

Geometric model – Geometry– Topology– Attributes

Constraint solver – Numerical– Symbolic– Graph-based / Constructive– Rule-based reasoning

Visualization

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Fixed-Value Parameter vs. Interval-Value Parameter

Fixed-value parameters may generate inconsistency errors from floating-point arithmetic.

Fixed-value constraints bring up conflicts easily at later design stages.

Fixed-value parameters make the development of Computer-Aided Conceptual Design difficult.

Interval parameters improve robustness of geometry computation.

Interval parameters capture the uncertainty and inexactness. Interval parameters directly represent boundary information

for optimization. Intervals provide a generic representation for geometric

constraints.

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N National Science Foundation Industry/University Cooperative Research Center for e-Design

University of Pittsburgh UMassAmherst

Application of IA in CAD/CAE Computer graphics: rasterizing [Mudur and Koparkar], ray tracing [Toth,

Kalra and Barr], collision detection [Moore and Wilhelms, Von Herzen et al., Duff, Snyder et

al.]. CAD: curve approximation [Sederberg and Farouki, Patrikalakis et al., Chen and Lou,

Lin et al.], shape interrogation [Maekawa and Patrikalakis], robust boundary evaluation [Patrikalakis et al., Wallner et al.]

CAE: finite element formulation [Muhanna and Mullen]

System design: set-based modeling [Finch and Ward], structural analysis [Rao et al.]

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N National Science Foundation Industry/University Cooperative Research Center for e-Design

University of Pittsburgh UMassAmherst

Given that A = [aL, aN, aU], B = [bL, bN, bU],

Nominal Intervals in IGM

Display Interactivity Tolerance

A: B: A ~> B A ~ B A ~< B A ~ B A: B: A = B A := B A B A B A: B: A B A B A B A B *Notation: xL xN xU

equivalence: nominal equivalence: strictly greater than or equal to: strictly greater than: strictly less than or equal to: strictly less than: inclusion:

UULL babaBA

UUNNLL bababaBA :

UL baBA ~

UL baBA ~

LU baBA ~

LU baBA ~

LLUU babaBA

LLUU babaBA

2D Point: 3D Point: p(X, Y) = p([xL, xN, xU],[yL, yN, yU]) p(X, Y, Z) = p([xL, xN, xU],[yL, yN, yU],[zL, zN, zU])

zL zU

zN yL

yN

yU

xL xU xN

yL

yN

yU

xL xU xN

UNLUNLULUNL xxxxxxxxxxxxxX ,,,,],,[

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N National Science Foundation Industry/University Cooperative Research Center for e-Design

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Sampling Relation between Real Number and Interval Number

strict equivalence: strictly greater than or equal to: strictly greater than: strictly less than or equal to: strictly less than:

yxByAxBA ,,~

yxByAxBA ,,~

yxByAxBA ,,~

yxByAxBA ,,~

yxByAxBA ,,~

Strict relations

Global relations global equivalence: greater than or equal to: greater than: less than or equal to: less than:

yxByAxBA ,,

LL baBA yxByAxBA ,,

LL baBA yxByAxBA ,,

UU baBA yxByAxBA ,,

UU baBA yxByAxBA ,,

yxYyXxYX ,,

yxYyXxYX ,,

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N National Science Foundation Industry/University Cooperative Research Center for e-Design

University of Pittsburgh UMassAmherst

Set vs. Individuals

Global relations are default relations in IA. Global relations ensure the feasibility of interval

arithmetic operations and solutions. Global relations make global solution and

optimization of interval analysis possible.

Strict relations exhibit the rigidity of RA. Strict relations specify constraints between variables

directly.

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N National Science Foundation Industry/University Cooperative Research Center for e-Design

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Preference, Specification, & Interval Constraint

Improve specification interoperability for design life-cycle

Represent soft constraint Capture the uncertainty of design Model incompleteness and

inexactness especially during conceptual design

Model a set of design alternatives Represent tolerance and boundary

information for global optimization Improve robustness of computation

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N National Science Foundation Industry/University Cooperative Research Center for e-Design

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Under-, Over-, & Well-Constrained ( a ) ( b )

0

0

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P 3 P 2

d 0

L 3 L 1

L 2

L 0

(a) (b)

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d0

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L2

L0

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d2

h

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N National Science Foundation Industry/University Cooperative Research Center for e-Design

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Special Considerations of Interval Linear Equations for CAD

Matrix-based methods are not for under- or over-constrained problems

Iteration-based methods (e.g. Jacobi iteration, Gauss-Seidel iteration) are more general and useful in CAD constraint solving

miYXA i

n

jjij ,...2,1

1

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X

A Y

Extended Gauss-Seidel Method

INPUT: Interval matrix A Interval vector Y OUTPUT: Interval vector X Interval V int i, j, k REPEAT until stop criterion is met FOR each 1 <= i <= m FOR each 1 <= j <= n IF Aij=0 continue next j iteration ENDIF V = 0 FOR each 1<=k<j V = V+Aik*Xk ENDFOR FOR each j+1<=k<=n V = V+Aik*Xk ENDFOR V = (Yi – V)/Aij Xj = Xj V ENDFOR ENDFOR

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N National Science Foundation Industry/University Cooperative Research Center for e-Design

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Solving Interval Nonlinear Equations based on Linear Enclosure

1. Transform to separable form;

2. Find linear enclosure;

3. Solve linear enclosure equations;

4. Update variable values

5. If stop criteria not satisfied, go to step 2; otherwise stop.

liCF ii ,...2,1XStart

Transform to Separable Form

Stop Criteria Satisfied?

End

Y

N

Find Linear Enclosure

Solve LinearEnclosure Equations

Update Variable Values

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1. Separable Form Function f(x1, x2, …, xn) is said to be separable iff

f(x1, x2, …, xn) = f1(x1) + f2(x2) + … + fn(xn). Yamamura’s algorithm[Yamamura,1996]: +, , , /, sin, exp, log,

sqrt, ^, etc.

For example:

f = f1 f2 f = (y2 f12 f2

2)/2

y = f1 + f2

f = f1 / f2 f = (y2 f121/ f2

2)/2

y = f1 + 1/f2

f = (f1)f2 f =exp(y1)

y1= (y22 (log(f1))

2 f22)/2

y2 = log(f1) + f2

liCF ii ,...2,1X

miDXfn

jijij ,...,2,1

0

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2. Linear Enclosure

Let Xj0 = [xL

j, xNj, xU

j]

fij(xj)

xj

Xj0

Xj1

Dij

fijS

fijT

Bij

jLij

Sij xff j

UijT

ij xff

jL

jU

Sij

Tij

ij xx

ffa

0jijijij XxforxaBxE

Linear Enclosure is defined as:

such that 0

jijij XxforxExf

Extending Kolev’s work: miXfn

jjij ,...,2,10

0

miDXfn

jijij ,...,2,1

0

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3. Solve Linear Enclosure EquationsIf fij(x) is continuous within interval Xj

0, solve

using root isolation [Collins et al.] and Secant method.

Suppose xjp (p=1, 2, …, P) is the pth solution of the above

equation, and xj0=xLj. Let Bij=[bL

ij, bNij, bU

ij], where

0jijij Xxforaxf

Ppxaxfb jpijjpijp

ijU ,...,2,1,0,max

00 jijjijijN xaxfb

Ppxaxfb jpijjpijp

ijL ,...,2,1,0,min

miforDXaB i

n

jjijij ,...,2,1

1

miDXfn

jijij ,...,2,1

0

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N National Science Foundation Industry/University Cooperative Research Center for e-Design

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4. Update Variable Values

Suppose Yj is the jth variable solution of linear

enclosure equations in the kth iteration, update Xj for

(k+1)th iteration by

If an empty interval is derived, the original system has no solution within the given initial intervals.

If the stop criterion is not met, iterate.

njforYXX jk

jk

j ,...,2,1)()1(

11

)(

1

)1( )(wid)(wid

n

j

kj

n

j

kj XX 2

1

)( )(wid

n

j

kjX

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N National Science Foundation Industry/University Cooperative Research Center for e-Design

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Solving Interval Inequalities

Adding slack variables to translate inequalities into equalities.

Solving linear/nonlinear equations with previous methods.

liCF ii ,...2,1X

liCSF iii ,...2,1X

liCF ii ,...2,1X

liCSF iii ,...2,1X

liSi ,...2,1],0,0[ liSi ,...2,1]0,0,[

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Interval Subdivision

Subpaving divides a hyper-cube into multiple smaller hyper-cubes recursively

Implemented as order elevation of power interval

P(m, n) = [X1, X2, …, Xm]

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Constraint Re-Specification

Need to differentiate active and inactive constraints. For a constraint set p = {f(X) = Y and g(X) = Z}, the subset f(X) = Y with respect to a solution D X is inactive if f(D) Y and g(D) Z.

(a)

(b)

(c)

S1

S2

D1 S1 D2

D2 S2 D1

x-space

f Z Y

z-space y-space

g

D1 S1

x-space

f Z Y

z-space y-space

g D2

S2

x-space

f Z Y

z-space y-space

g

(a) f – inactive, g – active

(b) f – active, g – active

(c) f – active, g – inactive

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An Example (a) (b)

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L0

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00

uxx

dvu

vyy

uxx

dvu

vyy

uxx

dvu

yy

by

ax

cxx

wvv

wuu

owwvuvu

wvv

wuu

owwvuvu

vyy

uxx

dvu

vyy

10

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430

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222222

0

0

222222

0

0

0

Convergence of intervals

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 5 10 15 20 25Iterations

X0Y0X1Y1X2Y2X3Y3

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Refinement - subdivision subdivide up to Level 3, and some sub-regions

are eliminated.

(a) original solution (b) level 1 elevation

(c) level 2 elevation (d) level 3 elevation

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What can interval provide for design?

The decisions to fix values of parameters can be postponed to later design stages.

Variation and uncertain are inherent in the process of design.

Soft constraint-driven geometry modeling Support under- and over-constrained problem Integrated linear, nonlinear equations, and inequality

solving

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Thank you!