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01101010100101010010111101011010101001010010100100111
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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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N National Science Foundation Industry/University Cooperative Research Center for e-Design
University of Pittsburgh UMassAmherst
Parametric Geometric Modeling
Geometric model – Geometry– Topology– Attributes
Constraint solver – Numerical– Symbolic– Graph-based / Constructive– Rule-based reasoning
Visualization
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N National Science Foundation Industry/University Cooperative Research Center for e-Design
University of Pittsburgh UMassAmherst
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
University of Pittsburgh UMassAmherst
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 ,,
01101010100101010010111101011010101001010010100100111
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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
University of Pittsburgh UMassAmherst
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
University of Pittsburgh UMassAmherst
Under-, Over-, & Well-Constrained ( a ) ( b )
0
0
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yyyyxxxx
yyyyxxxx
dyyxx
P 0 P 1
P 3 P 2
d 0
L 3 L 1
L 2
L 0
(a) (b)
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00
00
0
xx
oyyyyxxxx
oyyyyxxxx
dyyxx
dyyxx
dyyxx
dyyxx
yy
by
ax
P0 P1
P3 P2
d0
L3 L1
L2
L0
d3 d1
d2
h
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N National Science Foundation Industry/University Cooperative Research Center for e-Design
University of Pittsburgh UMassAmherst
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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N National Science Foundation Industry/University Cooperative Research Center for e-Design
University of Pittsburgh UMassAmherst
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
University of Pittsburgh UMassAmherst
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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N National Science Foundation Industry/University Cooperative Research Center for e-Design
University of Pittsburgh UMassAmherst
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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N National Science Foundation Industry/University Cooperative Research Center for e-Design
University of Pittsburgh UMassAmherst
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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N National Science Foundation Industry/University Cooperative Research Center for e-Design
University of Pittsburgh UMassAmherst
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
University of Pittsburgh UMassAmherst
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
University of Pittsburgh UMassAmherst
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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N National Science Foundation Industry/University Cooperative Research Center for e-Design
University of Pittsburgh UMassAmherst
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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N National Science Foundation Industry/University Cooperative Research Center for e-Design
University of Pittsburgh UMassAmherst
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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N National Science Foundation Industry/University Cooperative Research Center for e-Design
University of Pittsburgh UMassAmherst
An Example (a) (b)
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P0 P1
P3 P2
d0
L3 L1
L2
L0
d3 d1
d2
h
0
0
0
0
0
0
332
22
23
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221
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110
110
20
21
21
10
00
00
uxx
dvu
vyy
uxx
dvu
vyy
uxx
dvu
yy
by
ax
cxx
wvv
wuu
owwvuvu
wvv
wuu
owwvuvu
vyy
uxx
dvu
vyy
10
421
321
224
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22
22
21
21
241
141
122
21
24
24
21
21
430
430
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24
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0
0
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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N National Science Foundation Industry/University Cooperative Research Center for e-Design
University of Pittsburgh UMassAmherst
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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N National Science Foundation Industry/University Cooperative Research Center for e-Design
University of Pittsburgh UMassAmherst
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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N National Science Foundation Industry/University Cooperative Research Center for e-Design
University of Pittsburgh UMassAmherst
Thank you!