011010101001010010100100111
01101010100101010010111101
Real Time Collaboration and Sharing
0101010010101001011110001
011010101001010100101110
National Science Foundation Industry/University Cooperative Research Center for
01111100101101010100101010
e-Design: IT-Enabled Design and Realization of Engineered Products and Systems
Solving Interval Constraints in
Computer-Aided Design
Yan Wang
Department of Industrial Engineering
NSF Center for e-Design
University of Pittsburgh
University of Pittsburgh
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ENTER FOR e –DESIGN
National Science Foundation Industry/University Cooperative Research Center for e-Design
Outline



Parametric geometric modeling
Interval geometric modeling
Constraint
solving
011010101001010010100100111
01101010100101010010111101
0101010010101001011110001
011010101001010100101110
01111100101101010100101010
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Parametric Geometric Modeling

Geometric model
Geometry
– Topology
011010101001010010100100111
– Attributes
01101010100101010010111101
 Constraint
solver
0101010010101001011110001
– Numerical
011010101001010100101110
01111100101101010100101010
– 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.
011010101001010010100100111
Fixed-value
parameters make the development of Computer01101010100101010010111101
Aided
Conceptual Design difficult.
0101010010101001011110001
Interval
parameters improve robustness of geometry
011010101001010100101110
computation.
01111100101101010100101010
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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ENTER FOR e –DESIGN
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
011010101001010010100100111
01101010100101010010111101
boundary
evaluation [Patrikalakis et al., Wallner et al.]
0101010010101001011110001
CAE: finite
element formulation [Muhanna and Mullen]
011010101001010100101110
System
design: set-based modeling [Finch and Ward], structural
01111100101101010100101010
[Rao et al.]
analysis
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ENTER FOR e –DESIGN
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Nominal Intervals in IGM
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])
yU
yU
yN
yN
yL
yL
xL
xN
xU

zL zN
xL
xN

zU

xU
X  [ xL , xN , x011010101001010010100100111
U ]  x xL  x  xU , xL  x N  xU , xL  , x N  , xU  
Display
Interactivity
Tolerance
Given that01101010100101010010111101
A = [aL, aN, aU], B = [bL, bN, bU],







0101010010101001011110001
equivalence:
A  B  a  b   a  b 
011010101001010100101110
A : B  a  b   a  b   a
nominal
equivalence:
01111100101101010100101010
strictly
greater than or equal to: A ~ B  a  b
A ~ B  a  b
strictly greater than:
strictly less than or equal to:
A ~ B  a  b
strictly less than:
A ~ B  a  b
A  B  a  b   a  b 
inclusion:
L
L
U
U
L
L
N
N
U
 bU 
A:
B:
L
U
L
U
U
U
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:
L
L
*Notation:
xL xN xU
U
U
L
L
A  B  aU  bU   aL  bL 
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Sampling Relation between
Real Number and Interval Number
Strict relations





XY  x  X , y  Y , xy
A ~  B  x  A, y  B, x  y
strict equivalence:
A ~ B  x  A, y  B, x  y
strictly greater than or equal to:
A ~  B  x  A, y  B, x  y
strictly greater than:
A ~ B  x  A, y  B, x  y
strictly
less than or equal to:
011010101001010010100100111
A ~ B  x  A, y  B, x  y
strictly
less than:
01101010100101010010111101
0101010010101001011110001
Global relations XY  x  X , y  Y , xy
011010101001010100101110
A  B  x  A, y  B, x  y
 global equivalence:
01111100101101010100101010




greater than or equal to:
greater than:
less than or equal to:
less than:
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A  B  aL  bL
A  B  x  A, y  B, x  y
A  B  aL  bL
A  B  x  A, y  B, x  y
A  B  aU  bU
A  B  x  A, y  B, x  y
A  B  aU  bU
A  B  x  A, y  B, x  y
ENTER FOR e –DESIGN
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Set vs. Individuals



Global relations are default relations in IA.
Global relations ensure the feasibility of interval
arithmetic operations and solutions.
011010101001010010100100111
Global
relations make global solution and
01101010100101010010111101
optimization
of interval analysis possible.
0101010010101001011110001
011010101001010100101110
01111100101101010100101010


Strict relations exhibit the rigidity of RA.
Strict relations specify constraints between variables
directly.
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Preference, Specification, & Interval Constraint
Improve specification
interoperability for design life-cycle
 Represent soft constraint
 Capture the uncertainty of design
011010101001010010100100111
 Model incompleteness and
01101010100101010010111101
inexactness especially during
0101010010101001011110001
conceptual design
011010101001010100101110
 Model a set of design alternatives
01111100101101010100101010
 Represent tolerance and boundary
information for global optimization
 Improve robustness of computation

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Under-, Over-, & Well-Constrained
P3
P2
L2
L3
L1
x1  x0 2   y1  y0 2  d 02
x0  x3 x1  x0    y0  y3  y1  y0   0
x2  x1 x1  x0    y2  y1  y1  y0   0
L
011010101001010010100100111
P
P
d
01101010100101010010111101
(a)
(b)
0101010010101001011110001
x a
011010101001010100101110
y b
d
P
P
y y 0
01111100101101010100101010
L
0
0
1
0
2
3
2
2
d3
L3
P0
d1
L1
L0
h
P1
d0
0
0
0
0
1
x1  x0 2   y1  y0 2  d 02
x2  x1 2   y2  y1 2  d12
x3  x2 2   y3  y2 2  d 22
x0  x3 2   y0  y3 2  d 32
x0  x3 x1  x0    y0  y3  y1  y0   o1
x2  x1 x1  x0    y2  y1  y1  y0   o2
x0  x1
(a)
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Special Considerations of Interval Linear
Equations for CAD
n
A X
j 1
ij
j
 Yi
i  1,2,...m
011010101001010010100100111
Matrix-based
methods are not for under- or over01101010100101010010111101
constrained
problems
0101010010101001011110001
 Iteration-based methods (e.g. Jacobi iteration, Gauss011010101001010100101110
Seidel iteration) are more general and useful in CAD
01111100101101010100101010
constraint solving

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Extended Gauss-Seidel Method
X
A
INPUT:
Interval matrix A
Interval vector Y
OUTPUT: Interval vector X
Y
Interval V
011010101001010010100100111
int i, j, k
REPEAT until stop criterion is met
FOR each 1 <= i <= m
01101010100101010010111101
FOR each 1 <= j <= n
IF A =0
0101010010101001011110001
continue next j iteration
011010101001010100101110 VENDIF
= 0
FOR each 1<=k<j
01111100101101010100101010
V = V+A *X
ij
ik
k
ENDFOR
FOR each j+1<=k<=n
V = V+Aik*Xk
ENDFOR
V = (Yi – V)/Aij
X j = Xj  V
ENDFOR
ENDFOR
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Solving Interval Nonlinear Equations based on
Linear Enclosure
Fi X  Ci
i  1,2,...l
Transform to separable form;
011010101001010010100100111
2. Find
linear enclosure;
01101010100101010010111101
3. Solve linear enclosure
0101010010101001011110001
equations;
011010101001010100101110
4. 01111100101101010100101010
Update variable values
5. If stop criteria not satisfied, go
to step 2; otherwise stop.
Start
Transform to
Separable Form
1.
Find
Linear Enclosure
Solve Linear
Enclosure Equations
Update
Variable Values
N
Stop Criteria
Satisfied?
Y
End
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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.
011010101001010010100100111
For example:
01101010100101010010111101
f = f1  0101010010101001011110001
f2  f = (y2 f12 f22)/2
Fi X  Ci i  1,2,...l
011010101001010100101110
y = f 1 + f2
01111100101101010100101010
f=
f1 / f2  f = (y2 f121/ f22)/2
n
y = f1 + 1/f2
f ij X j  Di i  1,2,..., m

j 0
f = (f1)f2  f =exp(y1)
y1= (y22 (log(f1))2 f22)/2
y2 = log(f1) + f2
 
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2. Linear Enclosure
Extending Kolev’s work:
 f X   0
n
j 0
ij
j
 f X   D
n
i  1,2,..., m
j 0
ij
j
i
i  1,2,..., m
Let Xj0 = [xLj, xNj, xUj]
 
f (x )
f ijS  f ij 011010101001010010100100111
xLj
f ijT  f ij xUj 
ij
j
01101010100101010010111101
f  f ijS
0101010010101001011110001
aij 
x011010101001010100101110
 x Lj
D
01111100101101010100101010
f
Linear Enclosure is defined as:
B
T
ij
j
U
ij
S
ij
Eij x  Bij  aij x
such that
f ij x Eij x
fijT
for x  X 0j
ij
for x  X 0j
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X j1
Xj 0
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3. Solve Linear Enclosure Equations
If fij(x) is continuous within interval Xj0, solve
f ij x  aij for x  X 0j
using root isolation [Collins et al.] and Secant method.
011010101001010010100100111
Suppose xjp (p=1, 2, …, P) is the pth solution of the above
01101010100101010010111101
equation,
and xj0=xLj. Let Bij=[bLij, bNij, bUij], where
0101010010101001011110001
bUij  max f ij x jp   aij x jp , p  0,1,2,..., P
011010101001010100101110
p
01111100101101010100101010
bNij  fij x j 0   aij x j 0
bLij  min f ij x jp   aij x jp , p  0,1,2,..., P
p
 f X   D
n
j 0
ij
j
i
i  1,2,..., m
 B
n
j 1
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ij
 aij X j   Di
for i  1,2,..., m
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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

011010101001010010100100111
X (j k 1)  X (j k )  Y j for j  1,2,..., n
01101010100101010010111101
If an 0101010010101001011110001
empty interval is derived, the original system has
no011010101001010100101110
solution within the given initial intervals.
01111100101101010100101010

If the stop criterion is not met, iterate.
n
 wid ( X
j 1
( k 1)
j
n
)  wid ( X
(k )
j
)  1
j 1
University of Pittsburgh
n
(k )
wid
(
X

j )  2
j 1
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Solving Interval Inequalities


Adding slack variables to translate inequalities
into equalities.
Solving linear/nonlinear equations with previous
methods.
011010101001010010100100111
01101010100101010010111101
0101010010101001011110001
X   C i  1,2,...l
F011010101001010100101110
F X   C
01111100101101010100101010
i
i
i
i  1,2,...l
Fi X   Si  Ci i  1,2,...l
Si  [,0,0] i  1,2,...l
Fi X   Si  Ci i  1,2,...l
Si  [0,0,] i  1,2,...l
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Interval Subdivision


Subpaving divides a hyper-cube into multiple smaller
hyper-cubes recursively
Implemented as order elevation of power interval
011010101001010010100100111
01101010100101010010111101
P(m, n) = [X1, X2, …, Xm]
0101010010101001011110001
011010101001010100101110
01111100101101010100101010
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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)
 Y011010101001010010100100111
and g(D)  Z.
01101010100101010010111101
0101010010101001011110001
(a) f – inactive, g – active
(a)
011010101001010100101110
01111100101101010100101010
y-space
x-space
f
Y
D1
S1
S2
D2
Z
g
x-space
y-space
Y
z-space
z-space
g
D2
f
D1
Z
S2
S1
(b) f – active, g – active
(b)
y-space
Y
x-space
z-space
f
D1
S2
S1
D2
g
Z
(c)
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(c) f – active, g – inactive
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An Example
x0  a 0
y0  b0
d2
P3
y0  y1  0
P2
L2
d3
L3
d1
L1
L0
P0
h
x1  x0 2   y1  y0 2  d 02
x2  x1 2   y2  y1 2  d12
x3  x2 2   y3  y2 2  d 22
x0  x3 2   y0  y3 2  d 32
x0  x3 x1  x0    y0  y3  y1  y0   o1
x2  x1 x1  x0    y2  y1  y1  y0   o2
P1
d0
011010101001010010100100111
01101010100101010010111101
y  y v  0
x a
0101010010101001011110001
y b
u v  d
y y 0
 x  x u  0
011010101001010100101110
 y  y v  0
u v  d
u 2v 2u 2v 2w 2w 2  o
x  x01111100101101010100101010
u  0
x0  x1
(a)
(b)
0
2
3
0
0
2
4
2
4
0
1
2
1
2
1
0
0
1
2
0
2
1
1
3
2
3
0
3
4
0
3
4
2
1
2
4
y 0  y1  v1  0
 u1  u 4  w1  0
u v  d
 v1  v 4  w2  0
2
2
2
2
2
1
2
4
2
1
2
2
1
x1  x 2  u 2  0
u12 2  v12 2  u 22 2  v 22 2  w32 2  w42 2  o 2
y1  y 2  v 2  0
 u 1  u 2  w3  0
u 32  v32  d 22
 v1  v 2  w4  0
x 2  x3  u 3  0
 x 0  x1  c
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Refinement - subdivision

subdivide up to Level 3, and some sub-regions
are eliminated.
011010101001010010100100111
01101010100101010010111101
0101010010101001011110001
(a) original solution
(b) level 1 elevation
011010101001010100101110
01111100101101010100101010
(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
011010101001010010100100111
design.
01101010100101010010111101
 Soft
constraint-driven geometry modeling
0101010010101001011110001
011010101001010100101110
 Support
under- and over-constrained problem
01111100101101010100101010
 Integrated linear, nonlinear equations, and inequality
solving

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UMassAmherst
ENTER FOR e –DESIGN
National Science Foundation Industry/University Cooperative Research Center for e-Design
Thank you!
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University of Pittsburgh
UMassAmherst