1. 01101010100101010010111101 011010101001010010100100111 011010101001010100101110 0101010010101001011110001 Real Time Collaboration and Sharing 01111100101101010100101010 National Science Foundation Industry/University Cooperative Research Center for 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

2. 01101010100101010010111101 011010101001010010100100111 011010101001010100101110 0101010010101001011110001 01111100101101010100101010 C E N T E R F O R e – D E S I G N National Science Foundation Industry/University Cooperative Research Center for e-Design Outline Parametric geometric modeling Interval geometric modeling Constraint solving

3. 01101010100101010010111101 011010101001010010100100111 011010101001010100101110 0101010010101001011110001 01111100101101010100101010 C E N T E R F O R e – D E S I G N National Science Foundation Industry/University Cooperative Research Center for e-Design Parametric Geometric Modeling Geometric model – Geometry – Topology – Attributes Constraint solver – Numerical – Symbolic – Graph-based / Constructive – Rule-based reasoning Visualization

4. 01101010100101010010111101 011010101001010010100100111 011010101001010100101110 0101010010101001011110001 01111100101101010100101010 C E N T E R F O R e – D E S I G N National Science Foundation Industry/University Cooperative Research Center for e-Design 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.

5. 01101010100101010010111101 011010101001010010100100111 011010101001010100101110 0101010010101001011110001 01111100101101010100101010 C E N T E R F O R e – D E S I G N National Science Foundation Industry/University Cooperative Research Center 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 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.]

6. 01101010100101010010111101 011010101001010010100100111 011010101001010100101110 0101010010101001011110001 01111100101101010100101010 C E N T E R F O R e – D E S I G N National Science Foundation Industry/University Cooperative Research Center for e-Design Given that A = [a L, a N, a U ], B = [b L, b N, b U ], Nominal Intervals in IGM Display Interactivity Tolerance  equivalence:  nominal equivalence:  strictly greater than or equal to:  strictly greater than:  strictly less than or equal to:  strictly less than:  inclusion:

7. 01101010100101010010111101 011010101001010010100100111 011010101001010100101110 0101010010101001011110001 01111100101101010100101010 C E N T E R F O R e – D E S I G N National Science Foundation Industry/University Cooperative Research Center for e-Design 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: Strict relations Global relations  global equivalence:  greater than or equal to:  greater than:  less than or equal to:  less than:

8. 01101010100101010010111101 011010101001010010100100111 011010101001010100101110 0101010010101001011110001 01111100101101010100101010 C E N T E R F O R e – D E S I G N National Science Foundation Industry/University Cooperative Research Center for e-Design 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.

9. 01101010100101010010111101 011010101001010010100100111 011010101001010100101110 0101010010101001011110001 01111100101101010100101010 C E N T E R F O R e – D E S I G N National Science Foundation Industry/University Cooperative Research Center for e-Design 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

10. 01101010100101010010111101 011010101001010010100100111 011010101001010100101110 0101010010101001011110001 01111100101101010100101010 C E N T E R F O R e – D E S I G N National Science Foundation Industry/University Cooperative Research Center for e-Design Under-, Over-, & Well-Constrained

11. 01101010100101010010111101 011010101001010010100100111 011010101001010100101110 0101010010101001011110001 01111100101101010100101010 C E N T E R F O R e – D E S I G N National Science Foundation Industry/University Cooperative Research Center for e-Design 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

12. 01101010100101010010111101 011010101001010010100100111 011010101001010100101110 0101010010101001011110001 01111100101101010100101010 C E N T E R F O R e – D E S I G N National Science Foundation Industry/University Cooperative Research Center for e-Design X AY Extended Gauss-Seidel Method

13. 01101010100101010010111101 011010101001010010100100111 011010101001010100101110 0101010010101001011110001 01111100101101010100101010 C E N T E R F O R e – D E S I G N National Science Foundation Industry/University Cooperative Research Center for e-Design 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. Start Transform to Separable Form Stop Criteria Satisfied? End Y N Find Linear Enclosure Solve Linear Enclosure Equations Update Variable Values

14. 01101010100101010010111101 011010101001010010100100111 011010101001010100101110 0101010010101001011110001 01111100101101010100101010 C E N T E R F O R e – D E S I G N National Science Foundation Industry/University Cooperative Research Center for e-Design 1. Separable Form  Function f(x 1, x 2, …, x n ) is said to be separable iff f(x 1, x 2, …, x n ) = f 1 (x 1 ) + f 2 (x 2 ) + … + f n (x n ). Yamamura’s algorithm[Yamamura,1996]: +, , , /, sin, exp, log, sqrt, ^, etc. For example: f = f 1  f 2  f = (y 2  f 1 2  f 2 2 )/2 y = f 1 + f 2 f = f 1 / f 2  f = (y 2  f 1 2  1/ f 2 2 )/2 y = f 1 + 1/f 2 f = (f 1 ) f2  f =exp(y 1 ) y 1 = (y 2 2  (log(f 1 )) 2  f 2 2 )/2 y 2 = log(f 1 ) + f 2

15. 01101010100101010010111101 011010101001010010100100111 011010101001010100101110 0101010010101001011110001 01111100101101010100101010 C E N T E R F O R e – D E S I G N National Science Foundation Industry/University Cooperative Research Center for e-Design 2. Linear Enclosure Let X j 0 = [x L j, x N j, x U j ] Linear Enclosure is defined as: such that Extending Kolev’s work:

16. 01101010100101010010111101 011010101001010010100100111 011010101001010100101110 0101010010101001011110001 01111100101101010100101010 C E N T E R F O R e – D E S I G N National Science Foundation Industry/University Cooperative Research Center for e-Design 3. Solve Linear Enclosure Equations If f ij (x) is continuous within interval X j 0, solve using root isolation [Collins et al.] and Secant method. Suppose x jp (p=1, 2, …, P) is the p th solution of the above equation, and x j0 =x L j. Let B ij =[b L ij, b N ij, b U ij ], where

17. 01101010100101010010111101 011010101001010010100100111 011010101001010100101110 0101010010101001011110001 01111100101101010100101010 C E N T E R F O R e – D E S I G N National Science Foundation Industry/University Cooperative Research Center for e-Design 4. Update Variable Values Suppose Y j is the j th variable solution of linear enclosure equations in the k th iteration, update X j 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.

18. 01101010100101010010111101 011010101001010010100100111 011010101001010100101110 0101010010101001011110001 01111100101101010100101010 C E N T E R F O R e – D E S I G N National Science Foundation Industry/University Cooperative Research Center for e-Design Solving Interval Inequalities Adding slack variables to translate inequalities into equalities. Solving linear/nonlinear equations with previous methods.

19. 01101010100101010010111101 011010101001010010100100111 011010101001010100101110 0101010010101001011110001 01111100101101010100101010 C E N T E R F O R e – D E S I G N National Science Foundation Industry/University Cooperative Research Center for e-Design Interval Subdivision Subpaving divides a hyper-cube into multiple smaller hyper-cubes recursively Implemented as order elevation of power interval P (m, n) = [X 1, X 2, …, X m ]

20. 01101010100101010010111101 011010101001010010100100111 011010101001010100101110 0101010010101001011110001 01111100101101010100101010 C E N T E R F O R e – D E S I G N National Science Foundation Industry/University Cooperative Research Center for e-Design 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) f – inactive, g – active (b) f – active, g – active (c) f – active, g – inactive

21. 01101010100101010010111101 011010101001010010100100111 011010101001010100101110 0101010010101001011110001 01111100101101010100101010 C E N T E R F O R e – D E S I G N National Science Foundation Industry/University Cooperative Research Center for e-Design An Example

22. 01101010100101010010111101 011010101001010010100100111 011010101001010100101110 0101010010101001011110001 01111100101101010100101010 C E N T E R F O R e – D E S I G N National Science Foundation Industry/University Cooperative Research Center for e-Design Refinement - subdivision subdivide up to Level 3, and some sub-regions are eliminated.

23. 01101010100101010010111101 011010101001010010100100111 011010101001010100101110 0101010010101001011110001 01111100101101010100101010 C E N T E R F O R e – D E S I G N National Science Foundation Industry/University Cooperative Research Center for e-Design 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

24. 01101010100101010010111101 011010101001010010100100111 011010101001010100101110 0101010010101001011110001 01111100101101010100101010 C E N T E R F O R e – D E S I G N National Science Foundation Industry/University Cooperative Research Center for e-Design Thank you!