Solving Interval Constraints in Computer-Aided Design

1. Solving Interval Constraints in Computer-Aided Design Yan Wang Department of Industrial Engineering NSF Center for e-Design University of Pittsburgh

2. Outline • Parametric geometric modeling • Interval geometric modeling • Constraint solving

3. Parametric Geometric Modeling • Geometric model • Geometry • Topology • Attributes • Constraint solver • Numerical • Symbolic • Graph-based / Constructive • Rule-based reasoning • Visualization

4. 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. 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. 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: Nominal Intervals in IGM Given that A =[aL, aN, aU], B =[bL, bN, bU],

7. Sampling Relation between Real Number and Interval Number Strict relations • strict equivalence: • strictly greater than or equal to: • strictly greater than: • strictly less than or equal to: • strictly less than: Global relations • global equivalence: • greater than or equal to: • greater than: • less than or equal to: • less than:

8. 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. 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. Under-, Over-, & Well-Constrained

11. 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. X A Y Extended Gauss-Seidel Method

13. Solving Interval Nonlinear Equations based on Linear Enclosure • Transform to separable form; • Find linear enclosure; • Solve linear enclosure equations; • Update variable values • If stop criteria not satisfied, go to step 2; otherwise stop. Start Transform to Separable Form Find Linear Enclosure Solve Linear Enclosure Equations Update Variable Values N Stop Criteria Satisfied? Y End

14. 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 f22)/2 y = f1 + f2 f = f1 / f2 f = (y2 f121/ f22)/2 y = f1 + 1/f2 f = (f1)f2f =exp(y1) y1= (y22 (log(f1))2 f22)/2 y2 = log(f1)+ f2

15. 2. Linear Enclosure Extending Kolev’s work: Let Xj0 = [xLj, xNj, xUj] Linear Enclosure is defined as: such that

16. 3. Solve Linear Enclosure Equations If fij(x) is continuous within interval Xj0, 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=[bLij, bNij, bUij], where

17. 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.

18. Solving Interval Inequalities • Adding slack variables to translate inequalities into equalities. • Solving linear/nonlinear equations with previous methods.

19. 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]

20. 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. An Example

22. subdivide up to Level 3, and some sub-regions are eliminated. Refinement - subdivision

23. 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. Thank you!