1. INTERSECTIONS -- TOPOLOGY, ACCURACY, & NUMERICS FOR GEOMETRIC OBJECTS I-TANGO III NSF/DARPA

2. Intellectual Integration of Project Team New conceptual model (Stewart - UConn) Intersection improvements (Sakkalis – MIT) Polynomial evaluation (Hoffmann – Purdue) Industrial view (Ferguson – DRF Associates) Key external interactions (Peters, UConn)

3. Stability Proofs with Uncertain Data Neil Stewart, Université de Montréal with L.-E. Andersson and M. Zidani Thanks: T. J. Peters and J. Bisceglio

4. Representation: Geometric Data Two trimmed patches. The data is inconsistent, and inconsistent with the associated topological data. The first requirement is to specify the set defined by these inconsistent data.

5. Forward + Backward Analysis The second requirement is to do an error analysis. We seek to show we’ve found “a slightly wrong solution to a slightly wrong problem’’ [Kahan 1971]. We have the luxury of associating all or part of the error with the problem because we assume that there is uncertainty in the input data.

6. Specification and Error Analysis The first requirement corresponds to defining what the arrows mean. The second requirement corresponds to showing that the distance between the pairs of dots is small. (We had better define the metric.)

7. Well- and ill-conditioned problems To see why it might be advantageous to associate all or part of the error with the problem, we must distinguish between ill-condition of the problem, and instability of the numerical method. The problem is ill-conditioned if small perturbations of the data can lead to large changes in the solution. P S Ill-conditioned P S Well-conditioned

8. Stable numerical methods A method is stable if it gives a slightly wrong solution to a slightly wrong problem. A stable method does not necessarily provide us with a small error: it just provides us with a solution that is as good as the data warrants. P S Ill-conditioned P S Well-conditioned

9. Appropriate goals An often-used criterion for robustness is whether the program crashes. This is not sensible: we need an error metric. Even with a metric, in the presence of uncertainty it is futile to use exact arithmetic, or other similar means, to provide a solution better than one that is already the exact answer to a slightly perturbed problem.

10. 1. Specification of defined set We have used the Whitney extension theorem to perturb the patch, throughout the domain, by an amount that is smaller than the discrepancy along the boundary. (This much is inherent in the given data.)

11. 2. Backward Error Analysis We have given such an analysis (cf Song-Sederberg- Zheng-Farouki-Hass) for the above example, motivated by an example of Hoffmann. DT-NURBS was used to obtain the numerical intersection. The perturbation analysis is particular to this example, and there is no explicit boundary curve b(t).

12. NSF/DARPA Project I-TANGO: Intersections-Topology, Accuracy and Numerics for Geometric Objects N. M. Patrikalakis T. Maekawa T. Sakkalis K. H. Ko H. Mukundan Massachusetts Institute of Technology May 2004

13. Surface to Surface Intersection Obtaining an accurate starting point in each component –Roots of polynomials with accurate error bounds Multiple roots with accurate error bounds Tracing the given intersection correctly –An accurate estimate of error in 3D model space –Transversal and tangential intersections

14. Achievements Validated error bounds in 3D model space which enclose the true curve of intersection : Interval Solid Modeling Prevention of the phenomenon of straying or looping. The scheme can accommodate the errors in: –Initial condition –Rounding during digital computation Reduction of error bounds –Error bound reduction in parametric space. –Error bound reduction in model space. Robust computation of roots and multiplicities for univariate and bivariate polynomial equations

15. Tracing Intersection Curves Tracing using validated ODE solvers to obtain validated error bounds in parametric space of both surfaces Input error bound on the starting point in both parametric spaces Mapping into 3D model space using rounded interval arithmetic Output Validated error bounds in 3D model space The intersection problem can be written in an interval vector ODE system of the form Validated nonlinear ODE system solvers are used. –Interval Taylor Series Method –Interval Hermite-Obreschkoff Method

16. Error Bounds in 3D Model Space Mapping into 3D Model Space Mapping from parametric space to 3D model space –using corresponding surfaces –coupled with rounded interval arithmetic evaluation Ensures continuous error bounds in 3D model space guaranteed to contain the true curve of intersection.

17. Reduction of Enclosure In Parametric Space –Find two linear polynomials at t j and two at t j+1 such that a root is enclosed by those four linear polynomials. Lohner’s polynomial enclosure method* using a polynomial of degree one. In Model Space –Collections of boxes obtained from the mapping from each of the surfaces, contain the true solution. Take union of the set of boxes obtained from each surface. Take intersection of the two previously constructed sets. * R. J. Lohner, “Step size and Order Control in the Verified Solution of IVP with ODE’s”, SciCADE’95. International Conference on Scientific Computation and Differential Equations, Stanford, CA, 1995  – t u – v After Intersection

18. Transversal intersection of rational parametric surfaces Results & Examples Error Bounds in 3D Model Space (Transversal Intersection) 0.02 Self intersection of a bi-cubic surface Intersection of a hyperbolic surface with a plane

19. Tangential intersections of parametric surfaces Results & Examples Error Bounds in 3D Model Space (Tangential Intersection)

20. Validated ODE solver can correctly trace the intersection curve segment even through closely spaced features, where standard methods fail. Results & Examples Preventing Straying or Looping Adams-Bashforth Runge-Kutta Result from a validated interval scheme

21. Computation of Starting Points Starting points are the initial conditions for solving the nonlinear ODE system. -IPP algorithm; –Case A : Simple and isolated roots IPP can handle this case efficiently. –Case B : Multiple or not sufficiently isolated roots IPP is inefficient. Objective –Robust calculation of multiple roots. Method –Application of the topological degree of the Gauss map defined by polynomials in the plane. T. Sakkalis, “The Euclidean algorithm and the degree of the Gauss map”, SIAM J. of Computing, Vol. 19, No. 3, 538-543 (1990).

22. Gauss Map Let h(z) be a polynomial in a single variable z. We write h(z) = h(x+iy) = p(x,y) + i q(x,y). Let also F = (p, q). Observe that the zeros of h(z) are precisely the real zeros of F. We shall locate the latter zeros by invoking the notion of the local degree of F. To achieve this, we introduce the Gauss Map:

23. Illustration of the Gauss Map

24. Use the map F directly. Computation Algorithm for Degree of Gauss Map (Principle of Argument)

25. Use of complex rounded interval arithmetic in the evaluation of F. -Maintenance of the angle between two vectors F m and F m+1 below . -If the map includes zero in pq domain, then we refine the discretization.

26. Robust Algorithm for Solving Univariate Polynomial Equations Univariate polynomial in complex variable z. (Substitute x with a complex variable z = x+iy) Input : –initial domain : –a complex polynomial : h(z) –tolerance, number of sample points Output : real and complex roots, multiplicities Algorithm –Quadtree decomposition –Degree of Gauss map computation Verification of the existence of roots –Complex interval arithmetic s1s1 s2s2 s3s3 s4s4 b2b2 a1a1 b1b1 a2a2

27. Conclusions FutureWork Strict error control code. Research on the general problem of multiple roots of nonlinear polynomial systems in 3 or 4 variables. Classification of the interval intersection curves.

28. Accurate Polynomial Evaluation Christoph Hoffmann, Purdue University Neil Stewart, University of Montreal Thanks Gahyun Park, Purdue J.-R. Simard, Montreal

29. Motivation If evaluation is inaccurate, then surface intersection has a weakened foundation Exact evaluation may be too costly Traditional evaluation may be too loose What is a good balance?

30. Evaluation Methods Simple Horner Residual iteration –Using fp arithmetic –Using accurate inner product Compensating direct fp arithmetic Distillation

31. Residual Iteration Wilkinson: Method converges to the true solution even when the residuals are inaccurately computed Block floating-point arithmetic used in Wilkinson’s proof Accurate evaluation may require more when there are multiple roots, and the accurate inner product is essential in those situations.

32. FP Techniques Priest and others have ways to compute accurate polynomial values using ordinary fp arithmetic. Key method depends on computation such as a+b=c+d Distillation:

33. Polynomials p 1 thru p 3 : root with high multiplicity but low bit complexity Others: coefficients with high bit complexity, simple roots

34. p1p2p3p4p5p6p7 degree4686789 iterations3451111 Horner637894787994 Hammer39011251938328375406453 Direct19065563107207137103601393817517 Distill15311192232062609672766859 Timing

35. Accuracy Achieved p 1 : Horner = 0, others = 10 -32, enc=3ulp p 2 : Horner = 8·10 -14, others = 10 -48, enc=1ulp p 3 : Horner = 1.02·10 -12, others = 10 -64, enc=1ulp p 4 : Horner within 1 ulp of accurate value p 5 – p 7 : Horner within 2 ulp of accurate value

36. Industrial Perspective David R Ferguson Historical perspective Where we are now Why (really!) do we care What is being done and What should be done

37. Descriptive Geometry Gaspard Monge (Circa 1800) Catalog of Ship Lines David Taylor (~1900) NACA Airfoils (~1915)

38. Liming’s Work Goal: Develop a “practical system of analytic calculation techniques for direct use in the aircraft, automotive, and marine industries” Benefits: More “complete control of developments from basic conditions [means] intended performance characteristics are more nearly realized The mathematics form a “permanent...authority”, and “constitute an easily established permanent source of dimensions.” Calculation of additional data with the same basic analytic equations..., assures the highest practical degree of dimensional integrity”

39. Roy Liming would boast, with pardonable hyperbole, that the Britain-based Mustangs could fly to Berlin and back because their surface contours did not deviate from the mathematical ideal.

40. Roy Liming, developed a mathematical system (conic lofting) for the definition of a compound-curved aerodynamic surface for the P-51 Mustang. Liming would boast, with pardonable hyperbole, that the Britain-based Mustangs could fly to Berlin and back because their surface contours did not deviate from the mathematical ideal.

41. Where we are now! CAD systems getting better – perhaps but Industry and government still reporting poor performance. Examples –Boeing CFD still reports 40-60% loss of engineering productivity –US Auto industry spends in excess of a billion dollars / year mitigating the effects of bad geometry. –See report on an interdisciplinary workshop at UC Davis in April, 1999. But Industry and government not interested. As long as they can do their job with current means they are willing to bumble along.

42. So, Why do we care? Simply put: The job is changing The real issue is automation supporting advanced design and manufacture. It’s all about –Virtual prototyping (driving out cost and time) –Studying families of designs for good solutions and –Finding feasibility where none is known at the time. These mean seamless integration of CAD/CAM/CAE tools and robust, automatic, continuous variation of geometry is required.

43. Intersection plays a critical role Surface intersections are the single greatest source of non-operator error in CAD and in grid generation for CFD and other analyses. –Tolerances vary from CAD system to CAD system –More importantly, tolerances vary from application to application Algorithmic characteristics of surface intersection will wreck havoc with continuous variation (smooth morphing) of geometry in current CAD systems. –Nonlinear aspects may dictate different computational paths as the geometry changes.

44. For a approximately 1, the approximation fluctuates between a constant spline and a spline with one interior knot, i.e., changing the parameter a may cause the model to change discontinuously. Approximating f(x) = ax 2 Geometric variation presents a new challenge for geometry systems and intersections: Smooth Morphing

45. What’s being done and what should be done Geometry repair –Farouki et al: Perturbation schemes for surface repair –Klein (and others): Practical tools –Commercial products: CADFix Fundamentals of the intersection process –I-TANGO team Unfortunately …

46. I believe there is little or no chance these efforts will be embraced by present-day CAD vendors. Major beneficiaries of better geometry paradigms will be government and industrial programs dependent on advanced design through modeling and simulation. This leads me to the following: Is there a way of forming a consortium of a small number of interested players to investigate a new way of doing geometry by Building a system based on and emphasizing –Sound mathematical and scientific principles –Families of designs rather than single designs –CAE and geometry algorithms that support each other –Optimization, especially multi-disciplinary

47. Previously, geometry was used solely to document what was built. Geometry had to match the product. Now, it’s the product that has to match the geometry: Liming’s goal. A Final Thought: We are seeing a fundamental change in the use of geometry

48. International Visibility 1. Invited tutorial: Effective Computational Geometry 2. Talk at ICIAM (Australia, 2003) 3. Upcoming a. Three papers, Italy, Solids & Shapes b. Dagstuhl Seminar

49. Technology Transfer I-TANGO I –Existing GK interface in parametric domain –Taylor’s theorem for theory –New model space error bound prototype CAGD paper Transfer to Boeing through GEML

50. Computational Topology for Regular Closed Sets (within the I-TANGO Project) –Invited article, Topology Atlas –Entire team authors (including student) –I-TANGO interest from theory community Topology

51. Diversity (Scientific & Human) Sabbatical with Wesleyan (Chemistry) –Transition to molecular modeling –Mentoring through seminar –Submitted article CCCG04 Diversity recruitment –Majority women –Mixed post-docs, graduates & undergraduates

52. Broad Dissemination –Internationally –Industrially Summary Effective intellectual integration & synergy Diversity recruitment Poster tomorrow