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INTERSECTIONS -- TOPOLOGY, ACCURACY, & NUMERICS FOR GEOMETRIC OBJECTS

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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)

StabilityProofswithUncertainData

Neil Stewart, Université de Montréal

with L.-E. Andersson and M. Zidani

Thanks: T. J. Peters and J. Bisceglio

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.

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.

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.)

S

P

Ill-conditioned

S

Well-conditioned

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-conditionof the problem, and instabilityof the numerical method.
- The problem is ill-conditioned if small perturbations of the data can lead to large changes in the solution.

S

P

Ill-conditioned

S

Well-conditioned

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.

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.

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.)

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).

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

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

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

Results & Examples Error Bounds in 3D Model Space (Transversal Intersection)

Transversal intersection of rational parametric surfaces

0.02

Self intersection of a bi-cubic surface

Intersection of a hyperbolic surface with a plane

Results & Examples Error Bounds in 3D Model Space (Tangential Intersection)

Tangential intersections of parametric surfaces

Results & Examples Preventing Straying or Looping

Validated ODE solver can correctly trace the intersection curve segment even through closely spaced features, where standard methods fail.

Adams-BashforthRunge-Kutta

Result from a validated interval scheme

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).

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

b2

s1

s2

a1

b1

s3

s4

a2

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.

Accurate Polynomial Evaluation

Christoph Hoffmann, Purdue University

Neil Stewart, University of Montreal

Thanks

Gahyun Park, Purdue

J.-R. Simard, Montreal

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?

Evaluation Methods

- Simple Horner
- Residual iteration
- Using fp arithmetic
- Using accurate inner product
- Compensating direct fp arithmetic
- Distillation

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.

FP Techniques

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

Polynomials

- p1 thru p3:root with high multiplicity but low bit complexity
- Others:coefficients with high bit complexity, simple roots

Accuracy Achieved

p1: Horner = 0, others = 10-32, enc=3ulp

p2: Horner = 8·10-14, others = 10-48, enc=1ulp

p3: Horner = 1.02·10-12, others = 10-64, enc=1ulp

p4: Horner within 1 ulp of accurate value

p5 – p7: Horner within 2 ulp of accurate value

Industrial PerspectiveDavid R Ferguson

- Historical perspective
- Where we are now
- Why (really!) do we care
- What is being done and
- What should be done

Descriptive Geometry

Gaspard Monge

(Circa 1800)

Catalog of

Ship Lines

David Taylor

(~1900)

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.

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.

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.

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.

Geometric variation presents a new challenge for geometry systems and intersections: Smooth Morphing

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) = ax2

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 …

A Final Thought: We are seeing a fundamental change in the use of geometry

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.

International Visibility

I-TANGO

1. Invited tutorial: Effective Computational Geometry

2. Talk at ICIAM (Australia, 2003)

3. Upcoming

a. Three papers, Italy, Solids & Shapes

b. Dagstuhl Seminar

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

Topology

- 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

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

Summary

- Broad Dissemination
- Internationally
- Industrially

- Effective intellectual integration & synergy
- Diversity recruitment
- Poster tomorrow

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