unifying topology and numerical accuracy in geometric modeling surface to surface intersections

Unifying Topology and Numerical Unifying Topology and Numerical Accuracy in Accuracy in

Geometric ModelingGeometric ModelingSurface to Surface IntersectionsSurface to Surface Intersections

K. H. KoK. H. KoSchool of MechatronicsSchool of Mechatronics

Gwangju Institute of Science and Gwangju Institute of Science and TechnologyTechnology

MotivationsMotivations

Digital Laboratory, digital Digital Laboratory, digital manufacturing, computer simulation, manufacturing, computer simulation, etc.etc.Physical mockup creation and evaluationPhysical mockup creation and evaluation

Time consuming and expensiveTime consuming and expensiveEvaluation of models with various changes is Evaluation of models with various changes is

not practical.not practical.Evaluation using virtual computer modelsEvaluation using virtual computer models

Allows us to evaluate the effects of small Allows us to evaluate the effects of small changes of the model.changes of the model.

Leads to an optimum design in an efficient Leads to an optimum design in an efficient manner.manner.


CAD is the key technology in virtual CAD is the key technology in virtual environment.environment.Geometric models are in the center of Geometric models are in the center of

such a design and evaluation process.such a design and evaluation process.HoweverHowever……....

IntroductionIntroduction

SIAM News, June, 1999 by R. FaroukiSIAM News, June, 1999 by R. Farouki “…“… Although modern CAD systems have Although modern CAD systems have

attained a certain degree of maturity, their attained a certain degree of maturity, their efficiency, reliability, and compatibility with efficiency, reliability, and compatibility with subsequent analysis tools fall far short of subsequent analysis tools fall far short of what was envisaged at their inception, what was envisaged at their inception, some 25 years ago. some 25 years ago. At the heart of this At the heart of this problem lie some deep mathematical problem lie some deep mathematical issues, concerned with the issues, concerned with the computation, representation and computation, representation and manipulation of complex geometries, manipulation of complex geometries, …”…”


The fundamental issue in geometric The fundamental issue in geometric modeling is to create a (watertight) modeling is to create a (watertight) model which is model which is Topologically consistentTopologically consistentNumerically accurateNumerically accurate

Intersection computation is the main Intersection computation is the main process to create such a model. process to create such a model. But this is not easy!!!But this is not easy!!!


Why is this difficult???Why is this difficult??? It is impossible to compute the exact It is impossible to compute the exact

intersections in general.intersections in general.Intersections of bicubic tensor-product: the Intersections of bicubic tensor-product: the

degree of the algebraic curve of intersection degree of the algebraic curve of intersection could reach up to 324!!!could reach up to 324!!!

Limitation of precision in floating point Limitation of precision in floating point arithmetic.arithmetic.Due to rounding operation, the Due to rounding operation, the

mathematically exact equation may yield an mathematically exact equation may yield an approximation value.approximation value.


In general, they use a low degree In general, they use a low degree curve to approximate the exact one curve to approximate the exact one of high degree.of high degree.

However, this approach inevitably However, this approach inevitably contains errors at the intersection!!!.contains errors at the intersection!!!.No rigorous consideration of topological No rigorous consideration of topological

consistencyconsistencyLack of numerical accuracyLack of numerical accuracy

Geometry Flaws - ExamplesGeometry Flaws - Examples

Gaps may be generated at intersections during Gaps may be generated at intersections during model construction or trimming process.model construction or trimming process.

In defining a solid, it causes a serious topological In defining a solid, it causes a serious topological problem.problem. Confusion in determining the inside/outside of a solid.Confusion in determining the inside/outside of a solid.


Problems in mesh generation.Problems in mesh generation. In most cases, the mesh generation In most cases, the mesh generation

software assumes that a perfect model software assumes that a perfect model be given.be given.Defect detection and correction capability is Defect detection and correction capability is

not provided.not provided.Therefore without manual corrections, Therefore without manual corrections,

we run into problems.we run into problems.


Complex geometries are used in CAD models.Complex geometries are used in CAD models. Flaws are detected and repaired manually.Flaws are detected and repaired manually.

Critical bottleneck for smooth integration of CAD and CFD Critical bottleneck for smooth integration of CAD and CFD analysis systems. analysis systems.

Generation of geometrically and topologically correct Generation of geometrically and topologically correct models will save time and money in product design.models will save time and money in product design.

MotivationsMotivations(Summary)(Summary)

Such problems prevent CAD systems from Such problems prevent CAD systems from being more powerful tools in the design being more powerful tools in the design and development stage.and development stage. They are rooted in the generation of They are rooted in the generation of

numerically and topologically inappropriate numerically and topologically inappropriate models.models.

Efficient computation of intersections Efficient computation of intersections considering topological and numerical aspects considering topological and numerical aspects is important.is important.

ThereforeTherefore……

VisionVision From the practical view point, a curve From the practical view point, a curve

which is topologically consistent and which is topologically consistent and ““sufficiently closesufficiently close”” to the exact one can to the exact one can avoid any error due to approximation.avoid any error due to approximation. We can achieve a We can achieve a ““watertightwatertight”” solid model. solid model. For this consider the topology and accuracy For this consider the topology and accuracy

together and then combine them.together and then combine them.

Objectives of this work focusing on Objectives of this work focusing on intersection curvesintersection curves Determination of an approximation curve for Determination of an approximation curve for

consistent topology.consistent topology. Low degree curve approximation and error Low degree curve approximation and error

minimization.minimization.

Surface-to-Surface Surface-to-Surface IntersectionsIntersections

In most cases, Rational Parametric In most cases, Rational Parametric Polynomial surface intersections are Polynomial surface intersections are common.common.

Solution MethodsSolution MethodsLattice MethodLattice MethodSubdivision MethodSubdivision MethodMarching Method (Tracing method)Marching Method (Tracing method)

Marching Method is a popular choice.Marching Method is a popular choice.


Surface to Surface Intersection ProblemsSurface to Surface Intersection Problems Formulated as a set of nonlinear ODE systemsFormulated as a set of nonlinear ODE systems

vut qqvuQpptP

vuQvuQ

vuQcuq

vvuQvuQ

vuQvqc

u

tPtP

tPcpt

tPtP

tPtpc

vuqtp

),(,),(

),().,(

)],(,',det['

),().,(

)],(,,'det['

),().,(

)],(,',det['

),().,(

)],(,,'det['

),(),(

)])(([ sds

dyf

y


Surface to Surface Intersection ProblemsSurface to Surface Intersection Problems Transversal IntersectionTransversal Intersection

vut qqvuQpptP

vuQtP

vuQtPsc

),(,),(

,),(),(

),(),()('


Surface to Surface Intersection Surface to Surface Intersection ProblemsProblemsTangential IntersectionTangential Intersection

Overall PictureOverall Picture

Error Bound

Curve Approximatio

n

Curve Perturbation /

Error Minimization against the

exact solution

Exact Solution

VisionVision

Topological

Configura-tion of

Intersection

Curves

Topological

Configura-tion of

Intersection

Curves

Curve Tracing

and Error Bound

Determina-tion

Curve Tracing

and Error Bound

Determina-tion

Approxima-tion of Curves

Approxima-tion of Curves

Error Minimiza

-tion

Error Minimiza

-tion

Step 1Step 1 Step 2Step 2 Step 3Step 3 Step 4Step 4

Flow of Proposed SolutionFlow of Proposed Solution

Intersection Curve Topology

Geometric Model Topology

Topological Configuration of Topological Configuration of Intersection CurvesIntersection Curves

Computation of critical pointsComputation of critical pointsUse a robust and stable root solver.Use a robust and stable root solver.

Interval Projected Polyhedron AlgorithmInterval Projected Polyhedron Algorithm

Resolving the topological Resolving the topological configuration of each curve segment configuration of each curve segment in the parametric space.in the parametric space.Loop, branch, etc.Loop, branch, etc.

u=0,v=0 u=1

v=1

Curve Tracing and Error Bound Curve Tracing and Error Bound DeterminationDetermination

Ordinary Numerical Solution Ordinary Numerical Solution Methods: Runge-Kutta, Adams-Methods: Runge-Kutta, Adams-bashforth, etc.bashforth, etc.Generally, they work fine.Generally, they work fine.But they may suffer from But they may suffer from ““straying or straying or

loopinglooping”” problems. problems.Near singular points, the behavior of the Near singular points, the behavior of the

methods may not be defined.methods may not be defined.


Tracing intersection curvesTracing intersection curves

Use the Validated ODE SolverUse the Validated ODE Solver Phase I: Step size selection and a priori enclosure Phase I: Step size selection and a priori enclosure

computationcomputation Determination of a region where existence and Determination of a region where existence and

uniqueness of the solution is validated.uniqueness of the solution is validated.

Phase II: Tight enclosure computationPhase II: Tight enclosure computation Given an a priori enclosure, a tight enclosure at the Given an a priori enclosure, a tight enclosure at the

next step is computed minimizing the wrapping effect.next step is computed minimizing the wrapping effect. Using compute a tight enclosureUsing compute a tight enclosure


si si+1 si+2

Parameter

True solution curveS

olu

tion

Error bounds

Conceptual Illustration of Validated ODE Conceptual Illustration of Validated ODE Solver.Solver.


Phase I: Step size selection and a priori Phase I: Step size selection and a priori enclosure computationenclosure computation To compute a step-size To compute a step-size hhjj and an a priori and an a priori

enclosure such that,enclosure such that,

]~[])~([])([][

],[],~[)(

1

0

][][

1

j

k

ij

kkjj

iijj

jjj

yyfhyfhy

tttallforyty

]~[ jy

Tight Enclosure

Ex. Constant Enclosure Method


Phase II: Tight enclosure computationPhase II: Tight enclosure computation Avoid the wrapping effect.Avoid the wrapping effect.

QR decomposition methodQR decomposition method


ExampleExample

Transversal intersection of rational parametric Transversal intersection of rational parametric surfacessurfaces

Results & Examples Results & Examples

Self intersection of a bi-cubic surface

Intersection of a hyperbolic surface with a plane

Tangential intersectionsTangential intersections of parametric surfaces of parametric surfaces

Results & Examples Results & Examples

Results & Examples Results & Examples Preventing Straying or LoopingPreventing Straying or Looping

Adams-Bashforth Runge-Kutta

Result from a validated interval

scheme

t

t

t


u

t v

s

s s

s


Determination of error boundDetermination of error bound

u

v

r2

r1

r1(u(s),v(s))

β(s)=(u(s),v(s))

오차 범위매핑


– tu – v

Mapping of a priori enclosures in parametric space to 3D space


Intersection Curve RepresentationIntersection Curve Representation IdeallyIdeally C(s) = r C(s) = r11(u(s),v(s)) = r(u(s),v(s)) = r22((σσ(s),t(s))(s),t(s))

u

v

σ

t

r2

r1

C(s)

r2(σ(s),t(s))

r1(u(s),v(s))


Intersection Curve RepresentationIntersection Curve RepresentationHowever in practice,However in practice,

rr11(u*(s),v*(s)) ≠ r(u*(s),v*(s)) ≠ r22((σσ*(s),t*(s)) *(s),t*(s))

Approximation of CurvesApproximation of Curves

It is almost impossible to compute the It is almost impossible to compute the exact intersection curve.exact intersection curve.

We do not have an instance of We do not have an instance of intersection to work with by using the intersection to work with by using the Validated ODE solver.Validated ODE solver.

Any curve in the validated region can Any curve in the validated region can be considered as an intersection be considered as an intersection curve.curve.

We need one curve!!!We need one curve!!!

m

iii tNPtP

14, )()(


Naïve Piecewise Linear

Topologically Equivalent Approximation


Tubular Neighborhood by Offsets


Small neighborhood

1 intersection


We take the center points at each We take the center points at each step and approximate them using a step and approximate them using a cubic B-spline curve in each cubic B-spline curve in each u-vu-v and and σσ––tt parametric space. parametric space.rr11(u*(s),v*(s)), and (u*(s),v*(s)), and

rr22((σσ*(s),t*(s)) in 3D space*(s),t*(s)) in 3D spaceBut rBut r11 ≠ r ≠ r22..But each should be But each should be

within the validated within the validated region. region.

u

v

σ

t

r2

r1

C(s)

r2(σ(s),t(s))

r1(u(s),v(s))


si si+1 si+2

Parameter

Solu

tion

Curve approximationCurve approximation

Error MinimizationError Minimization

The approximated B-spline curve is The approximated B-spline curve is perturbed to minimize the error between perturbed to minimize the error between two space curves two space curves rr11(u*(s),v*(s)) (u*(s),v*(s)) andand rr22((σσ*(s),t*(s)).*(s),t*(s)).

Curve perturbation through minimizationCurve perturbation through minimizationGiven points Given points YYkk (k=1,2,(k=1,2,……,n),n) and the and the

approximated B-spline curve approximated B-spline curve P(t)P(t)..Then find the control points Then find the control points PPii (i=1,2, (i=1,2,……,m),m)

which minimize the following.which minimize the following.

m

iii tNPtP

14, )()(

Error MinimizationError Minimization

Adjust the Adjust the curves in curves in u-vu-v and and σσ––tt parametric parametric spaces such that spaces such that the error in 3D the error in 3D space becomes space becomes minimal.minimal. In this case, the In this case, the

objective objective function needs to function needs to be reformulated.be reformulated. u

v

σ

t

r2

r1

C(s)

r2(σ(s),t(s))

r1(u(s),v(s))

Error minimizatio

n

ConclusionsConclusions

Compute an approximate curve which is Compute an approximate curve which is ““sufficiently closesufficiently close”” to the exact one. to the exact one. Topologically consistentTopologically consistent Numerically accurate.Numerically accurate.

Theoretical and practical foundation on Theoretical and practical foundation on watertight solid model creation.watertight solid model creation.

Provide a smooth interface between CAD Provide a smooth interface between CAD systems and downstream applications.systems and downstream applications. Mesh creationMesh creation

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unifying topology and numerical accuracy in geometric modeling surface to surface intersections

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