Unifying Topology and Unifying Topology and Numerical Accuracy in Numerical Accuracy in Geometric Modeling Geometric Modeling Surface to Surface Surface to Surface Intersections Intersections K. H. Ko K. H. Ko School of Mechatronics School of Mechatronics Gwangju Institute of Science Gwangju Institute of Science and Technology and Technology
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Unifying Topology and Numerical Accuracy in Geometric Modeling Surface to Surface Intersections
Unifying Topology and Numerical Accuracy in Geometric Modeling Surface to Surface Intersections. K. H. Ko School of Mechatronics Gwangju Institute of Science and Technology. Motivations. Digital Laboratory, digital manufacturing, computer simulation, etc. - PowerPoint PPT Presentation
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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.
MotivationsMotivations
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, …”…”
MotivationsMotivations
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!!!
MotivationsMotivations
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.
MotivationsMotivations
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
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,
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
In most cases, Rational Parametric In most cases, Rational Parametric Polynomial surface intersections are Polynomial surface intersections are common.common.
Surface to Surface Intersection ProblemsSurface to Surface Intersection Problems Formulated as a set of nonlinear ODE systemsFormulated as a set of nonlinear ODE systems
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.
Curve Tracing and Error Bound Curve Tracing and Error Bound DeterminationDetermination
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
Curve Tracing and Error Bound Curve Tracing and Error Bound DeterminationDetermination
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.
Curve Tracing and Error Bound Curve Tracing and Error Bound DeterminationDetermination
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
Curve Tracing and Error Bound Curve Tracing and Error Bound DeterminationDetermination
Phase II: Tight enclosure computationPhase II: Tight enclosure computation Avoid the wrapping effect.Avoid the wrapping effect.
QR decomposition methodQR decomposition method
Curve Tracing and Error Bound Curve Tracing and Error Bound DeterminationDetermination
ExampleExample
Curve Tracing and Error Bound Curve Tracing and Error Bound DeterminationDetermination
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
Curve Tracing and Error Bound Curve Tracing and Error Bound DeterminationDetermination
u
t v
s
s s
s
Curve Tracing and Error Bound Curve Tracing and Error Bound DeterminationDetermination
Determination of error boundDetermination of error bound
u
v
r2
r1
r1(u(s),v(s))
β(s)=(u(s),v(s))
오차 범위매핑
Curve Tracing and Error Bound Curve Tracing and Error Bound DeterminationDetermination
Curve Tracing and Error Bound Curve Tracing and Error Bound DeterminationDetermination
– tu – v
Mapping of a priori enclosures in parametric space to 3D space
Curve Tracing and Error Bound Curve Tracing and Error Bound DeterminationDetermination
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, )()(
Approximation of CurvesApproximation of Curves
Naïve Piecewise Linear
Topologically Equivalent Approximation
Approximation of CurvesApproximation of Curves
Approximation of CurvesApproximation of Curves
Tubular Neighborhood by Offsets
Approximation of CurvesApproximation of Curves
Small neighborhood
1 intersection
Approximation of CurvesApproximation of Curves
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))
Approximation of CurvesApproximation of Curves
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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