ROBUSTNESS IN NUMERICAL COMPUTATION IIVALIDATED ODE SOLVERKWANG HEE KOSCHOOL OF MECHATRONICSGWANGJU INSTITUTE OF SCIENCE AND TECHNOLOGY

SOLVING ODESTraditional ODE solvers: Runge-Kutta, Adams-Bashforth, etc.Work well in general.When two solution features are close to each otherthe step size selection becomes complexIncorrect step size may lead to a critical problemLooping or straying

SOLVING ODESTraditional ODE solvers: Runge-Kutta, Adams-Bashforth, etc.Such problems happen since they control the size of each step solely based on controlling just the error alone.Do not consider the existence and uniqueness of solution.

VALIDATED INTERVAL ODE SOLVERTracing intersection curves

Use the Validated ODE SolverPhase I: Step size selection and a priori enclosure computationDetermination of a region where existence and uniqueness of the solution is validated.

Phase II: Tight enclosure computationGiven an a priori enclosure, a tight enclosure at the next step is computed minimizing the wrapping effect.Using compute a tight enclosure

VALIDATED INTERVAL ODE SOLVERConceptual Illustration of Validated ODE Solver.

VALIDATED INTERVAL ODE SOLVERPhase I: Step size selection and a priori enclosure computationTo compute a step-size hj and an a priori enclosure such that,

Tight EnclosureEx. Constant Enclosure Method

VALIDATED INTERVAL ODE SOLVERPhase II: Tight enclosure computationAvoid the wrapping effect.

QR decomposition method

VALIDATED INTERVAL ODE SOLVERExample

APPLICATION TO SURFACE-TO-SURFACE INTERSECTIONSIn most cases, Rational Parametric Polynomial surface intersections are common.Solution MethodsLattice MethodSubdivision MethodMarching Method (Tracing method)Marching Method is a popular choice.

DERIVATION!!!

EXAMPLE: BICUBICBEZIER INTERSECTION Two rational bicubic-bezier patches.

Starting point found by interval projected polyhedron(IPP) algorithm."Computation of the Solutions of Nonlinear Polynomial Systems" by E. C. Sherbrooke and N. M. Patrikalakis, Computer Aided Geometric Design, 10, No. 5, (1993) 379-405

OUTPUT FROM THE VALIDATED ODE SOLVERWith respect to the arc length parameter the a priori enclosures of the pre-images of the surfaces are connected.

3D MAPPING OF THE PARAMETER BOXESs tu v

INTERSECTION OF THE BOXES IN 3DCollections 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.

During the intersection, we can in general obtain a substantial reduction in model space error.The above is related to, and nicely complements, our previous work on interval solids.Topological and Geometric Properties of Interval Solid Models" by T. Sakkalis, G. Shen and N. M. Patrikalakis, Graphical Models. Vol. 63, No. 3, pp. 163-175, May 2001

INTERSECTION APPROXIMATED BY AN INTERVAL B-SPLINEThe result can be expressed as an interval B-spline curve.Substantial reduction of data storage.Essentially expressed as two B-spline curves representing,Spine curve.An error curve representing half-width.Slight increase in the width of the model space bound.Approximation of measured data with interval B-splines. S. T. Tuohy, T. Maekawa, G. Shen and N. M. Patrikalakis. Computer-Aided Design, Vol. 29, No. 11, pp. 791-799, 1997.

VARIATION WITH TOLERANCERel. Tolerance = 1.4x10-2Rel. Tolerance = 2.6x10-4