1 / 44

OUTSTANDING PROBLEMS IN GEOMETRIC CONSTRAINT SOLVING FOR CAD

Explore current challenges and novel solutions in geometric constraint solving for CAD applications, including state-of-the-art techniques and unsolved CAD problems. Our research delves into variational constraint representation, feature hierarchies, and formal problem classifications in geometric modeling. Learn about decomposition-recombination plans and performance measures for intricate constraint systems.

ernestovaughan
Download Presentation

OUTSTANDING PROBLEMS IN GEOMETRIC CONSTRAINT SOLVING FOR CAD

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. OUTSTANDING PROBLEMS IN GEOMETRIC CONSTRAINT SOLVING FOR CAD Meera Sitharam, University of Florida Partially supported by NSF grants CCR 99-02025, EIA 00-96104

  2. ORGANIZATION • CAD motivation and state of the art • Suite of Formal Problems • Our contribution-- FRONTIER • Unsolved Problems

  3. CAD MOTIVATION 1/4 Variational constraint representation and feature hierarchy

  4. CAD MOTIVATION 2/4 Another Assembly constraint representation and subassembly hierarchy

  5. CAD MOTIVATION 3/4 A geometric (variational) constraint representation with feature hierarchy is: • Generated declaratively. • Easily updated and maintained. • Minimal, complete.

  6. CAD MOTIVATION 4/4 The Catch: implicit representation. How to • Want explicit geometric realization(s): • Navigate conformation of each feature consistent with subfeatures. • Derive implied geometric properties/invariants. • Eliminate inconsistencies in requirements. • Independently manipulate features and interface with other representations.

  7. STATE OF THE ART 1/3 • 2 dimensions :Small, simple, no feature hierarchy, stand- alone. • 3 dimensions : 2d views; CSG; history of sweeps, extrusions; parametric constraint solving Hoffman et al (EREP), Bruderlin et al, Bronsvoort et al, Kramer et al, Michelucci et al,Owen et al (D-cubed), Latham, Middleditch et al

  8. STATE OF THE ART: 3 Dimensions 2/3 Pictures of 2d views of 3d part

  9. STATE OF THE ART: 3D 3/3 D-cubed's pipe routing

  10. GOAL

  11. FORMAL BASIC PROBLEM 1/7 Input1:Primitive geometric objects: (id, type) (type chosen from repertoire)

  12. FORMAL BASIC PROBLEM 2/7 Input2:Geometric constraints: (object1, object2, .., objectk, type) (type chosen from repertoire) constraint types include some inequalities

  13. FORMAL BASIC PROBLEM 3/7

  14. FORMAL BASIC PROBLEM 4/7 • Input3 : Feature hierarchies: • (more than one) partial order or DAG of subsets of objects • partial realization (output) information for the nodes of DAG.

  15. FORMAL BASIC PROBLEM 5/7

  16. FORMAL BASIC PROBLEM6/7

  17. SUITE OF FORMAL PROBLEMS 1/12 • Existence:of realization • Conformation: One conformation (if it exists)for each node in feature hierarchy, represented as a rigid transformation applied to each child's conformation.

  18. FORMAL BASIC PROBLEM 7/7 • For conformation, need to solve polynomial system over the reals. d2=((x2-x1)2 + (y2-y1)2 Problem classification Red: Algebraic; Blue: Combinatorial; Purple: Mixture

  19. SUITE OF FORMAL PROBLEMS 2/12 • Generic, parameter-freeversion of existence • Approached combinatorially using only the geometric constraint graph, object and constraint types.

  20. SUITE OF FORMAL PROBLEMS 3/12 • Generic answer holds • For all but a small set of forbidden parameter values that satisfy discriminant/resultant (in)equalities.

  21. SUITE OF FORMAL PROBLEMS 4/12 • Generic Classification:some information on how many conformations exist? • finitely many (rigid or wellconstrained) • infinitely many(flexible or underconstrained) • none(inconsistently overconstrained)

  22. SUITE OF FORMAL PROBLEMS 5/12 • Navigation:A well-defined set of conformations for each node in feature hierarchy, represented as a set of transformations applied to each child's set of conformations? • Meaning of well-defined: complete in some formal sense, systematically navigable. • Invariant:Does a given geometric property hold for all conformations?

  23. SUITE OF FORMAL PROBLEMS 6/12

  24. SUITE OF FORMAL PROBLEMS 7/12

  25. SUITE OF FORMAL PROBLEMS 8/12

  26. SUITE OF FORMAL PROBLEMS9/12

  27. SUITE OF FORMAL PROBLEMS 10/12 • Generic Overconstraint correction: a well-defined set of removable constraint-sets for each node in feature hierarchy.

  28. SUITE OF FORMAL PROBLEMS 11/12 • Generic underconstraint navigation: a well-defined set of addable constraint-sets for each node in feature hierarchy.

  29. SUITE OF FORMAL PROBLEMS 12/12 • Combinatorial complete generic solution: Big open question. Gives rise to a combinatorial theory of rigidity. Whiteley et al. • Laman's theorem: complete combinatorial classification for 2D points and distances. Simple dof analysis.

  30. OUR CONTRIBUTIONS 1/12 • (1) Formalizing decomposition problem and performance measures.

  31. OUR CONTRIBUTIONS 2/12 A Decomposition-Recombination plan (DR-plan) for an input constraint system G, consistent with an input feature hierarchy F is a DAG: • nodes are subsets of primitive objects of G such that their induced subsystems are well-over-constrained 1 • nodes include the nodes of F • each leaf/source is a primitive object in S; • each root/sink represents a maximal well-over-constrained subsystem of G1 1 more generally, they possess atmost a specified number of degrees of freedom

  32. OUR CONTRIBUTIONS 3/12

  33. OUR CONTRIBUTIONS 4/12 • Other performance measures on DR-planners • An optimal DR-planner minimizes the maximum fan-in (size of the largest subsystem in DR-plan)

  34. OUR CONTRIBUTIONS 5/12 • (2) Partial-generic characterization of DR-plan based on degree of freedom analysis of constraint graphs: minimal dense subgraph usually corresponds to well-over-constrained subsystem. • Algorithm for construction of DR-plan: using • network flows to iteratively find the minimal dense subgraphs in current graph • graph transformations that repeatedly simplify them.

  35. OUR CONTRIBUTIONS 6/12

  36. OUR CONTRIBUTIONS 7/12

  37. OUR CONTRIBUTIONS 8/12 Optimal DR Planning problem (Partial-generic version) • Already finding smallest well-constrained graph is NP-complete. Polynomial time algorithms known for special cases. Approximation status unknown.

  38. OUR CONTRIBUTIONS 9/12 • (3) Towards a more complete generic solution

  39. OUR CONTRIBUTIONS 10/12

  40. OUR CONTRIBUTIONS 11/12 • (4) Decomposition gives partial-generic solution to: • Existence • Classification • Overconstraint Correction • Generic underconstraint Navigation • Dealing with mixed representations, multiple input feature hierarchies • (5) Plus additional work on equation and conformation management gives: • Well-constrained Conformations • Well-constrained Navigation • Easy updates of constraint repertoire • Easy updates of constraint representation, feature hierarchy and realizations • Online constraint solving

  41. OUR CONTRIBUTIONS 12/12 • (6) Software architecture and implementation

  42. REITERATING UNSOLVED PROBLEMS 1/3 • Isolation of Conformation: Chirality, Semi-global constraints, Symmetries, Forces. • Efficiently solving polynomial systems for rigid transformations : physically based semi-numerical algorithms are welcome. • Invariant problem. • Inverse problem of finding minimal constraint representation

  43. REITERATING UNSOLVED PROBLEMS 2/3 • Underconstrained Conformation and Navigation: in addition to addable constraint sets, need forbidden parameter regions.

  44. REITERATING UNSOLVED PROBLEMS 3/3 • Complete generic solution to original problems-- combinatorial geometry, geometric graphs. • Approximation algorithm for Optimal DR-plan problem, even the partial-generic version based on dof analysis. • Complexity of existence problem NP-hard; not known to be in NP; in DNPR (partial algebraic version of NP); not known to be DNPR-hard. • Algebraic description of generic describe the semi-algebraic set of forbidden parameter values when generic solution does not hold

More Related