Abstract
Current parametric CAD systems require geometric parameters to have fixed values. Specifying fixed parameter values implicitly adds rigid constraints on the geometry, which have the potential to introduce conflicts during the design process. This paper presents a soft constraint representation scheme based on nominal interval. Interval geometric parameters capture inexactness of conceptual and embodiment design, uncertainty in detail design, as well as boundary information for design optimization. To accommodate under-constrained and over-constrained design problems, a double-loop Gauss-Seidel method is developed to solve linear constraints. A symbolic preconditioning procedure transforms nonlinear equations to separable form. Inequalities are also transformed and integrated with equalities. Nonlinear constraints can be bounded by piecewise linear enclosures and solved by linear methods iteratively. A sensitivity analysis method that differentiates active and inactive constraints is presented for design refinement.
Similar content being viewed by others
References
Aldefeld, B.: Variation of Geometries Based on a Geometric-Reasoning Method, Computer- Aided Design 20 (3) (1988), pp. 117–126.
Aldfeld, B.: Rule-Based Approach to Variational Geometry, in: Smith, A. (ed.), Knowledge Engineering and Computer Modelling in CAD, Proceedings of the 7th International Conference on the Computer as a Design Tool, September 2–5, 1986, London, pp. 59–67.
Alefeld, G.: Bounding the Slope of Polynomial Operators and Some Applications, Computing 26 (1981), pp. 227–237.
Alefeld, G.: On the Convergence of Some Interval-Arithmetic Modifications of Newton's Method, SIAM Journal on Numerical Analysis 21 (2) (1984), pp. 363–372.
Alefeld, G. and Herzberger, J.: Introduction to Interval Computations, Academic Press, New York, 1983.
Alefeld, G. and Mayer, G.: Interval Analysis: Theory and Application, Journal of Computational and Applied Mathematics 121 (1–2) (2000), pp. 421–464.
Alefeld, G. and Platzoder, L.: A Quadratically Convergent Krawczyk-Like Algorithm, SIAM Journal on Numerical Analysis 20 (1) (1983), pp. 210–219.
Anantha, R., Kramer, G. A., and Crawford, R. H.: Assembly Modelling by Geometric Constraint Satisfaction, Computer-Aided Design 28 (9) (1996), pp. 707–722.
Benhamou, F. and Granvilliers, L.: Automatic Generation of Numerical Redundancies for Non- Linear Constraint Solving, Reliable Computing 3 (3) (1997), pp. 335–344.
Berz, M.: Modern Map Methods in Particle Beam Physics, Academic Press, San Diego, 1999.
Berz, M. and Hoffstatter, G.: Computation and Application of Taylor Polynomials with Interval Remainder Bounds, Reliable Computing 4 (1) (1998), pp. 83–97.
Bliek, C.: Computer Methods for Design Automation, unpublished Ph.D. thesis, Massachusetts Institute of Technology, 1992.
Bouma, W., Fudos, I., Hoffmann, C., Cai, J., and Paige, R.: Geometric Constraint Solver, Computer-Aided Design 27 (6) (1995), pp. 487–501.
Buchberger, B., Collins, G., and Kutzler, B.: Algebraic Methods for Geometric Reasoning, Annual Review of Computer Science 3 (1988), pp. 85–120.
Ceberio, M. and Granvilliers, L.: Horner's Rule for Interval Evaluation Revisited, Computing 69 (1) (2002), pp. 51–81.
Chen, F. and Lou, W.: Degree Reduction of Interval Bezier Curves, Computer-Aided Design 32 (10) (2000), pp. 571–582.
Chen, X.: A Verification Method for Solutions of Nonsmooth Equations, Computing 58 (1997), pp. 281–294
Chiu, C.-K. and Lee, J. H.-M.: Efficient Interval Linear Equality Solving in Constraint Logic Programming, Reliable Computing 8 (2) (2002), pp. 139–174.
Chyz, W.: Constraint Management for CSG, Unpublished Master Thesis, Massachusetts Institute of Technology, 1985.
Collins, G. E. and Akritas, A. G.: Polynomial Real Root Isolation Using Descarte's Rule of Signs, in: Proceedings of the Third ACM Symposium on Symbolic and Algebraic Computation, August 10–12, 1976, Yorktown Heights, New York, pp. 272–275.
Collins, G. E. and Johnson, J. R.: Quantifier Elimination and the Sign Variation Method for Real Root Isolation, in: Proceedings of the ACM-SIGSAM 1989 International Symposium on Symbolic and Algebraic Computation, July 17–19, 1989 Portland, Oregon, pp. 264–271.
Duff, T.: Interval Arithmetic and Recursive Subdivision for Implicit Functions and Constructive Solid Geometry, Computer Graphics 26 (2) (1992), pp. 131–138.
Finch, W. W. and Ward, A. C.: A Set-Based System for Eliminating Infeasible Designs in Engineering Problems Dominated by Uncertainty, in: ASME Proceedings of DETC97/dtm-3886, Sept. 14–17, 1997, Sacramento, CA, USA.
Fudos, I. and Hoffmann, C. M.: A Graph-Constructive Approach to Solving Systems of Geometric Constraints, ACM Transactions on Graphics 16 (2) (1997), pp. 179–216.
Gao, X.-S. and Chou, S.-C.: Solving Geometric Constraint Systems II: A Symbolic Approach and Decision of Rc-constructibility, Computer Aided-Design 30 (2) (1998), pp. 115–122.
Garloff, J.: The Bernstein Algorithm, Interval Computations (2) (1993), pp. 154–168.
Ge, J.-X., Chou, S.-C., and Gao, X.-S.: Geometric Constraint Satisfaction Using Optimization Methods, Computer-Aided Design 31 (14) (1999), pp. 867–879.
Gossard, D. C., Zuffante, R. P., and Sakurai, H.: Representing Dimensions, Tolerances, and Features in MCAE Systems, IEEE Computer Graphics & Applications 8 (3) (1988), pp. 51–59.
Hansen, E.: Bounding the Solution of Interval Linear Equations, SIAM Journal on Numerical Analysis 29 (5) (1992), pp. 1493–1503.
Hansen, E.: Interval Arithmetic in Matrix Computations, SIAM Journal on Numerical Analysis 2 (1965), pp. 308–320.
Hansen, E. R.: Interval Forms of Newton's Method, BIT 20 (1978), pp. 153–163.
Hansen, E.: Preconditioning Linearized Equations, Computing 58 (2) (1997), pp. 187–196.
Hansen, E. R. and Greenberg, R. I.: An Interval Newton Method, Applied Mathematics and Computation 12 (2–3) (1983), pp. 89–98.
Hansen, E. and Sengupta, S.: Bounding Solutions of Systems of Equations Using Interval Analysis, BIT 21 (1981), pp. 203–211.
Hansen, E. and Smith, R.: Interval Arithmetic in Matrix Computations, Part II, SIAM Journal on Numerical Analysis 4 (1) (1967), pp. 1–9.
Hansen, E. and Walster, G. W.: Global Optimization Using Interval Analysis, 2nd Edition, Marcel Dekker, New York, 2004.
Hansen, E. R. and Walster, G. W.: Sharp Bounds on Interval Polynomial Roots, Reliable Computing 8 (2) (2002), pp. 115–122.
Hillyard, R. C. and Braid, I. C., Analysis of Dimensions and Tolerances in Computer-Aided Mechanical Design, Computer-Aided Design 10 (3) (1978), pp. 161–166.
Hoffmann, C. M., Lomonosov, A., and Sitharam, M.: Decomposition Plans for Geometric Constraint Systems, Part I: Performance Measures for CAD, Journal of Symbolic Computation 31 (4) (2001), pp. 367–408.
Hoffmann, C. M., Lomonosov, A., and Sitharam, M.: Decomposition Plans for Geometric Constraint Systems, Part II: New Algorithms, Journal of Symbolic Computation 31 (4) (2001), pp. 409–427.
Hong, H. and Stahl, V.: Bernstein Form Is Inclusion Monotone, Computing 55 (1995), pp. 43–53.
Hsu, C. Y. and Bruderlin, B.: Constraint Objects—Integrating Constraint Definition and Graphical Interaction, in: ACM Proceedings of the Second Symposium on Solid Modeling and Applications, 1993, Montreal, Quebec, Canada, pp. 467–468.
Hu, C. Y., Maekawa, T., Patrikalakis, N. M., and Ye, X.: Robust Interval Algorithm for Surface Intersections, Computer-Aided Design 29 (9) (1997), pp. 617–627.
Hu, C. Y., Patrikalakis, N. M., and Ye, X.: Robust Interval Solid Modeling, Part II: Boundary Evaluation, Computer-Aided Design 28 (10) (1996), pp. 819–830.
Hungerbuhler, A. R. and Garloff, B. J.: Bounds for the Range of a Bivariate Polynomial over a Triangle, Reliable Computing 4 (1) (1998), pp. 3–13.
Jaulin, L., Kieffer, M., Didrit, O., and Walter, E.: Applied Interval Analysis, Springer, London, 2001.
Kalra, D. and Barr, A. H.: Guaranteed Ray Intersections with Implicit Surfaces, Computer Graphics 23 (3) (1989), pp. 297–304.
Kaucher, E.: Interval Analysis in the Extended Interval Space IR, Computing Supplementum 2, Springer, Heidelberg, 1980, pp. 33–49.
Kearfott, R. B.: On Existence and Uniqueness Verification for Non-Smooth Functions, Reliable Computing 8 (4) (2002), pp. 267–282.
Kearfott, R. B.: Preconditioners for the Interval Gauss-Seidel Method, SIAM Journal on Numerical Analysis 27 (3) (1990), pp. 804–822.
Kearfott, R. B.: Rigorous Global Search: Continuous Problems, Kluwer Academic Publishers, Dordrecht, 1996.
Kearfott, R. B. and Dian, J.: Existence Verification for Higher Degree Singular Zeros of Nonlinear Systems, SIAM Journal on Numerical Analysis 41 (6) (2003), pp. 2350–2373.
Kearfott, R. B., Dian, J., and Neumaier, A.: ExistenceVerification for Singular Zeros of Complex Nonlinear Systems, SIAM Journal on Numerical Analysis 38 (2) (2000), pp. 360–379.
Kearfott, R. B. and Walster, G. W.: Symbolic Preconditioning with Taylor Models: Some Examples, Reliable Computing 8 (6) (2002), pp. 453–468.
Kolev, L. V.: A New Method for Global Solution of Systems of Non-Linear Equations, Reliable Computing 4 (2) (1998), pp. 125–146.
Kolev, L. V.: An Improved Method for Global Solution of Non-Linear Systems, Reliable Computing 5 (2) (1999), pp. 103–111.
Kolev, L. V.: Automatic Computation of a Linear Interval Enclosure, Reliable Computing 7 (1) (2001), pp. 17–28.
Kolev, L.: Use of Interval Slopes for the Irrational Part of Factorable Functions, Reliable Computing 3 (1) (1997), pp. 83–93.
Kolev, L. V. and Nenov, I.: Cheap and Tight Bounds on the Solution Set of Perturbed Systems of Nonlinear Equations, Reliable Computing 7 (5) (2001), pp. 399–408.
Kondo, K.: Algebraic Method for Manipulation of Dimensional Relationships in Geometric Models, Computer-Aided Design 24 (3) (1992), pp. 141–147.
Kondo, K.: PIGMOD: Parametric and Interactive Geometric Modeller for Mechanical Design, Computer-Aided Design 22 (10) (1990), pp. 633–644.
Kramer, G. A.: A Geometric Constraint Engine, in: Freuder, E. C. and Mackworth, A. K. (eds), Artificial Intelligence: Constraint-Based Reasoning 58, Elsever, 1992, pp. 327–360.
Krawczyk, R.: Newton-Algorithmen zur Bestimmug von Nullstellen mit Fehlerschranken, Computing 4 (1969), pp. 187–201.
Krawczyk, R. and Neumaier, A.: An Improved Interval Newton Operator, Journal of Mathematical Analysis and Applications 118 (1) (1986), pp. 194–207.
Krawczyk, R. and Neumaier, A.: Interval Slopes for Rational Functions and Associated Centered Forms, SIAM Journal on Numerical Analysis 22 (3) (1985), pp. 604–615.
Lamure, H. and Michelucci, D.: Solving Geometric Constraints by Homotopy, in: Proceedings of the Third ACM Symposium on Solid Modeling and Applications, May 17–19, 1995, Salt Lake City, Utah, pp. 263–269.
Latham, R. S. and Middleditch, A. E.: Connectivity Analysis: A Tool for Processing Geometric Constraints, Computer-Aided Design 28 (11) (1996), pp. 917–928.
Lee, J.Y. and Kim, K.:A2-D Geometric Constraint Solver Using DOF-Based Graph Reduction, Computer-Aided Design 30 (11) (1998), pp. 883–896.
Lee, J. Y. and Kim, K.: Geometric Reasoning for Knowledge-Based Parametric Design Using Graph Representation, Computer-Aided Design 28 (10) (1996), pp. 831–841.
Light, R. and Gossard, D.: Modification of Geometric Models Through Variational Geometry, Computer-Aided Design 14 (4) (1982), pp. 209–214.
Lin, H., Liu, L., and Wang, G.: Boundary Evaluation for Interval Bezier Curve, Computer-Aided Design 34 (9) (2002), pp. 637–646.
Maekawa, T. and Patrikalakis, N. M.: Computation of Singularities and Intersections of Offsets of Planar Curves, Computer Aided Geometric Design 10 (5) (1993), pp. 407–429.
Maekawa, T. and Patrikalakis, N. M.: Interrogation of Differential Geometry Properties for Design and Manufacture, The Visual Computer 10 (4) (1994), pp. 216–237.
Makino, K. and Berz, M.: Efficient Control of the Dependency Problem Based on Taylor Model Methods, Reliable Computing 5 (1) (1999), pp. 3–12.
Makino, K. and Berz, M.: Remainder Differential Algebras and Their Applications, in: Berz, M., Bischof, C., Corliss, G., and Griewank, A. (eds.), Computational Differentiation: Techniques, Applications, and Tools, SIAM, Philadelphia pp. 63–74.
Mayer, G.: Epsilon-Infiation in Verification Algorithms, Journal of Computational and Applied Mathematics 60 (1–2) (1995), pp. 147–169.
Modares, M., Mullen, R., Muhanna, R. L., and Zhang, H.: Buckling Analysis of Structures with Uncertain Properties and Loads Using an Interval Finite Element Method, in: Muhanna, R. L. and Mullen, R. L. (eds), Proceedings of the 2004 NSF Workshop on Reliable Engineering Computing, September 15–17, 2004, Savannah, GA, USA, pp. 317–327.
Moore, M. and Wilhelms, J.: Collision Detection and Response for Computer Animation, Computer Graphics 22 (4) (1988), pp. 289–298.
Moore, R. E.: Methods and Applications of Interval Analysis, SIAM, Philadelphia, 1979.
Moore, R. E. (ed.): Reliability in Computing: The Role of Interval Methods in Scientific Computing, Academic Press, Boston, 1988.
Moore, R. E.: Sparse Systems in Fixed Point Form, Reliable Computing 8 (4) (2002), pp. 249–265.
Moore, R. E. and Qi, L.: A Successive Interval Test for Nonlinear Systems, SIAM Journal on Numerical Analysis 19 (4) (1982), pp. 845–850.
Mudur, S. P. and Koparkar, P. A.: Interval Methods for Processing Geometric Objects, IEEE Computer Graphics and Applications 4 (2) (1984), pp. 7–17.
Muhanna, R. L. and Mullen, R. L.: Formulation of Fuzzy Finite-Element Methods for Solid Mechanics Problems, Computer-Aided Civil and Infrastructure Engineering 14 (1999), pp. 107–117.
Muhanna, R. L. and Mullen, R. L.: Uncertainty in Mechanics Problems—Interval-Based Approach, ASCE Journal of Engineering Mechanics 127 (6) (2001), pp. 557–566.
Muhanna, R. L., Mullen, R. L., and Zhang, H.: Interval Finite Element as a Basis for Generalized Models of Uncertainty in Engineering Mechanics, in: Muhanna, R. L. and Mullen, R. L. (eds), Proceedings of the 2004 NSFWorkshop on Reliable Engineering Computing, September 15–17, 2004, Savannah, GA, USA, pp. 353–370.
Mullineux, G: Constraint Resolution Using Optimisation Techniques, Computers & Graphics 25 (3) (2001), pp. 483–492.
Neumaier, A.: A Simple Derivation of the Hansen-Bliek-Rohn-Ning-Kearfott Enclosure for Linear Interval Equations, Reliable Computing 5 (2) (1999), pp. 131–136.
Neumaier, A.: Interval Methods for Systems of Equations, Cambridge University Press, 1990.
Neumaier, A.: On Shary's Algebraic Approach for Linear Interval Equations, SIAM Journal on Matrix Analysis and Applications 21 (4) (2000), pp. 1156–1162.
Neumaier, A.: Taylor Forms – Use and Limits, Reliable Computing 9 (1) (2003), pp. 43–79.
Ning, S. and Kearfott, R. B.: A Comparison of Some Methods for Solving Linear Interval Equations, SIAM Journal on Numerical Analysis 34 (4) (1997), pp. 1289–1305.
Owen, J. C.: Algebraic Solution for Geometry from Dimensional Constraints, in: ACM Proceedings of the First Symposium on Solid Modeling Foundations and CAD/CAM Applications, 1991, Austin, Texas, pp. 397–407.
Perez, A. and Serrano, D.: Constraint Based Analysis Tools for Design, in: ACM Proceedings on the 2nd Symposium on Solid Modeling and Applications, 1993, Montreal, Quebec, Canada, pp. 281–291.
Rao, S. S. and Berke, L.: Analysis of Uncertain Structural Systems Using Interval Analysis, AIAA Journal 35 (4) (1997), pp. 727–735.
Rao, S. S. and Cao, L.: Optimum Design of Mechanical Systems Involving Interval Parameters, ASME Journal of Mechanical Design 124 (2002), pp. 465–472.
Ratschek, H. and Rokne, J.: Computer Methods for the Range of Functions, Ellis Horwood, Chichester, 1984, Ch. 6.
Ratschek, H. and Rokne, J.: New Computer Methods for Global Optimization, Ellis Horwood, New York, 1988.
Rohn, J.: Cheap and Tight Bounds: the Recent Result by E. Hansen Can Be Made More Efficient, Interval Computations 4 (1993), pp. 13–21.
Rokne, J. G.: A Note on the Bernstein Algorithm for Bounds for Interval Polynomials, Computing 21 (1979), pp. 159–170.
Roller, D.: An Approach to Computer-Aided Parametric Design, Computer-Aided Design 23 (5) (1991), pp. 385–391.
Rump, S. M.: A Note on Epsilon-Infiation, Reliable Computing 4 (4) (1998), pp. 371–375.
Rump, S. M.: Inclusion of Zeros of Nowhere Differentiable n-Dimensional Functions, Reliable Computing 3 (1) (1997), pp. 5–16.
Sederberg, T.W. and Farouki, R. T.: Approximation by Interval Bezier Curves, IEEE Computer Graphics and Applications 12 (5) (1992), pp. 87–95.
Shary, S. P.: A New Technique in Systems Analysis under Interval Uncertainty and Ambiguity, Reliable Computing 8 (5) (2002), pp. 321–418.
Shary, S. P.: Algebraic Approach in the “Outer Problem” for Interval Linear Equations, Reliable Computing 3 (2) (1997), pp. 103–135.
Snyder, J.: Generative Modeling for Computer Graphics and CAD: Symbolic Shape Design Using Interval Analysis, Academic Press, Cambridge, 1992.
Snyder, J. M., Woodbury, A. R., Fleischer, K., Currin, B., and Barr, A. H.: Interval Methods for Multi-Point Collisions Between Time-Dependant Curved Surfaces, in: ACMProceedings of the 20th Annual Conference on Computer Graphics and Interactive Techniques, New York, 1993, pp. 321–334.
Solano, L. and Brunet, P.: Constructive Constraint-Based Model for Parametric CAD Systems, Computer-Aided Design 26 (8) (1994), pp. 614–621.
Stahl, V., Interval Methods for Bounding the Range of Polynomials and Solving Systems of Nonlinear Equations, unpublished Ph.D. thesis, University of Linz, 1995.
Sunde, G.: Specification of Shape by Dimensions and Other Geometric Constraints, in: Wozny, M. J., McLaughlin, H. W., and Encarnacao, J. L. (eds), Geometric Modeling for CAD Applications, IFIP WG 5.2 Working Conference on Geometric Modeling for CAD Applications, May 12–16, 1986, Rensselaerville, New York, North-Holland, Amsterdam, 1988, pp. 199–213.
Toth, D. L.: On Ray Tracing Parametric Surfaces, Computer Graphics 19 (3) (1985), pp. 171–179.
Tuohy, S. T., Maekawa, T., Shen, G., and Patrikalakis, N. M.: Approximation of Measured Data with Interval B-Splines, Computer-Aided Design 29 (11) (1997), pp. 791–799.
Von Herzen, B., Barr, A. H., and Zatz, H. R.: Geometric Collisions for Time-Dependent Parametric Surfaces, Computer Graphics 24 (4) (1990), pp. 39–48.
Wallner, J., Krasauskas, R., and Pottmann, H.: Error Propagation in Geometric Constructions, Computer-Aided Design 32 (11) (2000), pp. 631–641.
Wolfe, M. A.: A Modification of Krawczyk's Algorithm, SIAM Journal on Numerical Analysis 17 (3) (1980), pp. 376–379.
Wolfe, M. A.: On Bounding Solutions of Underdetermined Systems, Reliable Computing 7 (3) (2001), pp. 195–207.
Yamaguchi, Y. and Kimura, F.: A Constraint Modeling System for Variational Geometry, in: Wozny, M. J., Turner, J. U., and Preiss, K. (eds), Geometric Modeling for Product Engineering, IFIP WG 5.2/NSF Working Conference on Geometric Modeling, September 18–22, 1988, Rensselaerville, New York, North-Holland, Amsterdam, 1990, pp. 221–233.
Yamamura, K.: An Algorithm for Representing Functions of Many Variables by Superpositions of Functions of One Variable and Addition, IEEE Transactions on Circuits and Systems–I: Fundamental Theory and Application 43 (4) (1996), pp. 338–340.
Zhang, D., Li, W., and Shen, Z.: Solving Underdetermined Systems with Interval Methods, Reliable Computing 5 (1) (1999), pp. 23–33.
Zuhe, S. and Wolfe, M. A.: On Interval Enclosures Using Slope Arithmetic, Applied Mathematics and Computation 39 (1) (1990), pp. 89–105.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Wang, Y., Nnaji, B.O. Solving Interval Constraints by Linearization in Computer-Aided Design. Reliable Comput 13, 211–244 (2007). https://doi.org/10.1007/s11155-006-9023-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11155-006-9023-4