Classical numerical methods in engineering: A note on existential quantifier elimination under parametric inequality constraints

Communications in Numerical Methods in Engineering

Volumn 14, Issue 2, 1998, Pages 103-134

  Ioakimidis, Nikolaos I ^a,b  

^a UNIVERSITY OF PATRAS   (Greece)

^b NONE   (Greece)

Author keywords

Computational quantifier elimination; Edge crack; Feasibility (consistency) conditions; Fracture mechanics; Numerical methods in engineering; Parametric inequality constraints

Indexed keywords

ALGEBRA; ALGORITHMS; APPROXIMATION THEORY; BOUNDARY CONDITIONS; COMPUTATIONAL METHODS; CONSTRAINT THEORY; CRACK PROPAGATION; FRACTURE MECHANICS; INTEGRATION; STRESS INTENSITY FACTORS;

COMPUTATIONAL QUANTIFIER ELIMINATION (CQE) TECHNIQUE; PARAMETRIC INEQUALITY CONSTRAINTS; WEISPFENNING ELIMINATION METHODS;

NUMERICAL METHODS;

ALGORITHM; CRACK; FRACTURE MECHANICS;

EID: 0032007199     PISSN: 10698299     EISSN: None     Source Type: Journal    
DOI: 10.1002/(SICI)1099-0887(199802)14:2<103::AID-CNM133>3.0.CO;2-V     Document Type: Article

References (62)

1. O. C. Zienkiewicz and K. Morgan, Finite Elements and Approximation, Wiley, New York, 1983.
2. C. A. Brebbia and A. J. Ferrante, Computational Methods for the Solution of Engineering Problems, 3rd edn., Pentech Press, London, 1986.
3. O. C. Zienkiewicz and R. L. Taylor, The Finite Element Method, 4th edn., Vois. 1 and 2, McGraw-Hill, London, 1989 and 1991, respectively.
4. P. K. Banerjee, The Boundary Element Methods in Engineering, McGraw-Hill, London, 1994.
5. J. H. Davenport, Y. Siret and E. Tournier, Computer Algebra: Systems and Algorithms for Algebraic Computation, 2nd edn., Academic Press, London, 1993.
6. B. W. Char, K. O. Geddes, G. H. Gonnet, B. L. Leong, M. B. Monagan and S. M. Watt, Maple V Library Reference Manual, Springer, New York, 1991.
7. R. M. Corless, Essential Maple: an Introduction for Scientific Programmers, Springer, New York, 1995.
8. K. M. Heal, M. L. Hansen and K. M. Rickard, Maple V: Learning Guide, Springer, New York, 1996.
9. M. B. Monagan, K. O. Geddes, K. M. Heal, G. Labahn and S. Vorkoetter, Maple V: Programming Guide, Springer, New York, 1996.
10. R. L. Greene, Classical Mechanics with Maple, Springer, New York, 1995.
11. E. Kreuzer (Ed.), Computerized Symbolic Manipulation in Mechanics (CISM Courses and Lectures, No. 343, International Centre for Mechanical Sciences, Udine), Springer, Wien, 1994.
12. W. Gander and J. Hrebicek (Eds.), Solving Problems in Scientific Computing Using Maple and MATLAB, 2nd edn., Springer, Berlin, 1995.
13. A. M. Cohen (Ed.), Computer Algebra in Industry: Problem Solving in Practice, Wiley, Chichester, 1993.
14. A. M. Cohen, L. van Gastel and S. V. Lunel (Eds.), Computer Algebra in Industry 2: Problem Solving in Practice, Wiley, Chichester, 1995.
15. A. I. Beltzer, Variational and Finite Element Methods: a Symbolic Computation Approach, Springer, Berlin, 1990.
16. N. I. Ioakimidis, 'Chebyshev approximations to stress intensity factors: an application of 'Derive' ', Commun. appl. numer. methods, 7, 289-293 (1991).
17. N. I. Ioakimidis, 'Application of computer algebra to the iterative solution of singular integral equations', Comput. Methods Appl. Mech. Eng., 94, 229-237 (1992).
18. D. S. Arnon and B. Buchberger (Eds.), Algorithms in Real Algebraic Geometry, Academic Press, London, 1988 (reprinted from J. Symbolic Comput., 5(1-2), (1988)).
19. D. S. Arnon and B. Buchberger (Eds.), Algorithms in Real Algebraic Geometry, Academic Press, London, 1988 (reprinted from J. Symbolic Comput., 5(1-2), (1988)).
20. H. Hong (Ed.), Computational Quantifier Elimination, special issue of Computer J., 36(5), (1993).
21. B. F. Caviness and J. R. Johnson (Eds.), Quantifier Elimination and Cylindrical Algebraic Decomposition, Springer, Wien, 1997 (in press).
22. V. Weispfenning, 'The complexity of linear problems in fields', J. Symbolic Comput., 5, 3-27 (1988) (Reference 18 above).
23. R. Loos and V. Weispfenning, 'Applying linear quantifier elimination', Computer J., 36, 450-462 (1993) (Reference 19 above).
24. V. Weispfenning, 'Parametric linear and quadratic optimization by elimination', Technische Berichte No. MIP-9404, Fakultät für Mathematik und Informatik, Universität Passau, Passau, Germany, 1994.
25. V. Weispfenning, 'Solving parametric polynomial equations and inequalities by symbolic algorithms', Technische Berichte No. MIP-9504, Fakultät für Mathematik und Informatik, Universität Passau, Passau, Germany, 1995.
26. N. I. Ioakimidis, 'Kahan's ellipse problem, quantifier elimination and computational mechanics', IACM Bull., 10, 7-11 (1995).
27. N. I. Ioakimidis, 'Elementary quantifier-free formulae in boundary elements', Bound. Elem. Commun., 6, 107-112 (1995).
28. N. I. Ioakimidis, 'Application of quantifier elimination to a simple elastic beam finite element below a straight rigid obstacle', Mech. Res. Commun., 22, 271-278 (1995).
29. N. I. Ioakimidis, 'An inequality constraint for the deflection of an elastic beam under a uniform distributed loading', Strojn. Čas., 46, 229-242 (1995).
30. N. I. Ioakimidis, 'Computer-aided quantifier elimination in crack problems under constraints for the stress intensity factors', Eng. Fract. Mech., 52, 571-574 (1995).
31. N. I. Ioakimidis, 'Lack-of-contact conditions for a penny-shaped crack under a polynomial normal loading', Acta Mech., 117, 229-235 (1996).
32. N. I. Ioakimidis, 'Beams on tensionless elastic foundation: approximate quantifier elimination with Chebyshev series'. Int. j. numer. methods eng., 39, 663-686 (1996).
33. N. I. Ioakimidis, 'Inequality constraints in rectangular finite/boundary elements', Comput. Struct., 60, 415-431 (1996).
34. N. I. Ioakimidis, 'Inequality constraints in one-dimensional finite elements for an elastic beam on a tensionless Winkler foundation', Finite Eiern. Anal. Des., 24, 67-75 (1996).
35. N. I. Ioakimidis, 'Conditions for contact/lack of contact along a loaded simple straight crack in plane isotropic elasticity'. Eng. Fracture Mech., 56, 675-689 (1997).
36. N. I. Ioakimidis, 'Quantifier elimination in applied mechanics problems with cylindrical algebraic decomposition', Int. J. Solids Struct., 34, 4037-4070 (1997).
37. D. A. Kohler, Translation of a report by Fourier on his work on linear inequalities', Opsearch, 10, 38-42 (1973).
38. H. P. Williams, 'Fourier's method of linear programming and its dual', Am. Math. Mon., 93, 681-695 (1986).
39. L. M. Kutt and M. P. Bieniek, 'Elasto-plastic constitutive equations of stiffened plates', ASCE J. Eng. Mech., 114, 656-670 (1988).
40. W. L. Winston, Introduction to Mathematical Programming: Applications and Algorithms, 2nd edn., Duxbury Press, Belmont, CA, 1995.
41. N. Kikuchi and J. T. Oden, Contact Problems in Elasticity: a Study of Variational Inequalities and Finite Element Methods, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1988.
42. H. Antes and P. D. Panagiotopoulos, The Boundary Integral Approach to Static and Dynamic Contact Problems: Equality and Inequality Methods, Birkhäuser, Basel, 1992.
43. R. T. Haftka and Z. Gürdal, Elements of Structural Optimization, 3rd edn., Kluwer, Dordrecht, 1992.
44. U. Kirsch, Structural Optimization: Fundamentals and Applications, Springer, Berlin, 1993.
45. B. H. V. Topping and M. Papadrakakis (Eds.), Advances in Structural Optimization, Civil-Comp Press, Edinburgh, 1994.
46. R. J. Yang and C. H. Chuang, Optical topology design using linear programming', Comput. Struct., 52, 265-275 (1994).
47. S. M. Fereig, 'An application of linear programming to bridge design with standard prestressed girders', Comput. Struct., 50, 455-469 (1994).
48. S. M. Sun, H. S. Tzou and M. C. Natori, 'Parametric quadratic programming method for dynamic contact problems with friction', AIAA J., 32, 371-378 (1994).
49. T. C. Yang and C. H. Tseng, 'Effective optimization-based approach for designing active noise control systems in enclosures', Finite Elem. Anal. Des., 15, 303-316 (1994).
50. S.-S. Lee, 'A computational method for frictional contact problem using finite element method', Int. j. numer. methods eng., 37, 217-228 (1994).
51. R. Barauskas, 'Dynamic analysis of structures with unilateral constraints: numerical integration and reduction of structural equations', Int. j. numer. methods eng., 37, 323-342 (1994).
52. P. D. Panagiotopoulos, E. S. Mistakidis and O. K. Panagouli, 'Fractal interfaces with unilateral contact and friction conditions', Comput. Meth. Appl. Mech. Eng., 99, 395-412 (1992).
53. M. H. Aliabadi and D. P. Rooke, Numerical Fracture Mechanics, Computational Mechanics Publications, Southampton, and Kluwer, Dordrecht, 1991, pp. 193-274.
54. H. Tada, P. C. Paris and G. R. Irwin, The Stress Analysis of Cracks Handbook, Del Research Corporation, Hellertown, PA, 1973, pp. 8.3-8.4.
55. R. J. Hartranft and G. C. Sih, 'Alternating method applied to edge and surface crack problems', in G. C. Sih (Ed.), Methods of Analysis and Solutions of Crack Problems (Mechanics of Fracture, Vol. 1), Noordhoff, Leyden, 1973, Chap. 4, pp. 179-238.
56. P. J. Davis and P. Rabinowitz, Methods of Numerical Integration, 2nd edn., Academic Press, Orlando, FL, 1984.
57. D. A. Hills, P. A. Kelly, D. N. Dai and A. M. Korsunsky, Solution of Crack Problems: the Distributed Dislocation Technique, Kluwer, Dordrecht, 1996.
58. F. Winkler, Polynomial Algorithms in Computer Algebra, Springer, Wien, 1996.
59. M. Fitting, First-Order Logic and Automated Theorem Proving, 2nd edn., Springer, New York, 1996.
60. W. K. Grassmann and J.-P. Tremblay, Logic and Discrete Mathematics: a Computer Science Perspective, Prentice Hall, Upper Saddle River, NJ, 1996.
61. H. Hong, 'Quantifier elimination for formulas constrained by quadratic equations via slope resultant', Computer J., 36, 439-449 (1993) (Reference 19 above).
62. V. Weispfenning, 'Quantifier elimination for real algebra - the cubic case', in J von zur Gathen and M. Giesbrecht (Eds.), ISSAC '94: Proceedings of the International Symposium on Symbolic and Algebraic Computation, Oxford, July 1994, ACM Press, New York, 1994, pp. 258-263.