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

In this paper, an attempt is made to show the usefulness of computational quantifier elimination (CQE) techniques in computer algebra inside classical numerical methods in engineering for the derivation of feasibility (consistency) conditions in problems with weakly parametric linear inequality cons...

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Veröffentlicht in:Communications in numerical methods in engineering Jg. 14; H. 2; S. 103 - 134
1. Verfasser: Ioakimidis, Nikolaos I.
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Sussex John Wiley & Sons, Ltd 01.02.1998
Wiley
Schlagworte:
computational quantifier elimination
edge crack
Exact sciences and technology
feasibility (consistency) conditions
fracture mechanics
Fracture mechanics (crack, fatigue, damage...)
Fracture mechanics, fatigue and cracks
Fundamental areas of phenomenology (including applications)
Mathematical methods in physics
Numerical approximation and analysis
Numerical differentiation and integration
numerical methods in engineering
parametric inequality constraints
Physics
Solid mechanics
Structural and continuum mechanics
Stress intensity factors
Numerical integration
Ruptures
Numerical method
Quantifier
Computer algebra
Structural analysis
Edge crack
ISSN:1069-8299, 1099-0887
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Zusammenfassung:In this paper, an attempt is made to show the usefulness of computational quantifier elimination (CQE) techniques in computer algebra inside classical numerical methods in engineering for the derivation of feasibility (consistency) conditions in problems with weakly parametric linear inequality constraints (with the parameters appearing only in their right‐hand sides). A simple, but non‐trivial, straight edge‐crack problem in fracture mechanics under linear inequality constraints both on the applied loading along the crack faces and on the value of the stress intensity factor at the crack tip (associated with the Green/weight function method, numerical approximations and classical numerical integration) is used for an elementary illustration of the proposed approach. In this application, the method used tries to imitate the theoretical principle of the linear programming methods. The manually obtained related result is expressed as a disjunction of conjunctions of inequalities (as is frequently the case in similar CQE problems), and concrete numerical results are also displayed. The related influence of various approximations and the application of the trapezoidal quadrature rule are also considered in some detail. Further possibilities could concern the application of the approach to other numerical methods in engineering (such as to the finite and the boundary element methods, to singular and hypersingular integral equation methods, etc.) combined with efficient algorithms for linear inequality constraints such as the old Fourier and the recent Weispfenning elimination methods. © 1998 John Wiley & Sons, Ltd.
Bibliographie:istex:ED30C67D21E248475F87C1045D8C88DA640EED48
ArticleID:CNM133
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ISSN:1069-8299
1099-0887
DOI:10.1002/(SICI)1099-0887(199802)14:2<103::AID-CNM133>3.0.CO;2-V