Redlog-based derivation of feasibility conditions for singular integral equations of crack problems under parametric inequality constraints

• Κύριος συγγραφέας: Ioakimidis, Nikolaos
• Άλλοι συγγραφείς: Ιωακειμίδης, Νικόλαος
• Μορφή: Technical Report
• Γλώσσα: English
• Έκδοση: 2018
• Θέματα: Cracks; Periodic arrays of cracks; Stress intensity factors; Parametric inequality constraints; Feasibility conditions; Cauchy-type singular integral equations; Numerical methods; Numerical integration; Lobatto–Chebyshev method; Computer algebra; Computational quantifier elimination; Weispfenning's algorithm; Reduce computer algebra system; Redlog; Ρωγμές; Περιοδικές διατάξεις ρωγμών; Συντελεστές εντάσεως τάσεων; Παραμετρικοί ανισοτικοί περιορισμοί; Συνθήκες εφικτότητας; Ιδιόμορφες ολοκληρωτικές εξισώσεις τύπου Cauchy; Αριθμητικές μέθοδοι; Αριθμητική ολοκλήρωση; Μέθοδος των Lobatto–Chebyshev; Υπολογιστική άλγεβρα; Υπολογιστική απαλοιφή ποσοδεικτών; Αλγόριθμος του Weispfenning; Σύστημα υπολογιστικής άλγεβρας Reduce

Crack problems in the classical theory of two-dimensional static elasticity are frequently reduced to singular integral equations with Cauchy-type kernels further approximately solved by classical numerical techniques such as those based on the Gauss– and Lobatto–Chebyshev numerical integration rules. The derivation of approximate necessary and sufficient feasibility conditions for the existence of an approximate solution under the simultaneous validity of parametric inequality constraints of either a geometrical or a loading or even a strength/fracture nature is also of interest. This possibility was recently studied in detail for singular/hypersingular integral equations/inequalities by using the powerful quantifier elimination algorithm implemented in the computer algebra system Mathematica. Here a related alternative possibility is also suggested with respect to singular integral equations. This possibility is based on the use of the Reduce computer algebra system and, mainly, of the powerful Redlog (Reduce Logic) computer logic package of Dolzmann and Sturm, which employs the Weispfenning quantifier elimination algorithm for the related existential computational quantifier elimination and is a standard package of Reduce. The problem of a periodic array of straight cracks (either collinear or parallel) is used again as the vehicle for the illustration of the present alternative possibility here applied to singular integral equations under parametric inequality constraints. The present results are directly applicable essentially to any type of singular integral equations under parametric inequality constraints and they constitute an interesting alternative possibility for the derivation of feasibility conditions by using the method of computational quantifier elimination here applied to singular integral equations under parametric inequality constraints appearing in crack problems. The present alternative approach may also be found useful in several further computational mechanics and engineering problems.