Modified fundamental solutions for the scattering of elastic waves by a cavity: Numerical results

Abstract

The boundary integral equation method is very often used to solve exterior problems of scattering of waves (elastic waves, acoustic waves, water waves and electromagnetic waves). It is known, however, that this method fails to provide a unique solution at the so-called irregular frequencies. This difficulty is inherent to the method used rather than the nature of the problem. In the context of elastodynamics, we proposed, in a recent work', two methods for eliminating these irregular frequencies. Both are based on modifying the fundamental solution. Here we present numerical results pertaining to the solutions of the modified and unmodified integral equations. I . INTRODUCTlONConsider a homogeneous isotropic elastic solid of unbounded extent, containing an infinite cylindrical cavity of smooth cross-section. Suppose that the cavity is irradiated by time-harmonic stress-waves, of frequency w, propagating perpendicular to the generators of the cylinder so that the material is in a state of plane strain. We wish to determine the scattered waves when the boundary of the cavity is free from applied tractions. This leads to a linear two-dimensional boundary-value problem in any plane perpendicular to the generators of the cylinder.Let D -denote the interior of the cylinder, with smooth boundary curve 6D. Let D denote the unbounded region outside i?D. Choose Cartesian co-ordinates (xl, x2) with origin 0 in D _ . We shall also use the following notation.Capital letters P, Q denote points of D; lower-case letters p , q denote points of dD; and P -, Q-denote points of D -; b(p) is the unit normal at p~6 D pointing into D; uinc and tinc are the incident displacement and stress, respectively.The decomposition of the total displacement u and thus of the total stress T in the form = ,,ino ' + ~S C and t = tint + TSC defines the scattered displacement us' and stress t P ' , respectively. Mathematically, the scattering boundary-value problem can be formulated as follows. Boundary-value problem S(uinc): determine a function us' for P E D , satisfying with k2 = pw2/(% + 2p) and K 2 = pw2/p, where p is the mass density of the solid and , i and 11 are the Lame constants.

Low frequency scattering of elastic waves by a cavity using a matched asymptotic expansion method

“…In Table 1 C, = (1 + nm 2 )ft; C 2 = ft; C 4 = mft; D 7 = <5 2 ; with = yfc'(\ -nm 2 ){kL) 2 /%; 8 2 = ^}-J^^± -(kL) 2 We do not record the lengthy expressions for the null-potentials here; they do not contribute to the displacement field itself.…”

“…Let us recall the formulation of the boundary value problem as it is stated in [1]: Determine a function & x for P e D, satisfying 1. Elastodynamic equations of motion in D k- 2 .V(y.u sc (P))-K- 2 .Vx(Vxn sc (P))+u sc (P) = 0, PzD. (2.1)…”

Operators of Minimal Norm via Modified Green’s Function for the Elastic Two-Dimensional Case Using the Simple Multipole Coefficients

“…The first work which adopts the modified Green's function technique in elasticity is due to Jones [8], who examined the cavity in IR³. In [5,7] consideration is also given to elastic problems in IR². In [2,4] The exterior Dirichlet problem in IR³ is investigated by Argyropoulos, Kiriaki and Roach, and the nonuniqueness of the boundary integral equation is overcome with a suitable choice of simple mutipole coefficients in the modification.…”

Η Τροποποιημενη Συναρτηση Green Στη Γραμμικη Ελαστικοτητα: Κριτηρια Βελτιστοποιησης

“…x>y Ετσι η θεμελιώδης λύση της φασματικής εξίσωσης Navier δέχεται, μέσω των Μια τέτοια τροποποίηση κατ' αναλογία με την ακουστική περίπτωση[49,50,51] και τις εργασίες[44,47,48,37,38] στην ελαστικότητα, συνίσταται στη θεώρηση ως θεμελιώδους λύσης του μερικού διαφορικού » όρους N* m (χ) ® LL (y) και L e nm (χ) ® N" m (y) των μη ορθογωνίων μεταξύ των διανυ σμάτων N^m και L e nm , που υπάρχουν στην έκφραση r D (x,y) Αλλωστε και στην εργασία[37] χρησιμοποιούνται σταυρωτοί όροι για την αντιμετώπιση όμως διδιάστατου προ βλήματος της γραμμικής ελαστικότητας Με την ανωτέρω τροποποιημένη θεμελιώδη λύση θα θέσουμε στη συνέχεια, πέραν των συνθηκών που εξασφαλίζουν την σύγκλιση της σειράς,…”