ELSEVIER Wave Motion 20 ( 1994) 165-176 Finite element analysis of wave scattering from singularities Dan Givoli Department zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA of Aerospace Engineering. Technion-Israel Institute of Technology, Haifa 32000, Israel Shmuel Vigdergauz Israel Electric Corporation Ltd., Research and Development Division, P.O.B. IO, Haifa 31000, Israel Received 28 February 1994 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONM Abstract A combined analytic-finite element method is used for the efficient solution of the Helmholtz equation in the presence of geometrical singularities. In particular, time-harmonic waves in a membrane which contains one or more fixed-edge cracks (stringers) are investigated. The Dirichlet-to-Neumann ( DtN) map is used in the procedure, to enable the replacement of the original singular problem by an equivalent regular problem, which is then solved by a finite element scheme. The method yields the solution in the entire membrane, as well as the dynamic “stress intensity factor.” Numerical results are presented for a circular membrane containing an edge stringer, two edge stringers and an internal stringer. The first few critical wave numbers of the membrane are also found. 1. Introduction One important branch of fracture mechanics deals with the interaction of elastic waves with geometrical singu- larities such as cracks. Exact analytic solutions for the stress fields around cracks, even in the static case, are available only in very simple configurations, mainly for problems in infinite domains [ zyxwvutsrqponmlkjihgfedcbaZYXW 11. Exact solutions in the dynamic case are even more scarce; see [2] and references therein. On the other hand, the numerical treatment of such problems is not free of difficulties, since most standard numerical schemes fail to give meaningful results in the region near the singularity. If not specially treated, singularities are known to lead to larger errors in the numerical solutions than the errors which occur without singularities, and thus to slower rates of convergence [ 31. In the realm of the finite element method, there are two well known general procedures to treat geometrical singularities, each having many variants. The first method uses a specially refined mesh in the singularity region [ 4-71, while the second uses special singular finite element shape functions [S-l 11. A third approach to treat crack and sharp corner problems is based on the boundary element method [ 12-141. Particularly relevant to the present investigation are the works by Achenbach et al. [ 15-171 and other authors [ 18-211, which deal with boundary integral solution methods for problems involving wave scattering from cracks in elastic bodies. In all these works, the singularity is handled as part of the numerical solution process. Another method has been recently devised in Ref. [ 221. In this method the singularity is first eliminated in an exact manner. In other words, the original singular problem is first replaced by an equivalent regular problem. This regular problem is then solved by a finite element scheme. The use of the Dirichlet-to-Neumann (DtN) map is the main mathematical tool in the proposed method, which is therefore called the DtN Finite Element Method. In Ref. [ 221, the method was introduced 0165-2125/94/$07.00 0 1994 Elsevier Science B.V. All rights reserved SSD10165-2125(94)00021-V