AIMETA ‘03 XVI Congresso AIMETA di Meccanica Teorica ed Applicata 16 th AIMETA Congress of Theoretical and Applied Mechanics VECTOR LEVEL SET REPRESENTATION OF PROPAGATING CRACKS IN FINITE ELEMENTS G. Ventura 1 , E. Budyn 2 and T. Belytschko 2 Dipartimento di Ingegneria Strutturale e Geotecnica, Politecnico di Torino, Torino Department of Mechanical Engineering, Northwestern University, Evanston, IL, U.S.A. SOMMARIO Viene sviluppato un nuovo metodo Level Set che consente di descrivere superfici invariabili dietro il loro fronte di evoluzione, come ad esempio fessure. Nella formulazione proposta, in due dimensioni, la funzione che definisce il level set ` e data da una terna formata dal segno della distanza dalla superficie e dal vettore di minima distanza dalla superficie stessa. L’aggiornamento del level set avviene attraverso formule geometriche, e risultati numerici dimostrano l’accuratezza del metodo nella descrizione della superficie. Il metodo si combina naturalmente con l’extended finite element method, dove l’arricchimento dis- continuo per la frattura ` e efficacemente descritto da funzioni level set. ABSTRACT A new level set method is developed for describing surfaces that are frozen behind a moving front, such as cracks. In this formulation, the level set is described in two dimensions by a three-tuple: the sign of the level set function and the components of the closest point projection to the surface. The update of the level set is constructed by geometric formulas, which are easily implemented. Results for growth of lines in two dimensions show the method is very accurate. The method combines very naturally with the extended finite element method where the discontinuous enrichment for cracks is best described in terms of level set functions. 1 INTRODUCTION This paper presents a vector level set method for describing the growth of cracks in two dimensional problems. The vector level set method provides a very simple and efficient tool for describing the geometry of evolving cracks. In conjunction with the extended finite element method, it allows for problems of crack growth to be solved without remeshing. The eXtended Finite Element Method (XFEM) enriches the standard finite element basis through a local partition of unity (PU), Babuska and Melenk [1]. This was first applied to fracture in Belytschko and Black [2], where the asymptotic nearfield for a crack was incorporated by a local PU and the discontinuity in this field was used to represent the crack discontinuity independent of the mesh. By adding a step function, Mo¨ es et 1