METHODOLOGY OF COMBINED APPLICATION OF DIRECTIONAL DERIVATIVES AND THE EXTENDED FINITE ELEMENT METHOD (X-FEM) FOR SOLVING VIBRATION EIGENVALUE PROBLEMS

Abstract

This paper presents a new methodology for solving the eigenvalue problem for time dependent structures. The time dependent structures of interest are structures with a moving discontinuity such as crack or structures with moving free (external/internal) surfaces. For the last case, they can result from a removal of material during a machining process or from a deterioration of the structure’s geometry. The methodology that we developed, is based on a combination of the eXtended Finite Element Method (X-FEM) and the Directional Derivatives method. X-FEM enables to overcome the drawbacks of conformity and remeshing: indeed, using standard FEM, a moving discontinuity in time within a structure requires not only that the mesh must conform to the discontinuity geometry but also to fully remesh the structure as much as necessary to follow the discontinuity in time. In order to alleviate this last point, the directional derivatives are a powerful tool because they allow to estimate the evolution of quantities from on reference domain to another one. In our case, they will allow to estimate the solutions of the eigenvalue problem. We suggest on the first sections to remind the main keys of both methods and we present then the combined methods in order to solve an eigenvalue problem. The application will be done on a one-dimensional eigenvalue problem and the numerical results will be presented to demonstrate the accuracy and the advantages of selected approaches. We conclude on the future prospects of the current work that mainly consist of to develop the methodology at the second order in order to increase the accuracy and to find a criteria in order to automatize the combined methods.