Vibration-based diagnostic methods are used for the detection of the presence of cracks in beams and other structures. To simulate such a beam with an edge crack, it is necessary to model the beam using finite elements. Cracked beam finite elements, being one-dimensional, cannot model the stress field near the crack tip, which is not one-dimensional. The change in neutral axis is also not modeled properly by cracked beam elements. Modeling of such beams using two-dimensional plane elements is a better approximation. The best alternative would be to use three-dimensional solid finite elements. At a sufficient distance away from the crack, the stress field again becomes more or less one-dimensional. Therefore, two-dimensional plane elements or three-dimensional solid elements can be used near the crack and one-dimensional beam elements can be used away from the crack. This considerably reduces the required computational effort. In the present work, such a coupling of dissimilar elements is proposed and the required transition element is formulated. A guideline is proposed for selecting the proper dimensions of the transition element so that accurate results are obtained. Elastic deformation, natural frequency and dynamic response of beams are computed using dissimilar elements. The finite element analysis of cracked rotating shafts is complicated because of the fact that elastic deformations are superposed on the rigid-body motion (rotation about an axis). A combination of three-dimensional solid elements and beam elements in a rotating reference is proposed here to model such rotors.
Adaptive-multilevel BDDC algorithm for three-dimensional plane wave Helmholtz systems