PSI - Issue 42

Nikola Milovanovic et al. / Procedia Structural Integrity 42 (2022) 362–367 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

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methods of structural integrity assessment are based on application of NDT methods, on real structures in real operating conditions [6, 7]. Beside cracks, it is known that the type of loading also has an influence on the structural integrity and remaining life. Working condition to which machine is exposed during service can be simulated using software which enables application of the finite element method (FEM). The FEM provide great possibilities in determining stress states around cracks, [8, 9], studies concerning multiple errors in welded joints of structural steels. Using FEM, as well as principles of FAD, it is also possible to perform integrity assessment, [10-13]. However, more complex problems, such as fatigue, require the application of the extended finite element method (xFEM), to simulate crack growth under amplitude loading, [16-25]. In the study [23], a comparison of both mentioned methods was presented on the example of a pressure vessel with crack under the fatigue loading. The obtained results of crack stress state using these two methods were significantly different (Fig. 1). The xFEM approach have provided more conservative results, which proves to be better from the aspect of structural integrity assesment, because more extreme results were placed in the function of higher safety in comparison to the classical FEM. In papers [12-14], the xFEM was used to determine the remaining service life of Kaplan turbines in the presence of a crack, which is the topic of this paper, as well. The used concept and software package enabled the introduction of a dynamic loading into the proposed model. However, the application of this method has its problems, which concern mostly meshing, loading settings and boundary conditions defining, which is the focus of this paper, especially xFEM meshing.

Fig. 1. (left) Stress distribution in the welded joint area (FEM); (right) Stress distribution and crack propagation in welded joint are (XFEM) [23].

The xFEM has enabled of solving problems that involve the presence of defects in discretized structure, which was impossible with the classical FEM approach, [26-29]. The essence of xFEM represents a feature called "partition of unity" [28], i.e. the property of the finite element that the sum of all interpolation functions in it is equal to 1. This means that there is the possibility of adding new interpolation functions to the FE model, as long as the sum of them and pre-existing functions remain equal to 1. In that way so-called improvement functions have been introduced, which have to be selected in a way that allows the presentation of discontinuities, e.g. Heaviside jump function. The advantage of such approach is reflected in the fact that there is no need of finite elements remeshing around the crack, which is inherent feature in other similar methods due simulating crack growth. Equation (1) shows the general form of the improvement function: (1) Some problems of xFEM application are presented in this paper, using the example of a Kaplan turbine, as well as simplification, which iteratively led to the possibility of simpler and faster calculation during simulation of working condition. The original set of finite element mesh has undergone certain changes, and the classical approach of the refined generated FE mesh has been replaced by more iterative one, which leads of obtaining more realistic and satisfactory results concerning fatigue crack growth. ( ) ( ) ( ) j h j i i i i j u x N x u v x a = +  