| Abstract [eng] |
The simulation of propagating waves is of primary importance in many engineering applications, such as planning ultrasonic measurement procedures, monitoring of the structural integrity of pipelines by analyzing pressure pulses propagation, earthquake waves propagation and many other types of real-life applications. Traditionally, the computational methods of wave propagation analysis in geometrically complex structures and environments use the finite element or the finite difference approaches. However, inherent shortcomings arise due to huge dimensionalities of the models in cases when the length of the analyzed waves is much lesser than the linear dimensions of the structure. The limit situation is the infinite wave propagation environment in one or several directions, which actually is extremely difficult to simulate by traditional FEM because fictitious wave reflections form artificially introduced boundaries of the computational domain. The existing techniques within traditional FEM which enable to cope with the infinite domains are the non-reflecting boundary condition (suitable only for acoustic waves), scaled-boundary FEM techniques and perfectly matched layers. However, the latter approaches are approximate, and they also require significant computational resources, anyway. The semi-analytical approaches such as the SAFE method seem to be promising as they enable their users to avoid the discretization of the structures along the infinite direction. Even though the principles of SAFE have been well-known for several decades, the approach is still underdeveloped as perfectly as the traditional FEM – therefore, further research is still necessary. The guided waves in the waveguide (plate, bar, pipe, etc.) are described by their dispersion curves. The dispersion curves present the relationships of phase, group and energy velocities of the waves against the wave frequency. The SAFE method facilitates the calculation of dispersion curves for waveguides having uniform cross-section geometries along at least one direction. The finite element covers the discretization of the waveguide cross-section only. Along the wave propagation direction, the harmonic solutions in space and time are used. The expressions of such solutions use the exponential functions in the space of complex numbers. Similarly to the conventional FEM, the SAFEM enables to express forced time-dependent wave response analysis as a superposition of modal responses. While there are many researches addressing waveguides in vacuum, SAFE modeling of traveling waves in dissipative environments is still a challenging task. It is caused by the fact that the traditional SAFE analyses presume the amplitude decay of the traveling wave as negligible and, therefore, certain mathematical simplifications of the FE formulation are possible. In the case of higher damping, such simplifications would lead to considerable errors of the solution. In this research, the SAFE formulation is extended in order to treat the wave propagation problems in viscous environments. The energy dissipation model is presented via Rayleigh damping (i.e., energy dissipation caused by material damping) and via the leaky wave, where the waveguide immersed into the perfect fluid is considered. |
