High performance machines rely on fast moving parts and generally avoid resonance for improved accuracy. To improve the dynamic properties of these high performance machines, their parts are optimized via a lengthy iterative process. Topology optimization for vibrations problems could shorten this time consuming design process and provide a more optimal design compared to the manual iteration process.

In the field of topology optimization for vibration problems there are various methods to solve a given problem. The two most commonly used methods in recent research are the density approach and the level-set approach. The characteristics of the density and level-set approach are well understood in context of topology optimization for vibration problems, however a direct comparison between these two methods has not yet been conducted. Several crucial aspects of topology optimization for vibration problems will be investigated, such as localized eigenmodes, mode multiplicity, grey areas and efficiency for practical applications. Additionally, the applicability of these aspects will be tested in the academic and industrial field to determine their values when applied in industry. This thesis provides an extensive study of various design cases in which the density and level-set topology optimization approaches are compared on their ability to solve vibration problems. These design cases are based on frequently used design cases in literature which are generally seen as benchmark problems.

For this thesis it is opted to have as many similarities between the density and level-set approach as possible, to ensure a fair comparison between the two methods. To accomplish this, the level-set approach uses a density based mapping in combination with material parameter sensitivities and the method of moving asymptotes (MMA) to update the design variables. Furthermore, the level-set function is parameterized with compactly supported radial basis functions (CSRBF). This leaves the difference that the density approach uses element densities as design variables, whereas the level-set approach uses expansion coefficients as design variables.

The design cases indicate that the density approach is versatile as it is able to solve a wide variety of problems. Additionally, there are less parameters, which makes this method easier and faster to work within an industrial setting. Furthermore, the method produces well-performing designs even with more difficult tasks, such as a coarse mesh.
Although occasionally localized eigenmodes occurred whilst using this method, they do not seem to interfere with the final result. Thus, the density approach is less time consuming to setup and needs less tuning of the method specific parameters.
On the other hand, results from the design cases also indicated that the level-set approach is able to produce designs with an improved objective function at the cost of possibly more tuning of method specific parameters. Furthermore, the level-set approach is able to solve all the design cases without the occurrence of localized eigenmodes. Although the level-set approach is less optimal for coarse meshes, it outperforms the density approach at more refined mesh sizes. Additionally, it features a crisp geometry description by the zero level-set contour. Thus, the level-set approach is able to produce more optimal designs without the occurrence of local eigenmodes at the cost of more complexity and possibly more tuning of the method specific parameters.

To conclude, both approaches have unique properties to be able to solve vibration problems. The density approach is more applicable as a standard approach in an industrial setting due to it being more robust and the method is less time consuming. However, the level-set approach should be opted for more complex vibration problems due to the crisp geometry definition of complex geometric features and its ability to outperform the density approach.
A practical application has been solved with an optimization run, where the use of a set of predefined parameters that solved the benchmark cases has been used. Additionally, an optimization run where all parameters are optimized for the specific design case was performed to see the ultimate performance. The level-set approach was able to outperform the density approach in the predefined parameter case, whereas the ultimate performance case gave usable results for both methods. The differences came down to a more improved objective function for the density approach, or a more simplistic and lighter design for the level-set approach.