Computational analysis and design of phononic crystals

Immersed boundary analysis and level set based topology optimization

Sanne van den Boom (PhD candidate), Alejandro M. Aragón (supervisor), and Fred van Keulen (supervisor)

Phononic crystals have an interesting effect on waves traveling waves. For instance, they show bandgaps—ranges of frequencies that are prevented from propagating through the crystal. This is a useful property in many engineering applications, such as vibration isolation and energy harvesting. However, the design of these materials tailored to a specific application and frequency range is challenging, and as such, accurate and efficient modeling techniques are invaluable.

Wave traveling through a unit cell of a phononic crystal
Immersed periodic unit cell, where both the interior and exterior boundaries are non-conforming to the mesh

The performance of phononic crystals depends on both the topology of the inclusion and the lattice type, which is reflected in the shape of the PUC. Therefore, both the exterior and the interior boundaries are decoupled from the FE mesh. To that end, the PUC is analyzed in a fully immersed boundary setting, and Bloch-Floquet periodic boundary conditions are prescribed strongly on non-matching edges. With this approach the same performance as in standard FEM is achieved, while the geometry can be changed without changing the underlying analysis mesh.

As the behavior of phononic crystals is sensitive to the boundary description, the standard density-based topology optimization approaches become prohibitive, as extremely fine meshes would be required. Instead, the Interface-enriched Generalized Finite Element Method combined with level set-based topology optimization is more suitable for their numerical design than standard density based approaches.

Initial design of the periodic unit cell and its corresponding band structure (top) and topology optimized unit cell and band structure with band gap (bottom)
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