Abstract
Band gaps, i.e., frequency ranges in which waves cannot propagate, can be found in elastic structures for which there is a certain periodic modulation of the material properties or structure. In this paper, we maximize the band gap size for bending waves in a Mindlin plate. We analyze an infinite periodic plate using Bloch theory, which conveniently reduces the maximization problem to that of a single base cell. Secondly, we construct a finite periodic plate using a number of the optimized base cells in a postprocessed version. The dynamic properties of the finite plate are investigated theoretically and experimentally and the issue of finite size effects is addressed.
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References
Bendsøe MP, Sigmund O (2003) Topology optimization—theory, methods and applications. Springer, Berlin Heidelberg New York
Borel PI, Harpøth A, Frandsen LH, Jensen M, Jensen JS, Sigmund O (2004) Topology optimization and fabrication of photonic crystal structures. Opt Express 12:1996–2001
Borrvall T, Petersson J (2001) Topology optimization using regularized intermediate density control. Comput Methods Appl Mech Eng 190:4911–4928
Cox SJ, Dobson DC (1999) Maximizing band gaps in two-dimensional photonic crystals. SIAM J Appl Math 59(6):2108–2120
Diaz AR, Haddow AG, Ma L (2005) Design of band-gap grid structures. Struct Multidisc Optim 29(6):418–431
Graff KF (1991) Wave motion in elastic solids. Dover, Boulder, CO
Halkjær S, Sigmund O, Jensen JS (2005) Inverse design of phononic crystals by topology optimization. Z Kristallogr 220(9–10):895–905
Jensen JS, Sigmund O (2004) Systematic design of photonic crystal structures using topology optimization: low-loss waveguide bends. Appl Phys Lett 84:2002–2024
Kao CY, Osher S, Yablonovitch E (2005) Maximizing band gaps in two-dimensional photonic crystals by using level set methods. Appl Phys B Lasers Opt 81(2):235–244
Leissa A (1993) Vibration of plates. Acoustical Society of America, Melville, NY
Matthews J, Walker R (1964) Mathematical methods of physics. Addison-Wesley, USA
Mindlin RD (1951) Influence of rotary inertia and shear on flexural motions of isotropic, elastic plates. J Appl Mech 18:31–38
Sigalas MM, Economou EN (1994) Elastic waves in plates with periodically placed inclusions. J Appl Phys 75(6):2845–2850
Sigalas M, Kushwaha MS, Economou EN, Kafesaki M, Psarabas I, Steurer W (2005) Classical vibrational modes in phononic lattices: theory and experiment. Z Kristallogr 220(9–10):765–809
Sigmund O, Jensen JS (2003) Systematic design of photonic band-gap materials and structures by topology optimization. Philos Trans R Soc Lond A 361:1001–1019
Svanberg K (1987) The method of moving asymptotes: a new method for structural optimization. Int J Numer Methods Eng 24:359–373
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Halkjær, S., Sigmund, O. & Jensen, J.S. Maximizing band gaps in plate structures. Struct Multidisc Optim 32, 263–275 (2006). https://doi.org/10.1007/s00158-006-0037-7
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DOI: https://doi.org/10.1007/s00158-006-0037-7