zbMATH — the first resource for mathematics

Topological material layout in plates for vibration suppression and wave propagation control. (English) Zbl 1274.74359
Summary: We propose a topological material layout method to design elastic plates with optimized properties for vibration suppression and guided transport of vibration energy. The gradient-based optimization algorithm is based on a finite element model of the plate vibrations obtained using the Mindlin plate theory coupled with analytical sensitivity analysis using the adjoint method and an iterative design update procedure based on a mathematical programming tool. We demonstrate the capability of the method by designing bi-material plates that, when subjected to harmonic excitation, either effectively suppress the overall vibration level or alternatively transport energy in predefined paths in the plates, including the realization of a ring wave device.

74P15 Topological methods for optimization problems in solid mechanics
74M05 Control, switches and devices (“smart materials”) in solid mechanics
Full Text: DOI
[1] Auld BA (1973) Acoustic fields and waves in solids, vol I. Wiley, New York
[2] Bendsøe MP, Kikuchi N (1988) Generating optimal topologies in structural design using a homogenization method. Comput Methods Appl Mech Eng 71(2):197–224 · Zbl 0671.73065 · doi:10.1016/0045-7825(88)90086-2
[3] Bendsøe MP, Sigmund O (2003) Topology optimization - theory, methods and applications. Springer, Berlin Heidelberg New York
[4] Cox SJ, Dobson DC (1999) Maximizing band gaps in two dimensional photonic crystals. SIAM J Appl Math 59(6):2108–2120 · Zbl 1027.78521 · doi:10.1137/S0036139998338455
[5] Du J, Olhoff N (2007) Minimization of sound radiation from vibrating bi-material structures using topology optimization. Struct Multidisc Optim 33(4–5):305–321 · doi:10.1007/s00158-006-0088-9
[6] Guest J, Prevost J, Belytschko T (2004) Achieving minimum length scale in topology optimization using nodal design variables and projection functions. Int J Numer Methods Eng 61(2):238–254 · Zbl 1079.74599 · doi:10.1002/nme.1064
[7] Halkjær S, Sigmund O, Jensen JS (2005) Inverse design of phononic crystals by topology optimization. Z Kristallogr 220:895–905
[8] Halkjær S, Sigmund O, Jensen JS (2006) Maximizing band gaps in plate structures. Struct Multidisc Optim 32:263–275 · doi:10.1007/s00158-006-0037-7
[9] Hussein MI, Hulbert GM, Scott RA (2007) Dispersive elastodynamics of 1d banded materials and structures: design. J Sound Vib 307:865–893 · doi:10.1016/j.jsv.2007.07.021
[10] Jensen JS (2007a) Topology optimization of dynamics problems with Padé approximants. Int J Numer Methods Eng 72:1605–1630 · Zbl 1194.74270 · doi:10.1002/nme.2065
[11] Jensen JS (2007b) Topology optimization problems for reflection and dissipation of elastic waves. J Sound Vib 301:319–340 · doi:10.1016/j.jsv.2006.10.004
[12] Jensen JS, Sigmund O (2005) Topology optimization of photonic crystal structures: a high bandwidth low loss T-junction waveguide. J Opt Soc Am B 22(6):1191–1198 · doi:10.1364/JOSAB.22.001191
[13] Jog CS (2002) Topology design of structures subjected to periodic loading. J Sound Vib 253(3):687–709 · doi:10.1006/jsvi.2001.4075
[14] Ma ZD, Kikuchi N, Cheng HC (1995) Topological design for vibrating structures. Comput Methods Appl Mech Eng 121:259–280 · Zbl 0849.73045 · doi:10.1016/0045-7825(94)00714-X
[15] Olhoff N, Du J (2008) Topological design for minimum dynamic compliance of continuum structures subjected to forced vibration. Struct Multidisc Optim (in press)
[16] Rupp C, Evgrafov A, Maute K, Dunn ML (2007) Design of phononic materials/structures for surface wave devices using topology optimization. Struct Multidisc Optim 34(2):111–121 · Zbl 1273.74405 · doi:10.1007/s00158-006-0076-0
[17] Sigmund O (2007) Morphology-based black and white filters for topology optimization. Struct Multidisc Optim 33(4–5):401–424 · doi:10.1007/s00158-006-0087-x
[18] Sigmund O, Jensen JS (2003) Systematic design of phononic band-gap materials and structures by topology optimization. Philos Trans R Soc Lond Ser A Math Phys Eng Sci 361:1001–1019 · Zbl 1067.74053 · doi:10.1098/rsta.2003.1177
[19] Stolpe M, Svanberg K (2001) An alternative interpolation scheme for minimum compliance topology optimization. Struct Multidisc Optim 22:116–124 · doi:10.1007/s001580100129
[20] Svanberg K (1987) The method of moving asymptotes - a new method for structural optimization. Int J Numer Methods Eng 24:359–373 · Zbl 0602.73091 · doi:10.1002/nme.1620240207
[21] Uchino K (1998) Piezoelectric ultrasonic motors: overview. Smart Mater Struct 7:273–285 · doi:10.1088/0964-1726/7/3/002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.