Systematic design of phononic band–gap materials and structures by topology optimization

Sigmund, Ole; Jensen, Jakob Søndergaard

doi:10.1098/rsta.2003.1177

Cited by 630 publications

(311 citation statements)

References 16 publications

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“…Multiphysics problems, where coupled analyses are required, are promising applications due to the inherent difficulties in obtaining good designs intuitively, and thus recently more research has been conducted in this area [2,3,4,5,6]. These applications, however, have mostly been restricted to problems where the multi-physics behavior is limited to the material-part of the design.…”

Section: Introductionmentioning

confidence: 99%

Topology optimization of acoustic–structure interaction problems using a mixed finite element formulation

Yoon

Jensen

Sigmund

2006

Numerical Meth Engineering

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184

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SUMMARYThe paper presents a gradient based topology optimization formulation that allows to solve acousticstructure (vibro-acoustic) interaction problems without explicit boundary interface representation. In acoustic-structure interaction problems, the pressure and displacement fields are governed by Helmholtz equation and the elasticity equation, respectively. Normally, the two separate fields are coupled by surface-coupling integrals, however, such a formulation does not allow for free material redistribution in connection with topology optimization schemes since the boundaries are not explicitly given during the optimization process. In this paper we circumvent the explicit boundary representation by using a mixed finite element formulation with displacements and pressure as primary variables (a u/p-formulation). The Helmholtz equation is obtained as a special case of the mixed formulation for the elastic shear modulus equating zero. Hence, by spatial variation of the mass density, shear and bulk moduli we are able to solve the coupled problem by the mixed formulation. Using this modeling approach, the topology optimization procedure is simply implemented as a standard density approach.Several two-dimensional acoustic-structure problems are optimized in order to verify the proposed method.

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Section: Introductionmentioning

confidence: 99%

Topology optimization of acoustic–structure interaction problems using a mixed finite element formulation

Yoon

Jensen

Sigmund

2006

Numerical Meth Engineering

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184

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“…Usually, the post-processed structure has a performance that is very close to the optimized structure. However, recently the author's research group has applied the topology optimization to phononic and photonic crystal design (Sigmund and Jensen, 2003;Jensen and Sigmund, 2004) where structures with intricate semi-periodic patterns appear as the result of the topology optimization process. An example of the design of a nano-scale optical splitter from Borel et al (2005) is shown in Fig.1b.…”

Section: Introductionmentioning

confidence: 99%

Morphology-based black and white filters for topology optimization

Sigmund

2007

Struct Multidisc Optim

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1,411

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In order to ensure manufacturability and mesh-independence in density-based topology optimization schemes it is imperative to use restriction methods. This paper introduces a new class of morphology based restriction schemes which work as density filters, i.e. the physical stiffness of an element is based on a function of the design variables of the neighboring elements. The new filters have the advantage that they eliminate grey scale transitions between solid and void regions. Using different test examples, it is shown that the schemes in general provide black and white designs with minimum length-scale constraints on either or both minimum hole sizes and minimum structural feature sizes. The new schemes are compared with methods and modified methods found in the literature.

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“…The resonance peaks can be removed or reduced by including some damping, and we also found that there is no significant change of the band-gap behavior for relatively small damping. The effect of smoothing by including damping is often used in the topology optimization of band-gap structures [7].…”

Section: Free Wave Propagation and Forced Vibration In The Optimized mentioning

confidence: 99%

“…Due to a wealth of potential applications in vibration protection, noise isolation, waveguiding, etc., the study and development of bandgap rod, mass-spring, beam, grillage, disk and plate structures, in most cases by topology optimization, have attracted increasing attention in recent years, see e.g. [5,[7][8][9][10][11][12][13][14][15][16][17][18]. Elastodynamics of finite or infinite periodic 1D rod or beam structures has been studied in [6,[19][20][21].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On Optimum Design and Periodicity of Band-Gap Structures

Olhoff¹,

Niu²,

Cheng³

2014

Proceedings of the 4th International Conference on Computational Methods in Structural Dynamics and Earthquake Engineering (COM

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Abstract. A band-gap structure usually consists of a periodic distribution of elastic materials or segments, where the propagation of waves is impeded or This paper extends earlier optimum shape design results for transversely vibrating Bernoulli-Euler beams by determining new optimum band-gap beam structures for (i) different combinations of classical boundary conditions, (ii) much larger values of the orders n (n>1) and n-1 of adjacent upper and lower eigenfrequencies of maximized band-gaps, and (iii) different values of a minimum beam cross-sectional area constraint. In the present paper, instead of maximizing band-gaps between frequencies of propagating waves or forced vibrations excited by external time-harmonic loads, the closely relatedproblem of maximizing the gap between two adjacent eigenfrequencies ω n and ω n-1 of any given consecutive orders n (n>1) and n-1, is considered. This is justified by the fact that external time-harmonic dynamic loads cannot excite resonance with high vibration levels of standing waves, if the eigenfrequencies of the structure are moved outside the range of the external excitation frequencies by the optimization.Finally, the present study shows that if an infinite beam structure is constructed by repeated translation of an inner beam segment obtained by the aforementioned frequency gap optimization, then a band-gap of traveling waves in this infinite beam is found to correlate almost perfectly with the maximized frequency gap in the finite structure.

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Systematic design of phononic band–gap materials and structures by topology optimization

Cited by 630 publications

References 16 publications

Topology optimization of acoustic–structure interaction problems using a mixed finite element formulation

Topology optimization of acoustic–structure interaction problems using a mixed finite element formulation

Morphology-based black and white filters for topology optimization

On Optimum Design and Periodicity of Band-Gap Structures

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