Dynamics of Phononic Materials and Structures: Historical Origins, Recent Progress, and Future Outlook

The notion of a generalized finite element method is introduced. This class of methods is analyzed and their relation to mixed methods is discussed. The class of generalized finite element methods offers a wide variety of computational procedures from which particular procedures can be selected for particular problems. A particular generalized finite element method which is very effective for problems with rough coefficients is discussed in detail.

Generalized Finite Element Methods: Their Performance and Their Relation to Mixed Methods

Note: Univ maryland,dept math,college pk,md 20742. ecole polytech,ctr math appl,f-91128 palaiseau,france. univ paris 6,f-75230 paris 05,france. Mercier, b, cea limeil,f-94190 villeneuve,france.ISI Document Delivery No.: LQ584Times Cited: 29Cited Reference Count: 44 Reference ASN-ARTICLE-1981-005doi:10.2307/2007651 Record created on 2006-08-24, modified on 2017-05-12

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Eigenvalue approximation by mixed and hybrid methods

A method with which standard finite element programs can be used to compute dispersion relations in periodic composites is proposed. The method is applied to two composite microstructures: a two-phase laminate and a fiber composite. The dispersion relations computed for the laminate are compared with a known analytical solution and the agreement is very good. The dispersion relations computed for the fibrous composite are compared with an existing approximate model and experimental results from the literature. The agreement between the approximate model, the experiments, and the computations is very good in the wave guide case and satisfactory for the wave reflect case.

The usage of standard finite element codes for computation of dispersion relations in materials with periodic microstructure

A method for homogenization of an elastic composite with periodic microstructure is presented, focusing on the Floquet-type elastic waves. The resulting homogenized frequency-dependent elasticity and mass density then automatically satisfy the overall conservation laws and by necessity produce the exact dispersion relations. It is also shown that the dispersion relations and the associated field quantities can be accurately calculated using a mixed variational approach, based on the microstructure of the associated unit cell. The method is used to calculate the dynamic effective parameters for a layered composite by using both the exact solution and the results of the mixed variational formulation. The exact and approximate results are shown to be in close agreement, which makes it possible to use the approximate method for the proposed type of homogenization in cases where an exact solution does not exist. The homogenized frequency-dependent effective parameters give rise to the concept of dynamic Ashby charts that can be used to illustrate the effect of the microstructural architecture on the dynamic properties of a composite. In particular, the charts vividly display how this effective stiffness and density vary with frequency and may attain negative values within certain frequency ranges which can be changed as desired using the microarchitecture while keeping the volume fraction of the unit cell's constituents constant.

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S. Minagawa

Papers

On Harmonic Waves in Layered Composites

Harmonic waves in one-, two- and three-dimensional composites: Bounds for eigenfrequencies

Harmonic waves in three-dimensional elastic composites

Harmonic Waves in Layered Composites: Comparison Among Several Schemes

Dispersion of waves in two-dimensional layered, fiber-reinforced, and other elastic composites