the surrounding fluid media. The radiation and scattering of waves in such fluid-structure interaction has been the subject of research for many decades.

Mathematically, the problem can be adequately modeled by a system of linear conservation laws, which, with further manipulations, can be reduced to the wave equation for the acoustical pressure coupled to the equations of elastodynamics for the structure. Traditionally, approaches for the computer simulations of structural acoustics phenomena have been primarily based on frequency-domain formulations of such problems, in which a Fourier transformation of the dynamic equations is made, and this results in a system of partial-differential equations in complex-valued field variables with frequency-dependent coefficients. The fluid may be modeled with the exterior Helmholtz equation or, as is commonly done, by an equivalent boundary-integral equation. These classical approaches have been very successful for a large and important class of problems, particularly those involving scales which are not too small in relation to characteristic dimensions of the structure. In recent times, attempts at large-scale simulations involving high ka values (k being the wave number and a the characteristic dimension) have encountered formidable computational difficulties and have exposed serious and fundamental limitations of some classical frequency-domain approaches.

In the present work, some new high-order time-domain methods are presented that have proven to be effective for simulating certain classes of problems in structural acoustics and that may have potential for overcoming many of the shortcomings of the frequency-domain methods for high ka. These techniques are built around several special ideas and algorithms:

  1. the structural acoustics problem is formulated as a hyperbolic system of conservation laws and this leads to an abstract Cauchy problem in common Hilbert space settings in which the spectral theory of linear operators is applicable;

  2. new, high-order, multistage Taylor-Galerkin (TG) approximations of these equations in the time-domain are constructed which deliver high-order temporal accuracy with unconditional stability;

  3. spatial approximations are constructed using hp-finite element methods in which local mesh-size h and local spectral order p are varied in a way to adaptively control error, which provides multi-level approximations of the unknowns;

  4. a posteriori error estimates are constructed over each element and time step to provide a running account of solution quality and to provide a basis for the adaptive control strategy;

  5. applications to representative model problems reveal that the methods are robust and provide remarkably accurate results for a class of two-dimensional cases.

In addition to the time-domain methods, a brief account of a coupled boundary element/finite element method is also given. This approach also employs hp-methods and is based on a new version of the Burton-Miller integral equations for the acoustical fluid. Results obtained by using large-scale parallel computers for such problems are presented.

We also make comparisons of time-domain and frequency-domain approaches for a model class of problems. In these computations, some advantages of the time-domain approach over traditional frequency-domain approaches are observed.



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