Helmholtz Equation

The present work concerns the new capability of LS-DYNA® in solving acoustic and vibroacoustic problems. In vibroacoustic problems, which are assumed to be weak acoustic-structure interactions, the transient structural response is computed first. By applying the FFT, it is transformed into a frequency response. The obtained result is taken as boundary condition for the acoustic part of the vibroacoustic problem.

The equations of motion for straight fluid filled pipes are greatly simplified. It is found, for frequencies below a third of the ring frequency, that the radial-axial waves in cylinders are as if the circumferential motion were inextensional. This is the fundamental assumption for the analysis. The derivation is also based on the assumption of long axial wavelengths, resulting in the axial inertia of the fluid and the axial flexural stiffness of the pipe wall being negligible. The formulation is restricted to frequencies well below the cut-on of higher order fluid modes. For such frequencies, the compressibility of the fluid is neglected and the internal fluid loading, on the pipe, is approximated as an increase in the radial inertia. Upon this basis, the equations of motion, for each circumferential mode, are similar to those for a Timoshenko beam on a Winkler foundation. Numerical experiments are made, comparing the approximate theory with results from calculations from the Helmholtz equation for the fluid and accurate thin-walled cylinder theory. Criteria of the application of the simplified theory are formulated.

In the numerical solution of some boundary value problems by the finite element method (FEM), the unbounded domain must be truncated by an artificial absorbing boundary or layer to have a bounded computational domain. The perfectly matched layer (PML) approach is based on the truncation of the computational domain by a reflectionless artificial layer which absorbs outgoing waves regardless of their frequency and angle of incidence. In this paper, we present the near-field numerical performance analysis of our new PML approach, which we call as locally-conformal PML, using Monte Carlo simulations. The locally-conformal PML method is an easily implementable conformal PML implementation, to the problem of mesh truncation in the FEM. The most distinguished feature of the method is its simplicity and flexibility to design conformal PMLs over challenging geometries, especially those with curvature discontinuities, in a straightforward way without using artificial absorbers. The method is based on a special complex coordinate transformation which is ‘locally-defined’ for each point inside the PML region. The method can be implemented in an existing FEM software by just replacing the nodal coordinates inside the PML region by their complex counterparts obtained via complex coordinate transformation. We first introduce the analytical derivation of the locally-conformal PML method for the FEM solution of the two-dimensional scalar Helmholtz equation arising in the mathematical modeling of various steady-state (or, time-harmonic) wave phenomena. Then, we carry out its numerical performance analysis by means of some Monte Carlo simulations which consider both the problem of constructing the two-dimensional Green’s function, and some specific cases of electromagnetic scattering.

A preconditioning iterative algorithm is proposed for solving electromagnetic scattering from an open cavity embedded in an infinite ground plane. In this iterative algorithm, a physical model with a vertically layered medium is employed as a preconditioner of the model of general media. A fast algorithm developed in (SIAM J. Sci. Comput. 2005; 27:553–574) is applied for solving the model of layered media and classical Krylov subspace methods, restarted GMRES, COCG, and BiCGstab are employed for solving the preconditioned system. Our numerical experiments on cavity models with large numbers of mesh points and large wave numbers show that the algorithm is efficient and the number of iterations is independent of the number of mesh points and dependent upon the wave number. Copyright © 2008 John Wiley & Sons, Ltd.