Causal and passive interpolation of structural acoustic matrices

Abstract

The present work investigates the causal and passive interpolation of structural acoustic matrices in the frequency domain. Interpolations significantly ease computational burdens involved in computing these matrices from models. For example, one may wish to compute a small admittance matrix, relating forces to velocities at points on a structure, from a large finite element model with over one million degrees of freedom. In many structural acoustic systems, the elements of such matrices vary smoothly with frequency and therefore interpolation can be accurate and effective. In the time domain, causality requires no effect before its cause. In the frequency domain, this takes the form of Hilbert transform relations that relate the real and imaginary parts of scalar transfer functions. Passivity requires that the net power dissipated by the structure over a cycle of vibration be positive semidefinite. The present work develops series expansions in which these conditions may be implicitly satisfied by the proper choice of basis functions. Coefficients of in the series expansion are found by matching the series to the known values at the interpolation points. Examples demonstrate situations in which the satisfaction of these physical conditions yields more accurate interpolations.