Analytical expressions are derived for fast calculations of time-harmonic and transient near field pressures generated by triangular pistons. These fast expressions remove singularities from the impulse response, thereby reducing the computation time and the peak numerical error with a general formula that describes the nearfield pressure produced by any triangular piston geometry. The time-domain expressions are further accelerated by a time-space decomposition approach that analytically separates the spatial and temporal components of the numerically computed transient pressure. Analytical 2D integral expressions are derived for fast calculations of time-harmonic and transient nearfield pressures generated by apodized rectangular pistons. The 2D expressions eliminate the numerical singularities that are otherwise present in numerical models of pressure fields generated by apodized rectangular pistons. A simplified time space decomposition method is also described, and this method further reduces the computation time for transient pressure fields. The results, compared with the Rayleigh-Sommerfeld integral, the Field II program and the impulse response method, indicate that the FNM achieves smallest errors for the same computation time among those methods. A 1D FNM for calculating the pressure generated by a polynomial apodized rectangular piston is also obtained. The fast method is based on the instantaneous impulse response. A trigonometric transform of the integrand is performed and the order of integration is exchanged to obtain the ID integral for the apodized FNM for both apodization functions. The time and error comparisons are performed among the 1D polynomial apodized FNM, the 2D apodized FNM and the Rayleigh-Sommerfeld integral. The results show that the 1D polynomial apodized FNM has the fastest convergence. Analytical expressions are derived for fast calculations of potential integrals. These potential integrals inculde uniformly excited volume potential integrals, polynomial apodized surface integrals and polynomial apodized volume potential integrals. The derivation starts with the fast near-field method (FNM), which originates from ultrasound pressure calculations generated by polygonal pistons. For potential integrals evaluated over a volume source, the volume source is first subdivided into subdomains about the observation points. The total potential is the summation of the potential over each submain which can be reduced to 1D integrals. Those calculation methods remove the singularities from the Rayleigh-Sommerfeld integral by subtracting sigularities in the integrand and thus can achieve rapid convergence. Simulations results are compared with the Rayleigh-Sommerfeld integral and the singularity cancellation method evaluated on a 3D grid. The results indicate that the 1D FNM expressions reduces the computation time or the number of sample point needed significantly than the Rayleigh-Sommerfeld integral and the singularity cancellation method for a given number signifcant digits.
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