A preconditioning technique has been developed for numerical solution of the Helmholtz equation as applied to the steady-state propagation of sound in a semi-infinite, two-dimensional (2-D) rigid duct. As explained below, the technique involves the use of two pseudo-time parameters in a finite-difference approximation of the equation that describes the propagation of sound. The use of these parameters makes the solution proceed much faster than in older transient- and steady-state-analysis-based numerical solution methods.
Figure 1 illustrates the 2-D duct geometry and the finite-difference mesh. The governing differential equation for the propagation of sound in nondimensionalized form is
f 2φ'tt = φ'xx + φ'yy
where f is a dimensionless frequency, φ' is a dimensionless transient potential or amplitude of the propagating signal, t is dimensionless time, x and y are the dimensionless coordinates indicated in the figure, and subscripts represent partial differentiation with respect to the noted variables. The spatial unit for nondimensionalization is the width (y dimension) of the duct; the temporal unit for nondimensionalization is the width of the duct divided by the speed of sound; and the unit for nondimensionalization of transient potential is the width of the duct multiplied by the speed of sound.
Suppose that either the transient potential comprises a single frequency or else the time dependence of the transient potential has been Fourier-transformed into the frequency domain. One can express the transient potential in the form
φ'(x,y,t) = ψ(x,y)e-i2πt
where y(x,y) contains the spatial dependence of the Fourier component or single-frequency signal. Using this expression, one can transform the governing equation into the classical Helmholtz equation
ψxx + ψyy + ω2ψ = 0
where ω= 2πf. In this formulation, the Fourier amplitude ψ(x,y) is independent of time, as in the case of steady excitation of the duct with sound of a single frequency.
In general, what one seeks is the steady-state solution ψ(x,y). This can be accomplished indirectly via a time-dependent formulation. The Helmholtz equation is preconditioned by expressing the transient potential in the form
φ'(x,y,t) = φ(x,y,t)e-i2πt
The time dependence in φ(x,y,t) can be used to represent what happens if the duct is initially quiet and then the source of sound is turned on at t = 0. In this formulation, the governing equation becomes
f 2φtt - 2ifωφt = φxx + φyy + ω2φ
This is a preconditioned Helmholtz equation. In the steady state, φ'(x,y,t) becomes ψ(x,y), causing the left side of this equation to vanish and thus causing reversion to the classical Helmholtz equation.
In previous research, it was found that in the special case (called the "parabolic" preconditioner) in which one neglects φ2ftt, the numerical solution for φ(x,y,t) converges to ψ(x,y) for long times; that is,
lim φ'(x,y,t) = ψ(x,y)
In effect, the finite-difference solution for the steady-state spatial dependence is found by iterating in time or pseudo-time.
One can generalize the preconditioned Helmholtz equation by incorporating pseudo-time parameters a and b as follows:
αf2φtt - β2ifωφt = φxx + φyy + ω2φ
The choice of α=0, β=1 yields the parabolic approximation. The more general case of α≠0 and β≠0 (the "mixed" preconditioner) is the basis of the present technique. In formulating the finite-difference approximation of the preconditioned Helmholtz equation, one can choose nonzero values of α and β, in conjunction with increments of t, x, and y, to accelerate convergence to a steady-state solution. The acceleration achievable by use of optimum values of α and β is considerable; for example, in one test case (see Figure 2), the number of iterations needed for convergence with optimum a and b was about one-tenth the number of iterations needed to achieve the same degree of convergence with a parabolic approximation.
This work was done by Kenneth J. Baumeister of Glenn Research Center and Kevin L. Kreider of the University of Akron. For further information, access the Technical Support Package (TSP) free on-line at www.nasatech.com/tsp under the Information Sciences category.
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