1.4890195

A three-dimensional, longitudinally-invariant finiteelement model for acoustic propagation in shallow

water waveguidesMarcia J. Isakson, Benjamin Goldsberry, and Nicholas P. Chotiros

Applied Research Laboratories, The University of Texas at Austin, Austin,Texas 78713-8029

[email protected], [email protected], [email protected]

Abstract: A three-dimensional, longitudinally-invariant finite element(FE) model for shallow water acoustic propagation is constructedthrough a cosine transform of a series of two-dimensional FE models atdifferent values of the out-of-plane wavenumber. An innovative wave-number sampling method is developed that efficiently captures theessential components of the integral as the out-of-plane wave numberapproaches the water wavenumber. The method is validated by compar-ison with benchmark solutions of two shallow water waveguide environ-ments: a flat range independent case and a benchmark wedge.VC 2014 Acoustical Society of AmericaPACS numbers: 43.30.Dr [AL]Date Received: December 20, 2013 Date Accepted: June 24, 2014

1. Introduction

High fidelity models of acoustic propagation in shallow water waveguides are necessary forsonar performance predictions and acoustic communication. However, the waveguides areoften very complex including interface roughness, sediment patchiness, sound speed pro-files, and complex bathymetry. The finite element method (FEM) provides an excellenttool to model these environments because it is fully customizable and provides a full wavesolution. One drawback of the method is its intense computational load. There are cur-rently no fully three-dimensional finite element shallow water waveguide acoustic propaga-tion models although two-dimensional models do exist.1 Because two-dimensional modelscannot be compared with experimental data, a longitudinally-invariant three-dimensionalmodel is proposed to bridge the gap. In this scheme, a cosine transform is performed onthe three-dimensional Helmholtz equation, producing a sum of two-dimensional compo-nents that can be solved using current methods. Although this produces a fully three-dimensional model, one constraint is that the geometry must be invariant in one direction.Many environments including continental slopes, ridges, and canyons can be approxi-mated as longitudinally invariant. Within this constraint, the method can still include rangedependent effects such as sediment and sound speed variation.

2. The longitudinally-invariant finite element model

In this FEM, the variational form of the Helmholtz equation is solved over small sub-domains or elements using polynomial basis sets. The solution is obtained by consider-ing the boundary conditions between the elements that lead to a linear system of equa-tions. The collection of interconnected elements, know as a mesh, represents the totalfield. A rigorous mathematical description of the finite element method is given inRef. 2, while a description of its application to underwater acoustics is given in Ref. 3.The FEM makes no approximations to the Helmholtz equation, and the system geom-etry can be expressed accurately to the size of the element. This makes the modelextremely versatile. For the calculations in this paper, a commercially available finiteelement program, COMSOL, is used for meshing and solving.4

Full three-dimensional FEMs for ocean waveguides are still beyond the com-putation capabilities available. However, the physics of three-dimensional propagation

EL206 J. Acoust. Soc. Am. 136 (3), September 2014 VC 2014 Acoustical Society of America

Isakson et al.: JASA Express Letters [http://dx.doi.org/10.1121/1.4890195] Published Online 5 August 2014

Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 103.23.224.131 On: Wed, 03 Sep 2014 03:04:54

can be captured if the geometry modeled is invariant in one direction. In this scheme,a cosine transform is performed on the three-dimensional Helmholtz equations alongone spatial axis. The resulting two-dimensional pressure components can be calculatedusing two-dimensional FEM.

Following Ref. 5, the derivation begins with the three-dimensional Helmholtzequation, appropriate for a time harmonic acoustic propagation problem,

r2P k2P sxdx xody yodz zo: (1)Here P is the pressure field, k is the acoustic wavenumber, kx/c(x, y, z) where x isthe radial acoustic frequency and c is the sound speed that can vary with position. Thesource frequency spectrum is s(x). In this study, only single frequencies are considered.Last, (xo, yo, zo) is the point source location.

If the sound speed and other environmental factors are only functions ofrange, x, and depth, z, the three-dimensional Helmholtz equation can be reduced to

r22D P k2 k2y

P 12s x d x xo d z zo ; (2)

where

Px; x; y; z 10

Px; x; ky; z cos kyydky: (3)

The integral can be written discretely as

Px; x; y; z Xkymaxky0

Px; x; ky; z cos kyydky: (4)

Here ky is the out-of-plane wavenumber and Px; x; ky; z is the transformedfield. Using Eq. (2), the two-dimensional (2D) pressure fields are computed for a seriesof out-of-plane wavenumbers using 2D FE and summed to determine the 3D field.

Two parameters in this derivation must be addressed: The limits of the sum inEq. (4) and the discretization of the sum. To determine these parameters, it is helpful toconsider the canonical problem of a point source in an unbounded medium at 500Hz.Shown in Fig. 1(a) is the integrand of Eq. (3) as a function of the normalized out-of-planewavenumber and range from the source. Note that the value of the integrand goes tozero smoothly past ky k where the effective wavenumber in Eq. (2) is complex. Thisevanescent part of the integral describes the curving wavefronts near the source and isonly significant at short ranges. Also, note that the integrand is smooth as ky ! 0. Thissuggests that the integrand should be sampled more finely near ky k. Two differentsampling schemes are shown in Fig. 1(b). The straight line has constant discretization inky. The curved line uses a slightly offset gamma cumulative distribution function (CDF)coupled with a linear function just after ky k. This results in a much finer sampling nearky k for the same number of samples. For ky> k the samples are discretized much morecoarsely because the integrand is smooth. The sampling scheme is shown for the integralat range 10m from the source in Fig. 1(c). Note how the variable discretization samplesthe integrand more finely near ky k where the function is more variable. Both the vari-able and the constant discretization schemes used 300 points.

The effect of the different discretization schemes is shown in Fig. 1(d) wherethe results of the two schemes are compared with the analytic solution. The constantscheme did not resolve the integrand near ky k and results in an unstable solution.The variable discretization scheme, on the other hand, results in a near perfect solutionthat deviates less than the width of the analytic solution line. This example illustrateshow a variable sampling scheme can provide a stable solution with fewer samples.

The structure of the integrand for an ocean waveguide is much more complex.However, it retains many of the same qualities as the free space solution. It approaches


J. Acoust. Soc. Am. 136 (3), September 2014 Isakson et al.: Finite element propagation model EL207


zero smoothly for values of ky> k and only has significant values for the evanescentwavenumbers at short ranges. It is much more complex near ky k and becomessmooth as ky ! 0. Therefore a similar scheme in which the discretization is determinedby the gamma CDF can be used. For propagation in a range dependent waveguidedescribed in the following text, the variable discretization method resulted in a 30-folddecrease in the number of evaluations of the 2D model relative to the equivalent con-stant method. To put this in perspective, running the fully parallelized model on theTexas Advanced Computing Center LoneStar Cluster took 3 days with the variablediscretization method compared with a projected 90 days using the constant method.6

At first glance, it may seem that the integral could be evaluated with quadra-ture schemes. However, the integrand suffers from many of the same problems dis-cussed by Jensen, Kuperman, Porter, and Schmidt in Ref. 7, Sec. 4.5.2, with respect tothe evaluation of the inverse Hankel transform for axially symmetric problems. In par-ticular, the oscillatory nature of the integral is highly dependent on range and environ-ment requiring different quadrature points for each instance. The authors of Ref. 7rejected quadrature scheme in favor of the simple equidistant discretization. However,as discussed previously, this is impractical for this problem. Through trial and error,the discretization based on the cumulative CDF has been found to be the most robustand efficient for this problem.

Fig. 1. (Color online) (a) The integrand for the cosine transform of a point source in free space. (b) Two differ-ent methods of sampling the out-of-plane wavenumber. (c) An example of how the integral is sampled for arange of 10m. (d) A comparison of the two different sampling methods and the analytic solution.


EL208 J. Acoust. Soc. Am. 136 (3), September 2014 Isakson et al.: Finite element propagation model


Table 1. Waveguide parameters for validation.

Range independent Range dependent

Waveguide depth (m) 200 200-0Source depth (m) 150 100Frequency (Hz) 100 25Receiver depth (m) 150 100Water sound speed (m/s) 1500 1500Water density (kg/m3) 1024 1000Water attenuation (dB/k) 0 0Sediment sound speed (m/s) 1700 1700Sediment density (kg/m3) 1500 1500Sediment attenuation (dB/k) 0.5 0.5Wedge angle () 2.86

Fig. 2. (Color online) A comparison of the finite element approach and the wavenumber integration approachfor transmission loss in a range independent shallow water waveguide.

Fig. 3. (Color online) Transmission loss calculated using finite elements for a wedge environment. The blackline is the ocean bottom. The gray line denotes the cut shown in Fig. 4.




3. Validation

To validate the model, range independent and dependent test cases were selected tocompare with existing models. The model parameters are given in Table I. The rangeindependent case consisted of a 200m deep waveguide with a sandy bottom. Thisproblem required 2500 evaluations of the 2D field components. The solution was com-pared with a wavenumber integration solution provided by OASES (Ocean Acoustic andSeismic Exploration Synthesis).8 Results are shown in Fig. 2. The models agree withexcellent precision.

The range dependent case is the ASA benchmark wedge, case III, with a lossypenetrable bottom.9 This case required 6100 evaluations of the 2D field components.The pressure field along the source axis up the wedge is shown in Fig. 3. This can bedirectly compared with Fig. 8 in Ref. 9. It should be noted that unlike the calculationsprovided by Jensen and Ferla in Ref. 9, the FE model includes all orders of couplingand scattering making it an excellent benchmark solution. To display the out-of-planefeatures of the model, a cut was taken along the gray line in Fig. 3 and shown inFig. 4. Note the modal cut-offs as the field approaches the wedge apex. This can becompared qualitatively with Fig. 7 of Ref. 10 although in Ref. 10, the source was con-siderably closer to the wedge apex. Many of the same physical features such as shadowzones and mode cut-offs are evident.

4. Conclusion

A 2D finite element propagation model has been extended to three dimensions througha cosine transform that is appropriate for scenarios in which the geometry along onespatial coordinate is invariant. The model requires that a 2D field be calculated foreach out-of-plane wavenumber. An inverse cosine transform yields the final solution. Itwas determined that a constant sampling interval scheme to compute the transformwas computationally prohibitive. A non-uniform spacing based on an offset gammacumulative distribution function was found to be robust and efficient.

The model was validated by comparison with range independent and range de-pendent benchmark problems. The range independent case was a shallow water wave-guide of water over sand, appropriate for a continental shelf environment. The rangedependent case was a wedge environment appropriate for propagation near shore. Inboth instances, the finite element model agreed well with available solutions.

Because the FEM solves the Helmholtz equation exactly to the degree of thediscretization and is fully customizable, it can be applied to complex range dependent

Fig. 4. (Color online) The out-of-plane transmission loss for the wedge environment along the cut shown inFig. 3.


EL210 J. Acoust. Soc. Am. 136 (3), September 2014 Isakson et al.: Finite element propagation model


environments such as continental shelf regions where interface roughness, range de-pendent sediment parameters, and water column sound speed profiles are important.Unlike many other models, it provides the entire pressure field, both forward andbackward propagation. Therefore it can be used for both propagation and reverbera-tion modeling.

Acknowledgments

This work was supported by ONR, Ocean Acoustics under the direction of RobertHeadrick.

References and links1M. J. Isakson and N. P. Chotiros, Finite element modeling of reverberation and transmission loss inshallow water waveguides with rough boundaries, J. Acoust. Soc. Am. 129(3), 12731279 (2011).2J. N. Reddy, An Introduction to the Finite Element Model (McGraw-Hill, New York, 2006).3F. B. Jensen, W. A. Kuperman, M. B. Porter, and H. Schmidt, Computational Ocean Acoustics, 2nd ed.(Springer, New York, 2011), Chap. 7.4Information available on Comsol Multi-Physics at http://www.comsol.com/ (Last viewed June 4, 2012).5B. Zhou and S. A. Greenhalgh, Composite boundary-valued solution of the 2.5-d greens function forarbitrary acoustic media, Geophysics 63(5), 18131823 (1998).6Information on Texas Advanced Computing Center Lonestar Cluster available at https://www.tacc.utexas.edu (Last viewed December 17, 2013).7F. B. Jensen, W. A. Kuperman, M. B. Porter, and H. Schmidt, Computational Ocean Acoustics, 2nd ed.(Springer, New York, 2011), Chap. 4.8H. Schmidt, OASES Version 2.1 User Guide and Reference Manual (Department of Ocean Engineering,Massachusetts Institute of Technology, Cambridge, MA, 1997).9F. B. Jensen and C. M. Ferla, Numerical solutions of range-dependent benchmark problems in oceanacoustics, J. Acoust. Soc. Am. 87(4), 14991510 (1990).

10G. B. Deane and M. J. Buckingham, An analysis of the three-dimensional sound field in a penetrablewedge with a stratified fluid or elastic basement, J. Acoust. Soc. Am. 93(3), 13191328 (1993).




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