Boundary element modeling of the external human auditory system

Boundary element modeling of the external human auditory system Timothy Walsh Sandia National Laboratories, P.O. Box 5800, MS 0835, Albuquerque, New Mexico 87185

Leszek Demkowicza) Texas Institute for Computational and Applied Mathematics, The University of Texas at Austin, ACES 6.332, Austin, Texas 78712

Richard Charles Sulzer Carbomedics, 1300 E. Anderson Lane, Austin, Texas 78752

共Received 20 February 2002; revised 22 November 2003; accepted 1 December 2003兲 In this paper the response of the external auditory system to acoustical waves of varying frequencies and angles of incidence is computed using a boundary element method. The resonance patterns of both the ear canal and the concha are computed and compared with experimental data. Specialized numerical algorithms are developed that allow for the efficient computation of the eardrum pressures. In contrast to previous results in the literature that consider only the ‘‘blocked meatus’’ configuration, in this work the simulations are conducted on a boundary element mesh that includes both the external head/ear geometry, as well as the ear canal and eardrum. The simulation technology developed in this work is intended to demonstrate the utility of numerical analysis in studying physical phenomena related to the external auditory system. Later work could extend this towards simulating in situ hearing aids, and possibly using the simulations as a tool for optimizing hearing aid technologies for particular individuals. © 2004 Acoustical Society of America. 关DOI: 10.1121/1.1643360兴 PACS numbers: 43.20.Fn, 43.64.Ha 关LLT兴

I. INTRODUCTION

The external human auditory system consists of the head, the pinna, the ear canal, and the eardrum, which collect the sound waves from the exterior domain, filter the spectral properties, and provide an input signal to the middle and inner ears through oscillations of the eardrum. In this work numerical simulations, augmented with the capabilities of parallel computing, are used as tools to study the overall functionality of the outer ear. By comparing the results to experimental data from the literature, it is demonstrated that the simulations are capable of capturing the essential physics of the external auditory system. This suggests that such simulations could be used to assist in the development of relevant technologies, such as hearing aids, hearing protection devices, and virtual acoustic simulators. In the context of the external auditory system, an important parameter is the head-related transfer function 共HRTF兲, which measures the pressure at the eardrum as a function of the frequency and angle of incidence of the incident acoustical wave. It gives a measure of the filtering characteristics of the outer ear, as well as the directional dependence, which is critical for localization ability. The directionality information provided by an HRTF is useful in localization research 关Katz 共1998兲兴, and in virtual acoustic simulators 关Moller 共1996兲, Takeuchi 共1998兲, Kahana 共1999兲兴. The HRTF also provides information on the frequency dependence of the sound amplification, as well as the a兲

Author to whom correspondence should be addressed.

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resonance frequencies of the ear canal, and this can be useful for hearing aid designs 关Sandlin 共1995兲兴. The ability to compute this quantity could alleviate the need for experimental generation, and provide a tool to be used by all of the technologies above for phenomenological studies. Although the need for careful, accurate laboratory measurements will never be replaced, numerical simulations provide a useful alternative in cases when the experimental procedures are time-consuming or expensive, or when parametric studies are necessary which would require many repeated measurements. The measurement of an HRTF is an illustrative example. The eardrum pressures must be measured in an anechoic chamber at varying frequencies and angles of incidence, for each individual of interest. The determination of ear-canal resonances is another example. In this paper, numerical computations of both quantities are presented. There have been some results reported in the literature regarding numerical simulations of the external auditory system. The literature is very limited, and this is part of the motivation for the current paper. Ciskowski 共1991兲 studied both the plug and the earmufftype hearing protection devices 共HPD兲 in an attempt to characterize the best possible design for attenuating strong pulses in the ear canal. He modeled the plugged ear canal as a structure/acoustic cavity that coupled an elastodynamic finite-element model for the plug to a pressure boundary element model for the cavity. The resonances of the acoustic cavity were reproduced and compared with experimental measurements.

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© 2004 Acoustical Society of America

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Katz 共1998兲 computed the HRTF of real subjects using a boundary element method applied to the Helmholtz integral equation. In this study the finite impedance of the skin and hair was experimentally measured and incorporated into the numerical models. The frequency range of the simulations was 1–5 kHz. Only linear triangulations were used, and a decimation scheme was employed to provide appropriate mesh densities for the desired range of frequencies. Kahana 共1999兲 used a boundary element method to compute the HRTF up to a maximum frequency of 15 kHz. The principle of reciprocity was used to deal with sources at variable locations, simply by locating the source at the blocked meatus position, and computing the response at the original locations of the sources. A drawback to this approach was that the source could not be located directly on the surface, and had to be lifted off of the surface a small amount. This introduced an additional error in the approximation. An alternative approach would be to construct an adaptive mesh that is well-resolved for all incident waves, and then solve the linear system with multiple right-hand sides. This is certainly a more general approach, and would be useful for problems in which the principle of reciprocity is not applicable. Such an approach is taken in this paper. The two previous references 关Katz 共1998兲, Kahana 共1999兲兴 use the ‘‘blocked meatus’’ configuration in the simulations, meaning that the ear canal is not included in the mesh. Conversely, the first reference 关Ciskowski 共1991兲兴 involves modeling the ear canal, but the external head/ear geometry is not included. This eliminates the directional dependence imparted by the head and external ear. Also, in this reference the canal is modeled using simplified geometries, instead of actual canal data. In the current study a mesh of an actual ear canal is extracted from the literature 关Stinton 共1989兲兴, and connected to a separate mesh of the head and pinna. This allows for simulations on the entire external auditory system, and with more realistic geometries. For completeness, simulations are also conducted on the original blocked meatus configuration, so that they can be compared to those with ear canal. Although in this paper the internal parts of the hearing system are not included in the model, it is recognized that a long-term goal of this research is to model the entire auditory system, including the middle and inner ears. A full auditory simulation would involve coupling the external boundary element model with finite-element models for both the middle and inner ear. Some attempts have been made to model separately the inner parts of the auditory system. Ladak 共1996兲 used the finite-element method to study the middle ear of the cat. Wada 共1992兲 did a similar study, but on the human middle ear. Several new investigations have been undertaken to simulate the main structure of the inner ear, the cochlea: Parthasarathi 共2000兲, Kolston 共1996兲, Duncan 共1997兲. The latter reference models to the level of the hair bundles on the cochlea, using a micromechanical model. Models of this type are the first step towards understanding hearing system damage on a particular individual, since in many cases the cause of hearing impairment can be traced to the cochlea. 1034

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FIG. 1. The model scattering problem in R3 .

II. THE BOUNDARY ELEMENT TECHNOLOGIES

In this section we give brief remarks regarding the various simulation technologies used in this work. These remarks are not intended to be complete descriptions, but instead only introductions that give the reader the main ideas of the approaches taken. Complete details on the numerical formulation and implementation can be found in Walsh 共2003兲. A. BEM for scattering problems in acoustics

The boundary element method 共BEM兲 is a very attractive technique for modeling acoustic scattering/radiation problems, and in particular the external auditory system. By reducing the exterior boundary value problem to a boundary integral equation, it eliminates the need for a volume grid, and the problems associated with discretizing the unbounded domain. The Galerkin approach is taken in this work, since it has the following advantages over collocation. 共i兲

共ii兲 共iii兲

The collocation approach can simply be viewed as an underintegrated Galerkin method, and so pursing the Galerkin method is the most general approach one can take. Unlike collocation, the Galerkin method comes with a complete mathematical convergence theory. The Galerkin method allows for an ‘‘integration by parts’’ of the hypersingular part of the Burton–Miller formulation, thus eliminating the need for Hadamard finite part integration.

B. Burton–Miller formulation for BEM

For exterior acoustic scattering problems, the two widely used classical formulations are the Helmholtz and hypersingular integral formulations. However, neither one is fully equivalent to the original Helmholtz differential equation with Sommerfeld radiation condition, over the entire range of frequencies. Both exhibit the so-called fictitious frequencies, where the integral formulations contain spurious modes. Burton 共1971兲 presents a new formulation that consists of a complex combination of the Helmholtz and hypersingular integral equations, and proved that this is equivalent to the original problem. For this reason it is chosen as the formulation for the work in this paper. The problem of interest is shown schematically in Fig. 1. Given an incident wave p inc with wave number k, and a bounded domain ⍀ with a boundary ⌫ 共in our case ⍀ would Walsh et al.: Boundary elements for external auditory system

pose of error estimation is trivially parallelizable and thus is simply distributed across the processors. All computations are performed on the CRAY T3E distributed memory parallel architecture using between 4 and 64 processors, depending on the frequency.

E. Goal-oriented error estimation and mesh adaptivity

FIG. 2. The parallelization of the boundary element method.

be the human head, with or without the ear canal兲 the goal is to find a scattered pressure p s satisfying the Helmholtz differential equation ˆ ⫽R3 ⫺⍀, ⌬p s ⫹k 2 p s ⫽0 in ⍀

共1兲

Sommerfeld radiation condition

冏

冏

冉 冊

⳵p 1 ⫺ikp s ⫽O 2 ⳵R R s

for R→⬁,

III. NUMERICAL COMPUTATION RESULTS

共2兲

and rigid boundary condition on ⌫

⳵ 共 p s ⫹p inc兲 ⫽0 on ⌫. ⳵n

It is always desirable to have an estimate of the discretization error involved in numerical computations. Without such an estimate, it would be difficult to separate errors due to other sources, such as geometrical representation. Following the results derived in Walsh 共2003兲, a local, or goaloriented error estimation and mesh adaptivity scheme is used in this work, rather than a classical scheme based on a global energy norm. Details on the formulation and implementation can be found in this reference, as well as an exposition on various error estimation and adaptivity techniques. These techniques allow the pressure on the eardrum to be computed with high accuracy.

共3兲

The rigid condition can be replaced by a finite impedance condition, which would produce more accurate results, especially in the vicinity of the eardrum. However, the results presented in this paper only model rigid scattering. The exterior boundary-value problem is next replaced with the Burton–Miller integral formulation, a weak formulation is constructed, and a boundary element discretization is applied. More details are given in Walsh 共2003兲.

This section presents the results of the acoustical simulations. Using the boundary element technology described in the previous section, the simulation results are generated and compared to experimental data from the literature. All of the numerical results in this section involve rigid scatterers, with infinite impedance. This is, of course, an additional simulation error, but its consequences are somewhat obvious in the results and are pointed out when appropriate. Due to its simple geometry and well-known exact solution, results are first presented on a sphere. A. Sphere

The parallelization of the hp boundary element method involves three separate tasks: formation of the linear system, dense solve, and error estimation. Figure 2 shows a schematic of the adaptive process in the parallel environment. Both the formation and the factorization of the 共dense兲 linear system utilize the PLAPACK software of van de Geijn 共1997兲. The computation of the elementwise residuals for the pur-

The simplest model one can consider for the head is a sphere with a small microphone on either side. Since the exact solution is known for this case, the transfer function can be computed exactly. This model, though very simplified compared with the head, can be used to gain intuition for the types of responses that can be expected in more complicated simulations. For the case of normal incidence, Fig. 3 shows the exact transfer function 共obtained from the exact solution for scattering of a plane wave on a rigid sphere兲. The abscissa is ka, the wave number k multiplied by the sphere radius a, and the ordinate is dB⫽20 log(p/p0), where p 0 is the magnitude of the pressure of the incident wave, and p is the pressure at the detection point. The asymptotic limits on the transfer function in Fig. 3 have very simple physical explanations, which are useful when interpreting the results for the head. We consider the case where the sphere has a fixed radius, and the wavelength of the incident field is allowed to vary. When the wavelength becomes large relative to the radius of the sphere, the interference that the sphere has on the incident field becomes less and less significant. In the limit, the amplitude of the scattered pressure is negligible compared to the that of the incident field. At the point of normal incidence

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C. hp boundary element discretizations

The present work is a continuation of earlier work on h p-adaptive boundary elements for the solution of exterior and structure–fluid interaction problems in underwater acoustics; see Demkowicz 共1991, 1992兲, Chang 共1999兲, and Geng 共1996兲. The present boundary element code is implemented within a new FORTRAN 90 implementation of a twodimensional hp-adaptive package described in Demkowicz 共1998兲, allowing for full hp adaptivity. D. Parallel computations

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FIG. 3. Exact solution for scattering on a rigid sphere at the point of normal incidence.

冉 冊

p gain共dB)⫽20 log ⫽20 log共 1 兲 ⫽0, p0

B. Computational mesh of the ear canal

共4兲

where p⫽ p s ⫹p 0 is the total pressure field, and p s and p 0 are the scattered and incident pressure fields, respectively. Thus, when ka→0 the limit in Fig. 3 is zero. On the other hand, when the wavelength becomes small relative to the radius of the sphere, the local surface curvature at the point of normal incidence effectively disappears. In the limit, the solution at the point of normal incidence should be the same as for scattering of a plane wave on a planar, rigid surface. The well-known solution to this problem requires that the scattered pressure be equal to the incident pressure. Thus gain共dB)⫽20

冉 冊

p ⫽20 log共 2 兲 ⬇6, p0

共5兲

which is the upper limit in Fig. 3. A much more extensive discussion of the scattering of sound from a sphere is given by Duda 共1998兲. The results presented for the sphere provide a starting point for the understanding of the acoustics of the head/ear. In the case when the wavelength is large relative to the size of the pinna and canal, these structures do not contribute significantly to the acoustical gain, and the transfer function resembles that of a head without pinna, which one would expect to have a very similar transfer function as the sphere. This is especially true for normal incidence at very low frequencies, where the pinna has little effect on the pressure at the blocked meatus position. In the case when the wavelength is on the order of or smaller than the pinna/ear canal, however, the situation is markedly different. Here, the pinna, concha, and ear canal dramatically change the attainable gain as well as the directional dependence of the pressure field. This is a result of the structure of the pinna and ear canal, which are cavities that resonate within certain frequency ranges, and in the process amplify the incident waves more than was possible with just a convex surface. More detail on the frequency response of the head/ear is given in the next section. 1036

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FIG. 4. Cross-sectional slices of the ear canal, before connection.

Before presenting results of the acoustical simulations, a word is needed on the procedure for constructing the mesh of the ear canal and joining it to the head mesh. The starting point is a mesh of the human head with blocked meatus configuration from the Center for Computational Visualization at the University of Texas at Austin. This mesh is constructed from a CT scan. Second, a mesh of an actual ear canal is reconstructed from the detailed experimental measurements given in the paper by Stinton 共1989兲. This consists of joining several cross-sectional slices, as shown in Fig. 4. Next, this mesh is scaled, rotated, and joined to the original head mesh by cutting a small opening on the ear, attaching the canal, and redefining the local connectivities. Figure 5 shows images of the ear mesh with the ear canal included. C. Comparisons of numerical results with experimental data

In order to isolate the influence of the ear canal and other parts of the external auditory system, the simulations are conducted on meshes of the human head with and without 共blocked meatus兲 the ear canal, so that the two could later be distinguished and the overall function of the canal could be isolated. An initial word is needed on the level of agreement that can be expected. Of course, the ideal procedure would be to perform the necessary acoustical measurements in a laboratory on a fixed geometry, scan that same geometry, and construct a corresponding boundary element mesh on which the simulations would be conducted. In this study, however, the numerical simulations are made using a fixed representation of the human head. Thus, there is some difference between this geometry and those drawn from the literature for comparison, and this introduces some uncertainty in the comparisons. Even if the geometries were the same, exact agreement between the numerical and the experimentatal data sets could not be expected, and thus even more so in this case. However, reasonably good agreement has been obtained with regard to the main features of the response, and this is the intended goal of this work. Walsh et al.: Boundary elements for external auditory system

FIG. 7. Comparison of experimental data from Shaw 共1974兲 and numerical simulations showing the variation of sound-pressure level at the eardrum for various frequencies, angles of incidence.

FIG. 5. Views of the ear canal connected to the ear mesh.

Also, a word is needed on terminology. Figure 6 shows the horizontal plane around the head divided into three sectors: frontal 共⫺60° to 60°兲, lateral 共60° to 135°兲, and posterior 共135° to ⫺150°兲 关terminology taken from Sandlin 共1995兲兴. Note that the angles are measured from the midfrontal position. Also note that the view in this figure is from the top of the head, and that only the horizontal plane is considered.

FIG. 6. Terminology used in describing HRTF data for the human head. J. Acoust. Soc. Am., Vol. 115, No. 3, March 2004

Finally, the term ‘‘resonance’’ is frequently used with different intended meanings, and it would be wise to clarify this point. Since the simulations in this paper are formulated with respect to the exterior problem, the domain is unbounded from the standpoint of the boundary element simulations 共i.e., the domain includes both the ear canal as well as the infinite space surrounding the head兲. Standing waves can only exist in a bounded domain, and thus the term resonance is a bit unclear in this case. However, the response of the ear canal is highly frequency dependent, and in this paper the peak amplitude responses are denoted as resonances, simply to be consistent with most of the literature on the subject. 1. Directional dependence

The first set of experimental data that is considered is that of Shaw 共1974兲. This work presents the response of a generic head/ear by combining data from 12 separate studies. These data and the corresponding computational results are presented in Figs. 7 and 8, with the angle of incidence as the abscissa, and normalized so that the gain at frontal incidence is zero. That is, the ordinate is computed as dB ⫽20 log(p/p0), where the denominator p 0 ⫽ p inc⫹p s0 is the total pressure generated at the eardrum by a plane wave impinging from the front 共0° according to the definitions in Fig. 6兲. Also, the numerator is p⫽ p inc⫹ p s , which implies that at 0° the ratio is unity and thus the gain is zero. This is the same format that was used in the original paper by Shaw 共1974兲, and has the advantage that the gain at 0° is zero, which facilitates interpretation of the results. The experimental data are available in graphical format. In order to plot these data simultaneously with the numerical simulation results, it is necessary to extract this data from the graphs presented in Shaw 共1974兲. This is accomplished using graphical extraction software. The data is sampled at approximately every 2°, giving a total of 180 points for the entire range of interest 共⫺180° to 180°兲. Walsh et al.: Boundary elements for external auditory system

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FIG. 8. Comparison of experimental data from Shaw 共1974兲 and numerical simulations showing the variation of sound-pressure level at the eardrum for various frequencies, angles of incidence.

The numerical simulation results presented in Figs. 7 and 8 are generated using the boundary element procedure described earlier. For each curve, 24 right-hand sides are used, at increments of 15°, so that the entire 360° is spanned. These data are normalized in the same way as the experimental data 共described above兲. Reasonably good agreement is seen between the experimental and numerically generated data sets. At 300 and 500 Hz, two minimums and one maximum at negative angles of incidence are apparent, and one maximum at positive angles. At 1 and 1.6 KHz, there is a similar response, only that the peaks are more pronounced. Remarkably, the pronounced peak and valleys at 500 Hz and 1 KHz for negative angles of incidence are reproduced almost identically by the numerics. A point of interest in the comparisons shown in Figs. 7 and 8 is that for 2500, 3200, and 4000 Hz, and at negative angles of incidence, the numerical results show small oscillations that are not present in the data of Shaw. This is likely due to the fact that Shaw’s data were obtained by averaging the results from several studies. Any slight shifting of these peaks from one subject to another could cause an artificial cancellation effect. For example, the data of Mehrgardt 共1977兲 also show oscillations at 2500, 3200, and 4000 Hz, and at negative angles of incidence, which makes artificial cancellation in Shaw’s data seem more likely. Also at these higher frequencies, and at positive angles, the maximum occurs earlier than at the lower frequencies, and is followed by a sharp downturn between 90° and 120°. The numerics reproduce this feature remarkably well. Views of the pressure field on the blocked meatus configuration at 30° increments and the frequency of 500 Hz are shown in Fig. 9. In general, the pressure is higher on the receiving side of the wave than in the shadow zone, as expected. However, a small pressure peak around the ear at ⫺90° is seen, and this is identified as the ‘‘bright spot,’’ which occurs when the incident wave is directed along the 1038

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contralateral direction. This is also evident in the curves in Fig. 7 at 500 Hz, 1 kHz, and 1.6 kHz at ⫺90°, where a local maximum in pressure occurs, in between two local minima. The bright spot is contrary to intuition which would predict the shadow zone to receive the lowest pressure in a scattering situation. It may be partially explained in the case of a sphere, however, by considering the constructive interference between the waves that travel around the sphere and meet on the other side. The bright spot effectively results in a peak in pressure on the opposite side. In the case of a sphere, the bright spot is well-documented by Duda 共1998兲. In the case of the human head, the paths around either side are not identical, but presumably the same phenomenon is responsible for the peak at ⫺90°. The same numerical experiments that are conducted on the head mesh with ear canal are also conducted on the mesh with blocked meatus. In the latter case, the results are normalized as dB⫽20(p/ p 0 ), where p 0 is the pressure generated at the entrance to the 共blocked兲 ear canal, instead of at the eardrum, again by a plane wave impinging from the front. This is the same normalization procedure that was used for Figs. 7 and 8, only that the reference is taken at the blocked meatus position rather than at the eardrum. Thus, this allows for a relative comparison of the angular dependence with and without the ear canal. Using this consistent normalization, Figs. 10 and 11 show a direct comparison of the computational results with and without the ear canal. The comparison reveals that the directional dependence is indistinguishable. Thus, as eluded to earlier, the presence of the canal does not appear to influence the directional dependence of the pressures, and this is consistent with the experimental data of Hammershoi 共1996兲. When normalized with the same denominator, an absolute comparison of the cases with and without the canal is obtained. The results are shown in Figs. 12 and 13, where the ordinate is dB⫽20(p/ p inc). That is, in this case the denominator is the same with and without the ear canal. The comparison of the cases with and without the ear canal on an absolute basis 共Figs. 12 and 13兲 reveals that the eardrum pressures are only slightly higher than those at the blocked meatus for low frequencies, but near the resonance frequency of the canal 共2.5 kHz兲, the gain is higher at the eardrum. This difference between the eardrum pressure and blocked meatus pressure near the resonance of the canal is also documented in the experimental data of Shaw 共1968兲. When compared on an absolute basis 共Figs. 12 and 13兲, the eardrum pressure at 2.5 kHz is much higher than the blocked meatus pressure at this same frequency, even though at lower frequencies the difference between the two configurations is less dramatic. As mentioned in the previous paragraph, this can be attributed to the fact that the first resonance mode of the canal is around 2.5 kHz, whereas the first mode of the concha 共which would be the mode of interest in the blocked meatus case兲 peaks around 5 kHz, as shown in Shaw 共1968兲. At resonance the canal amplifies the soundpressure level significantly, whereas at 2.5 kHz the concha has not achieved full resonance. This accounts for the difference seen at 2.5 kHz in Fig. 13. The fact that the ear canal does not influence the direcWalsh et al.: Boundary elements for external auditory system

FIG. 9. Magnitude of surface pressure resulting from plane waves at various inclinations at 500 Hz. The legend in the first figure is the same for the remaining figures, showing red as maximum pressure and blue as minimum.

tionality of the acoustic signal that reaches the eardrum 共as demonstrated in Figs. 10 and 11兲 appears to imply that simplified models of the ear canal could be used if directional information is of interest. The simplest such model would be a cylindrical tube, and thus would imply neglecting the distinctive kinks that real ear canals have along their length. At high enough frequency this simplification would certainly break down, though it would be interesting to see how high into the frequency range such a model could be used. 2. Frequency dependence and resonance modes

Figure 14 shows the computed pressure at the entrance to the ear canal for the blocked meatus configuration, along with the exact solution for the sphere, which was given in Fig. 3. In this case, the abscissa is the wave number multiplied by the radius of the scatterer. For low wave numbers, the responses are similar, but significantly higher gain is observed for the head at higher wave numbers, and this behavior deserves an explanation. In Fig. 15, the pressure field is examined in the vicinity of the ear for the case of normal incidence, again for the

FIG. 10. Numerical computation of the pressure at the eardrum vs that at the blocked meatus position. Ordinate is dB⫽20 log(p/p0), where p 0 is the eardrum 共with ear canal兲 or blocked meatus 共without ear canal兲 pressure induced by frontal incidence.

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FIG. 11. Numerical computation of the pressure at the eardrum vs that at the blocked meatus position. Ordinate is dB⫽20 log(p/p0), where p 0 is the eardrum 共with ear canal兲 or blocked meatus 共without ear canal兲 pressure induced by frontal incidence.

blocked meatus configuration. Note that for this configuration the ear canal is not present. For low frequencies the wave is not sensitive to the presence of the ear, and a very uniform pressure field is observed in its vicinity of the ear. The second frequency 共5.0 kHz兲 shows a marked change. In this case the pressure is greatest at the base of the cavity 共concha兲, and decreases near the entrance of the cavity. As suggested from the experimental measurements of Sandlin 共1995兲 and Crocker 共1995兲, this pressure pattern fits the description of the first resonant frequency of the concha, which is a quarter-wave ‘‘depth’’ resonance, with maximum value at the bottom. Thus, both experimentation and numerics show that the broad resonance seen in the blocked meatus case around 5 kHz is directly attributable to the first mode of

FIG. 12. Numerical computation of the pressure at the eardrum vs that at the blocked meatus position. In both cases the ordinate is dB⫽20 log(p/pinc), where p inc is the incident pressure. 1040

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FIG. 13. Numerical computation of the pressure at the eardrum vs that at the blocked meatus position. In both cases the ordinate is dB⫽20 log(p/pinc), where p inc is the incident pressure.

the concha, and this explains the fact that a much higher gain is obtained with the head/ear than in the case of a sphere. Figure 16 shows the numerical computations of the gain versus frequency for the case with ear canal. In this case, the location of the resonance frequency 共2.5 kHz兲 is much lower than in the blocked meatus configuration, and this is consistent with experimental measurements of the first resonance of the ear canal 关Shaw 共1974兲兴. However, the overall gain is much higher in the numerical computations than in typical experimental data 关see, e.g., Shaw 共1974兲兴, and the resonance mode is much more narrow. This is actually expected since in this simulation the entire head, along with the ear canal and eardrum, is considered to be rigid, i.e., of infinite impedance. This assumption is most erroneous at the flexible eardrum, where the pressures are actually measured. The nu-

FIG. 14. A comparison of the sound-pressure level for a sphere and the human head at the blocked meatus position for normal incidence. Of significance is the dramatically higher gain experienced by the head, as a result of concha resonance. Walsh et al.: Boundary elements for external auditory system

FIG. 15. The variation of pressure near the blocked meatus position for increasing frequency, at normal incidence. The final figure shows the first 共quarter-wave兲 resonance mode of the concha.

merical roundoff error is also likely to increase as one approaches the approximate resonance condition of the ear canal, which will further decrease the accuracy of the computations. A finite impedance condition would help alleviate the roundoff problem, and help to model the actual physics more accurately. This would decrease the amplitude of the curve in Fig. 16, and thus would bring the results into better agreement with the experiments. Indeed, the data of Shaw 共1968兲 show an almost 10-dB reduction in maximum gain when going from the rigid eardrum to the case of a typical eardrum impedance, and a significant broadening of the region of resonance. Figures 17 and 18 show images of the magnitude of the

FIG. 17. Magnitude of surface pressure on the human head and within ear canal resulting from plane waves at normal incidence on head mesh with ear canal.

total pressure on the head mesh with ear canal, corresponding to a sequence of increasing frequencies. For each frequency two images are shown, the one on the left showing the pressure distribution on the impact zone on the head, and the one on the right showing the pressure distribution within the ear canal. The legends are the same in the two cases. Note that the images of the ear canal are taken from inside the head so that the canal is clearly visible, but that from this viewpoint the pinnas appear inverted. As frequency increases, the areas of highest pressure become more and more localized within the ear canal, in a consistent manner as was seen in Fig. 15. However, a dramatic change is seen at a much lower frequency 共2.9 kHz兲 than in the blocked meatus case, since this is the first resonant mode for the canal. In the 2.9-kHz case the pressure around the head is small compared with that at the eardrum, and Fig. 18 shows a distinctive quarter-wave resonant mode within the ear canal, as expected. The legends also show significant increases of the maximum pressure as the frequency approaches 2900 Hz. IV. CONCLUSIONS

FIG. 16. Numerical computations showing the variation of sound-pressure level at the eardrum for various frequencies, angles of incidence.

This paper presents computations of the head-related transfer function 共HRTF兲 and typical resonance behaviors of a typical human subject with and without the ear canal. The computations are compared with various sets of experimental data and showed reasonable agreement. The resonance frequencies are deduced from the numerical computations, and the graphical display of the pressure distributions in these

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共iii兲

etry兲. This would result in better overall designs for hearing aids, as well as greater comfort for the listener. Undesirable feedback is common in behind the ear 共BTE兲 hearing aids, especially when the hearing mold bore does not completely seal the ear canal. Numerical simulations could be used to simulate the feedback phenomena, which could help to understand and to control the phenomena.

It is believed that these results demonstrate that numerical computations can capture the essential physics involved in the external auditory system. Consequently, the use of such simulations in helping to advance technologies for the hearing impaired seems to be an exciting next step.

ACKNOWLEDGMENTS

FIG. 18. Magnitude of surface pressure on the human head and within ear canal resulting from plane waves at normal incidence on head mesh with ear canal.

cases shows the same quarter-wave patterns that are reported in the literature. While the absolute pressure at the eardrum is generally greater than that at the blocked meatus position, the relative response with respect to the angle of incidence in both cases is shown to be the same. This is consistent with experimental data and confirms that, up to a certain frequency, the ear canal does not influence spatial cues. There are several possibilities for the use of numerical simulations for the design of individualized hearing aids. A few such possibilities are described here. 共i兲

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As shown in this paper, an approximate resonance frequency of the ear canal can be computed simply by sweeping through discrete frequencies and computing the maximum eardrum amplification. Such a technology could be used to match the resonance frequencies of a hearing aid to those of the ear canal of the particular individual. We note that such a calculation should probably be done on the closed canal cavity only. Currently, canal bores have several possible designs, including single diameter, multiple diameter, tapered, etc. In most cases, a trial and error process is used to select the best design for a given individual. An additional variable is the depth of insertion of the bore into the canal. The resonant frequency of the canal is affected by the position of the bore, and the internal canal acoustics will be affected by the shape of insertion device. Numerical simulations could be used to study the acoustical affects of different bore geometries, and optimal insertion depths 共for a given geomJ. Acoust. Soc. Am., Vol. 115, No. 3, March 2004

The authors are grateful for funding for this project through the National Partnership for Advanced Computational Infrastructure 共NPACI兲. Also, the authors thank Professor Chandra Bajaj and Dr. Guoliang Xu of the Center for Computational Visualization at the University of Texas for the mesh of the human head, along with assistance in graphical visualization of the results. Burton, A., and Miller, G. 共1971兲. ‘‘The application of integral equation methods to the numerical solution of some exterior boundary-value problems,’’ Proc. R. Soc. London, Ser. A 323, 201–210. Chang, Y., and Demkowicz, L. 共1999兲. ‘‘Solution of viscoelastic scattering problems in linear acoustics using hp boundary/finite element method,’’ Int. J. Numer. Methods Eng. 44共12兲, 1885–1907. Ciskowski, R. 共1991兲. ‘‘Applications in bioacoustics,’’ in Boundary Element Method in Acoustics, edited by R. Ciskowski and C. Brebbia 共Computational Mechanics, Southhampton兲. Crocker, M. 共1995兲. 共Editor兲, Handbook of Acoustics 共Wiley, New York兲. Demkowicz, L., Oden, J., Ainsworth, M., and Geng, P. 共1991兲. ‘‘Solution of elastic scattering problems in linear acoustics using hp boundary element methods,’’ J. Comput. Appl. Math. 36, 29– 63. Demkowicz, L., Karafiat, A., and Oden, J. T. 共1992兲. ‘‘Solution of elastic scattering problems in linear acoustics using hp boundary element method,’’ Comput. Methods Appl. Mech. Eng. 101, 251–282. Demkowicz, L., Gerdes, K., Schwab, C., Bajer, A., and Walsh, T. 共1998兲. ‘‘HP90: A general and flexible FORTRAN 90 hp-FE code,’’ Comput. Vis. Sci. 1, 145–163. Duda, R. O., and Martens, W. L. 共1998兲. ‘‘Range dependence of the response of a spherical head model,’’ J. Acoust. Soc. Am. 104共5兲, 3048 – 3058. Duncan, R. K., and Grant, J. W. 共1997兲. ‘‘A finite-element model of the inner ear hair bundle micromechanics,’’ Hear. Res. 107, 15–26. Geng, P., Oden, J. T., and Demkowicz, L. 共1996兲. ‘‘Solution of exterior acoustic problems by the boundary element method at high wave number, error estimation, and parallel computation,’’ J. Acoust. Soc. Am. 100共1兲, 335–345. Hammershoi, D., and Moller, H. 共1996兲. ‘‘Sound transmission to and within the human ear canal,’’ J. Acoust. Soc. Am. 100共1兲, 408 – 427. Kahana, Y., Nelson, P., Petyt, M., and Choi, S. 共1999兲. ‘‘Numerical modeling of the transfer functions of a dummy-head and of the external ear,’’ AES 16th International Conference on Spatial Sound Rep., Finland. Katz, B. 共1998兲. ‘‘Measurement and calculation of individual head related transfer functions using a boundary element model including the measurement and effect of skin and hair impedance,’’ Doctoral dissertation, The Pennsylvania State University. Kolston, P. J., and Ashmore, J. F. 共1996兲. ‘‘Finite element micromechanical modeling of the cochlea in three dimensions,’’ J. Acoust. Soc. Am. 99共1兲, 455– 465. Ladak, H. M., and Funnell, W. R. J. 共1996兲. ‘‘Finite-element modeling of the Walsh et al.: Boundary elements for external auditory system

normal and surgically repaired cat middle ear,’’ J. Acoust. Soc. Am. 100共2兲, 933–944. Mehrgardt, S., and Mellert, V. 共1977兲. ‘‘Transformation characteristics of the external human ear,’’ J. Acoust. Soc. Am. 61共6兲, 1567–1576. Moller, H., Sorenson, M. F., Jenson, C. B., and Hammershoi, D. 共1996兲. ‘‘Binaural technique: Do we need individual recordings?,’’ J. Audio Eng. Soc. 44共6兲, 451– 469. Parthasarathi, A. A., Grosh, K., and Nuttall, A. L. 共2000兲. ‘‘Threedimensional numerical modeling for global cochlear dynamics,’’ J. Acoust. Soc. Am. 107共1兲, 474 – 485. Sandlin, R. 共1995兲. Handbook of Hearing Aid Amplification 共Singular Publishing, San Diego兲, Vol. 1. Shaw, E. A. G. 共1974兲. ‘‘Transformation of sound pressure level from the free field to the eardrum in the horizontal plane,’’ J. Acoust. Soc. Am. 56共6兲, 1848 –1861. Shaw, E. A. G., and Teranishi, R. 共1968兲. ‘‘Sound pressure generated in an

external ear replica and real human ears by a nearby point source,’’ J. Acoust. Soc. Am. 44共1兲, 240–249. Stinton, M. R., and Lawton, B. W. 共1989兲. ‘‘Specification of the geometry of the human ear canal for the prediction of sound-pressure level distribution,’’ J. Acoust. Soc. Am. 85共6兲, 2492–2503. Takeuchi, T., Nelson, P. A., Kirkeby, O., and Hamada, H. 共1998兲. ‘‘Influence of individual head-related transfer functions on the performance of virtual acoustic imaging,’’ Audio Eng. Soc. 104th Convention, preprint 4700. van de Geijn, R. A. 共1997兲. Using PLAPACK 共The MIT Press, Cambridge, MA兲. Wada, H., Metoki, T., and Kobayashi, T. 共1992兲. ‘‘Analysis of dynamic behavior of human middle ear using a finite element method,’’ J. Acoust. Soc. Am. 92共6兲, 3157–3168. Walsh, T. F., and Demkowicz, L. 共2003兲. ‘‘hp boundary element modeling of the external human auditory system—Goal oriented adaptivity with multiple load vectors,’’ Comput. Methods Appl. Mech. Eng. 192共1–2兲, 125–146.

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