TLM method and acoustics

Transcription

1 INTERNATIONAL JOURNAL OF NUMERICAL MODELLING: ELECTRONIC NETWORKS, DEVICES AND FIELDS Int. J. Numer. Model. 2001; 14:171}183 (DOI: /jnm.405) TLM method and acoustics Jorge A. PortmH * and Juan A. Morente Department of Applied Physics, Faculty of Sciences, University of Granada, Granada, Spain SUMMARY In this paper, the application of the transmission line modelling method (TLM) to acoustic problems is considered. The formal coupling of a parallel and three series circuits is used to present a new threedimensional symmetrical condensed node for acoustics. Some considerations about the analogy, "eld excitation, and di!erent boundary conditions are also included. Finally, examples are given demonstrating the suitability of the proposed TLM node to describe acoustic phenomena. Copyright 2001 John Wiley & Sons. Ltd. 1. INTRODUCTION Since its appearance at the beginning of the 1970s, the transmission line modelling (TLM) method has been closely related to electromagnetic applications, primarily to microwave problems. Despite the excellent results provided by the TLM method in electromagnetic problems, incursion of TLM into acoustics have been very scarce. For instance, a two-dimensional (2D) node is presented in Reference [1] to predict the radiation pattern of acoustic radiators. More recently, a three-dimensional (3D) acoustic node without stubs was reported to study the human vocal tract [2]. This node identi"es voltage and current at the centre of the node with the pressure and velocity of particles for acoustic phenomena. Although interesting and valuable, this 3D node is well suited for nodes of equal length along the three Cartesian directions but becomes problematic while dealing with nodes of di!erent sizes. This occurs because the speed of pulses at the lines is identical for all directions in spite of the di!erent distances they must travel. The aim of our paper is to present a new 3D symmetrical condensed node with stubs for acoustics and also discuss the main modi"cations involving important points, such as boundary conditions, excitation, etc., needed for a successful application of the TLM method to acoustic problems. The work begins with the derivation of a new 3D mode with stubs for acoustic problems. In the particular case for which no stubs are present, the topology of the node is similar to the one presented in * Correspondence to: Jorge A. PortmH, Department of Applied Physics, Faculty of Sciences, University of Granada, Granada, Spain. jporti@goliat.ugr.es Received 15 March 2000 Copyright 2001 John Wiley & Sons, Ltd. Revised 15 May 2000

2 172 J. A. PORTID AND J. A. MORENTE Reference [2], but a di!erent analogy is established that allows the addition of stubs to control impedances and velocities. After presenting the corresponding scattering matrix, some aspects concerning the electric}acoustic analogy, excitation and boundary conditions are also discussed. The suitability and versatility of the node is demonstrated in a "nal part that includes practical numerical applications. 2. A 3D SYMMETRICAL-CONDENSED NODE FOR ACOUSTICS Acoustic propagation in a #uid medium is described by the following equations: ) u"!σ p t p"!ρ u t (1a) (1b) where p stands for dynamic pressure and u for particle velocity, ρ is the equilibrium density of the medium and σ its compressibility factor. The "eld information contained in Equation (1a) for a portion of the medium with a size of ( l, l, l ) can be described by means of the parallel node in Figure 1(a), while the series node in Figure 1(b) contains all the information concerning the x-component of Equation (1b). In these circuits, the following equivalences can be established: p,<, u,i l, σ,c, ρ( l ), (2) Figure 1. Parallel and series nodes.

3 TLM METHOD AND ACOUSTICS 173 where C is the total capacitance of the parallel node and is the total inductance of the series node. The full set of equations can be described by means of the 3D symmetrical-condensed node shown in Figure 2. It is formed by assembling one parallel and three series nodes that share 6 main or link lines, responsible for the acoustic-"eld propagation, with four extra stubs to adjust the density and the compressibility factor. It is important to note that it is impossible to establish a physical connection between these circuits. Instead, a formal connection described by Equations (1) is considered and represented as a black box at the centre of the node in Figure 2. With this formal node, the full analogy must be set by considering the analogy described in Equation (2) for each line. Thus, line 1, for instance, contributes with a pressure < and a particle velocity I l propagating along the x-direction. Its contribution is > t/2 to the compressibility factor and Z t/(2 l ) to the density, where t is the time step, and Z "1/> is the characteristic impedance of lines 1}6. Similar considerations are valid for the rest of the link lines. Line 7, of characteristic admittance >>, only de"nes a pressure of value < and adds a compressibility factor of value >> t/2. The short-circuited lines 8}10, of characteristic impedance Z Z, Z Z, and Z Z, respectively, only de"ne particle velocity adding density to the acoustic medium. By the usual procedure involving Equations (1) or the circuits in Figure 1, the following preliminary form of the scattering matrix can be obtained: a b c c c c f!h 2 b a c c c c f h 3 c c a b c c f!h 4 c b a c c f h 5 c c c c a b f!h (3) 6 c c c c b a f h 7 d d d d d d g 8!e e j 9!e e j 10!e e j The explicit value for the coe$cients appearing in Equation (3) can be obtained by using the concept of common and uncommon lines for a parallel and a series node [3]. By doing so, these parameters are found to be a" 1 Z!2 2 Z#2!1 >#2 2 >#6, b"!1 Z!2 2 Z#2!1 >#2 2 >#6 c"d" 2 2Z, e" >#6 Z#2, f"c> g" >!6 >#6, h" e Z, j"2!z 2#Z (4)

4 174 J. A. PORTID AND J. A. MORENTE Figure 2. 3D symmetrical condensed node. It must be noted that, as usual, the relative impedance of a given parameter must take the appropriate value according to the line associated with it. 3. PARAMETER DEFINITION AND EXCITATION TECHNIQUE The problem of de"ning the acoustic medium appropriately is reduced to "xing only two parameters: the compressibility factor and the equilibrium density for all the Cartesian directions. The total capacitance of the parallel node and the total inductance of three series nodes de"ne these values, so, from the analogy de"ned by Equation (2), the relative admittance of line 7 and the impedance of line 8 must be >" 2σ > t!6, Z "2( l ) ρ!2 (5) Z t Similar values can be obtained for Z and Z. It is interesting to note that the stability of the method requires a nonnegative value for admittances and impedances, so Equation (5) provides a means for obtaining the maximum allowable timestep, t, corresponding to a given node size and characteristic admittance Z. The dispersion characteristics of the TLM method and most low-frequency numerical methods recommend that the node size in each Cartesian direction be chosen so as to take at least 10 samples of the wavelength corresponding to the maximum valid frequency. Regarding Z, there is no additional restriction, so in theory any arbitrary value could be chosen. A suitable choice seems to be that for which, in the case of equal length in all the Cartesian directions and maximum allowable timestep, no capacitive nor inductive stubs are required. The value for Z and the corresponding t are Z "Z 3h, t " h (6) c 3 where h is the minimum length of the node along the Cartesian directions and Z and c are the acoustic impedance and the speed of sound for this medium, respectively.

5 TLM METHOD AND ACOUSTICS 175 As regards the "eld de"nition, Equation (2) relates acoustic pressure to common voltage at the parallel node in Figure 1(a) and particle velocity along a certain direction to the common current at a series node like that in Figure 1(b). Thevenin's Theorem applied to these circuits and simple calculations provide the following expressions for p and u : p,<" 2 >#6 <#>< u, l I " l (<!< #< ) Z (1#Z /2) from which the set of pulses needed to excite a certain incident "eld can be obtained. When the source "eld is known over an imaginary closed surface surrounding all the scatterers and located between nodes, a useful excitation scheme is achieved by proceeding as follows [4]; (i) at intermediate timesteps, the voltage pulses corresponding to the incident "eld are calculated over the closed surface and (ii) once the incident pulses corresponding to a pre-existing are calculated, extra incident pulses are added to adjacent lines entering the closed surface and subtracted from adjacent lines outside this surface. In this manner, the total incident and scattered "eld is present inside the closed surface but, as outgoing pulses corresponding to the incident "eld are predicted and subtracted, only the scattered "eld is present at the outer region. It is worth mentioning that numerical dispersion is an important error source that must be taken into account when imposing this excitation technique. The reason for the appearance of this error is that, for a given imaginary surface, the incident "eld enters the surface and propagates a certain distance before being subtracted at an opposing point. As numerical dispersion causes the speed of the numerical signal to be slightly di!erent from the expected one, this subtraction may produce signi"cant errors for high frequencies if the distance covered is too long. To demonstrate this undesired e!ect, a Gaussian-shaped plane wave propagating along the x-direction is excited by using two di!erent surfaces of 10 and 150 nodes width, respectively. In the "rst case, the incident wave is added at the x"10 plane and subtracted at the x"20 plane, and the output pressure is sampled at the x"25 plane. In the second case, the wave is added at the x"10 plane and subtracted at the x"160 plane, taking the output pressure at x"165. Figure 3 is a plot of the output pressure normalized to the input pressure versus frequency. This "gure shows that a signi"cant erroneous non-zero pressure may appear near the usual limit λ (0.1 l if the distance being covered by the incident "eld is too long before being subtracted from the exciting surface. Therefore, if a large surface is to be used, it must kept in mind that the range of valid frequencies may be reduced. It is often useful to consider that certain parts of a system can be regarded as sources of pressure or velocity because their presence imposes a particular value for these magnitudes. For instance, a rigid piston moving at a velocity of u (t) is a way to "x this velocity for the particles adjacent to it. In other cases, the element "xes a particular ratio of pressure to velocity. In this case, the element can be substituted by an acoustic impedance. Using equivalences introduced in Equation (2), all these cases can be considered by an electrical circuit including the equivalent ideal source, the electrical impedance corresponding to the acoustic element and those lines interacting with the source and/or impedance. The case of a rigid piston normally orientated to the x-direction located at (i#1/2, j, k) is shown in Figure 4(a). The pulse re#ected at the piston, that is, the incident pulse at the adjacent line 1, can be easily obtained from the circuit in Figure 4(b). (7)

6 176 J. A. PORTID AND J. A. MORENTE Figure 3. E!ect of the dispersion on the excitation. Figure 4. Rigid piston and equivalent circuit. 4. ABSORBING BOUNDARY CONDITIONS The procedure described in the previous section is useful for imposing a simple as well as natural type of absorbing boundary condition (ABC) for the TLM method that we will refer to as

7 TLM METHOD AND ACOUSTICS 177 a matching condition. This condition consists of substituting the eliminated medium by its acoustic impedance normal to the limiting boundary, Z /cos φ, φ being the angle de"ned by the incident wave and the boundary. The ABC for the x-plane located at i"1 is < (1, j, k)" Z l /cos φ!z Z /cos φ#z l < (1, j, k) (8) Another type of ABC that has been successfully used in TLM calculations applied to electromagnetics is a set of discrete conditions originally derived by Higdon for the discrete form of the wave equation [5]. Both matching and Higdon conditions produce theoretical null re#ection only for certain incident angles. These angles can be predicted from the adjacent "eld values, so, in nondispersive problems, both sets of conditions usually provide acceptable results. Nevertheless, in highly dispersive systems, the incident angle depends on frequency and so these conditions provide acceptable results only around the frequency for which the condition has been devised, and therefore, other more elaborate conditions, such as a perfect matching layer (PML) conditions [6], are required. An acoustic PML medium is basically a medium that meets a modi"ed form of Equations (1) derived by substituting the spatial variables appearing in these equations by stretched coordinates. These new co-ordinates are obtained by multiplying each normal spatial co-ordinate x by a stretching factor s "a!iω /ω, where the subscript i"1, 2, and 3 de"nes the x, y, and z co-ordinates [7]. A normal medium is a particular case of a PML medium with a "1 and ω "0. The terms a are scaling factors that allow appropriate PMLs for absorptive media or evanescent waves to be de"ned; in this work, losses and evanescent waves are not considered, so a "1. The imaginary term controlled by ω '0 produces attenuation of the acoustic wave propagating through the PML region. In order to avoid undesired numerical integrals introduced by the application of the stretching factors to Equations (1), the pressure must be separated into three components, p. The total pressure is de"ned as the sum of these components and the PML region is described by the following equations for i"1, 2, and 3: u "!a σ p x t!ω σp p "!a ρ u x t!ω ρu (9) Let us now consider two di!erent PML regions with a common interface at the x"0 plane. It can be proved [7] that, if both regions are chosen so that ρ "ρ, σ "σ, s "s, s "s (10) no re#ected wave is originated at the interface between the two regions, irrespective of the incident angle, φ. Therefore, if medium 1 is the standard acoustic medium, ρ "ρ, σ "σ, s "s "s "1, and for region 2, of width δ, we choose ρ "ρ, σ "σ, s "s "1, and s "1!iω /ω with ω '0, then any wave reaching the interface will pass without re#ection to region 2 where, due to ω, it will be attenuated. The only re#ection returning to region 1 will be caused by the wave re#ected at the limit of the second PML region and, except for grazing incidence, this can be made arbitrarily small by choosing an appropriate value for ω. In order to prevent numerical re#ections at the interface of the two regions, ω is made to increase gradually from a zero value at x"0 to a maximum value, ω, at the limit of the PML region, x"δ.

8 178 J. A. PORTID AND J. A. MORENTE A quadratic variation of ω produces a theoretical re#ection, R"e (11) Figures 5 and 6 show the re#ection coe$cient obtained when a plane wave propagates with normal and 453 incidence to a boundary where matching and "rst-order Higdon conditions are applied. In addition, a PML region 10 nodes in width, with a theoretical re#ection of R"!80 db at normal incidence (!56.6 db at 453 incidence) is also considered by using a hybrid TLM-FDTD scheme [8]. Although re#ection is acceptable for matching and Higdon conditions, it becomes clear from these "gures that PML conditions are much better at preventing arti"cial re#ections at the mesh boundaries. In order to test the performance of the ABCs in dispersive problems, the modelling of a high-order mode in a long duct of dimensions (250, 26 and 1 cm) has been considered. The node size is 1 cm in the three Cartesian directions. The "rst non-fundamental mode, the p mode, with a cuto! frequency of f "636 Hz, has been excited by using a modulated Gaussian pulse with spectral content ranging from 600 Hz at 9 khz, to avoid the excitation of evanescent modes. The re#ection coe$cient has been calculated by using matching and PML conditions with a theoretical re#ection coe$cient of R"!80 db at normal incidence. Figure 7 shows the better performance of the PML condition for the range of frequencies that have been excited. 5. NUMERICAL RESULTS The capability of the TLM method to model acoustic situations is demonstrated in this section by its application to practical problems. The "rst case considered is the calculation of the resonant frequencies of a rectangular box "lled with air. The walls are considered to be perfectly rigid, of length 20, 30 and 40 cm, respectively, and the environmental conditions are such that the speed of sound is 331 m/s. The box is modelled with nodes of dimensions (2, 3 and 4 cm) so that the box length is 10 nodes in all directions. This node size requires the use of inductive stubs and provides Figure 5. Re#ection coe$cient for normal incidence.

9 TLM METHOD AND ACOUSTICS 179 Figure 6. Re#ection coe$cient for 453 incidence. Figure 7. Re#ection coe$cient for the p mode in a long rectangular duct. valid results for frequencies below approximately 800 Hz. The timestep is s, >"0, Z "0, Z "2.5, and Z "6. A delta pulse is added to all the link lines of node (2, 2, 2) and the output pressure is calculated at point (9, 9, 9). Ten-thousand time calculations have been carried out which give a frequency precision of 3 Hz in the Fourier transform algorithm which has been applied to obtain the output pressure shown in Figure 8, while Table I compares numerical and theoretical resonant frequencies. In all cases, the theoretical value is in agreement with the frequency precision. The second example is the 2D duct of height h"10 cm with an expansion area of height H"40 cm and length "5 cm shown in Figure 9(a). The system is modelled with a mesh of

10 180 J. A. PORTID AND J. A. MORENTE Figure 8. Output pressure for a rectangular cavity. Table I. Resonant frequencies for a rectangular cavity. Mode Theory TLM Error (Hz) (Hz) (%) } nodes with a side of 1 cm in all directions. A Gaussian plane wave is excited at the duct and the transmission coe$cient for the pressure is calculated with the TLM method. Figure 10 shows this coe$cient together with results obtained by using the "nite di!erences in the time-domain (FDTD) method under similar conditions of spatial and temporal sampling and two common theoretical approximations. The "rst one is based on the circuit description shown in Figure 9(b), which behaves as a low-band pass "lter [9]. This description is valid for very low frequencies for which pressure and velocity can be considered identical on both sides of the expansion area. The second approximation is obtained by considering plane-wave propagation and imposing continuity of pressure and velocity at the interfaces [9]. The agreement between the two numerical methods is excellent, while, as expected, the theoretical predictions are only valid at very low frequencies. The resonant behaviour of the transmitted pressure predicted by the TLM and the FDTD methods can be qualitatively explained by considering the system as long widening both above and below the duct. For the widening above the duct, pressure must be maximum at the upper rigid wall and approximately null at the junction with the duct. This condition implies resonant frequencies of f "0.5c (2n!1)/(H!h), with n"1, 2,2,R. For the example in

11 TLM METHOD AND ACOUSTICS 181 Figure 9. 2D acoustic "lter: (a) geometry and "rst resonance at the expansion area, (b) circuit approximation. Figure 10. Transmission coe$cient for the 2D acoustic "lter. Figure 9(a), f "552 Hz and f "1655 Hz, values that are qualitative in good agreement with the TLM and FDTD results. The "rst resonance corresponds to 0.25λ at the expansion area, as sketched in Figure 9(a).

12 182 J. A. PORTID AND J. A. MORENTE Figure 11. Long cylindrical duct loaded with a cylindrical Helmholtz resonator. Figure 12. Transmission loss factor for the long duct with a cylindrical Helmholtz resonator. The "nal example concerns the cylindrical Helmholtz resonator connected to a long cylindrical duct of diameter d "4.859 cm sketched in Figure 11. The resonator comprises a cylindrical cavity of diameter d "15.33 cm and height h "24.38 cm, and is connected to the long duct through a cylindrical neck of diameter d "4.044 cm and height h "8.5 cm. The resonator is modelled with acoustic nodes of equal side, cm, and the maximum allowable timestep, t" ns, is chosen so that no stubs are required. The long duct is modelled by using 400 nodes along its axial direction to ensure the elimination of higher-order modes at the output point. A plane-wave Gaussian pulse is excited at the beginning of the duct and PML ABCs are imposed at both ends of the duct. A theoretical value for the transmission loss of the system, de"ned as the ratio in dbs of the incident to the transmitted pressure, can be obtained by considering axial plane-wave propagation at the duct, the neck and the cavity and an empirical correction to take into account the volume of #uid near the neck [10]. Figure 12 is a plot of the transmission loss for both the TLM and the theoretical prediction for axial propagation. Excellent agreement is observed between the two results.

13 TLM METHOD AND ACOUSTICS CONCLUSIONS A 3D symmetrical condensed node with stubs has been presented for the TLM modelling of acoustic situations. After obtaining the scattering matrix, the parameter de"nition together with the "eld excitation and several boundary conditions are considered by using equivalent electrical quantities in parallel and series nodes. Finally, the application of the presented node to di!erent situations shows the good performance of the TLM method while dealing with acoustic-type phenomena. REFERENCES 1. Saleh AHM, Blanch"eldP. Analysis of acoustic radiation patterns of array transducers using the TLM method. International Journal of Numerical Modelling 1990; 3:39}56. 2 El-Masri S, Pelorson X, Saguet P, Badin P. Development of the transmission line matrix method in acoustic applications to higher modes in the vocal tract and other complex ducts. International Journal of Numerical Modelling 1998; 11:133} PortmH JA, Morente JA, CarrioH n MC. Simple derivation of scattering matrix for TLM nodes. Electronic etters 1998; 34:1763} German FJ. General electromagnetic scattering analysis by the TLM method. Electronic etters 1994; 30:689} Higdon RL. Numerical absorbing boundary conditions for the wave equation. Mathematics of Computation 1987; 49:65} Berenger JP. A perfectly matched layer for the absorption of electromagnetic waves. Journal of Computational Physics 1994; 144:185} Liu Q, Tao J. The perfectly matched layer for acoustic waves in absorptive media. Journal of the Acoustical Society of America 1997; 102:2072} Eswarappa C, Hoefer WJR. Implementation of Berenger absorbing boundary conditions in TLM by interfacing FDTD perfectly matched layers. Electronic etters 1995; 31:1264} Kinsler LE, Frey AR, Coppens AB, Sanders JV. Fundamentals of Acoustics. New York: Wiley, Selamet A, Radavich PM, Dickey NS, Novak JM. Circular concentric Helmholtz resonators. Journal of the Acoustical Society of America 1997; 101:41}51. AUTHORS' BIOGRAPHIES Jorge A. Port1H was born in Valle de Escombreras, Cartagena (Murcia), Spain, in He received the MS and PhD degrees in Physics from the University of Granada, Spain, in 1988 and 1993, respectively. From October 1988 to December 1990, he was with Fujitsu Espan a S.A., where he was engaged in datacommunication switching. Since October 1990, he has been with the Department of Applied Physics at the University of Granada, Spain, where he is now &Profesor Titular'. His current research activities deal with the numerical solution of transient electromagnetic and acoustic problems. Juan A. Morente was born in Porcuna (JaeH n), Spain, in He received the Licenciado and Doctor degrees in Physics from the University of Granada, Spain, in 1980 and 1985, respectively. He is presently &Profesor Titular' in the Department of Applied Physics at the University of Granada. His main "elds of interest include electromagnetic theory and applied mathematics. His current research activities deal with numerical analysis of physical systems and transient phenomena.