Slanted septum and multiple folded cavity liners for broadband sound absorption

The design of acoustic liners with complex cavities for a wide frequency range of attenuation using numerical method is investigated in this paper. Three novel liner concepts are presented, demonstrating predicted improvements in broadband sound absorption when compared with that for conventional designs. The liners include a slanted septum core, a slanted septum core with varying percentage open area, and a MultiFOCAL concept. A finite element model of a normal incidence impedance tube is developed using COMSOL Multiphysics modeling software to predict the acoustic properties (resistance and reactance) of liners at medium and high sound pressure levels, and to study the impact of variations in the liner design parameters. The impedance tube finite element model incorporates non-linear semi-empirical impedance equations, validated by comparing numerical results with measurements performed on a single-degree-of-freedom liner, with a perforated face sheet, at high sound pressure level. The design variables of the novel liner concepts are optimized using a hybrid automated optimisation procedure. The low-frequency optimum slanted septum core concept with an open area of 4.5% for the face sheet and 18% for the short slanted septum is predicted to have an absorption level of at least 14 dB in the frequency range of 400–1000 Hz for normally incident pure tone excitations at 150 dB. The slanted septum core concept with varying percentage open area, with broadband optimum design variables, is predicted to have good broadband sound absorption levels of at least 10 dB in the frequency range of 570–3800 Hz. Finally, the MultiFOCAL liner concept with optimised percentage open areas is predicted to have an excellent broadband sound absorption levels of at least 14 dB, for pure tone excitations at 150 dB, in the frequency range of 900–5300 Hz. This work will be followed by optimisation of the face sheet geometries of these novel liner designs in order to maximise lined duct attenuation for aircraft engine applications.


Introduction
Noise and CO 2 emissions are key characteristics of aero-engines, as they enable public acceptance of growing air traffic. For decades reductions of noise and CO 2 emissions were achieved by increasing the bypass ratio, based on Lighthill's pioneering work in 1952. Today, further reductions require optimisation of all components which contribute to the noise signature. This work will focus the optimisation potential of the acoustic treatment. In future aero-engines, even with the most advanced technologies for noise reduction at the source, there will be a continuing demand for passive noise reduction by acoustic liners. More effective designs of acoustic liners are required for passive noise control to achieve the envisaged noise and CO 2 emissions targets of the ACARE Flightpath 2050.
Theoretical and experimental studies have provided significant knowledge on the working principle of single-degree-of-freedom (SDOF) liners. [1][2][3][4][5][6] Engine duct and liner parameters such as mean flow/boundary layer thickness, liner resistive sheet geometry and cell depth, and incident sound pressure level, have all been recognized to affect the noise attenuation of acoustic liners. Empirical and semi-empirical models predicting the installed impedance of acoustic liners are documented in the scientific literature. 3,[6][7][8] However, there have been relatively few attempts to investigate the dedicated design of liner resistive sheets and cavity geometries for sound absorption over a broad range of frequencies. Jones and Parrot 9 developed a group of parallel-element, narrow-chamber liners constructed from ceramic tubular material with an effective percentage of open area (POA eff ) of 57%. The impedance of this novel concept liner was evaluated using the NASA normal incidence test set-up. Impedance prediction models were also developed based on the transmission line model. 9 These narrow-chamber liners derive their acoustic resistance from viscous losses within the narrow chambers. Jones et al. 10 developed a new liner treatment concept called a multi-layer liner, with mesh-caps. This concept allows the acoustic liner to be customized such that the impedance of each individual cell is independently controlled. This is achieved by selecting the combination of liner parameters to set the impedance in each cell. Schiller and Jones 11 subsequently developed a broadband liner design concept called a variable depth liner, composed of groups of resonators tuned for different frequencies.
Only recently has greater attention been devoted to the design of complex cavities to attain optimum acoustic impedance for aero-engine noise reduction at lower frequencies. Rie et al. 12 investigated the design and performance of folded cavity liners for turbofan engine inlets. The design of complex cavities appears to be an inevitable step in the evolution of acoustic liners for larger aero-engines, due to the large spread of source frequencies to be attenuated and the impact of cross section on drag and weight of the liner. Also today, additive manufacturing processes make complex acoustic liner designs feasible. 7,[13][14][15] The aim of this initial work is to develop novel acoustic liner concepts with improved normal incidence broadband absorption using a finite element (FE) impedance tube test model developed using the COMSOL Multiphysics modeling software. Particular interest will be on the design of the liner cavity with complex internal structures. The main advantage of the FE method lies in its ability to accurately simulate the physical processes involved in the sound absorption mechanism of acoustic liners, and offer detailed insights that can complement the knowledge gained from theoretical and experimental approaches. Also, the frequency domain approach is also much less expensive compared to the time domain approach in terms of the computational time, although only a single frequency can be computed at a time.

The computational model
A two-dimensional impedance tube of length L is modelled using COMSOL Multiphysics to predict the normal incidence impedance of acoustic liners of cavity height H c in the absence of grazing flow. Figure 1 shows a schematic of a 2-D impedance tube finite element model used in this study. A plane wave radiation boundary is defined at the top of the impedance tube, at y ¼ L þ H c . The amplitude of the incident pure tone plane waves is iterated until the target value is reached at the face sheet of the liner, at y ¼ H c . The impedance tube and the cavity is then meshed with quadratic elements. Considering the requirement for purely plane wave propagation inside the modelled 2-D impedance tube, the maximum frequency range of interest for a 29 mm diameter (D) tube is �6930 Hz. In order to minimize the influence of evanescent modes, this upper limit is reduced to 6400 Hz. The lower frequency limit is zero for traditional sample holder impedance tube measurements. When the model is used to represent flanged impedance tube measurements, a lower limit is identified which is driven by the mismatch in area between the tube and the 'visible' cavity (driven by the liner honeycomb cell width).
In this article the COMSOL FE model is used to design novel acoustic liners with complex geometries, which are optimized for maximum sound absorption at normal incidence. The frequency domain model employs semi-empirical impedance equations for resistive layers, as the primary aim is to establish the effect of complex designs on the impedance. It should also be noted that the acoustic properties of the liners modelled here for the normal impedance test set-up is a simple case without grazing flow. In practice, the impedance will be altered by the presence of a background grazing flow and a complex source consisting of a combination of tonal and broadband frequencies, typical of aero-engine operating conditions.
An assessment of the semi-empirical impedance equations used in the COMSOL FE model is also included. This is carried out by comparing the predicted normalised nonlinear impedance with measurements of SDOF liner treatments with different perforated face sheets that are typical of those used in aircraft engine ducts. The specific resistance and reactance at the liner surface are computed by solving the Helmholtz equation for a given liner geometry and frequency. Although the governing equations are derived for small disturbances under the assumption of linear acoustics, the numerical model has the potential to simulate the non-linear acoustic properties of liners at high sound pressure levels. The non-linear acoustic properties of the liners are included in the governing interior boundary condition equations.

Governing equations
In this study, the finite element modelling is based on linearisation of the equations governing the isentropic motion of an inviscid and adiabatic perfect gas. As a result, the mean speed of sound c 0 and density of air q 0 are constant. In the absence of background mean flow, i.e. u 0 ¼ 0, the resulting acoustic field equations, after ignoring the higher-order terms of small perturbations, are as follows: Acoustic continuity Acoustic energy @p 0 @t þ cp 0 r:u 0 ¼ 0 where p 0 ; q 0 , and u 0 are the acoustic pressure, density, and particle velocity fluctuations, respectively. In equation (3), c is the ratio of specific heats at constant pressure (C P ) and volume (C V) . Combining equations (1) to (3), the linearised acoustic wave equation describing the motion of acoustic pressure p 0 in the quiescent impedance tube model is given by, For a time harmonic (e ixt ) acoustic perturbation, the linearised acoustic wave equation can be rewritten in the form of the linearised Helmholtz equation in the frequency domain as, where k ¼ x=c 0 is acoustic free-field wave number. The linearised Helmholtz equation (5) is converted to an equivalent integrated or weak statement formulation and then the weak formulation is discretised into finite elements using COMSOL. 16 Liner impedance Model -Interior boundary condition Semi-empirical impedance equations are implemented, for perforated face sheets, into the FE model in the form of interior impedance boundary conditions. The linear part of the acoustic resistance is due to the viscous scrubbing frictional losses inside the perforate holes. This is given by 3 where the resistance R is normalised by q 0 c 0 . In equation (6), a is an empirical constant, l is the dynamic viscosity of air, t p is the face-sheet thickness, r is the percentage open area (POA) ratio or porosity of the face sheet, C D is the discharge coefficient, and d h is the face sheet perforate hole diameter. At medium and high sound pressure levels, the displacement of acoustic particles becomes comparable to the diameter of the holes, and vortices start shedding at the entrance and the exit of the perforation hole. This results in additional dissipation of acoustic energy, increasing the acoustic resistance of the perforate. Thus an additional non-linear resistance term is added to equation (6). The normalised non-linear acoustic resistance for the circular perforates is calculated as where ju n j is the surface acoustic particle velocity component normal to the perforated plate, 1 � r 2 is an experimentally derived dimensionless entrance and exit loss coefficient of flow at the perforate orifice, and C D is the hole discharge coefficient. Apart from the non-linear resistance response, clear evidence of non-linearity in the face sheet reactance with respect to the sound pressure level (SPL) was observed by Hersh et al., 4 Murray and Astley 6 and Serrano et al. 7 With increasing SPL, the face sheet reactance component reduces as the amplitude of the particle displacement through the perforate holes becomes greater than the hole diameter. The normalised non-linear perforate acoustic reactance is given by where d is the dimensionless non-linear correction for reactance deduced by using high amplitude impedance tube measurements, 6 and � is the dimensionless end correction proposed by Ingard. 17 For perforates, � is given by Combining equations (7) and (8) gives the total normalised specific acoustic impedance of a thin perforated face sheet or septum In the COMSOL FE model used for these studies, the boundary condition of the perforated face sheet is defined by equation (10).

Impedance calculation
For time-harmonic excitation, the total pressure field in the impedance tube can be expressed byP ðyÞ ¼Ãe jky þBe �jky (11) whereÃ andB are arbitrary complex constants which represent the amplitude and the relative phase of plane waves travelling in the positive and negative y direction, respectively. From the linearised equation of momentum conservation (equation (2)), the acoustic particle velocity is given byũ ðyÞ ¼Ã e jky �Be �jky q 0 c 0 (12) Therefore, the normalized specific acoustic impedance, from equations (11) and (12) is given by,Z ðyÞ ¼P ðyÞ uðyÞq 0 c 0 ¼Ã e jky þBe �jkỹ Ae jky �Be �jky (13) From equation (13) the normalised acoustic impedance at the facing sheet is given by, ZðH c Þ ¼Ã e jkH c þBe �jkH c Ae jkH c �Be �jkH c (14) The above expression for the impedance at the facing sheet is modified as the amplitudes of the incoming and outgoing waves cannot be measured at H c . The total pressure at the inlet plane can be calculated from COMSOL. Assuming that the amplitude of the incoming time-harmonic plane wave has a known input value P A and phase / A ¼ 0, the total pressure at the inlet (y ¼ L þ H c ) is given bỹ The incoming and outgoing wave terms in equation (14) can be re-written as, andB e �jkH c ¼Be �jkðLþH c Þ e jkL ¼ ½PðL þ H c Þ � P A e jkðLþH c Þ �e jkL (17) Substituting equations (16) and (17) in equation (14) gives the impedance at the liner face sheet,Z ðH c Þ ¼ P A e jky þ ½PðyÞ � P A e jky �e j2kL P A e jky � ½PðyÞ � P A e jky �e j2kL (18) Equation (18) is implemented in the COMSOL FE model to calculate the impedance on the liner surface by defining the value of P A and length of the impedance tube from the surface of the liner (L).

Validation for a Single Degree-of-Freedom liner with normal incidence
The model developed in COMSOL is now validated using the experimental results for SDOF liners without grazing flow from Murray and Astley. 6 The perforate liners were identified as low POA (in the range of 3-6%) and medium POA (in the range of 6-10%). In each case, the predicted value of the incident sound pressure level at the face sheet is iterated until it matches with that of the measured values.

SDOF liner with low POA
The predicted pure tone response of the SDOF liner with low open area, for sound pressure levels of 130 dB, 140 dB and 150 dB, and for frequencies upto 6.4 kHz are compared with the measurements and shown in Figure 2.
It can be seen from Figure 2 that both the measured and the predicted value of the resistance increases due to the increase in the incident sound pressure level. This verifies the results from the previous studies that at medium and high sound pressure levels acoustic dissipation is comprised of two components (as expressed in equation (10)): (1) the dominant shedding of small vortices for high hole acoustic velocities (non-linear part), and (2) the direct viscous dissipation near the liner opening (linear part).

SDOF liner with medium POA
In Figure 3, the predicted impedance of the SDOF liner with medium open area at 130 dB, 140 dB and 150 dB tonal sound amplitude, at frequencies upto 6.4 kHz, are compared with the pure tone response measured using a normal impedance tube. It is clear that the predicted resistance and reactance are in excellent agreement with the experimental results. The predicted value of the resistance increases due to the increase in the incident sound pressure level. It is also interesting to note that the non-linear resistance values of this medium POA SDOF liner are lower compared to those for the low POA SDOF liner. This is because the non-linear resistive part of a perforated face sheet is inversely related to the square of open area, as shown in equation (10).

Automated optimisation procedure
Early work on the liner optimisation problem focused primarily on the reduction of fan noise radiation from commercial aircraft engine nacelles. Specifically, the liner impedance is used as a control variable in order to estimate the optimal value that minimizes far-field radiated noise. Lafronza et al. 18 developed an optimisation procedure, based on a response surface model, to investigate the design of both a uniform and an axially-segmented acoustic liner for a simplified aero-engine inlet model. In the first instance, design of experimental methods were used to build an ensemble sample of points used to develop a response surface model. Later, this surrogate model (RSM) was used for searching for optimum design solutions instead of the real model. A semi-analytic mode-matching method was used for characterising the acoustic propagation in the lined and the hard-walled part of the aeroengine duct model. An approximate Perceived Noise Level (PNL) calculation was used as an objective function. Copiello et al. 19 developed an automated multi-objective procedure to optimize the design parameters of a zero-splice acoustic liner in a uniform cylindrical duct. The Non-Dominated Sorting Genetic Algorithm II 20 was applied to the optimisation problem to determine a set of candidate solutions for the final selection of the liner configuration. The component Effective Perceived Noise Levels (EPNL) at approach and flyover were calculated and used as cost functions. The cost functions were computed, assuming an engine fan model as a noise source, using analytical acoustic propagation and radiation models for each liner design.
Astley et al. 21 developed semi-automated and fully-automated computational aeroacoustic (CAA) procedures to optimize the barrel liner of an inlet at the approach condition. The finite element code ACTRAN/TM was used to predict the far-field sound pressure with and without the acoustic liner. The difference between the far-field sound pressure over a selected angular range, with and without the liner, was used as the cost function since propagation to selected directivity angles tends to dominate the EPNL at the approach condition. The fully automated optimisation procedure was based on a genetic algorithm (GA) multi-objective optimisation tool (ARMOGA) developed by Sasaki et al. 22 Chambers et al. 23 developed an optimisation procedure to design innovative 3-D folded core geometries for acoustic liners, targeting low-frequency (< 500 Hz) noise absorption. The relative performance of various 3-D folded core designs were compared using the lowfrequency performance (LFP) metric. The liner impedance was predicted using the Zwikker-Kosten Transmission Line (ZKTL) model developed by Jones and Parrott. 9 The main objective of this optimisation was to find the best packing of 3-D folded cavities based on the LFP metric.
Tissot et al. 24 developed a numerical shape optimisation strategy based on an ad-joint method to design the cavity of a SDOF liner for very low frequency sound absorption. The shape of the cavity was modified in order to match a given target normal incidence impedance Z ¼ 1 þ 0j, corresponding to perfect absorption, for a given normalised frequency range.
In this current work, a COMSOL FE model is used, along with the semi-empirical equations, for the prediction of the impedance of resistive layers to predict the acoustic properties of novel liners subjected to a normal incidence plane wave. The predictions are performed using a 2-D impedance tube test set-up.

Cost function
The objective of this optimisation problem is to determine the optimum percentage open area of the face sheet/sheets and septum/septa in order to minimize the average normal incidence sound pressure reflection coefficient ð� sÞ for a given frequency range Df ¼ f j � f i . At each frequency within the range, the normal incidence complex reflection coefficient is calculated froms whereZ is the normalised specific acoustic impedance of the liner calculated numerically from equation (18). Furthermore, the normal incidence absorption levels are calculated from the reflection coefficient as follows where positive values of DSPL s signify sound absorption, in dB. The average normal incidence reflection coefficient for a frequency range ½f i ; f j � is expressed using the cost function or fitness function, defined as follows where the integral is evaluated using the trapezium rule in MATLAB. 25 Optimisations are performed at a specific range of target frequencies by using the GA solvers. 25 It is noted that no weighting is applied to the objective function.

Design variables
One of the most important parameters that defines the impedance of a locally-reacting acoustic liner is the percentage open area (POA), r, which consequently has a key influence on the layer resistance and reactance and subsequently on the attenuation properties of the liner. As shown in Figure 4, the perforate holes of circular cross-section are uniformly distributed over the face sheet, and the POA is the ratio of the open area to the total area of the plate. For a regular square pitch pattern, the percentage open area is defined as where p is the centre to centre distance between the holes. The geometrical parameters of the liners such as the thickness of the face sheet (t p ) or septum, hole diameter (d h ), and overall thickness of the liner (H c ), are kept constant in this study.

Optimisation scheme
An automated hybrid optimisation procedure is used to determine the optimum percentage open area (POA) of the face sheet/septa for novel liner concepts at high SPLs. The flow chart of the optimisation procedure is presented in Figure 5. Acoustic calculations are performed using the COMSOL FE model where a pure tone plane wave is normally incident upon the liner surface with a sound pressure level of 150 dB. One of the most important inputs to the optimisation procedure is the liner configuration. Firstly, novel liners with complex cavities are designed using the COMSOL FE model to reduce noise over a relatively large bandwidth. The depth of the cavity (H c ) of the novel liners is maintained as low as possible (set nominally to 60 mm) to allow for potential installation inside an aero-engine nacelle. Secondly, optimum solutions are searched using the genetic algorithm (GA) and fmincon functions for single objective calculations, and gamultiobj function for multi-objective calculations, in MATLAB. 25 In future work, it is planned to build and test some of the optimum novel liner concepts that are presented in this article. Single objective optimisation. The average normal incidence sound pressure reflection coefficient, as given in equation (21), for a specific novel liner concept is calculated for frequencies between 400 Hz and 6 kHz(step 100 Hz), and set as the objective function. This singular objective function is then minimised by using a hybrid optimisation scheme which is a combination of an evolution based genetic algorithm (GA) and the gradient-based fmincon optimizer in MATLAB. In a similar approach, Astley et al. 21 has shown that the process of combining two optimisation solvers operated in sequence speeds the solution process.
The GA is an optimum solution search solver that evaluates the objective function for a set of random initial design variables, called the initial population, within the bounded limits. The initial population of the design variables in the current optimisation problem is uniformly distributed between the lower and upper limit. The lower limit of the design variable (POA) is set as 1%, approaching a hard-wall condition, and the upper limit is set as 20%, representing a highly-porous condition. Thus, the initial population of design variables includes a realistic range of porosities that may be used in acoustic liners for an aeroengine nacelle application.
The algorithm then creates a new set of populations, called "children", from a subset of individual design variables, called "parent", from the current population through mutation and cross-over processes. 25 In general, the algorithm tends to select parents that have the minimum cost function value, in the case of a minimization problem. This procedure is repeated iteratively until the populations of successive generations approach an optimum point/solution.
The accuracy of the optimum solution of a GA solver depends on the population size and the number of generations. The individual design variables in the population of the nth generation approaches the optimum design solution as the value of n increases. 25 As a result, if only a GA is used for the optimisation, the evaluation of the cost function will become computationally expensive. In this study, a hybrid optimisation is applied to search for the best possible solution in the entire design space bounded by the limits, using a GA solver followed by using the fmincon solver. 25 A population size of 50, 75, and 100 design points is used with a number of generations ranging from 1 to 3 depending on the number of design variables required to be optimized. The number of design variables depends on the complexity of the novel liner concepts. The other settings of the GA solver options are retained with their default values.
The best solution of the last generation from the GA solver is passed on to fmincon as the initial point to perform a more efficient local search near the optimum point. The fmincon algorithm is a non-linear multi-objective gradient-based method which uses the "trustregion reflective" algorithm [21]. The algorithm requires a gradient vector and a Hessian matrix at each iteration evaluated at a specific design point. 25 The nature of the cost function s in the neighbourhood of a specific design point is approximately modelled using these gradient matrices. This approximate model is called a 'trust-region'. It reflects the behaviour of the actual function s. The aim is then to minimise the cost function. Therefore a new design point at which the "trust-region" reaches the minimum value is obtained. For instance, if s(r j ) < s(r i ), then r j becomes the new point and the process continues. The functional tolerance value is calculated at each iteration and the process terminates when the tolerance value reaches a set minimum value.
Multi-objective optimisation. In this section, another GA based multi-objective optimizer is presented, called gamultiobj. 25 It performs a minimization of a vector of conflicting objectives subjected to constraints. Unlike a single-objective optimisation, in a multiobjective approach two conflicting objective functions, s 1 and s 2 , are evaluated at the same time to find the set of optimal solutions. The multi-objective optimisation problem solved in this section is defined as min s 1;2 ðrÞ; where s 1 is the average normal incidence reflection coefficient for a frequency range ½f i ; f j � and s 2 is the average normal incidence reflection coefficient for a frequency range ½f k ; f l �.
Here, r l is the lower bound and r u is the upper bound of the optimisation problem as shown in Figure 6. Figure 7 presents a schematic of ideal multi-objective optimal solutions. The search for optimum solutions starts from an initial population of p individuals. All of the individuals of  the initial population are evaluated. Each individual of a population is assigned a fitness or rank equal to its non-domination level and sorted on the basis of the non-domination level. The solutions/individuals which are not dominated by any other members of the set are called the non-dominated solutions. This set of non-dominated optimal solutions are called the Pareto optimum (dashed line in Figure 7) and it constitutes the basis in the multiobjective optimisation. Like a single-objective solver, a multi-objective algorithm creates a new set of populations from a subset of individuals with the best ranking of the current population through mutation and cross-over processes. 25 The children are ranked based on their objective function values and feasibility. Then, the current population and the children are combined into one matrix. Individuals of this matrix are sorted based on their rank and this procedure is iterated until the solver reaches one of the stopping conditions. 20,22,25 Due to the computational cost, a population of size 100 is evaluated for 3 generations. Earlier work has shown this number to be sufficient. 21 In this procedure, the Pareto-optimal solutions are processed further to arrive at a single preferred solution based on the liner absorption requirements.

Novel acoustic liners
In this section, geometries are presented of three novel liner configurations designed for broadband attenuation: configuration 1 -partly perforated slanted septum; configuration 2 -slanted septum with two different percentage open areas; and configuration 3 -Multiple FOlded CAvity Liner (MultiFOCAL).

Configuration 1: Slanted septum core with partial perforation
An acoustic liner with a partly perforated slanted septum is shown to increase the acoustic propagation path length for low frequency sound waves. A schematic of the partly perforated slanted septum liner is shown in Figure 8. The baseline for this liner geometry is from a liner concept that was previously developed by Pongratz et al. 26 with an significant improvement of the broadband attenuation by applying a porous slanted septum suggested by Murray. 27 By using a partly perforated slanted septum, the effective cavity depth is increased. The POA of the face sheet and septum are optimized for maximum sound absorption for both the low and high frequency regime. The optimum effective POA of the face sheet and septum is observed to approach the required panel resistance and reactance for maximum normal incidence absorption at the targeted frequency regions (approaching R ¼ 1, X ¼ 0).
As shown in Figure 8, L 1 and L 2 denote the length of the resistive layer and interior hardwall along the liner diagonal, respectively. The ratio L 1 =L 2 is denoted by r and set to 0.5. The depth (H c ) of the liner is set to 60 mm. The width (w) of the liner is set to 29 mm. The hole diameter and sheet thickness are set to 0.8 mm and 1 mm, respectively.
Once the novel design is set, the gamultiobj optimisation solver is used to investigate the acoustic performance of this novel liner with variations of the porosity of the face sheet (r FS ) and diagonal septum (r 1 and diagonal septum (soptimisation procedure. In the first step, the total number of populations and generations are set to 100 and 2, respectively. With a large population size, the genetic algorithm searches the solution space thoroughly, thereby reducing the chance that the algorithm returns a local minimum that is not the global minimum. However, a large population size also causes the algorithm to run more slowly. Pure tone plane waves with a frequency in the range from 400 to 6000 Hz (in step of 100 Hz), and at a sound pressure level of 150 dB, are normally incident on the liner face sheet. The open area of the liner face sheet and the diagonal septum are constrained within the range of 1% (approaching hard-wall) to 20% (highly-porous surface). In this optimisation, the hole diameter and the thickness of the face sheet and the diagonal layer are chosen to cover the range of interest applicable for aero-engine nacelle integration.
The objective functions for the low frequency regime (� s 1 ) is and for the high frequency regime (� s 2 ), are calculated using the COMSOL FE model. The results obtained from the GA are superimposed on independently computed contours of the cost function in the r fs -r 1 design space for validation. Figure 9(a) shows the result of a multi-objective GA optimisation in the low frequency range for the partly perforated slanted septum liner. The contour plot indicates the evaluation of the objective function given by equation (24), carried out independently in the (r FS ,r 1 ) design space. The design variables corresponding to Pareto-optimum solutions are represented by the solid red dots. The objective functions s 1 and s 2 are evaluated 200 times in a gamultiobj solver in order to converge to the optimum Pareto front. The blue hollow circle indicates the optimum design value for maximum sound absorption in the low frequency regime. The optimum high frequency solution is also shown for comparison. It is clear that different designs are defined for the two frequency ranges.
In Figure 9(b), results are presented of a multi-objective GA optimisation in the high frequency range for the partly perforated diagonal septum liner. The contour plot indicates the evaluation of the objective function s 2 carried out independently in the (r FS ,r 1 ) design space. The green hollow circle indicates the GA optimum design value for maximum sound absorption in the high frequency regime.
Configuration 2: Slanted septum core with variable POA As shown in Figure 10, configuration 2 is comprised of a slanted septum of total length L 1 þ L 2 , with different percentage of open areas in the two parts of the septum layer. The baseline for the geometry of the slanted septum core with different percentage open areas arises again from the liner concept previously patented by Murray. 27 The addition of perforations to the interior sound hard-wall part of the septum L 2 of configuration 1 provides an additional path for the incident sound. The introduction of an additional resistance and reactance are used to improve the sound absorption over a much broader frequency range compared to configuration 1. The length of the resistive layer 1 and 2 is equal to 22.2 mm and 44.4 mm respectively. The angle between the diagonal layer and the vertical wall are held constant and set to 25:8 � . The depth (H c ) and width of the liner, along with the hole diameters and sheet thickness, is kept the same as that of configuration 1.
The multi-objective optimisation procedure is applied to optimize the efffective POA of the face sheet and resistive layer 1 and 2 in order to maximise the sound absorption capability of this novel liner concept. The diameter of the perforations is fixed to 0.8 mm.
Similarly, the thickness of the face sheet and the resistive diagonal is fixed to 1 mm. In this case, the two objective functions to be minimized are In Figure 11, the result of the GA multi-objective optimisation is presented. The objective functions, as defined in equations (26) and (27), are plotted against each other. A population size of 100 and 2 generations are used in this GA multi-objective optimisation. As mentioned earlier, three design variables are optimised. The Pareto front of these 3 variables and 2 cost functions optimisation problem have 10 design points (represented by red solid circles). The solid blue circles represent the cost function values of all the individuals in the second generation. The hollow blue circle represents the optimum design solution in the low frequency regime, 400-1000 Hz, providing maximum weighting to the objective function � s 1 . The hollow green circle represents the optimum design solution in the high frequency regime, 2en kHz, providing maximum weighting to the objective function � s 2 . The hollow black circle represents the broadband optimum solution, which is a trade-off between the low and high frequency solutions (minimum distance between tangent of Pareto front and the axis origin). Figure 12 shows the geometry of a Multiple FOlded CAvity Liner, called the MultiFOCAL concept, based on the liner concept previously patented by Pool et al. 28 and Murray et al. 29 The folded part of the segments are inclined at an angle of 58:8 o primarily to minimise complexity in the (planned) 3 D printing. In addition to the folded segments, resistive septum layers are added inside the segments 1-5. The MultiFOCAL concept is expected to be better at all frequencies when compared to the previous liner configurations.

Configuration 3: Multiple FOlded CAvity liner-MultiFOCAL concept
The width of each segment is 4.83 mm and assumed to be large enough to neglect the thermal and viscous losses. The centreline path length of segments 1-6 is gradually decreased from 79.8 mm to 14.7 mm. The individual values are listed in Table 1. The overall width (w) and depth (H c ) of the liner is 29 mm and 60 mm, respectively. The open area of the septa inside segments 2 to 5 is set initially to 20%, with very low non-linearity, and hence low acoustic resistance, at high SPL. Future iterations of this design will also include the septa open areas as optimisation design variables. The hole diameter (d h ) and sheet thickness (t p ) are set to 0.8 mm and 1 mm, respectively.
In this study, pure tone plane waves at a reduced sound pressure level 140 dB are normally incident on the MultiFOCAL surface in the frequency range from 0.4 to 6 kHz. The percentage open area of the face sheet of segments 1 to 6 and the septum of segment 1 are optimized using the single objective optimisation method. The main objective of the optimisation is to minimise the average coefficient of reflection (� s) for frequencies between 0.4 and 6 kHz. Therefore, the objective function to be minimized is The optimisation results are presented in Figure 13. A population size of 50 with 3 generations are selected for this study. Figure 13(a) indicates the  evaluation points of the GA solver. The solid blue and black circle indicates the average and the best value of the objective function, given by equation (28), at each generation, respectively. In Figure 13(b), the pink solid circle indicates the value of the objective function at each iteration in the fmincon solver. The best solution in the 3 rd generation, represented by the black solid circle, is set as an initial value for the fmincon solver to perform an efficient local search. The blue solid circle indicates the final optimum value from the fmincon solver. In total, 200 evaluations are performed, in the GA optimiser, to reach the best solution and an additional 24 iterations, with 9 function evaluations for each iteration, are performed in fmincon 25 to converge to the final optimum value. The design variables (percentage open area) corresponding to the final optimum solution, represented by the blue solid circle in Figure 13(b), are listed in Table 2.

Acoustic properties of the novel liner concepts
Slanted septum core: Optimum impedance and absorption coefficient In Figure 14(a), the predicted resistance, reactance, and absorption coefficient spectrum of the partly perforated diagonal septum liner are presented for the low frequency regime optimum solution (Figure 9(a) -blue circle). For r FS ¼ 4:5% and r 1 ¼ 18%, normal incidence absorption coefficient of at least 0.8 are observed in the frequency range of 400-1000 Hz. The first peak absorption for this liner configuration is located at �700 Hz, which falls within the targeted low-frequency range 400-900 Hz. The high POA of the septum ensures the liner behaves like a deep single layer folded cavity. The peak at 700 Hz corresponds to the quarter wavelength of the centreline of the folded path length (11.1 cm). This observation is reasonable as the highly porous septum is expected to enhance the absorption performance for the lowest frequencies by maximizing the acoustic path length for such low frequency waves to propagate. Also, good absorption of sound (a � 0.8) is seen in another narrow bandwidth of frequencies in the range of 2000-2900 Hz. This is because the threequarter wavelength of the longest centreline path length (11.1 cm) is at approximately 2300 Hz. At other frequencies between 900-2000 Hz and 3500-5000 Hz, the sound absorption is significantly less due to the presence of a strong anti-resonance inside the liner complex cavity at �1500 Hz, 3300 Hz and 4700 Hz. The first two frequencies correspond to effective single layer path lengths of approximately 11.1 cm and 5.2 cm, respectively, while the third represents the three-half wavelengths anti-resonance for the longest path length. In Figure 14(b), the predicted resistance, reactance, and absorption coefficient spectrum of the partly perforated diagonal septum liner are presented for the high frequency regime optimum solution (Figure 9(b) -hollow green circle). For r FS ¼ 16:6 % and r 1 ¼ 3:4 %, a sound absorption coefficient of at least 0.8 is observed in the frequency range of 790 to 2800 Hz. The first peak of absorption for this high frequency optimum liner configuration is located at �2300 Hz corresponding to the three-quarter wavelength of the longest centreline path length (11.1 cm). Also, the low POA of the septum makes the cavity perform more like a double layer configuration. The absorption coefficient is good in another range of frequencies between 3400 and 5200 Hz. However, the anti-resonance corresponding to the initial 5.2 cm "cavity" remains, leading to a dip in the absorption spectra at 3300 Hz. The sound absorption level spectra of the slanted septum layer liner for the low and high frequency optimum solutions calculated from equation (20) are compared in Figure 15. It is shown that, generally, the higher the effective POA of the diagonal septum, the more the location of the first peak absorption is shifted towards lower frequencies. This is because the maximum available path length is used.  Table 3.  For optimum low frequency absorption, the percentage open area of the face sheet and the resistive layers 1 and 2 are equal to 4.7%, 4.2% and 14.6%, respectively. It has been shown previously that the non-linear resistance component at a given frequency is greatest for low open areas and high sound pressure levels. With the optimum low frequency design solution, the desired resistance for the liner geometry is mainly generated by the face sheet and resistive layer 1, as opposed to resistive layer 2. It can be seen in Figure 16(a), that the first peak in the absorption spectrum is at 920 Hz. This is because at 920 Hz the normalised reactance is 0 and the normalised resistance is �1:4, almost meeting the requirement for perfect absorption.
From the peak absorption frequency, the depth of an ideal quarter wave resonator (with R ¼ 1 and X ¼ 0) with a straight cavity (and zero face sheet inertance) can be directly estimated as where f is the first peak absorption frequency (taken from Figure 16(a)). Using c 0 ¼ 340:3m=s and f ¼ 920 Hz yields a cavity depth of 92.4 mm, which is 54% greater than the cavity depth (Hc). Figure 16(d) shows a comparison between the normal incidence sound absorption level for the slanted septum core with varying open areas optimised for low, high, and broadband frequency regimes. With the optimum low frequency liner, sound absorption of at least 14 dB is achieved in the frequency range of 570-1560 Hz, and the absorption decreases gradually due to the presence of an anti-resonance at 2500 Hz. By making layer 2 porous (upper diagonal), the strong anti-resonance observed in configuration 1 at 3300 Hz has been removed by the increased damping (compare with Figure 15). The acoustic properties of the liner with a high frequency optimum design solution are presented in Figure 16(b). For high frequency sound absorption, the optimum effective open areas obtained from the Pareto front of the GA multi-objective optimisation are r fs ¼ 19:2%, r 1 ¼ 2:5% and r 2 ¼ 4:2%. The criterion used for this solution is to minimize the average coefficient of reflection ( s < 0:2) in the frequency range of 2-6 kHz. Unlike the low frequency design, the total resistance of this liner geometry is mainly generated by the (septum) resistive layers 1 and 2 rather than the face sheet. From Figure 16(b) and (d), it can be noticed that the first absorption peak is at 1.5 kHz, outside the targeted bandwidth. This complements the performance of the high frequency liner to ensure the normal incidence absorption level is above 15 dB over the frequency range of 1000-5500 Hz.
In terms of obtaining a trade-off between a low and high frequency solution, a broadband optimum design solution, r fs ¼ 10:3 %, r 1 ¼ 8:6 % and r 2 ¼ 1:6 %, has been sought based on the criterion of maximum allowable average coefficient of reflection s 1;2 � 0:2, in both of the frequency ranges (0.4-1 kHz and 2-6 kHz). Figure 16(c) shows the acoustic impedance and absorption spectra of configuration 2 with a broadband optimum solution. In Figure 16(d), the normal incidence sound absorption level is shown for open areas optimised for broadband absorption. The resistive layer 2 is approaching a hard-wall condition. The first peak of absorption is at 920 Hz which is a characteristic feature of the optimum low frequency solution. Maximum sound absorption is at 920 Hz, 2500 Hz, 3500 Hz and 5200 Hz, where the normalised resistance and reactance values approach 1 and 0, respectively. The porous septum layer 2 helps to improve the overall panel absorption relative to that seen for configuration 1. It is noted, however, that the wide broadband requirement is difficult to achieve at all frequencies with this design, as evidenced by the dip in absorption at around 4300 Hz.  Table 2, are used in the calculations. The first absorption peak is located at 0.6 kHz. However, the absorption coefficient at 0.6 kHz is relatively low and equals 0.6. Although the panel's normalised resistance is close to unity, the normalised reactance is � � 2. The other peaks in the absorption spectra are located at 1.2, 2, 3.5, 4.3 and 5 kHz where the coefficient of absorption are atleast equal to 0.90 or close to 1. Also, there are two small dips in the absorption spectra at 2.8 kHz and 4 kHz. Figure 17(b) shows the acoustic properties predicted on the surface of the MultiFOCAL configuration when the liner is excited by pure tone plane waves at 150 dB. A higher level pure tone source is used to assess the robustness of the design. It can be seen that a high absorption coefficient (greater than 0.94) is maintained over a wide frequency band (0.9-5.3 kHz). The normalized resistance and reactance are close to 1 and 0, respectively, for most of the frequencies in the range of 0.9 to 5.3 kHz. The six closely packed peaks are broadened due to the general increase in the resistance, damping the anti-resonances, and flattening the normalised reactance curve towards 0. The resistance of the liner surface is mainly controlled by the septum (septum 1) of segment 1. This is because the optimum percentage open area of septum 1 is low compared to the face sheet of segments 1-6. Therefore, it has a significant non-linear impedance component at high sound pressure levels for such a low POA. Figure 18 shows the normal incidence absorption level comparison of the MultiFOCAL design at 140 and 150 dB, for frequencies between 0.1 kHz and 6 kHz. As mentioned earlier, the narrow absorption peaks seen with 140 dB excitation broaden with 150 dB excitation. For frequencies between 1 kHz and 5.1 kHz, the predicted MultiFOCAL absorption level is at least 14 dB, except for two small dips at 2.8 kHz and 4 kHz. For frequencies below 1 kHz, the acoustic performance of the MultiFOCAL configuration is relatively poor.

Conclusion and summary
This work presents a detailed numerical optimisation and investigation into the design of novel liner geometries for broadband sound absorption at normal incidence in the absence of grazing flow. A two-dimensional COMSOL FE model of a normal incidence impedance tube has been developed to predict the acoustic properties of the liners. In the FE model, semi-empirical equations are applied as interior boundary conditions to translate physical parameters of the face sheet and septum into normalised resistance and reactance. The semiempirical equations include only the effects of liner face sheet geometry and the incident sound pressure level of pure tones. It does not include the effects of grazing flow, as the initial aim of this research is to investigate the design of novel acoustic liners with maximised sound absorption at high SPL with normally incident, pure tone, acoustic plane waves. The design challenge is focussed on the cavity geometry needed to realise optimum reactance characteristics. Once the reactance spectrum is optimised, it is then a relatively straightforward task to adjust the face sheet geometry in order to realise the desired liner impedance under grazing flow.
A hybrid optimisation procedure has been presented where the objective functions evaluated by the COMSOL FE model are defined within a MATLAB optimisation solver. The optimisation technique involves a combination of a global GA search and a local gradient-based search algorithm. The percentage open areas of various resistive layers were optimised to minimize the normal incidence complex reflection coefficient at high SPLs for pure tone acoustic plane waves. The numerical results show the predicted benefit of using complex cavities with resistive septum layers for maximisation of sound absorption. It is also shown that the sound absorption capability of the optimised novel liners is predicted to be more broadband when the incident sound pressure level is increased, due to an increase in resistance and a reduction in reactance over a wide range of frequencies. This is due to the non-linear behaviour of the perforates attributed to the vortex shedding at the entrance and exit of the orifices, which were considered in this numerical modelling. This confirms that such effects cannot be neglected while designing liners for high SPLs. sound absorption over a broad range of frequencies. With the MultiFOCAL concept, significant amount of normal incidence absorption are predicted at high SPL excitation in the frequency range from 900 to 5300 Hz.
It is noted that these liners are designed for normal incidence without grazing flow. The face sheet properties may subsequently be altered in order to maximise engine nacelle duct attenuation.