Full numerical liner impedance eduction process

Abstract

To provide a method for producing a full numerical liner impedance eduction process.SOLUTION: A method performs a digital experiment of a liner in an environment using a defined 3D computer-based model of the liner. Results of performing the digital experiment include a reference transfer function. A 2D model of the environment is generated in the case where the liner is represented by an acoustic impedance boundary condition with an impedance value defined by a resistance value, a reactance value, and a non-linear coefficient. Iteratively, the impedance value is modified and a 2D simulation is performed using the generated 2D model of the environment with the acoustic impedance boundary condition with the modified impedance value, until a transfer function resulting from performing the 2D simulation matches the reference transfer function. The modified impedance value resulting in the transfer function matching the reference transfer function is acoustic impedance of the liner.SELECTED DRAWING: Figure 1

Description

[Technical field]

[0001] Many existing products and simulation systems are offered on the market for designing and simulating objects, such as vehicles. Such systems typically employ computer-aided design (CAD) and computer-aided engineering (CAE) programs. These systems allow users to build, manipulate, and simulate complex three-dimensional models of objects or assemblies of objects. These CAD and CAE systems provide model representations of objects, such as real-world objects, using edges or lines, and in certain cases edges or lines with faces. The lines, edges, faces, or polygons may be represented in various ways, such as, for example, non-uniform rational B-splines (NURBS). [Background technology]

[0002] Such systems manage parts or assemblies of parts of a modeled object, primarily a specification of the shape. In particular, a CAD file contains the specification from which the shape is generated. From the shape, a three-dimensional CAD model or model representation is generated. The specification, the shape, and the CAD model / representation may be stored in a single CAD file or multiple CAD files. The CAD system or other such CAE system includes graphical tools to visually represent the modeled object as it is represented in three-dimensional space to the designer, and these tools are dedicated to the display of complex real-world objects. For example, an assembly may include thousands of parts.

[0003] The advent of CAD and CAE systems allows for a wide range of representation possibilities, such as CAD models for objects. Computer-based models may be programmed such that they have the properties (e.g., physical, material, or other physics-based) of the underlying real-world object that the model represents. Exemplary properties include stiffness (ratio of force to displacement), plasticity (irreversible strain), and viscosity (resistance to flow of one layer over an adjacent layer), among others. When a CAD model or other such computer-based model known in the art is programmed in such a manner, it can be used to perform a simulation of the object that the model represents. For example, a mesh-based model may be used to represent the interior cavity of a vehicle, an acoustic fluid surrounding a structure, or any number of real-world objects. Additionally, CAD and CAE systems, along with computer-based models, may be utilized to simulate real-world physical systems, such as engineering systems such as automobiles, airplanes, buildings, and bridges, among other examples. Additionally, CAE systems may be used to simulate any variety and combination of the behavior of these physics-based systems, such as noise and vibration. Summary of the Invention

[0004] Simulation of noise is a task implemented by existing simulation methods, and in many cases these existing methods determine the properties, e.g., noise reduction capabilities, of real-world objects, e.g., liners. However, recently, objects of interest, e.g., liners, have become more and more complex, and improved methods for simulating objects and determining properties of said objects are needed. The embodiments provide such functionality.

[0005] One such embodiment is directed to a computer-implemented method for determining the acoustic impedance of a liner. The method defines a three-dimensional (3D) computer-based model of the liner and performs a digital experiment of the liner in an environment using the defined 3D computer-based model of the liner. The results of performing the digital experiment include a reference transfer function. To proceed, a two-dimensional (2D) model of the environment is generated, in which the liner is represented in the generated 2D model of the environment by acoustic impedance boundary conditions having impedance values, the impedance values ​​being defined by a resistance value, a reactance value, and a nonlinear coefficient value. The method then iteratively (i) modifies the impedance values, and (ii) performs a 2D simulation using the generated 2D model of the environment with the acoustic impedance boundary conditions having the modified impedance values, until a transfer function resulting from performing the 2D simulation matches the reference transfer function. The modified impedance values ​​used in performing the 2D simulation that result in a transfer function that matches the reference transfer function are the acoustic impedance of the liner.

[0006] According to one embodiment, defining the 3D computer-based model of the liner includes receiving a computer-aided design (CAD) model of the liner and identifying (i) one or more parts of the liner and (ii) dimensions of the one or more parts based on the received CAD model. A computational surface mesh representing the liner is then generated based on the identified one or more parts of the liner and dimensions of the one or more parts. In such an embodiment, the generated computational surface mesh is the defined 3D computer-based model of the liner.

[0007] Another embodiment generates a 3D model of the environment. According to such an embodiment, the generated 3D model of the environment includes a defined 3D computer-based model of the channel and a liner, the defined 3D computer-based model of the liner being disposed on a bottom surface of the channel. In an exemplary embodiment, generating the 3D model of the liner includes at least one of: (i) defining a length of the channel according to a wavelength of a pressure wave in the flow; and (ii) defining a location of a solid trip in the 3D model based on a velocity of the flow.

[0008] Yet another embodiment receives an indication of test conditions and performs digital experiments on the liner in the environment using (i) the defined 3D computer-based model of the liner, (ii) the generated 3D model of the environment, and (iii) the received indication of the test conditions. In one embodiment, the received indication of the test conditions includes flow conditions. Furthermore, the received indication of the test conditions may also include boundary conditions, such as an initial guess of impedance boundary conditions of the liner (used to create a 2D model of the environment). Furthermore, in another embodiment, performing digital experiments on the liner in the environment using (i) the defined 3D computer-based model of the liner, (ii) the generated 3D model of the environment, and (iii) the received indication of the test conditions includes collecting pressure data from one or more digital sensors in the channel while subjecting the defined 3D computer-based model of the liner in the generated 3D model of the environment to the test conditions. Such an embodiment may generate a reference transfer function by calculating a Fourier transform of the collected pressure data.

[0009] In one embodiment, the digital experimentation is a computational fluid dynamics (CFD) simulation. In such an embodiment, conducting a digital experimentation of the liner in the environment using (i) the defined 3D computer-based model of the liner, (ii) the generated 3D model of the environment, and (iii) the received representation of the test conditions includes (1) generating a CFD input file based on (a) the defined 3D computer-based model of the liner, (b) the generated 3D model of the environment, and (c) the received representation of the test conditions, and (2) conducting a CFD simulation using the generated CFD input file.

[0010] In one embodiment, the generated 2D model of the environment is a mesh-based model. In one such embodiment, the method further includes at least one of: defining pressure waves, setting a resolution of the mesh-based model as a function of a wave packet wavelength of the defined pressure waves, and performing a flow convergence simulation to determine the field data.

[0011] According to one embodiment, a transfer function resulting from performing a 2D simulation matches a reference transfer function when a difference metric between (i) the transfer function resulting from performing a 2D simulation and (ii) the reference transfer function is below a threshold value.

[0012] Yet another embodiment includes determining a modified impedance value at a given iteration. An embodiment determines the modified impedance value based on a difference (or differences) between (i) a given transfer function resulting from performing a 2D simulation and (ii) a reference transfer function. An embodiment may also determine the modified impedance value using an optimization algorithm. Furthermore, according to an embodiment, the given transfer function may be from an iteration prior to the given iteration. Still further, an embodiment determines the modified impedance value (i.e., the next impedance value of the test subject) by considering the difference between (i) a plurality of transfer functions, e.g., a subset of transfer functions that provided the best results, and (ii) the reference transfer function.

[0013] In an exemplary embodiment, modifying the impedance value includes modifying at least one of a resistance value, a reactance value, and a nonlinear coefficient value. Additionally, according to yet another embodiment, the nonlinear coefficient value is dependent on a first derivative of the local velocity of the flow.

[0014] Another embodiment is directed to a system for determining the acoustic impedance of a liner. In such an embodiment, the system includes a processor and a memory having computer code instructions stored thereon. The processor and memory are configured to use the computer code instructions to cause the system to implement any embodiment or combination of embodiments described herein.

[0015] Yet another embodiment is directed to a computer program product for determining the acoustic impedance of a liner, the computer program product including one or more non-transitory computer readable storage devices and program instructions stored in at least one of the one or more storage devices that, when loaded and executed by a processor, cause a device associated with the processor to implement any embodiment or combination of embodiments described herein.

[0016] One aspect of the embodiment includes automatically generating a 3D numerical experimental setup in which, according to one embodiment, a liner is placed in a planar channel in which traveling waves are generated, which represents the numerical equivalent of a real-world experimental test.

[0017] Another aspect includes collecting several measurements from a numerical simulation. In such an embodiment, the pressure signal above the liner is recorded with a series of microphones. A further aspect includes processing the simulation measurements and generating a reduced model to calculate the impedance of the liner. Yet another aspect in one embodiment is an optimization method to match the numerical test results and the reduced model for given target parameters by adjusting the impedance of the liner in the reduced model.

[0018] According to one embodiment, the liner numerical experiment setup is generated using a pre-processing tool that imports a given liner 3D model and generates a complete virtual test environment with ancillary geometric entities and measurement areas for storing the simulation data.

[0019] In one embodiment, the simulation data (i.e., data from the 3D digital experiment) is processed using a set of scripts that can read the time domain data, convert it to frequency domain space, and submit several reduced model analyses in an optimization loop to obtain the impedance (i.e., reference transfer function) of the liner. Yet another embodiment utilizes tools such as SIMULIA PowerACOUSTICS® by applicant-assignee Dassault Systemes Simulia Corporation to calculate the complex Fourier transform of a set of time domain pressure signals collected from the numerical tests.

[0020] In an exemplary workflow, a user prepares the liner geometry, e.g., surface mesh, according to a set of guidelines that specify the nomenclature to use for the relevant surfaces. According to one embodiment, the guidelines dictate given names to be assigned to liner surface groups in the liner CAD model and to the hole surfaces of the liner's perforated plate. This allows such an embodiment to automatically identify the key sections of the liner and generate a digital experimental equivalent of the liner. Additionally, in yet another embodiment, a set of input parameters including, e.g., flow conditions and target frequencies, are provided as inputs, e.g., specified by the user.

[0021] An embodiment uses inputs (e.g., liner geometry and flow parameters) to initiate the liner impedance estimation process. In such an embodiment, the process occurs in three stages. In stage 1, a numerical equivalent of the physical test of the liner is constructed. In other words, a digital twin of the real-world liner test is created. Stage 2 uses the digital twin to perform numerical testing, and some measurements are stored and used in the subsequent impedance calculation process. Stage 3 involves processing the 3D simulation results, i.e., the stored measurements, to obtain a reference transfer function. In one embodiment, stage 3 involves converting the time domain data to frequency domain data and using the frequency domain data to obtain a reference transfer function. In stage 3, an equivalent 2D reduced model of the 3D simulation is generated. An optimization loop is then performed that looks for a match between the 3D and 2D results to identify an equivalent impedance value for the liner.

[0022] Other aspects include a computer program product tangibly stored on a non-transitory computer-readable medium, and a computing system, such as a computer system and a computer server. [Brief description of the drawings]

[0023] The foregoing will become apparent from the following more particular description of exemplary embodiments, as illustrated in the accompanying drawings, in which like reference characters refer to the same parts throughout the different views. The drawings are not necessarily to scale, emphasis instead being placed upon illustrating embodiments.

[0024] [Figure 1] FIG. 1 is a flow chart of a method for determining the impedance of a liner according to one embodiment. [Figure 2A] FIG. 2A is a 3D computer-based model of a liner that may be analyzed using an embodiment. [Figure 2B] FIG. 2B shows the top plate of the liner of FIG. 2A. [Figure 2C] FIG. 2C shows a cell array of the liner of FIG. 2A. [Figure 3A] FIG. 3A is a visual depiction of a 3D numerical experiment setup according to one embodiment. [Figure 3B] FIG. 3B is a visual depiction of the 3D numerical experiment setup according to one embodiment. [Figure 4A] FIG. 4A is a visual depiction of a 2D reduced model of the experiment in FIG. 3A. [Figure 4B] FIG. 4B is a visual depiction of a 2D reduced model of the experiment in FIG. 3B. [Diagram 5] FIG. 5 is a flow chart of a process for liner impedance estimation according to one embodiment. [Figure 6] FIG. 6 is a flow chart of a method for determining the impedance of a liner according to one embodiment. [Figure 7A] FIG. 7A is a visualization of the grazing flow experimental setup according to one embodiment. [Figure 7B] FIG. 7B is a diagram of glazing flow on a liner. [Figure 8] FIG. 8 is a simplified block diagram of a computer system for determining the impedance of a liner according to one embodiment. [Figure 9]FIG. 9 is a simplified block diagram of a computer network environment in which embodiments may be implemented. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0025] A description of an exemplary embodiment follows.

[0026] As mentioned above, simulating noise is a task implemented by existing simulation methods, and in many cases, these existing methods determine the properties, e.g., noise reduction capabilities, of real-world objects, e.g., liners. However, recently, objects of interest, e.g., liners, have become more and more complex, and improved methods for simulating objects and determining properties of such objects are needed. The embodiments provide such functionality.

[0027] The embodiments relate to a new fully numerical process for indirect estimation of liner impedance. In embodiments, the liner may have an arbitrarily complex shape. The ability to determine the impedance of complex liners is becoming increasingly important in the aircraft industry, among other industries.

[0028] An aircraft turbofan engine generates thrust by compressing, burning, and expanding a certain amount of ingested air. At the same time, in the outer section of the engine, the fan draws in cold air and accelerates it without it undergoing thermal cycling. This secondary section is called bypass, and it generates thrust without increasing the engine exhaust.

[0029] In recent years, engines have evolved towards lower emission layouts. The general trend among major engine manufacturers implementing these lower emission layouts has been to move towards engines with larger bypass ratios (i.e., larger bypass sections and larger fans). This trend has not only reduced emissions but has also changed the primary noise generating mechanism of the engine. In older low bypass ratio engines, most of the noise is generated by the engine jets. However, in modern engines (i.e., lower emission engines with larger bypass ratios), the interaction of the engine fan and the fan wake with the outlet guide vanes (OGVs), a stage placed downstream of the fan to straighten the flow, is the primary source of engine noise.

[0030] To reduce the noise generated by the fan, engine manufacturers have tested several different solutions. One such solution is the placement of an acoustic liner on the inside surface of the engine nacelle. The liner is a passive element that can absorb pressure fluctuations near the wall to reduce the noise radiated by the engine. In its basic layout, the liner includes a perforated surface over a series of hollow cells. The passage of air in and out of the cells through a perforated plate creates a damping reaction to external pressure fluctuations that can reduce the intensity of the pressure fluctuations in a given frequency range. The shape and dimensions of the cells affect the range of frequencies over which the liner provides greater or less noise reduction.

[0031] In recent years, new liner layouts have emerged that offer enhanced noise reduction capabilities. Moving away from the more basic form of a perforated plate on top of a series of rectangular cells, new designs utilize non-uniform perforation patterns and cell layouts. These new liners have increased the range of frequencies over which they can reduce noise.

[0032] The ability to reduce noise at a given frequency is measured in terms of impedance. Impedance is equivalent to resistance in a complex space and can be calculated through experimental testing of simple liner geometries. However, with the appearance of more complex liner designs, standard empirical methods for measuring impedance have failed. For this reason, new indirect methods (both empirical and numerical) have gained popularity in recent years. These indirect methods generally replace local velocity and pressure measurements in the vicinity of the liner with an analysis of the change in pressure wave intensity as it travels over the top of the liner.

[0033] At the same time, several publications have shown in detail how the flow behavior inside the liner can be numerically modeled. Solvers based on the Lattice Boltzmann Method (LBM) are particularly well suited for this type of application due to their ability to deal with complex geometries and their inherent low numerical dissipation, a key factor in high-fidelity acoustic computations.

[0034] The embodiments propose a novel simulation-based process to calculate the equivalent impedance of an arbitrarily complex liner. One embodiment uses SIMULIA PowerFLOW® by applicant-assignee Dassault Systemes Simulia Corporation to obtain real-time measurements from the liner (replacing the need for real-world testing) and pairs the real-time measurements from the liner with an indirect impedance estimation method that determines the impedance of the liner by running several reduced simulations on an equivalent acoustic model of the original liner.

[0035] Being able to numerically calculate the impedance of a liner saves a lot of time in physical testing. Furthermore, acoustic measurements in real-world scenarios are inevitably subject to large uncertainties. The embodiments presented herein implement an automated process for calculating the equivalent impedance of a generic liner. The embodiments are independent of the specific liner layout and can be adopted for any complex liner geometry.

[0036] FIG. 1 is a flow chart of a method 100 for determining the impedance of a liner according to one embodiment. Method 100 is computer-implemented such that functions and operative operations, e.g., steps 101-104, may be implemented automatically by one or more digital processors. Additionally, method 100 may be implemented using any computer device or combination of computing devices known in the art. Among other examples, method 100 may be implemented using computer system 880, described herein below in connection with FIG. 8, and computer network environment 990, described below in connection with FIG. 9.

[0037] Method 100 begins at step 101 by defining a 3D computer-based model of the liner. Then, at step 102, a digital experiment of the liner is performed on the environment using the defined 3D computer-based model of the liner. One embodiment implements the digital experiment according to procedures known to those skilled in the art. For example, one such embodiment utilizes known procedures for setting mesh size and sampling frequency. The results of performing the digital experiment at step 102 include a reference transfer function. To proceed, at step 103, a 2D model of the environment is generated. The liner is represented in the generated 2D model of the environment by acoustic impedance boundary conditions having impedance values, the impedance values ​​being defined by a resistance value, a reactance value, and a nonlinear coefficient value. Then, at step 104, method 100 iteratively (i) modifies the impedance values, and (ii) performs a 2D simulation using the generated 2D model of the environment with the acoustic impedance boundary conditions having the modified impedance values, until the transfer function resulting from performing the 2D simulation matches the reference transfer function. In other words, the iterations in step 104 set impedance values ​​for the boundary conditions (representing the liner), perform a 2D simulation using the set impedance values ​​to determine a resulting transfer function, and compare the resulting transfer function to a reference transfer function (determined in step 102). This function is repeated until the two transfer functions (the transfer function determined in step 102 and the transfer function determined in step 104) match. The modified impedance values ​​used in method 100 to perform the 2D simulation that result in a transfer function that matches the reference transfer function is the acoustic impedance of the liner.

[0038] One embodiment implements the 2D simulation according to procedures known to those skilled in the art. For example, one such embodiment utilizes known procedures for setting the mesh size and sampling frequency of the 2D simulation.

[0039] According to one embodiment, defining the 3D computer-based model of the liner in step 101 includes receiving a computer-aided design (CAD) model of the liner and identifying (i) one or more parts of the liner and (ii) dimensions of the one or more parts based on the received CAD model. In one embodiment, the CAD model of the liner is formed, i.e., defined by entities such as points, lines, and surface elements. In such an embodiment, elements in the received model are grouped and the groups are named, for example, prior to receiving the model. One embodiment identifies the locations of the parts and liner parts, as well as dimensions of the parts, by accessing the grouping and group name data and calculating therefrom the minimum / maximum coordinates of the triangles that make up each group. To proceed, a computational surface mesh representing the liner is generated based on the identified one or more parts and dimensions of the one or more parts of the liner. In such an embodiment, the generated computational surface mesh is the defined 3D computer-based model of the liner in step 101. Further, an example of a CAD model 220 of the liner that may be used in an embodiment of the method 100 is described herein below in connection with FIG. 2A. Additionally, in another embodiment, dimensions of a real-world liner are measured and the dimensions are used in step 101 to generate a computer-based model of the liner.

[0040] Another embodiment of method 100 generates a 3D model of the environment. According to such an embodiment, the generated 3D model of the environment includes a defined 3D computer-based model of the channel and the liner (from step 101). In such an embodiment, the defined 3D computer-based model of the liner is disposed at the bottom surface of the channel. An example of such an environment model 330 is described herein below in relation to FIG. 3A. In one embodiment, generating the 3D model of the environment includes at least one of: (i) defining a length of the channel according to a wavelength of a pressure wave in the flow (e.g., in the channel); and (ii) defining a location of a solid trip in the 3D model based on the velocity of the flow (e.g., if a flow exists).

[0041] Yet another embodiment of the method 100 receives an indication of the test conditions. Examples of test conditions that may be received are listed in Table 1. Such an embodiment performs digital experimentation of the liner in the environment at step 102 using (i) the defined 3D computer-based model of the liner, (ii) the generated 3D model of the environment, and (iii) the received indication of the test conditions. In one embodiment, the received indication of the test conditions includes flow conditions. Furthermore, in another embodiment, performing digital experimentation of the liner in the environment at step 102 using (i) the defined 3D computer-based model of the liner, (ii) the generated 3D model of the environment, and (iii) the received indication of the test conditions includes collecting pressure data from one or more digital sensors in the channel while subjecting the defined 3D computer-based model of the liner in the generated 3D model of the environment to the test conditions. Such an embodiment may generate a reference transfer function by calculating a Fourier transform of the collected pressure data. An exemplary transfer function 662 that may be generated at step 102 is shown in FIG. 6 and described herein below.

[0042] In one embodiment, the digital experiment performed in step 102 is a computational fluid dynamics (CFD) simulation. In such an embodiment, the digital experiment of the liner in the environment is performed in step 102 using (i) a defined 3D computer-based model of the liner, (ii) a generated 3D model of the environment, and (iii) a received representation of the test conditions. Furthermore, performing the experiment in step 102 includes (1) generating a CFD input file based on (i) the defined 3D computer-based model of the liner, (ii) the generated 3D model of the environment, and (iii) the received representation of the test conditions, and (2) performing a CFD simulation using the generated CFD input file. One embodiment generates the CFD input file and performs the CFD simulation using procedures known to those skilled in the art. For example, one such embodiment utilizes known procedures for setting mesh size and sampling frequency of the CFD simulation.

[0043] In one embodiment of the method 100, the channel of the 3D environment is an elongated box, and in step 103, the 2D model is generated by replacing the elongated box with a rectangle of similar dimensions to the elongated box. Furthermore, the 3D liner is replaced by a line in the 2D domain, i.e., the 2D model, to which an impedance boundary condition is applied. In one embodiment, the impedance boundary condition coincides with the 3D liner axial position and length. In yet another embodiment of the method 100, the generated 2D model of the environment generated in step 103 is a mesh-based model. One such embodiment of the method 100 further includes at least one of: defining a pressure wave; setting a resolution of the mesh-based model as a function of the wave packet wavelength of the defined pressure wave; and performing a flow convergence simulation to determine field data. According to one embodiment, the flow convergence simulation is performed to simulate a period during which a stable velocity and pressure field in the channel is achieved using an automatically determined setup (e.g., a setup based on the time required for the flow to move from the trip to the liner). In one embodiment, when a flow convergence simulation begins, the pressure is gradually increased until a momentum balance between pressure and velocity is reached and the flow rate is stationary in time (steady flow condition). Further, according to one embodiment, exemplary field data includes pressure, temperature, density, velocity, and turbulence (K and omega).

[0044] According to one embodiment of method 100, the transfer function resulting from performing the 2D simulation (at step 102) matches the reference transfer function (at step 104) when a difference metric between (i) the transfer function resulting from performing the 2D simulation and (ii) the reference transfer function is below a threshold. Difference metrics that may be utilized in embodiments include least squared error and L2 norm, among other examples.

[0045] As described above, in method 100, the impedance value is defined by a resistance value, a reactance value, and a nonlinear coefficient value. Thus, in one embodiment, modifying the impedance value in step 104 includes modifying at least one of the resistance value, the reactance value, and the nonlinear coefficient value. According to an exemplary embodiment, the resistance and reactance are the real and imaginary parts of the complex impedance, respectively, and the nonlinear coefficient is an additional real parameter that affects the resistance. In one embodiment, the nonlinear coefficient value depends on the local velocity of the flow, e.g., the first derivative of the airflow across the liner. According to one embodiment, the local velocity of the flow is related to the velocity close to the wall of the liner.

[0046] FIG 2A shows an example of a liner shape 220 including a perforated plate 221 and an underlying array of cells 222, for which an embodiment can be used to determine impedance. Additionally, FIG 2B is a top view 223 of the plate 221, and FIG 2C is a top view of the array of cells 222. In one embodiment, the liner shape 220 is stored in a CAD file that includes a digital representation of a portion of a real-world liner. In one embodiment, a user prepares the shape 220 by isolating the holes in the perforated plate 221 from the rest of the shape. The shape 220 may be received in step 101 of the method 100 and used to define a 3D computer-based model of the liner.

[0047] FIG. 3A illustrates a 3D numerical experiment setup 330 according to one embodiment. The setup 330 is an example of a 3D numerical experiment setup that may be utilized in step 102 of the method 100. In one embodiment, the setup 330 is created by importing a 3D liner model 331 and placing the model 331 in a flat channel 332 with a perforated plate at the floor of the channel 332. In the channel 332, air can flow in a flow direction from the inlet section 333 to the outlet 334 (grazing flow case, i.e., flow moving on the liner and "scraping" the surface of the liner) or there is no average flow movement (no flow case). In one embodiment, a solid trip 335 may be placed upstream of the liner 331 to force a turbulent transition in the grazing flow case. According to one embodiment, the solid trip 335 is a solid element that is a stepped structure in the channel 332. The solid trip 335 creates turbulence in the flow near the channel 332 floor. In setup 330 , a traveling pressure wave 336 is generated upstream of liner 331 and moves in the flow direction during a simulation performed using setup 330 .

[0048] In the experimental setup 330, the liner 3D module 331 is a periodic shape along both the flow and span directions of the channel. This model 331 represents a portion of a larger liner. According to one embodiment, the liner module 331 models 8-10 cells in the flow direction, while a minimum of 1 cell is modeled along the span direction.

[0049] The setup 330 also includes a linear rack of microphones 337, or other such measurement equipment, positioned above the liner 331. According to one embodiment, the microphone numbers and locations may be modified by the user. Multiple racks of microphones 337 may also be used. One embodiment utilizes 100 microphones 337 arranged on a single linear rack at the height of the central channel 332. Additionally, it is noted that embodiments may utilize probes, e.g., microphones, on multiple linear racks.

[0050] FIG. 3B is a visual depiction of a 3D numerical experiment setup 340 according to another embodiment. Setup 340 includes a liner 341 and a channel 342. Channel 342 is composed of elements with various properties. For example, channel 342 includes an inlet 343, an outlet 344, free slip wall portions (walls with wear, i.e., no friction) 345a-c, a measurement plane 348, and a sponge region (region of high flow viscosity) 347a-b. For grazing flows, channel 342 is defined to include non-slip wall portions (walls with wear) portions 346a-b and a solid trip 349. Similar to setup 330, in setup 340, a traveling pressure wave 350 is generated upstream of liner 341 and travels in the flow direction 351 during a simulation performed using setup 340. Setup 340 also includes a linear rack of microphones 351.

[0051] In an embodiment, the length of the channels, e.g., 332 and 342, can be set to accommodate a certain number of waves (15+) before and after the liners, e.g., 331 and 341. Additionally, in an embodiment simulating grazing flows, a PowerFLOW® simulation is run at the beginning of the process to obtain a boundary layer developed on the liner. The setups 330 and 340 can be used to run a single frequency simulation at a time with a specific plane wave shape. An embodiment sets up a given pressure wave, e.g., 336, 350, upstream of the liner, e.g., 331, 341, via dedicated code (OptydB_fieldmod) that can access and manipulate the 3D simulation results files. Additionally, in an embodiment, a snapshot of the 3D flow field with added pressure waves is used as the initial condition for the acoustic run.

[0052] In one embodiment, after performing 3D numerical experiments (e.g., in step 102 of method 100), a reduced 2D model is derived (e.g., in step 103 of method 100). FIG. 4A shows an exemplary 2D model 440, according to one embodiment, derived from the setup 330 of FIG. 3A. The model 440 is an example of an environmental model that may be generated in step 103 of method 100 of FIG. 1. The model 440 includes a channel 441 with a channel length similar to the 3D model, and the 3D liner, e.g., 331, is replaced by a patch with the same flow direction extension of the 3D liner and boundary impedance conditions 442 that model the acoustic effects of the liner. The 2D model 440 uses the time-averaged velocity, pressure, temperature, and turbulence field datasets taken from the 3D simulation to account for the effects of the mean airflow (this dataset is not required for the no-flow case). The traveling waves 443 are set up similarly to the 3D case, with the same frequency. Additionally, channel 441 includes a linear rack of microphones 444 for collecting, for example, pressure data.

[0053] 4B shows an exemplary 2D model 450, according to one embodiment, derived from results generated using the setup 340 of FIG. 3B. Model 450 includes a channel 451, comprised of a standard portion 452 and a sponge region 453. Channel 451 has an inlet 454 and an outlet 455. In model 450, a pressure wave 458 is set at the inlet 454. 2D model 450 replaces filter 341 from 3D setup 340 with a wall patch impedance term 456. Additionally, channel 451 includes a linear rack of microphones 457 for collecting data.

[0054] In one embodiment, the 2D models, e.g., 440 and 450, are represented by rectangular element meshes. Furthermore, according to one embodiment, the reduced 2D models (e.g., 440, 450) are solved in the acoustic frequency domain by using a Finite Element Method (FEM) code (OptydB_gfd) that solves acoustic wave propagation in a 2D channel. Working in the frequency domain allows for the use of complex impedance boundary conditions on the liners, e.g., 442, 456. According to one embodiment, the simulation is run for a single frequency at a time, utilizing impedance boundary conditions (frequency domain) to model the liners. For grazing flows, the average flow field is interpolated on the 2D mesh using time-averaged results from a PowerFLOW® simulation (3D simulation). A plane pressure wave is set at the inlet of the domain. The mesh resolution is defined as the minimum between the channel height divided by 100 and 1 / 30 of the pressure wave wavelength.

[0055] 5 is a flow chart of a liner impedance estimation process 550 according to one embodiment. Process 550 includes three stages: stage 1, stage 2, and stage 3. Stage 1 is a pre-processing stage that includes steps 551-553. Stage 2 is a 3D simulation stage that includes step 554, and stage 3 includes steps 555-558.

[0056] During pre-processing (phase 1), 3D numerical experimental tests are automatically generated based on the liner geometry and a set of input parameters specified by the user. Specifically, phase 1 begins in step 551 by importing or otherwise defining the liner geometry, i.e., model, and flow conditions of the setup. In step 552, the liner model is meshed and part names of the model are assigned according to given guidelines. In one embodiment, the liner model is meshed and part names of the model are assigned using SIMULIA PowerDELTA® by applicant-assignee Dassault Systemes Simulia Corporation. Next, in step 553, solver input files are constructed. Constructing the solver input files in step 553 includes constructing a digital model (e.g., model 330) and generating solver input files based on the digital model. One embodiment builds a digital model and generates solver input files using functionality within SIMULIA PowerCASE® by applicant-assignee Dassault Systemes Simulia Corporation, where a Python-based environment is scripted to automatically import the liner model, generate simulation entities, set up measurement domains, and generate SIMULIA PowerFLOW® input files.

[0057] A 3D simulation is then performed on the local or remote cluster (Stage 2) at step 554. According to one embodiment, SIMULIA PowerFLOW® is used to perform the simulation at step 554. During the simulation (554), pressure signals are collected on a linear rack of microphones positioned above the liner, recorded and stored. One embodiment uses SIMULIA PowerACOUSTICS® by applicant-assignee Dassault Systemes Simulia Corporation to collect this data.

[0058] Once the 3D simulation is complete (step 554), stage 3 begins. At step 555, the 3D simulation data (resulting from the simulation at step 554) is processed. Processing at step 555 includes calculating the complex Fourier transform of each microphone signal. According to one embodiment, the Fourier transform is calculated using SIMULIA PowerACOUSTICS®. A reference transfer function is determined at step 555 from a reference of the ratio between the complex sound pressure of each microphone at a given frequency of interest and the first microphone of the rack (the one at the most upstream position). The resulting curve, i.e., the reference transfer function, measures the change in intensity versus the selected frequency of the pressure wave traveling through the top of the liner being simulated (performed at step 554).

[0059] Once the reference transfer function is obtained at step 555, process 550 moves to step 556 where a 2D reduced model (e.g., 440) is constructed from the 3D simulation data. An optimization loop is then initiated at step 557 where several simulations are submitted using the 2D reduced model (generated at step 556). According to one embodiment, the optimization loop runs a series of 2D simulations in the frequency domain to match the transfer function calculated from the PowerFLOW® results generated using the 3D model with the results from the 2D model.

[0060] The resistance and reactance are the real and imaginary parts of the complex impedance of the liner. The optimization loop implemented in step 557 works by independently modifying these two quantities together with a third parameter, i.e., a nonlinear coefficient value, to bias the impedance of the liner of the reduced model (i.e., 2D model). According to one embodiment, this third parameter is an additional impedance coefficient that depends on the derivative of the first velocity of the flow across the liner. According to one embodiment, the velocity is a local velocity, e.g., the velocity of the flow close to the wall of the liner. In one embodiment, the third parameter is used to take into account some degree of nonlinear effects that cannot be ignored for a particular liner layout. From the initial guesses of the resistance, reactance, and nonlinear coefficient, at each optimization step (of the optimization loop 557), the boundary conditions that model the liner are changed and the transfer function is re-evaluated. This process continues until the error between the reference transfer function (generated in step 555 based on the 3D simulation data from step 554) and the transfer function from the reduced model falls below a specified threshold. Once this occurs, the optimization loop 557 is considered converged in step 558 and the final value of the impedance used as the boundary condition in the reduced model is assumed to be the original 3D liner equivalent impedance.

[0061] It should be noted that the illustrated process 550 provides the impedance of the liner for a given frequency. Thus, process 550 may be repeated for each additional frequency of interest. Additionally, in embodiments of method 550, the standard SIMPLEX algorithm may be used in stage 3, for example, in step 557. While SIMPLEX is robust and simple, any multi-parameter optimization algorithm may be implemented in embodiments.

[0062] Embodiments, such as process 550, may utilize input parameters. Table 1 below lists exemplary input parameters that may be used by embodiments. Table 1 represents an exemplary set of parameters used in stage 1, such as steps 551-553 of process 550.

[0063] [Table 1]

[0064] FIG. 6 is a schematic diagram of an impedance estimation method 660 according to one embodiment. The process 660 begins by generating a reference transfer function 662 using a 3D numerical experiment setup 661. The 3D numerical experiment setup 661 is also used to generate a 2D reduced model 663. The 2D model 663 has boundary conditions with impedance values, and this model 663 is used to determine a transfer function 664 of the reduced model. The reference transfer function 662 and the transfer function 664 of the reduced model are compared 665 to determine whether the transfer functions 662 and 664 match. If the transfer functions (662 and 644) match, the impedance is found 666. Specifically, the impedance value of the reduced model 663 used to generate the transfer function 664 that matches the transfer function 662 is the impedance of the liner. However, if the impedances indicated by the functions 662 and 664 are determined not to match in step 665, the process 660 moves to step 667 where the impedance of the liner is modified. This modified 667 impedance is used in the model 663 and the process 660 continues until a match is found in step 665.

[0065] In one embodiment, the OptydB_gfd FEM solver can handle an additional parameter when modeling the liner. This additional parameter is a coefficient that adds a nonlinear contribution proportional to the first derivative of the velocity near the wall. Such an embodiment then works by minimizing the error between the reference transfer function (e.g., 662) and the reduced model transfer function (e.g., 664) by modifying three parameters: (1) the resistance of the liner, (2) the reactance of the liner, and (3) the nonlinear coefficient of the liner.

[0066] The embodiments can also handle cases with grazing flow. Figure 7A is a visualization of a grazing flow 2D experimental setup 770 according to one embodiment. The setup 770 includes a liner 771 and a channel 772. The channel 772 is composed of elements with various properties. More specifically, the channel 772 includes an inlet 773, an outlet 774, free slip wall portions 775a-c, non-slip wall portions 776a-b, sponge regions 777a-b, and a solid trip 778. A time-averaged wall boundary layer velocity profile 779 upstream from the liner 771 is provided at the input to the process.

[0067] In the case of grazing flow, an embodiment automatically calculates the distance of the solid trip 778 from the liner 771 to match a given velocity profile 779. Additionally, the mesh resolution near the channel floor is refined sufficiently to resolve the boundary layer between the trip 778 and the liner 771. Such an embodiment may also run additional flow convergence simulations at the beginning of the process to obtain initial field data.

[0068] 7B is a visualization 780 of grazing flow on a liner. Visualization 780 shows a free sliding top plate 781, a solid trip 782, and turbulent flow 783.

[0069] One embodiment provides a fully automated process for calculating the impedance of a generic liner. An exemplary process according to one embodiment is divided into three steps: (1) generating a virtual test model using a periodic 3D module of the liner, (2) performing a 3D high-fidelity CFD analysis, and (3) running an optimization loop using a 2D reduced model to match the impedance of the original liner. According to one embodiment, the match between the 3D model and the reduced 2D model is measured by the degree of similarity of a specific transfer function measured in both cases.

[0070] The embodiments allow for a fully automated execution of the three steps mentioned above. Moreover, the embodiments can be run on a computer cluster and replace real-world experimental testing with a high-fidelity acoustic simulation of the original liner. Such a fully computational process can replace the time-consuming and complex experimental testing that is typically required to determine the impedance of a liner.

[0071] Exemplary embodiments are directed to a computer-implemented automated methodology for calculating the equivalent impedance of an arbitrarily complex liner. Such embodiments import a file containing a digitized representation of a three-dimensional liner shape in a virtual test environment, where the liner is mounted on the floor of a flat channel, with grazing flow and traveling pressure waves. Such embodiments then perform a high-fidelity CFD simulation to calculate the transfer function along a series of microphones placed above the liner. A reduced 2D model is then generated that is equivalent to the previously simulated 3D numerical test environment. In the 2D model, the liner is replaced by an acoustic impedance boundary condition. To proceed, several short simulations are performed using the reduced model in an optimization loop, where the impedance of the liner is adjusted to match the reference transfer function obtained from the 3D simulation.

[0072] In one embodiment, importing the file includes reading the original liner CAD model, identifying the main parts of the liner by name (e.g., the liner surface and the holes in the perforated plate), generating a computational surface mesh, and obtaining the dimensions of said main parts of the liner by accessing the coordinates of the liner mesh elements. Furthermore, one embodiment generates ancillary virtual entities such as channels, measurement surfaces and points, and channel subdomains around the liner to create a 3D virtual test environment including a flat channel with the liner resting on the channel floor. As part of the virtual test setup, flow conditions can be set based on (i) user inputs and (ii) boundary conditions that can generate both grazing flow on the liner and traveling pressure waves in the channel. Furthermore, as part of the process during import, a CFD solver input file can be generated.

[0073] According to another embodiment, performing a CFD simulation comprises: Using a scheduling system to submit single or multiple runs on a remote or local cluster, and using a centralized storage memory to store data generated by the 3D engine simulation for subsequent post-processing.

[0074] Further, in one embodiment, generating the reduced 2D model includes executing a script or other such automated procedure to read the 3D solver input file and generate a 2D reduced model from the 3D solver input file, in which the liner is replaced by a surface patch having a given impedance boundary condition.

[0075] In one embodiment, the aforementioned 2D simulation performed by executing a script or alternative automated process uses a script or process that can run the simulation using a previously generated 2D reduced model, calculate a liner transfer function of the reduced model, implement an optimization algorithm that can identify a new temporary impedance value based on the error between the reference transfer function of the liner and the transfer function of the reduced model, and declare convergence of the optimization loop when the error falls below a given threshold to define the original 3D liner equivalent impedance.

[0076] Advantageously, the embodiments can perform liner impedance calculations in an automatic manner, which, according to one embodiment, results from the tools utilized to implement the embodiments and from the proposed methodology presented herein.

[0077] Additionally, the embodiments reduce the cost and time required to obtain accurate liner impedance values.

[0078] The embodiments may determine the impedance of a liner that exists in the real world. In such embodiments, the real world liner is analyzed, e.g., measured, and the results are used to build a model that is used in the embodiments. In this manner, such embodiments may determine the impedance of the real world liner.

[0079] Additionally, results of the embodiments can be used to select among multiple different candidate liners. For example, multiple real-world liners can be evaluated using the embodiments, and a given liner can be selected from the multiple based on the determined impedance of each liner, e.g., a liner that meets an impedance requirement can be selected. Furthermore, such a liner can be incorporated into another real-world object, e.g., a jet engine. Furthermore, after determining the impedance of a liner, the embodiments can manufacture said liner for a real-world application.

[0080] Computer Support FIG. 8 is a simplified block diagram of a computer-based system 880 that may be used to implement any of the various embodiments of the invention described herein. The system 880 comprises a bus 883. The bus 883 serves as an interconnect between the various components of the system 880. Connected to the bus 883 is an input / output device interface 886 for connecting various input and output devices, such as a keyboard, mouse, display, speakers, etc., to the system 880. A central processing unit (CPU) 882 is connected to the bus 883 and provides for the execution of computer instructions implementing the embodiments, such as methods 100, 550, 660, etc. The memory 885 provides volatile storage for data used to execute computer instructions implementing the embodiments described herein, such as the embodiments previously described herein above. The storage device 884 provides non-volatile storage for software instructions, such as an operating system (not shown) and embodiment configurations. The system 880 also comprises a network interface 881 for connecting to any of the various networks known in the art, including wide area networks (WANs) and local area networks (LANs).

[0081] Of course, the exemplary embodiments described herein may be implemented in many different ways. In some instances, the various methods and systems described herein may each be implemented by a physical, virtual, or hybrid general-purpose computer, such as computer system 880, or a computer network environment, such as computer environment 990 described herein below in connection with FIG. 9. Computer system 880 may be converted into a system that executes the methods described herein (e.g., 100, 550, 660), for example, by loading software instructions into either memory 885 or non-volatile storage 884 for execution by CPU 882. Those skilled in the art should further appreciate that system 880 and its various components may be configured to implement any embodiment or combination of embodiments of the invention described herein. Additionally, system 880 may utilize any combination of hardware, software, and firmware modules operably coupled internally or externally to system 880 to implement the various embodiments described herein.

[0082] 9 illustrates a computer network environment 990 in which embodiments of the present invention may be implemented. In the computer network environment 990, a server 991 is linked to clients 993a-n via a communication network 992. The environment 990 may be used to enable the clients 993a-n, alone or in combination with the server 991, to perform any of the embodiments described herein. As non-limiting examples, the computer network environment 990 may provide cloud computing embodiments, software as a service (SAAS) embodiments, and the like.

[0083] The embodiments or aspects thereof may be implemented in the form of hardware, firmware, or software. If implemented in software, the software may be stored on any non-transitory computer-readable medium configured to enable a processor to load the software, or a subset of its instructions. The processor is then configured to execute the instructions to operate an apparatus or cause it to operate in a method described herein.

[0084] Furthermore, firmware, software, routines, or instructions may be described herein as performing certain operations and / or functions of a data processor, however, it will be understood that such descriptions contained herein are merely for convenience and that such operations may in fact result from a computing device, processor, controller, or other device executing the firmware, software, routines, instructions, etc.

[0085] It will be understood that the flow diagrams, block diagrams, and network diagrams may include more or fewer elements, may be arranged differently, or may be represented differently, but it will also be understood that a particular implementation may implement the block diagrams and network diagrams, and the number of block diagrams and network diagrams illustrating the implementation of an embodiment, in a particular way.

[0086] Accordingly, further embodiments may also be implemented in various computer architectures, physical computers, virtual computers, cloud computers, and / or some combination thereof, and thus the data processors described herein are intended to be illustrative only and not limiting of the embodiments.

[0087] The teachings of all patents, published applications, and references cited herein are incorporated by reference in their entirety.

[0088] While exemplary embodiments have been particularly shown and described, it will be understood by those skilled in the art that various changes in form and details may be made therein without departing from the scope of the embodiments encompassed by the appended claims.

[0089] For example, the foregoing description and details of the illustrated embodiments refer to applicant-assignee (Dassault Systemes Simulia Corporation) and Dassault Systemes tools and platforms for purposes of illustration, but not limitation. Other similar tools and platforms are suitable.

Claims

1. 1. A computer-implemented method for determining the acoustic impedance of a liner, said method comprising: defining a three-dimensional (3D) computer-based model of the liner; conducting a digital experiment of the liner in an environment using the defined 3D computer-based model of the liner, wherein results of conducting the digital experiment include a reference transfer function; generating a two-dimensional (2D) model of the environment, wherein the liner is represented in the generated 2D model of the environment by an acoustic impedance boundary condition having an impedance value, the impedance value being defined by a resistance value, a reactance value, and a nonlinear coefficient value that depends on a first derivative of flow across the liner; iteratively (i) modifying the impedance values ​​until a transfer function resulting from performing a 2D simulation matches the reference transfer function; and (ii) performing a 2D simulation using the generated 2D model of the environment with the acoustic impedance boundary conditions having the modified impedance values, wherein the modified impedance values ​​used in the 2D simulation that result in the transfer function matching the reference transfer function are the acoustic impedance of the liner.

2. defining the 3D computer-based model of the liner; receiving a computer-aided design (CAD) model of the liner; (i) identifying one or more components of the liner and (ii) dimensions of the one or more components based on the received CAD model; and generating a computational surface mesh representing the liner based on the identified one or more parts of the liner and dimensions of the one or more parts, the generated computational surface mesh being the defined 3D computer-based model of the liner.

3. 2. The method of claim 1, further comprising generating a 3D model of the environment, wherein the generated 3D model of the environment includes the defined 3D computer-based model of a channel and the liner, and wherein the defined 3D computer-based model of the liner is positioned on a bottom surface of the channel.

4. generating the 3D model of the environment, defining the length of said channel according to the wavelength of pressure waves in the flow; and defining a location of a solid trip within the 3D model of the environment based on the velocity of the flow.

5. receiving an indication of test conditions; 4. The method of claim 3, further comprising: (i) conducting the digital experiment of the liner in the environment using the defined 3D computer-based model of the liner, (ii) the generated 3D model of the environment, and (iii) the received indications of test conditions.

6. The method of claim 5 , wherein the received indication of a test condition includes a flow condition.

7. conducting the digital experiment of the liner in the environment using (i) the defined 3D computer-based model of the liner, (ii) the generated 3D model of the environment, and (iii) the received indication of test conditions; 6. The method of claim 5, comprising collecting pressure data from one or more digital sensors in the channel while subjecting the defined 3D computer-based model of the liner in the generated 3D model of the environment to the test conditions.

8. The method of claim 7 , further comprising generating the reference transfer function by calculating a Fourier transform of the collected pressure data.

9. the digital experiment is a computational fluid dynamics (CFD) simulation, and conducting the digital experiment of the liner in the environment using (i) the defined 3D computer-based model of the liner, (ii) the generated 3D model of the environment, and (iii) the received indication of test conditions; generating a CFD input file based on (i) the defined 3D computer-based model of the liner, (ii) the generated 3D model of the environment, and (iii) the received indication of test conditions; and performing the CFD simulation using the generated CFD input file.

10. 2. The method of claim 1 , wherein the transfer function resulting from performing the 2D simulation matches the reference transfer function when a difference metric between (i) the transfer function resulting from performing the 2D simulation and (ii) the reference transfer function is below a threshold.

11. 10. The method of claim 1, further comprising: determining, at a given iteration, the modified impedance value based on a difference between (i) a given transfer function resulting from performing the 2D simulation and (ii) the reference transfer function.

12. modifying the impedance value The method of claim 1 , comprising modifying at least one of the resistance value, the reactance value, and the nonlinear coefficient value.

13. The method of claim 1 , wherein the flow across the liner is near a wall of the liner.

14. wherein the generated 2D model of the environment is a mesh-based model, and the method comprises: Defining a pressure wave; setting a resolution of the mesh-based model as a function of a wavelength of the defined pressure wave packet; and performing a flow convergence simulation to determine field data.

15. 1. A system for determining the acoustic impedance of a liner, the system comprising: a processor; and a memory having computer code instructions stored thereon, wherein said processor and said memory use said computer code instructions to cause said system to: defining a three-dimensional (3D) computer-based model of the liner; conducting a digital experiment of the liner in an environment using the defined 3D computer-based model of the liner, wherein results of conducting the digital experiment include a reference transfer function; generating a two-dimensional (2D) model of the environment, wherein the liner is represented in the generated 2D model of the environment by an acoustic impedance boundary condition having an impedance value defined by a resistance value, a reactance value, and a nonlinear coefficient value that depends on a first derivative of flow across the liner; a system configured to iteratively (i) modify the impedance values; and (ii) perform a 2D simulation using the generated 2D model of the environment with the acoustic impedance boundary conditions having the modified impedance values ​​until a transfer function resulting from performing a 2D simulation matches the reference transfer function, wherein the modified impedance values ​​used to perform the 2D simulation that result in the transfer function matching the reference transfer function are the acoustic impedance of the liner.

16. In defining the 3D computer-based model of the liner, the processor and the memory use the computer code instructions to instruct the system to: receiving a computer-aided design (CAD) model of the liner; (i) identifying one or more components of the liner and (ii) dimensions of the one or more components based on the received CAD model; 16. The system of claim 15, further configured to generate a computational surface mesh representing the liner based on the identified one or more parts of the liner and the dimensions of the one or more parts, wherein the generated computational surface mesh is the defined 3D computer-based model of the liner.

17. The processor and the memory use the computer code instructions to cause the system to:

16. The system of claim 15, further configured to generate a 3D model of the environment, the generated 3D model of the environment including the defined 3D computer-based model of a channel and the liner, the defined 3D computer-based model of the liner being positioned on a bottom surface of the channel.

18. The processor and the memory use the computer code instructions to cause the system to: receiving an indication of test conditions; 20. The system of claim 17, further configured to: (i) perform the digital experiment of the liner in the environment using the defined 3D computer-based model of the liner, (ii) the generated 3D model of the environment, and (iii) the received instructions of the test conditions.

19. When modifying the impedance value, the processor and the memory use the computer code instructions to cause the system to: The system of claim 15 , configured to modify at least one of the resistance value, the reactance value, and the non-linear coefficient value.

20. 1. A computer program product for determining the acoustic impedance of a liner, said computer program product comprising: one or more non-transitory computer-readable storage devices; and program instructions stored in at least one of the one or more storage devices, the program instructions, when read and executed by a processor, causing a device associated with the processor to: defining a three-dimensional (3D) computer-based model of the liner; conducting a digital experiment of the liner in an environment using the defined 3D computer-based model of the liner, wherein results of conducting the digital experiment include a reference transfer function; generating a two-dimensional (2D) model of the environment, wherein the liner is represented in the generated 2D model of the environment by an acoustic impedance boundary condition having an impedance value defined by a resistance value, a reactance value, and a nonlinear coefficient value that depends on a first derivative of flow across the liner; 1. A computer program product that iteratively (i) modifies the impedance values; and (ii) performs a 2D simulation using the generated 2D model of the environment with the acoustic impedance boundary conditions having the modified impedance values ​​until a transfer function resulting from performing a 2D simulation matches the reference transfer function, wherein the modified impedance values ​​used to perform the 2D simulation that result in the transfer function matching the reference transfer function are the acoustic impedance of the liner.

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