A fully numerical liner impedance estimation 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 available on the market for designing and simulating objects, such as vehicles. These 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, e.g., 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 specifications of the shape. In particular, a CAD file contains the specifications from which the shape is generated. From the shape, a three-dimensional CAD model or model representation is generated. The specification, shape, and CAD model / representation may be stored in a single CAD file or multiple CAD files. CAD systems or other such CAE systems include graphical tools to visually represent the modeled object as it appears in three-dimensional space to the designer; these tools are specialized for displaying complex real-world objects. For example, an assembly may contain thousands of parts.

[0003] The advent of CAD and CAE systems has enabled a wide range of representation possibilities for objects, such as CAD models. Computer-based models may be programmed to have the properties (e.g., physical, material, or other physics-based) of the underlying real-world object they represent. 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 it 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. Furthermore, CAD and CAE systems, along with computer-based models, may be utilized to simulate real-world physical systems, e.g., engineering systems such as automobiles, airplanes, buildings, and bridges, among other examples. Furthermore, CAE systems can 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] Noise simulation 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 increasingly complex, requiring improved methods for simulating objects and determining properties of said objects. 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 measured, with impedance values ​​corresponding to resistance values, reactance values, and Depends on the first derivative of the flow across the liner The generated 2D model of the environment is represented by acoustic impedance boundary conditions having impedance values ​​defined by the nonlinear coefficient values. The method then iteratively (i) modifies the impedance values ​​and (ii) performs a 2D simulation using the generated 2D model of the environment with 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 to perform 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 a 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 pressure waves in the flow, and (ii) defining a location of a solid trip within the 3D model based on the velocity of the flow.

[0008] Yet another embodiment is a test condition instructions receiving (i) a defined 3D computer-based model of the liner, (ii) a generated 3D model of the environment, and (iii) a received 3D model of the test conditions; instructions In one embodiment, the digital experiment of the liner is performed in an environment using the received instructions contains the flow conditions. In addition, the received test conditions instructions may also include boundary conditions, such as an initial guess for the liner's impedance boundary conditions (used to create a 2D model of the environment). Additionally, in another embodiment, the 3D model may be generated based on (i) a defined 3D computer-based model of the liner, (ii) a generated 3D model of the environment, and (iii) a received 3D model of the test conditions. instructions Conducting digital experiments of a liner in an environment using the method includes collecting pressure data from one or more digital sensors in the channel while subjecting a defined 3D computer-based model of the liner within the generated 3D model of the environment to test conditions. Such embodiments may generate a reference transfer function by calculating a Fourier transform of the collected pressure data.

[0009] In one embodiment, the digital experiment is a computational fluid dynamics (CFD) simulation. In such an embodiment, the digital experiment involves (i) a defined 3D computer-based model of the liner, (ii) a generated 3D model of the environment, and (iii) a received 3D model of the test conditions. instructionsConducting digital experiments of a liner in an environment using (1) (a) a defined 3D computer-based model of the liner, (b) a generated 3D model of the environment, and (c) a received 3D model of the test conditions. instructions and (2) performing 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 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.

[0012] Yet another embodiment includes determining a modified impedance value at a given iteration. One 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 one embodiment, the given transfer function may be from an iteration prior to the given iteration. Still further, one embodiment determines the modified impedance value (i.e., the next impedance value under test) by considering the difference between (i) multiple transfer functions, e.g., a subset of transfer functions that produced 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 non-linear coefficient value. Flow across the liner teeth , which is the flow close to the wall of the liner .

[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, the program instructions, when read and executed by a processor, causing 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, pressure signals above the liner are 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 that matches 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 simulation data.

[0019] In one embodiment, the simulation data (i.e., data from the 3D digital experiment) is processed using a series of scripts that can read the time domain data, transform it into 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 the series of time domain pressure signals collected from the numerical tests.

[0020] In an exemplary workflow, a user prepares the liner geometry, e.g., a surface mesh, according to a set of guidelines that specify the nomenclature to use for associated 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 enables such an embodiment to automatically identify 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] One 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 liner's physical testing is constructed. In other words, a digital twin of the real-world liner testing is created. Stage 2 uses the digital twin to perform numerical testing, and several 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 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 computer program products tangibly stored on non-transitory computer-readable media, and computing systems such as computer systems and computer servers. [Brief explanation of the drawings]

[0023] The foregoing will be 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 flowchart 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. [Figure 5] FIG. 5 is a flow chart of a process for estimating the impedance of a liner according to one embodiment. [Figure 6] FIG. 6 is a flowchart of a method for determining the impedance of a liner according to one embodiment. [Figure 7A] FIG. 7A is a visualization of a grazing flow experimental setup according to one embodiment. [Figure 7B] FIG. 7B is a diagram of glazing flow on the 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 INVENTION

[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 increasingly complex, and improved methods for simulating objects and determining the properties of such objects are needed. Embodiments provide such functionality.

[0027] 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 intake air. At the same time, in the outer section of the engine, the fan draws in cool air and accelerates it without subjecting it to thermal cycling. This secondary section, called a bypass, generates thrust without increasing engine exhaust.

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

[0030] To reduce fan-generated noise, engine manufacturers have tested several different solutions. One such solution is the placement of acoustic liners on the interior surface of the engine nacelle. Liners are passive elements that can absorb near-wall pressure fluctuations to reduce noise radiated by the engine. In its basic layout, a liner includes a perforated surface over a series of hollow cells. The passage of air into and out of the cells through a perforated plate creates a damping response to external pressure fluctuations that can reduce the intensity of pressure fluctuations in a given frequency range. The shape and dimensions of the cells affect the frequency range over which the liner provides significant or minimal 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 fail. 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 calculations.

[0034] 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 liner's impedance by running several reduced-size simulations on an equivalent acoustic model of the original liner.

[0035] The ability to numerically calculate liner impedance saves significant 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 shape.

[0036] FIG. 1 is a flowchart of a method 100 for determining the impedance of a liner, according to one embodiment. Method 100 is computer-implemented, such that the functions and operative operations, e.g., steps 101-104, may be implemented automatically by one or more digital processors. Furthermore, 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 herein below in connection with FIG. 9.

[0037] Method 100 begins in step 101 by defining a 3D computer-based model of the liner. Next, in step 102, a digital experiment of the liner is performed in 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 in step 102 include a reference transfer function. To proceed, in 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 ​​defined by resistance values, reactance values, and nonlinear coefficient values. Then, in step 104, method 100 iteratively (i) modifying the impedance values ​​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 ​​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 a 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 components of the liner and (ii) dimensions of the one or more components 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, before receiving the model. One embodiment identifies the locations of the components and liner components, as well as the dimensions of the components, by accessing the grouping and group name data and calculating the minimum / maximum coordinates of the triangles that make up each group therefrom. To proceed, a computational surface mesh representing the liner is generated based on the identified one or more components and dimensions of the one or more components 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 a liner that may be used in an embodiment of method 100 is described herein below in connection with FIG. 2A. Additionally, in another embodiment, the dimensions of the real-world liner are measured and 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 of the channel. An example of such an environment model 330 is described herein below in connection with 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 pressure waves in the flow (e.g., in the channel); and (ii) defining a location of a solid trip within the 3D model based on the velocity of the flow (e.g., if flow is present).

[0041] Yet another embodiment of the method 100 is to instructions Examples of test conditions that may be received are listed in Table 1. Such an embodiment may include: (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. instructions In one embodiment, the received test conditions are used to perform a digital experiment of the liner in the environment in step 102. instructions Further, in another embodiment, the 3D model includes a received 3D model of the test conditions, the 3D model includes a defined 3D computer-based model of the liner, the generated 3D model of the environment, and the received 3D model of the test conditions. instructions Conducting a digital experiment of the liner in the environment in step 102 using the liner-based modeling algorithm includes collecting pressure data from one or more digital sensors in the channel while subjecting a defined 3D computer-based model of the liner within the generated 3D model of the environment to 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 in 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 using (i) a defined 3D computer-based model of the liner, (ii) a generated 3D model of the environment, and (iii) a received 3D model of the test conditions. instructions Further, performing the experiment in step 102 includes: (1) using (i) a defined 3D computer-based model of the liner, (ii) a generated 3D model of the environment, and (iii) a received 3D model of the test conditions. instructionsand (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 the mesh size and sampling frequency of a CFD simulation.

[0043] In one embodiment of method 100, the channel of the 3D environment is a long, narrow box, and in step 103, a 2D model is generated by replacing the long, narrow box with a rectangle of similar dimensions to the long, narrow 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 corresponds to the axial position and length of the 3D liner. In yet another embodiment of method 100, the generated 2D model of the environment generated in step 103 is a mesh-based model. One such embodiment of method 100 further includes at least one of defining pressure waves, setting the resolution of the mesh-based model as a function of the wave packet wavelength of the defined pressure waves, 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 stable velocity and pressure fields in the channel are 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 squares 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. Therefore, 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 near the wall of the liner.

[0046] FIG. 2A shows an example 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 plate 221, and FIG. 2C is a top view of cell array 222. In one embodiment, 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 shape 220 by separating holes in perforated plate 221 from the rest of the shape. Shape 220 may be received in step 101 of 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. Setup 330 is an example of a 3D numerical experiment setup that can be utilized in step 102 of method 100. In one embodiment, 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 streamwise direction from the inlet section 333 to the outlet 334 (grazing flow case, i.e., flow moving over the liner and "scraping" the surface of the liner) or there is no mean flow movement (no flow case). In one embodiment, a solid trip 335 can be placed upstream of the liner 331 to force a turbulent transition in the grazing flow case. According to one embodiment, solid trip 335 is a stepped solid element within channel 332. Solid trip 335 generates turbulence in the flow near the floor of channel 332. 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] Setup 330 also includes a linear rack of microphones 337, or other such measurement equipment, positioned above liner 331. According to one embodiment, 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 level of central channel 332. Additionally, it should be noted that embodiments may utilize probes, e.g., microphones, on multiple linear racks.

[0050] FIG. 3B is a visual depiction of a 3D numerical experimental 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 sections (walls with wear, i.e., frictionless walls) 345a-c, a measurement plane 348, and sponge regions (regions of high flow viscosity) 347a-b. For grazing flow, channel 342 is defined to include non-slip wall sections (walls with wear) 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 simulations 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 specific number of waves (15+) before and after the liners, e.g., 331 and 341. Additionally, in an embodiment simulating grazing flow, a PowerFLOW® simulation is run at the beginning of the process to obtain a boundary layer developed on the liner. Setups 330 and 340 can be used to run a single-frequency simulation at a time with a specific plane wave shape. One embodiment sets up given pressure waves, e.g., 336 and 350, upstream of the liner, e.g., 331 and 341, via dedicated code (OptydB_fieldmod) that can access and manipulate 3D simulation result files. Additionally, in one embodiment, a snapshot of the 3D flow field with the 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 setup 330 of FIG. 3A. Model 440 is an example of an environmental model that may be generated in step 103 of method 100 of FIG. 1. Model 440 includes a channel 441 with a channel length similar to that of 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 liner's acoustic effects. The 2D model 440 uses 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). 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 setup 340 of FIG. 3B. Model 450 includes a channel 451 composed 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 established at 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 2D channels. Working in the frequency domain allows for the use of complex impedance boundary conditions on 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 liner. For grazing flow, the mean flow field is interpolated on the 2D mesh using time-averaged results from a PowerFLOW® simulation (3D simulation). A plane pressure wave is established at the inlet of the domain. The mesh resolution is defined as the minimum value between the channel height divided by 100 and 1 / 30 of the pressure wave wavelength.

[0055] 5 is a flowchart 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 including steps 551-553. Stage 2 is a 3D simulation stage including step 554, and Stage 3 includes steps 555-558.

[0056] During preprocessing (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 setup flow conditions. In step 552, the liner model is meshed and model part names are assigned according to given guidelines. In one embodiment, the liner model is meshed and model part names 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 a local or remote cluster (Stage 2) in step 554. According to one embodiment, SIMULIA PowerFLOW® is used to perform the simulation in 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), phase 3 begins. In step 555, the 3D simulation data (resulting from the simulation in step 554) is processed. Processing in 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 in step 555 from a reference ratio between the complex sound pressure of each microphone at a given frequency of interest and the first microphone in the rack (the one in the most upstream position). The resulting curve, i.e., the reference transfer function, measures the change in intensity with selected frequency of pressure waves traveling over the liner being simulated (performed in step 554).

[0059] Once the reference transfer function is obtained in step 555, process 550 moves to step 556, where a 2D condensed model (e.g., 440) is constructed from the 3D simulation data. An optimization loop then begins in step 557, where several simulations are submitted using the 2D condensed model (generated in step 556). According to one embodiment, the optimization loop runs a series of 2D simulations in the frequency domain and uses results from the 2D model to match a transfer function calculated from PowerFLOW® results generated using the 3D 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 along with a third parameter, i.e., a nonlinear coefficient value, to alter the impedance of the liner in a reduced model (i.e., a 2D model). According to one embodiment, this third parameter is determined by the flow across the liner. Speed degree First derivative of, an additional impedance coefficient that depends on the liner's component value. According to one embodiment, the velocity is a local velocity, e.g., the velocity of the flow near the liner wall. In one embodiment, a third parameter is used to account, to some extent, for nonlinear effects that may not be negligible for a particular liner layout. From the initial guesses for the resistance, reactance, and nonlinear coefficients, at each optimization step (in optimization loop 557), the boundary conditions modeling 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, optimization loop 557 is considered converged in step 558, and the final value of impedance used as a boundary condition in the reduced model is assumed to be the original 3D liner equivalent impedance.

[0061] Note 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. Furthermore, 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. 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 condensed 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 condensed model. The reference transfer function 662 and the transfer function 664 of the condensed model are compared 665 to determine whether transfer functions 662 and 664 match. If the transfer functions (662 and 664) match, the impedance is found 666. Specifically, the impedance value of the condensed model 663 used to generate transfer function 664 that matches transfer function 662 is the impedance of the liner. However, if the impedances represented by functions 662 and 664 are determined to not match in step 665, process 660 moves to step 667, where the impedance of the liner is modified. This corrected 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 near-wall velocity. 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 liner resistance, (2) the liner reactance, and (3) the liner nonlinear coefficient.

[0066] Embodiments can also handle cases with grazing flow. FIG. 7A is a visualization of a grazing flow 2D experimental setup 770 according to one embodiment. Setup 770 includes a liner 771 and a channel 772. Channel 772 is composed of elements with various properties. More specifically, channel 772 includes an inlet 773, an outlet 774, free-slip wall sections 775a-c, no-slip wall sections 776a-b, sponge regions 777a-b, and a solid trip 778. A time-averaged wall boundary layer velocity profile 779 upstream from liner 771 is provided as an input to the process.

[0067] In the case of grazing flow, one 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 specific transfer functions measured in both cases.

[0070] Embodiments allow for the fully automated execution of the three aforementioned steps. Furthermore, embodiments can be run on a computer cluster to replace real-world experimental testing with high-fidelity acoustic simulations of the original liner. Such a fully computational process can replace the time-consuming and complex experimental testing 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. These embodiments import a file containing a digitized representation of a three-dimensional liner geometry in a virtual test environment, where the liner is mounted on a flat channel floor and has grazing flow and traveling pressure waves. These embodiments then perform a high-fidelity CFD simulation to calculate a transfer function along a series of microphones positioned 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 liner's impedance 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 key parts of the liner by name (e.g., the liner surface and holes in the perforated plate), generating a computational surface mesh, and obtaining dimensions of the key parts of the liner by accessing the coordinates of the liner mesh elements. Furthermore, one embodiment generates additional virtual entities around the liner, such as channels, measurement surfaces and points, and channel subdomains, 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 input and (ii) boundary conditions that can generate both grazing flow on the liner and traveling pressure waves in the channel. Furthermore, a CFD solver input file can be generated as part of the import process.

[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 the 2D reduced model from the 3D solver input file, wherein 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 a simulation using a previously generated 2D reduced model; calculate a liner transfer function for 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, 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, embodiments reduce the cost and time required to obtain accurate liner impedance values.

[0078] 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] Furthermore, results of embodiments can be used to select among multiple different candidate liners. For example, multiple real-world liners can be evaluated using 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, embodiments can manufacture the liner for a real-world application.

[0080] Computer Support FIG. 8 is a simplified block diagram of a computer-based system 880 that can be used to implement any of the various embodiments of the invention described herein. System 880 includes a bus 883. Bus 883 serves as an interconnect between the various components of system 880. Connected to bus 883 is an input / output device interface 886 for connecting various input and output devices, such as a keyboard, mouse, display, and speakers, to system 880. A central processing unit (CPU) 882 is connected to bus 883 and provides for the execution of computer instructions that implement embodiments, such as methods 100, 550, and 660. Memory 885 provides volatile storage for data used to execute computer instructions that implement embodiments described herein, such as those previously described herein. Storage device 884 provides non-volatile storage for software instructions, such as an operating system (not shown) and embodiment configurations. System 880 also includes 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] It should be understood that 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) by, for example, 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. Furthermore, system 880 may implement the various embodiments described herein utilizing any combination of hardware, software, and firmware modules operably coupled internally or externally to system 880.

[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 communications 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, etc.

[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 a device 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, although it will be understood that such descriptions contained herein are merely for convenience and that such operations actually 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 therefore 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, those skilled in the art will understand that various changes in form and details can 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) cause 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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