A modal expansion based method for simulating transient acoustic field in a closed space

Abstract

The application discloses a closed space transient sound field simulation method based on modal expansion. The wave equation is solved based on the modal expansion theory, and finally the sound field is represented as superposition of a series of sound modes, that is, represented as a linear combination of the product of modal expansion coefficients and characteristic functions, wherein the modal expansion coefficients are quantities related to time information, and the characteristic functions are quantities related to space information. The application establishes a calculation format fusing boundaries and internal spaces based on the decomposition of the time-domain wave equation, thereby avoiding the large calculation amount problem of the traditional finite difference method in time and space. A time scale stable model is established through the solution of the transient modal expansion coefficient, thereby avoiding the instability problem of the time-domain boundary element method in the time scale. The application realizes efficient and accurate simulation of the closed space transient sound field through accurate modeling of the propagation and attenuation process of the sound field in the closed space.

Description

Technical Field

[0001] This invention belongs to the field of acoustic simulation technology, specifically relating to a method for simulating transient sound fields in enclosed spaces. Background Technology

[0002] Currently, numerical calculation methods in closed space sound field simulation are mainly divided into three categories: sound field simulation methods based on geometric acoustics (sound ray tracking method, virtual sound source method), sound field simulation methods based on wave acoustics (finite element method, boundary element method, meshless method, finite difference method in time domain, boundary element method in time domain, etc.), and sound field simulation methods based on energy analysis (radiance method, statistical energy analysis method, etc.).

[0003] The sound ray tracing method and virtual sound source method based on geometric acoustics are mainly aimed at simulation research of large halls and rooms. They can realize numerical calculations for spaces of arbitrary shapes, but they can only realize high-frequency analysis, and it is a statistical analysis that cannot obtain accurate frequency response.

[0004] The finite element method (FEM) is a standard tool for calculating acoustic wave equations. It is simple, effective, and widely applicable in acoustic numerical calculations, and has been widely used in engineering practice. While it is a good method for acoustic numerical calculations, the FEM also has several problems. For example, the analysis wavenumber and accuracy are severely limited by numerical dispersion, and the requirements for model mesh quality are relatively high. The high stiffness of the FEM model leads to increased approximation errors, and severe numerical dispersion causes the accuracy of acoustic numerical calculations to decrease sharply with increasing calculation frequency. Therefore, in practical applications, the FEM is generally only used to calculate acoustic problems in the low-frequency range; its calculation errors are larger for mid-to-high-frequency problems. The boundary element method (BEM) is developed based on the finite element method (FEM). The BEM can reduce the dimensionality of acoustic problems, making it suitable for solving complex acoustic models with high computational accuracy. Theoretically, it can calculate high-frequency acoustic problems, making it a good numerical computation method for acoustics. However, the BEM also has many drawbacks, such as an asymmetric full matrix system matrix, low computational efficiency, complex matrix calculations, and a critical problem size limit, making it unsuitable for large-scale acoustic problems. These shortcomings lead many acoustic engineers to prefer the aforementioned finite element method for acoustic problems. To improve the accuracy of acoustic numerical computation, researchers have applied meshless techniques to the calculation of acoustic wave equations. Compared to the finite element method, the meshless method offers significantly improved computational accuracy, greatly reduces contamination errors caused by dispersion effects, and eliminates the need for mesh generation, simplifying the preprocessing process and saving considerable workload. However, due to the complexity of the interpolation function in the meshless method, as well as the instability of direct nodal integration and the difficulty in applying direct boundary conditions, the computation of acoustic problems remains complex, and the computational accuracy is still not high. In addition, the aforementioned finite element method, boundary element method, and meshless method are mainly used for simulation in the frequency domain and are primarily aimed at steady-state sound fields.

[0005] Finite-difference time-domain (FDTD) directly displays time-domain characteristics, making it relatively simple to solve flow problems in spatial coordinate systems. Based on its excellent time and spatial discretization, FDTD can completely display the iterative process of the sound field. However, FDTD requires rectangular meshes and has a long computation time, limiting its application. Boundary element time-domain (BEM) can also simulate time-domain sound fields. Since BEM does not require Fourier transforms for processing time-domain signals, its computational accuracy is not affected by factors such as winding errors, thus achieving high accuracy when processing broadband signals, such as pulse signals, impact signals, and some short-duration non-periodic signals. Furthermore, because the coefficient matrix of BEM is sparse, its computation speed is relatively fast. However, when the computation time is long, BEM suffers from time-domain instability. In recent years, many scholars have studied this instability. Rizos et al. proposed a TBEM using a B-spline interpolation function as the fundamental solution. This method achieves high computational accuracy, and its stability is independent of the time step selection. Chappell et al., based on Ergin et al.'s time-domain Burton-Miller method, derived the time-domain Burton-Miller formula for radiation problems. Addressing the difficulty in handling hypersingular integrals, Chappell et al. transformed hypersingular integrals into weakly singular integrals using Taylor series expansion, making computation easier, and this method is applicable to high-order discrete units. Wang et al., based on Walker et al.'s analysis of the eigenvalues ​​of the TBEM single-iteration matrix, further studied the relationship between eigenvalues ​​and cavity resonant frequencies, proving from both theoretical and numerical analysis that eigenvalues ​​close to 1 correspond one-to-one with the cavity resonant frequencies. Soares et al. proposed a time-domain process to improve stability while reducing computation time. This process only modifies the convolution-related vectors in TBEM, thus easily applying it to existing code. Furthermore, this method introduces a stabilizing parameter, allowing near-time and far-time integrals to be implemented through matrix interpolation; choosing an appropriate stabilizing parameter achieves both stability and efficiency. Stutz and Ochmann proposed the time-domain CHIEF method. Similar to the frequency-domain CHIEF method, this method adds constraint points inside the cavity, thereby increasing the number of equations and stabilizing the time-domain solution. However, unlike the frequency-domain CHIEF method, the time-domain CHIEF method requires solving for all unknowns at all time points at once, rather than using the MOT method. This results in the computation of a very large matrix, and the dimension of this matrix increases exponentially with computation time, which is largely limited by computer performance in practical applications. Furthermore, while this method achieves stabilization, it cannot eliminate the influence of characteristic frequencies, which is also different from the frequency-domain CHIEF method.Falletta proposed a novel integration technique to improve the accuracy of integration in TBEM, reducing numerical errors in integration calculations and thus enhancing temporal stability. This method requires combination with a fine-grained temporal interpolation function to achieve high-precision results using only a minimal number of integration points. However, the methods described above only suppress temporal instability and do not fundamentally solve the problem. Summary of the Invention

[0006] To overcome the shortcomings of existing technologies, this invention provides a method for simulating transient sound fields in closed spaces based on modal expansion. It solves the wave equation based on modal expansion theory, ultimately representing the sound field as a superposition of a series of sound modes, i.e., a linear combination of the products of modal expansion coefficients and characteristic functions. The modal expansion coefficients are time-dependent quantities, while the characteristic functions are spatially dependent quantities. This invention establishes a computational format for fusing the boundary and internal space based on the decomposition of the time-domain wave equation, avoiding the large computational burden in both time and space aspects of the traditional finite difference method. Furthermore, it establishes a time-scale stable model by solving for the transient modal expansion coefficients, avoiding the instability issues that arise in the time-scale boundary element method. This invention achieves efficient and accurate simulation of transient sound fields in closed spaces by accurately modeling the propagation and attenuation processes of the sound field.

[0007] The technical solution adopted by this invention to solve its technical problem includes the following steps:

[0008] Step 1: Perform geometric modeling and mesh generation for the enclosed space;

[0009] Step 2: Modeling the sound field in a closed space. When there is no volume sound source inside the sound field, the spatiotemporal distribution of the sound field is determined by the wave equation and boundary conditions.

[0010] Wave equation:

[0011]

[0012] In the formula For the Laplace operator, in a three-dimensional Cartesian coordinate system, p(r,t) is the sound pressure at position r at time t; c is the speed of sound in the air; t0 is the time when the sound source emits sound; r0 is the position of the sound source;

[0013] Boundary conditions:

[0014]

[0015] In the formula For the gradient operator, in a three-dimensional Cartesian coordinate system, n is the outward normal vector of the closed space wall; β is the specific admittance.

[0016] Step 3: Solve the wave equation. According to modal expansion theory, the solution to the wave equation is:

[0017]

[0018] Where Φ m (r) is the modal function of the closed space, p m (t) represents the modal expansion coefficients; m represents the modal order;

[0019] Step 4: Determine the mode functions based on the room normal mode theory:

[0020]

[0021] In the formula, m is the modal order, and x, y, z are the coordinates of a point in space. Let l represent the nth mode in the x, y, and z directions, respectively. x l y l z Let x, y, and z represent the lengths in the x, y, and z directions, respectively, and V be the volume of the space.

[0022] The modal function has the following properties:

[0023] (1) The modal functions satisfy the orthogonality property in space Ω:

[0024]

[0025] (2) Modal function Φ m Its corresponding modal frequency ω m It has the following relationship:

[0026]

[0027] Step 5: Solve for the modal expansion coefficients. Solving for the modal expansion coefficients depends on the transformation of the wave equation (1). First, multiply both sides by Φ. m By performing a space Ω-integral on the equation, we obtain:

[0028]

[0029] According to Green's theorem:

[0030]

[0031] Transforming equation (8) yields:

[0032]

[0033] Substituting equation (9) into equation (7), we get:

[0034]

[0035] Substituting equations (2), (3), (5), and (6) into equation (10) yields the modal expansion coefficient p. m Differential equation:

[0036]

[0037] Solving equation (11) yields:

[0038]

[0039] In the formula S The surface area of ​​the enclosed space;

[0040] Step 6: Solve for the spatial impulse response. From equations (3), (4), and (12), we get:

[0041]

[0042] Step 7: Solve for the spatial sound field distribution. According to the sound propagation theory, the spatial sound field distribution is determined by the convolution integral of the sound source signal and the impulse response.

[0043]

[0044] In the formula, q(r0,t0) is the sound source signal; v0 is the sound source volume.

[0045] Preferably, the In a three-dimensional Cartesian coordinate system

[0046] The beneficial effects of this invention are as follows:

[0047] This invention represents a sound field as a superposition of a series of sound modes, enabling rapid and accurate simulation of transient sound fields in enclosed spaces. It also allows for the visualization of sound wave propagation, reflection, and attenuation phenomena in space. This invention provides accurate data for applications such as sound source localization, sound source identification, noise control, sound quality design, and sound environment monitoring in enclosed spaces, possessing significant theoretical and practical engineering value. Attached Figure Description

[0048] Figure 1 This is for spatial geometric modeling and sound source location in an embodiment of the present invention.

[0049] Figure 2 This is a schematic diagram of the spatial model mesh division in an embodiment of the present invention.

[0050] Figure 3The following are the sound field distributions at different times on the xy plane where the sound source is located in the embodiment of the present invention: (a) is the sound field distribution at 1ms, (b) is the sound field distribution at 1.8ms, (c) is the sound field distribution at 2.2ms, and (d) is the sound field distribution at 2.5ms. Detailed Implementation

[0051] The present invention will be further described below with reference to the accompanying drawings and embodiments.

[0052] This invention relates to the field of acoustic simulation, specifically to a method for simulating transient sound fields in enclosed spaces based on modal expansion. This method establishes a computational format for merging the boundary and internal space based on the decomposition of the time-domain wave equation, avoiding the large computational burden in both time and space aspects of the traditional finite difference method. By solving for the transient modal expansion coefficients, a time-scale stable model is established, avoiding the instability issues that arise in the time-scale boundary element method. Ultimately, the sound field is represented as a superposition of a series of sound modes, enabling efficient and accurate simulation of transient sound fields in enclosed spaces. This invention can provide accurate basis for applications such as sound source localization, sound source identification, noise control, sound quality design, and sound environment monitoring in enclosed spaces, possessing significant theoretical and practical engineering application value.

[0053] 1) Construct characteristic functions for a closed space;

[0054] The characteristic functions of a regular cubic space can be expressed in specific forms using normal mode theory:

[0055]

[0056] 2) Solve for the modal expansion coefficients;

[0057] Modal expansion coefficients are fundamental to constructing time-domain propagation models. Therefore, after constructing the characteristic functions, we derive the modal characteristic functions. The modal expansion coefficients are obtained by transforming and solving the wave equation. First, the wave equation is transformed into a volume integral equation; then, Green's theorem is applied to transform the volume integral equation into a boundary integral equation; next, boundary conditions and modal theory are used to simplify the boundary integral into a non-homogeneous differential equation; finally, the parameter transformation method proposed in mathematics is used to solve this equation to obtain the modal expansion coefficients.

[0058]

[0059] 3) Construct a closed-space temporal propagation model;

[0060] Based on the characteristic functions and modal expansion coefficients obtained in steps 1) and 2), construct a time-domain propagation model of sound waves in a closed space, and calculate the impulse response of this closed space:

[0061]

[0062] Finally, convolving the sound source and impulse response yields the spatial sound field distribution:

[0063]

[0064] Example:

[0065] The enclosed space is a box with a length, width and height of 1m. The sound source is located at the center of the space with coordinates (0.5, 0.5, 0.5). The sound source signal is a pulse signal.

[0066] Step 1: Perform geometric modeling and mesh generation for the enclosed space, as shown in the attached diagram. Figure 1 Appendix Figure 2 As shown.

[0067] Step 2: Modeling the sound field in a closed space. When there is no volume sound source inside the sound field, the spatiotemporal distribution of the sound field is determined by the wave equation and boundary conditions.

[0068] Wave equation:

[0069]

[0070] In the formula For the Laplace operator, in a three-dimensional Cartesian coordinate system,

[0071] Boundary conditions:

[0072]

[0073] Step 3: Solve the wave equation. According to modal expansion theory, the solution to the wave equation is:

[0074]

[0075] Where Φ m (r) is the modal function of the closed space, p m (t) represents the modal expansion coefficients.

[0076] Step 4: Obtain the modal functions. For a regular, closed space like the one in the simulation case, the modal functions are obtained based on the room normal mode theory:

[0077]

[0078] In the formula, m is the modal order, and x, y, z are the coordinates of a point in space. Let l represent the nth mode in the x, y, and z directions, respectively. x l y l zLet x, y, and z represent the lengths in the x, y, and z directions, respectively, and V be the volume of the space.

[0079] The modal function has the following properties:

[0080] (1) These modal functions satisfy the orthogonality property in space Ω:

[0081]

[0082] (2) Each mode function Φ m Its corresponding modal frequency ω m It has the following relationship:

[0083]

[0084] Step 5: Solve for the modal expansion coefficients. Solving for the modal expansion coefficients depends on a proper transformation of the wave equation (5). First, multiply both sides by Φ. m By performing a space Ω-integral on the equation, we obtain:

[0085]

[0086] According to Green's theorem:

[0087]

[0088] Transforming equation (12) yields:

[0089]

[0090] Substituting equation (13) into equation (11), we get:

[0091]

[0092] Substituting equations (6), (7), (9), and (10) into equation (14) yields the modal expansion coefficients p. m Differential equation:

[0093]

[0094] Solving equation (15) yields:

[0095]

[0096] In the formula t0 is the sound source emission time, and r0 is the sound source location coordinates.

[0097] Step 6: Solve for the spatial impulse response. From equations (7), (8), and (16), we get:

[0098]

[0099] Step 7: Solve for the spatial sound field distribution. According to the theory of sound propagation, the spatial sound field distribution can be determined by the convolution integral of the sound source signal and the impulse response.

[0100]

[0101] The numerical simulation results of this embodiment are attached. Figure 3 As shown in the figure, the present invention can intuitively observe the propagation process of sound waves in space and can achieve efficient and accurate simulation of transient sound fields in enclosed spaces.

Claims

1. A method for simulating transient sound fields in a closed space based on modal expansion, characterized in that, Includes the following steps: Step 1: Perform geometric modeling and mesh generation for the enclosed space; Step 2: Modeling the sound field in a closed space. When there is no sound source inside the sound field, the spatiotemporal distribution of the sound field is determined by the wave equation and boundary conditions. Wave equation: (1) In the formula For the Laplace operator, in a three-dimensional Cartesian coordinate system, ; Let r be the sound pressure at time t; The speed of sound in air; The moment when the sound source emits sound; Location of the sound source; Boundary conditions: (2) In the formula For the gradient operator, in a three-dimensional Cartesian coordinate system, ; The vector representing the outward normal to the wall of the closed space; For specific admittance; Step 3: Solve the wave equation. According to modal expansion theory, the solution to the wave equation is: (3) in It is a mode function of a closed space. is the modal expansion coefficient; m is the modal order; Step 4: Determine the mode functions based on the room normal mode theory: (4) In the formula m Let x, y, and z be the modal order, and x, y, and z be the coordinates of a point in space. , , These are represented as the nth order modes in the x, y, and z directions, respectively. , , Let x, y, and z represent the lengths in the x, y, and z directions, respectively, and V be the volume of the space. The modal function has the following properties: (1) Modal functions in space The following properties are satisfied: (5) (2) Modal function Its corresponding modal frequencies It has the following relationship: (6) Step 5: Solve for the modal expansion coefficients. Solving for the modal expansion coefficients depends on the transformation of the wave equation (1). First, multiply both sides by the transformation. Then perform a spatial operation on the equation. Integrating, we get: (7) According to Green's theorem: (8) Transforming equation (8) yields: (9) Substituting equation (9) into equation (7), we get: (10) Substituting equations (2), (3), (5), and (6) into equation (10) yields the modal expansion coefficients. Differential equation: (11) Solving equation (11) yields: (12) In the formula , , The surface area of ​​the enclosed space; Step 6: Solve for the spatial impulse response. From equations (3), (4), and (12), we get: (13) Step 7: Solve for the spatial sound field distribution. According to the sound propagation theory, the spatial sound field distribution is determined by the convolution integral of the sound source signal and the impulse response. (14) In the formula The sound source signal; Let V be the volume of the sound source.

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