A method for analyzing the characteristics of guided wave acoustic fields in fluid-saturated porous materials based on semi-analytical finite element method

Abstract

The present invention proposes a method for analyzing the guided wave acoustic field characteristics of fluid-saturated porous materials based on a semi-analytical finite element method. The method comprises the following steps: obtaining a stress-strain relationship based on the material parameters of the fluid-saturated porous material; obtaining a wave equation for the fluid-saturated porous material based on the stress-strain relationship; deriving expressions for open-pore boundary conditions and closed-pore boundary conditions based on the surface treatment method of the fluid-saturated porous material; obtaining the wave equation and boundary conditions for the fluid-saturated porous material in the form of an eigenvalue problem using the semi-analytical finite element method; converting the wave equation and boundary conditions in the eigenvalue problem form into the standard form of a coefficient-type partial differential equation in commercial finite element software; and solving the eigenvalue problem to obtain the guided wave acoustic field characteristics of the fluid-saturated porous material. The method can solve the guided wave acoustic field characteristics of fluid-saturated porous materials of arbitrary cross-sections with low computational effort, high computational speed, low computational cost, and low probability of missing solutions.

Description

Technical Field

[0001] The present invention relates to the technical field of guided wave detection, and in particular to a method for analyzing characteristics of guided wave acoustic fields of fluid-saturated porous materials. Background Art

[0002] In recent years, with the advancement of materials science, artificial fluid-saturated porous materials, such as nanomaterials, artificial bones, and lithium batteries, have been widely used. Due to their complex internal structures and service environments, fluid-saturated porous materials are highly susceptible to various types of combined damage, including corrosion, fracture, and porosity mutations. The gradual development of these damages can severely impact the service life of fluid-saturated porous materials, leading to unpredictable casualties and property losses, including environmental pollution, mechanical structural failure, and medical accidents. Therefore, health assessment and structural health monitoring of fluid-saturated porous materials are necessary to ensure their safe and stable service.

[0003] During their propagation, elastic waves can carry information such as the physical properties and geometric morphology of the waveguide. The rough characteristics of the waveguide can be preliminarily obtained by using information such as the attenuation, flight time, reflection and refraction of the body wave. Traditional body wave detection technology has played a positive role in pipeline transportation, aerospace, medical imaging and other fields. However, the morphology of new artificial fluid-saturated porous materials is becoming increasingly complex, and the application scenarios are becoming increasingly diverse. Body wave detection technology can no longer meet the detection needs of fluid-saturated porous materials. Ultrasonic guided wave detection technology breaks through the limitations of point-by-point scanning of body wave detection. It has the advantages of global scanning, multi-modal selection, large detection range and high sensitivity. It has broad application prospects in the field of detection of fluid-saturated porous materials.

[0004] Research on bulk wave propagation in porous materials has been widely applied in geophysical fields such as acoustic logging and seismic exploration, but research on guided waves is still in its infancy. The main difference between guided and bulk wave governing equations lies in the introduction of boundary conditions when solving the guided wave problem, resulting in a theoretically infinite number of guided wave modes. Before using guided waves for testing, it is necessary to determine the dispersion characteristics of ultrasonic guided waves propagating in fluid-saturated porous materials and then select the appropriate mode and frequency for testing. The success of ultrasonic guided wave testing depends largely on the choice of guided wave mode, so accurately solving the governing equations for guided waves is key to the successful application of ultrasonic guided wave testing technology in fluid-saturated porous materials. The interaction between pore fluid and the solid skeleton and the complex boundary conditions make the governing equations for guided waves in fluid-saturated porous materials complex and difficult to solve. Traditional root search methods, such as the global matrix method and the transfer matrix method, have been used to study guided wave propagation in fluid-saturated porous materials. However, solving the governing equations for guided waves in multi-layered and thick layers of fluid-saturated porous materials suffers from high computational complexity and poor stability. While spectral methods offer fast solutions and simple modeling, they are only applicable to simple structures such as layers and columns that can be accurately modeled using mathematical equations. They struggle with complex waveguide geometries. Wave finite element methods typically utilize the poroelastic wave frequency domain module of highly integrated commercial finite element software, which suffers from poor scalability and struggles to adapt to the complex and varied internal structures and geometries of fluid-saturated porous materials. Summary of the Invention

[0005] In response to the technical problems of existing guided wave acoustic field characteristic analysis methods for fluid-saturated porous materials, such as large computational complexity, slow calculation speed, poor stability, and a small scope of application, the present invention proposes a guided wave acoustic field analysis method for fluid-saturated porous materials based on the semi-analytical finite element method. The cross section of the waveguide is discretized into a finite element grid, and simple harmonic waves are used to describe the waveform in the propagation direction of the guided wave. The eigenvalue problem is established by introducing the displacement field into the control equation. This method not only reduces the computational complexity and is applicable to fluid-saturated porous materials of arbitrary cross-section, but also ensures calculation accuracy.

[0006] In order to achieve the above object, the technical solution of the present invention is achieved as follows:

[0007] A method for analyzing the characteristics of guided wave acoustic fields of fluid-saturated porous materials based on a semi-analytical finite element method comprises the following steps:

[0008] S1. Derivation of the stress-strain relationship of the fluid-saturated porous material based on the material parameters of the fluid-saturated porous material;

[0009] S2. Based on the stress-strain relationship of fluid-saturated porous materials, derive the wave equation of fluid-saturated porous materials;

[0010] S3. Based on the surface treatment of fluid-saturated porous materials, derive the expressions for open-pore boundary conditions and closed-pore boundary conditions;

[0011] S4. Establish a finite element model based on the semi-analytical finite element method, discretize the waveguide into a finite element network in the cross section, and use an analytical solution in the wave propagation direction to obtain the wave equation and boundary conditions of the fluid-saturated porous material in the form of an eigenvalue problem;

[0012] S5. Convert the wave equation and boundary conditions for fluid-saturated porous materials in the form of eigenvalue problems into the standard form of coefficient-type partial differential equations in commercial finite element software;

[0013] S6. Solve the eigenvalue problem and obtain the guided wave acoustic field characteristics of the fluid-saturated porous material.

[0014] Preferably, the method for deriving the stress-strain relationship of the fluid-saturated porous material is:

[0015] For a fluid-saturated and statistically isotropic porous material, the solid displacement u is defined as [u1,u2,u3] T , fluid displacement u f =[u f1 ,u f2 ,u f3 ] T , the relative displacement between fluid and solid w=φ(u f -u), where φ is the porosity, u1, u2 and u3 are the solid displacements in the x1, x2 and x3 directions respectively, and u f1 、u f2 and u f3 are the relative displacements of the fluid and the solid in the directions x1, x2, and x3, respectively. x1, x2, and x3 represent the three directions in the Cartesian coordinate system of three-dimensional space;

[0016] The poroelastic parameters M, H, and C are obtained from the known material parameters of the fluid-saturated porous material:

[0017]

[0018] Among them, K f , K d and K m are the bulk moduli of the pore fluid, the solid skeleton, and the particles that make up the skeleton, respectively. μ is the shear modulus of the fluid-saturated porous material. is the Biot-Willis coefficient;

[0019] The stress-strain relationship of the fluid-saturated porous material is expressed as:

[0020] σ ik =Cikjl ε jl -Cξδ ik ;

[0021] p f =-Cε ii +Mξ;

[0022] Among them, σ ik and ε jl are the stress and strain components of the solid skeleton, represents the increment of fluid content, ε ii =ε 11 +ε 22 +ε 33 is the intermediate parameter, ε 11 , ε 22 and ε 33 are the normal strains in the x1, x2 and x3 directions, p f is the stress of the pore fluid, is the Nabla operator, represents the divergence operation, δ ik is the Kronecker symbol, C ikjl is the generalized elastic constant of the solid skeleton of the fluid-saturated porous material. Due to the symmetry of the stress tensor and the strain tensor, the generalized elastic constant C of the solid skeleton of the fluid-saturated porous material is ikjl Converted into stiffness matrix C nm :

[0023]

[0024] Preferably, the method for deriving the wave equation of a fluid-saturated porous material is:

[0025] According to Biot's theory, the propagation of elastic waves in fluid-saturated porous materials can be expressed as:

[0026]

[0027] Where ρ = φρ f +(1-φ)ρ m is the density of the porous material, φ is the porosity, ρ m and ρ f are the solid density and the fluid density, respectively. q is the effective density of the fluid during relative motion between the fluid and the solid. The subscripts i=1, 2, and 3 represent the x1, x2, and x3 directions in the Cartesian coordinate system of three-dimensional space. is the Nabla operator, represents the divergence operation, σ represents the stress, express express t represents time;

[0028] Consider the simple harmonic oscillation of displacement and assume that the time-dependent term of displacement is e -Iωt , where I is the imaginary unit, ω = 2πf is the angular frequency, and f is the frequency. Therefore, the propagation mechanism of elastic waves in fluid-saturated porous materials is given by the wave equation, which is expressed as:

[0029]

[0030] q(ω) is the frequency-dependent effective density of the fluid during relative motion between the fluid and the solid.

[0031] Preferably, the frequency-dependent effective density of the fluid during relative motion between the fluid and the solid is in, η is the fluid viscosity, I is the imaginary unit, k0 is the low-frequency permeability, τ is the tortuosity, ω t is the transition frequency, Λ is the ratio of weighted volume to surface area, m is a dimensionless constant, and k(ω) is the dynamic permeability.

[0032] Preferably, the open-pore boundary condition represents the continuity of normal-direction motion at the boundary, and the pore fluid can flow through the surface; the closed-pore boundary condition corresponds to the closure of the surface pores, and the closure of the surface pores includes but is not limited to wrapping a waterproof layer, filling the surface pores, and polishing the surface.

[0033] Preferably, the method for deriving the expressions of the open hole boundary condition and the closed hole boundary condition is:

[0034] The stress vector T of the solid skeleton at any point on the boundary i Expressed as:

[0035]

[0036] Among them, n k is a unit vector perpendicular to the boundary and pointing outward, n1, n2, n3 are unit vectors in the directions of x1, x2, x3;

[0037] The stress vector p of the pore fluid at any point on the boundary is i Expressed as:

[0038]

[0039] Then the expression of the opening boundary condition is:

[0040] T i =0,p i =0;

[0041] The expression of the closed-cell boundary condition is:

[0042] Ti =0,w n =0;

[0043] Among them, w n It represents the normal component of the relative displacement between fluid and solid on the boundary.

[0044] Preferably, the method for obtaining the wave equation and boundary conditions of the fluid-saturated porous material in the form of an eigenvalue problem is:

[0045] The waveguide is discretized into a finite element mesh in the cross section using the semi-analytical finite element method, and an analytical solution is applied in the propagation direction. Assuming that the simple harmonic wave propagates along the x3 axis, the displacement in the waveguide is expressed as:

[0046]

[0047] Where I is the imaginary unit, K is the wave number, ω = 2πf is the angular frequency, and U i represents the spatial distribution function used to describe the solid displacement at any point in the waveguide medium, W i represents the spatial distribution function used to describe the relative displacement of the fluid and solid at any point in the waveguide medium. Therefore, the displacement gradient is expressed as:

[0048]

[0049] Substituting the above formula into the wave equation, we obtain the wave equation for fluid-saturated porous materials in the form of an eigenvalue problem:

[0050]

[0051] Where, the sum index j = 1, 2, 3, and the subscripts k, l = 1, 2;

[0052] Substitute the following equations into the expressions of the open hole boundary condition and the closed hole boundary condition respectively to obtain the boundary conditions in the form of eigenvalue problem;

[0053]

[0054] Preferably, the method for converting the wave equation and boundary conditions of the fluid-saturated porous material in the form of an eigenvalue problem into the standard form of a coefficient-type partial differential equation in commercial finite element software is:

[0055] In commercial finite element software, the standard form of the coefficient-type partial differential equation module is:

[0056]

[0057] where λ is the eigenvalue, is the independent variable required for the eigenvalue problem. For the two-dimensional model established by the semi-analytical method, the Nabla operator e a, d a ,c,α,γ,β,a,F are the coefficients of the coefficient-type partial differential equation. The flux boundary conditions and Dirichlet boundaries in the coefficient-type partial differential equation module are expressed as:

[0058]

[0059] Where n is the external unit vector of the solution domain, g is the boundary flux term, q is the boundary absorption term, and r is the specified value of the Dirichlet boundary condition;

[0060] To convert the wave equation and boundary conditions for a fluid-saturated porous material into standard form, the independent variable for:

[0061]

[0062] Among them, U k1 =KU1,U k2 =KU2,U k3 =KU3,W k1 =KW1,W k2 =KW2,W k3 =KW3;

[0063] The coefficients of the coefficient-type partial differential equation are:

[0064]

[0065] Among them, 0 is the zero matrix of the corresponding dimension;

[0066] The hole boundary condition corresponds to the zero flux boundary condition in the Coefficient Form PDE module:

[0067]

[0068] The closed-pore boundary condition involves constraints on stress and displacement, and requires the use of both flux boundary conditions and weak contribution boundary conditions; the pore fluid stress vector p on the boundary i = 0 is affected by the normal component w of the fluid-solid relative motion n =0 condition;

[0069] According to T i = 0, the boundary absorption term q and the boundary flux term g are:

[0070]

[0071]

[0072] According to the normal component w of the fluid-solid relative displacement n= 0, we can obtain w1n1 + w2n2 = 0;

[0073] w1n1 + w2n2 = 0 is implemented using the weak contribution boundary condition in the coefficient-type partial differential equation module, and the weak expression is:

[0074] Lam_a · [test(w1)n1 + test(w2)n2] + test(Lam_a) · (w1n1 + w2n2);

[0075] Among them, Lam_a is the auxiliary dependent variable, and test() is the test function built into the coefficient-type partial differential equation module.

[0076] Preferably, the sub-matrices in the coefficients of the coefficient-type partial differential equation are respectively:

[0077]

[0078]

[0079] Among them, C 11 、C 66 etc. are the elements at the corresponding positions in the stiffness matrix C nm in the corresponding positions.

[0080] Preferably, the method for solving the eigenvalue problem to obtain the guided wave sound field characteristics of the fluid-saturated porous material is: given the angular frequency ω, solving the eigenvalue equation can obtain the wave numbers K of all possible modes at the angular frequency ω; the wave numbers that conform to the physical laws are screened according to the following conditions:

[0081] real(K) > 0 and real(K) > abs(Im(K));

[0082] Among them, real() and Im() respectively represent the real part and the imaginary part of the complex number. In the required frequency range, repeating this process at a certain frequency step dω can obtain the corresponding K-ω relationship;

[0083] The phase velocity C p and the attenuation Att in the guided wave sound field characteristics are given by the following formula:

[0084]

[0085] The wave structure information in the guided wave sound field characteristics is given by the eigenvector corresponding to the wave number K under the given angular frequency ω. Compared with the prior art, the beneficial effects of the present invention are:

[0086] The present invention provides an innovative method for the analysis of guided wave acoustic field characteristics of irregular fluid-saturated porous materials; the semi-analytical finite element method proposed in the present invention has a small amount of calculation, fast calculation speed, and is not prone to missing solutions compared to traditional root search methods and spectral methods; a single solution can obtain the wave number and corresponding eigenvector at a given frequency, and wave structure information can be obtained without additional work; and it has strong scalability and is suitable for the analysis of guided wave acoustic field characteristics of irregularly shaped fluid-saturated porous materials; by encapsulating in finite element software, there is no need to write specific finite element codes for different waveguides; accurate guided wave acoustic field characteristic analysis promotes the application of guided wave detection technology in fluid-saturated porous materials. BRIEF DESCRIPTION OF THE DRAWINGS

[0087] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the following briefly introduces the drawings required for use in the embodiments or the description of the prior art. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying any creative work.

[0088] Figure 1 Flowchart of the present invention.

[0089] Figure 2 This is a schematic diagram of the semi-analytical finite element method modeling of the present invention.

[0090] Figure 3 Schematic diagram of the semi-analytical finite element model of the plate-like fluid-saturated porous material in Example 1 of the present invention.

[0091] Figure 4 The phase velocity dispersion curve and attenuation of the plate-like fluid-saturated porous material in Example 1 of the present invention are shown; wherein, Figure 4 -(a) is the phase velocity dispersion and attenuation characteristics of the plate-like fluid-saturated porous material under open-hole boundary conditions, Figure 4 -(b) Phase velocity dispersion and attenuation characteristics of plate-like fluid-saturated porous materials under closed-cell boundary conditions.

[0092] Figure 5 The displacement distribution near the position of the distorted narrow stop band of the S0 mode under the open boundary condition of the plate-like fluid saturated porous material in Example 1 of the present invention is shown; wherein, Figure 5 -(a) is Figure 4 - Partial magnification of the twisted narrow stopband in (a), Figure 5 -(b) Figure 5 -(c) and Figure 5 -(d) are Figure 5 -Displacement distribution of points A, B, and C in (a).

[0093] Figure 6 The solid displacement and fluid-solid relative displacement of the plate-like fluid-saturated porous material in the Lamb wave mode and high attenuation mode in Example 1 of the present invention are the changes and their ratios along the plate thickness direction; wherein, Figure 6 -(a) and Figure 6 -(b) are Lamb wave modes ( Figure 4 -(a) midpoint M) and high attenuation mode ( Figure 4 -(a) Displacement distribution of midpoint N).

[0094] Figure 7 Schematic diagram of the semi-analytical finite element model of the L-shaped columnar fluid-saturated porous material in Example 2 of the present invention.

[0095] Figure 8 The phase velocity dispersion curve and attenuation of the L-shaped columnar fluid-saturated porous material in Example 2 of the present invention are shown; wherein, Figure 8 -(a) is the phase velocity dispersion curve and attenuation of L-shaped columnar fluid saturated porous material under open boundary conditions, Figure 8 -(b) Phase velocity dispersion curve and attenuation of L-shaped columnar fluid-saturated porous materials under closed-pore boundary conditions. DETAILED DESCRIPTION

[0096] The following will clearly and completely describe the technical solutions in the embodiments of the present invention in conjunction with the accompanying drawings. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. All other embodiments obtained by ordinary technicians in this field based on the embodiments of the present invention without creative work are within the scope of protection of the present invention.

[0097] like Figure 1 As shown, a method for analyzing the guided wave acoustic field characteristics of fluid-saturated porous materials based on the semi-analytical finite element method is provided. The stress-strain relationship of the fluid-saturated porous material is derived according to the material parameters of the fluid-saturated porous material; the wave equation of the fluid-saturated porous material is derived according to the stress-strain relationship of the fluid-saturated porous material; the expressions of the open-pore boundary conditions and the closed-pore boundary conditions are derived according to the surface treatment method of the fluid-saturated porous material; according to the semi-analytical finite element method, a finite element model is established, the waveguide is discretized into a finite element network on the cross section, and an analytical solution is used in the wave propagation direction to establish a linear equation group containing unknown frequencies and wave numbers and boundary condition expressions; the guided wave control equation in the form of an eigenvalue problem is converted into a standard form of a coefficient-type partial differential equation in commercial finite element software; the eigenvalue problem is repeatedly solved in the required frequency range to obtain the guided wave acoustic field characteristics of the fluid-saturated porous material, such as phase velocity, attenuation and wave structure.

[0098] S1. Derivation of the stress-strain relationship of the fluid-saturated porous material based on the material parameters of the fluid-saturated porous material.

[0099] For a fluid-saturated and statistically isotropic porous material, the solid displacement u is defined as [u1,u2,u3] T , fluid displacement u f =[u f1 ,u f2 ,u f3 ] T , the relative displacement between fluid and solid w=φ(u f -u), where φ is the porosity, u1, u2 and u3 are the solid displacements in the x1, x2 and x3 directions respectively, and u f1 、u f2 and u f3 are the relative displacements of the fluid and the solid in the directions of x1, x2 and x3, respectively. x1, x2 and x3 represent the three directions in the Cartesian coordinate system of three-dimensional space.

[0100] The poroelastic parameters M, H, and C are obtained from the known material parameters of the fluid-saturated porous material:

[0101]

[0102] Among them, K f , K d and K m are the bulk moduli of the pore fluid, the solid skeleton, and the particles that make up the skeleton, respectively. μ is the shear modulus of the fluid-saturated porous material. is the Biot-Willis coefficient.

[0103] The stress-strain relationship of the fluid-saturated porous material is expressed as:

[0104] σ ik =C ikjl ε jl -Cξδ ik (2)

[0105] p f =-Cε ii +Mξ (3)

[0106] Among them, σ ik and ε jl are the stress and strain components of the solid skeleton, represents the increment of fluid content, ε ii =ε 11 +ε 22 +ε 33 is the intermediate parameter, ε 11 , ε 22 and ε33 are the normal strains in the x1, x2 and x3 directions, p f is the stress of the pore fluid, is the Nabla operator, represents the divergence operation, δ ik is the Kronecker symbol. C ikjl is the generalized elastic constant of the solid skeleton of the fluid-saturated porous material. Due to the symmetry of the stress tensor and the strain tensor, the fourth-order tensor C containing 81 elements ikjl Can be converted into a 6×6 stiffness matrix C nm :

[0107]

[0108] S2. Based on the stress-strain relationship of fluid-saturated porous materials, derive the wave equation of fluid-saturated porous materials.

[0109] According to Biot's theory, the propagation of elastic waves in fluid-saturated porous materials can be expressed as:

[0110]

[0111] Where ρ = φρ f +(1-φ)ρ m is the density of the porous material, φ is the porosity, ρ m and ρ f are the solid density and the fluid density, respectively. q is the effective density of the fluid during relative motion between the fluid and the solid. The subscripts i=1, 2, and 3 represent the x1, x2, and x3 directions in the Cartesian coordinate system of three-dimensional space. is the Nabla operator, represents the divergence operation, σ represents the stress, express express t represents time.

[0112] Consider the simple harmonic oscillation of displacement and assume that the time-dependent term of displacement is e -Iωt , where I is the imaginary unit, ω = 2πf is the angular frequency, and f is the frequency. Therefore, equation (5) is expressed as:

[0113]

[0114] q(ω) is the frequency-dependent effective density of the fluid during relative motion between the fluid and the solid, expressed as:

[0115]

[0116]

[0117] Where η is the fluid viscosity, I is the imaginary unit, k0 is the low-frequency permeability, τ is the tortuosity, and the transition frequency ω is t It is used to distinguish low-frequency viscous flow from high-frequency inertial flow. Λ represents the ratio of weighted volume to surface area, and the dimensionless constant m describes the spatial geometry of the pores. The dynamic permeability k(ω) describes the inertial coupling between the solid skeleton and the pore fluid in the general frequency-dependent model. As long as the wavelength of the wave in the pore fluid is much larger than the characteristic size of the pore, the dynamic permeability model (Eqs. (8) and (9)) is suitable for analyzing acoustic properties over the entire frequency range.

[0118] S3. In order to solve the wave equation of fluid-saturated porous materials, the expressions of open-pore boundary conditions and closed-pore boundary conditions are derived according to the surface treatment method of fluid-saturated porous materials.

[0119] The stress vector T of the solid skeleton on the boundary i Expressed as:

[0120]

[0121] Among them, n k is a unit vector perpendicular to the boundary and pointing outward, n1, n2, n3 are unit vectors in the x1, x2, x3 directions. The pore fluid stress vector p on the boundary i Expressed as:

[0122]

[0123] For porous materials in practical applications, two boundary conditions, open pore and closed pore, are considered.

[0124] The open pore boundary condition indicates the continuity of motion in the normal direction at the boundary, and the pore fluid can flow through the surface, which is expressed as:

[0125] T i =0,p i =0 (11)

[0126] The closed-pore boundary condition corresponds to the closure of surface pores, such as wrapping a waterproof layer, filling surface pores, or polishing the surface, and is expressed as:

[0127] T i =0,w n =0 (12)

[0128] Among them, w n It represents the normal component of the relative displacement between fluid and solid on the boundary.

[0129] S4. Based on the semi-analytical finite element method, a finite element model is established. The waveguide is discretized into a finite element network in the cross section. An analytical solution is used in the wave propagation direction to obtain the wave equation and boundary conditions of the fluid-saturated porous material in the form of an eigenvalue problem. That is, a set of linear equations and boundary conditions containing unknown frequencies and wave numbers is established.

[0130] like Figure 2 As shown in Figure 2, the semi-analytical finite element method discretizes the waveguide into a finite element grid on the cross section and applies an analytical solution in the propagation direction. Assuming that the simple harmonic wave propagates along the x3 axis, the displacement in the waveguide is expressed as:

[0131]

[0132] Where I is the imaginary unit, K is the wave number, ω = 2πf is the angular frequency, and U i represents the spatial distribution function used to describe the solid displacement at any point in the waveguide medium, W i represents the spatial distribution function used to describe the relative displacement of the fluid and solid at any point in the waveguide medium. Therefore, the displacement gradient is expressed as:

[0133]

[0134] Substituting Equation (14) into Equation (6), we obtain a set of linear equations including angular frequency ω and wave number K:

[0135]

[0136] Where the sum index j = 1, 2, 3 and the subscripts k, l = 1, 2. The stress vector T of the solid skeleton on the boundary i and the stress vector p of the pore fluid on the boundary i Expressed as:

[0137]

[0138] S5. Convert the wave equation and boundary conditions of fluid-saturated porous materials in the form of eigenvalue problem into the standard form of coefficient-type partial differential equations in commercial finite element software.

[0139] In commercial finite element software, the standard form of the coefficient-type partial differential equation module is:

[0140]

[0141] where λ is the eigenvalue, is the independent variable required for the eigenvalue problem. For the two-dimensional model established by the semi-analytical method, the Nabla operator e a, d a,c,α,γ,β,a,F are the coefficients of the coefficient-type partial differential equation. The flux / source boundary conditions and Dirichlet boundary in the coefficient-type partial differential equation module are expressed as:

[0142]

[0143] Where n is the exterior unit vector of the solution domain, g is the boundary flux term, q is the boundary absorption term, and r is the specified value of the Dirichlet boundary condition.

[0144] To convert the wave equation and boundary conditions for a fluid-saturated porous material into standard form, the independent variable for:

[0145]

[0146] Among them, U k1 =KU1,U k2 =KU2,U k3 =KU3,W k1 =KW1,W k2 =KW2,W k3 =KW3.

[0147] The coefficients of the coefficient-type partial differential equation are:

[0148]

[0149] Among them, 0 is the zero matrix of the corresponding dimension, and the remaining sub-matrices are:

[0150]

[0151]

[0152] Among them, C 11 、C 66 etc. is the stiffness matrix C nm The elements at the corresponding positions in .

[0153] The hole boundary condition corresponds to the zero flux boundary condition in the Coefficient Form PDE module:

[0154]

[0155] The closed-pore boundary condition involves constraints on stress and displacement, and requires the use of both flux / source boundary conditions and weak contribution boundary conditions. The pore fluid stress vector p on the boundary i = 0 is affected by the normal component w of the fluid-solid relative motion n = 0. According to T i = 0, the boundary absorption term q and boundary flux term g in formula (18) are:

[0156]

[0157] Among them, 0 is the zero matrix of the corresponding dimension.

[0158]

[0159] According to the normal component w of the relative displacement between the fluid and the solid n = 0, it can be obtained that

[0160] w1n1 + w2n2 = 0 (32)

[0161] Equation (32) is implemented using the weak contribution boundary condition in the coefficient-type partial differential equation module, and the weak expression is:

[0162] Lam_a·[test(w1)n1 + test(w2)n2] + test(Lam_a)·(w1n1 + w2n2) (33)

[0163] Among them, Lam_a is the auxiliary dependent variable, and test( ) is the test function built into the coefficient-type partial differential equation module.

[0164] S6. Solve the eigenvalue problem to obtain the guided wave acoustic field characteristics such as the phase velocity, attenuation, and wave structure of the fluid-saturated porous material.

[0165] After a given angular frequency ω is given, solving the eigenvalue equation can obtain the wave numbers K of all possible modes at this frequency. The wave numbers that conform to the physical laws are screened according to the following conditions:

[0166] real(K) > 0 and real(K) > abs(Im(K)) (34)

[0167] Among them, real( ) and Im( ) represent the real part and the imaginary part of the complex number respectively. In the required frequency range, repeating this process at a certain frequency step dω can obtain the corresponding K-ω relationship.

[0168] The phase velocity C p and the attenuation Att are given by the following formulas:

[0169]

[0170] The wave structure information is given by the eigenvector corresponding to the wave number K under the given angular frequency ω given.

[0171] Example 1

[0172] Taking a plate-like fluid-saturated porous material as an example, the guided wave acoustic field characteristics of the fluid-saturated porous material are analyzed based on the semi-analytical finite element method, and the results are compared with those of the traditional root search method to verify the correctness of the present invention.

[0173] In order to simulate the propagation of waves along an infinitely wide plate structure, a narrow strip structure with periodic boundary conditions is used to simulate the infinitely wide plate structure. The periodic boundary condition is a special case of the Neumann boundary condition, which is used to express the continuity of displacement and stress between two boundaries. The semi-analytical finite element model of a plate-like fluid-saturated porous material is as follows: Figure 3 As shown, the plate thickness is 10 mm and the width of the narrow strip structure is 1 mm.

[0174] like Figure 4 As shown, Figure 4 (a) and Figure 4 (b) shows the phase velocity dispersion and attenuation characteristics of a plate-like fluid saturated porous plate under open and closed boundary conditions, respectively. The color of the data points on the phase velocity dispersion curve reflects the corresponding attenuation. The results of the semi-analytical finite element method and the root search method are very consistent. Figure 4 Based on the attenuation information in the pore fluid, the guided wave modes propagating in the laminar fluid-saturated porous material are divided into two categories: high attenuation mode and low attenuation mode. The phase velocity dispersion curve of the low attenuation mode is similar to the Lamb wave mode in the elastic plate, and also has symmetric or antisymmetric properties, which is called Lamb-like wave. The interaction between the pore fluid and the solid skeleton leads to the existence of a high attenuation guided wave mode in the porous material that propagates at the Biot slow longitudinal wave velocity. The phase velocity dispersion curve of the high attenuation mode tends to infinity at the cutoff frequency and intersects with the Lamb-like wave mode. The interaction between the two modes causes the dispersion curve of the Lamb-like wave mode to be distorted. At the intersection, the Lamb-like wave mode is affected by the high attenuation mode and exhibits high attenuation in a small frequency range. In addition, with the increase of frequency, the attenuation phenomenon of the high attenuation mode becomes more severe.

[0175] For the aperture boundary, affected by the high attenuation mode, the S0 mode has a distorted narrow stop band around 0.4MHz·mm~0.5MHz·mm, showing high attenuation characteristics. Figure 5As shown, points A, B, and C are selected to analyze the distribution of solid displacement ||u|| and fluid-solid relative displacement ||w|| along the plate thickness near the narrow stop band. The frequency-thickness products of points A, B, and C are 0.1MHz·mm, 0.46MHz·mm, and 0.9MHz·mm, respectively. Points A and C are located at the two ends of the narrow stop band, and the attenuation is not severe. Therefore, the solid displacement is much larger than the fluid-solid relative displacement. Point B is located just in the narrow stop band, and the attenuation is relatively obvious, and the ratio of solid displacement to fluid-solid relative displacement decreases. Although the fluid-solid relative displacement increases at point B, the solid displacement still dominates. Furthermore, the distribution of solid displacement and fluid-solid relative displacement along the plate thickness of the Lamb wave mode and the high attenuation mode is analyzed when fd=3MHz·mm, as shown in the figure. Figure 6 As shown in Figure 2, the comparison shows that for the high attenuation mode, the fluid-solid relative displacement plays a dominant role. The above phenomenon shows that the fluid-solid relative displacement is the main reason for the high attenuation of guided waves in fluid-saturated porous materials.

[0176] In general, the results under closed-pore boundary conditions are similar to those under open-pore boundary conditions. The main difference is that the narrow stopband phenomenon of the S0 mode near 0.4MHz·mm~0.5MHz·mm is replaced by the missing band phenomenon of the A0 mode, which is also the result of the interaction between the pore fluid and the solid skeleton. In addition, under closed-pore boundary conditions, the high-attenuation mode has a greater impact on the lambda-like wave mode. For example, in addition to the obvious missing band in the low-frequency band, the A0 mode and the S0 mode also have different degrees of band missing at the intersection with other high-attenuation modes. Therefore, it is predicted that polishing and smoothing the surface of the porous material in the experiment will help to observe the slow longitudinal waves predicted by Biot theory.

[0177] Example 2

[0178] In Example 1, a plate-like structure is used as an example to verify the effectiveness of the proposed semi-analytical finite element method in the analysis of the acoustic field characteristics of fluid-saturated porous materials. In this example, the semi-analytical finite element method is applied to L-shaped columnar fluid-saturated porous materials with irregular interfaces and no analytical solution, such as Figure 7 shown. Figure 8 (a) and Figure 8 (b) Phase velocity dispersion curves and attenuation of an L-shaped columnar fluid-saturated porous material under open and closed boundary conditions, respectively. Under closed boundary conditions, the high-attenuation mode is more severely distorted, the guided wave modes are more numerous, and the acoustic field characteristics become more complex, consistent with the phenomena observed in plate-like structures. L-shaped columnar fluid-saturated porous materials are waveguides with complex geometry and material properties. The semi-analytical finite element method provides an efficient solution for resolving the acoustic field of this type of complex waveguide.

[0179] The semi-analytical finite element method proposed in the present invention performs finite element discretization on the cross section of fluid-saturated porous materials and adopts analytical solutions in the wave propagation direction. Compared with traditional methods, the present invention has small computational complexity, fast computational speed, is not prone to missed solutions, and is applicable to complex structures with arbitrary cross sections. It has significant advantages in solving the guided wave acoustic field characteristics of fluid-saturated porous materials.

[0180] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principles of the present invention should be included in the scope of protection of the present invention.

Claims

1. A method for analyzing the characteristics of guided wave acoustic fields of fluid-saturated porous materials based on a semi-analytical finite element method, characterized in that: The following steps are involved: S1. Derivation of the stress-strain relationship of the fluid-saturated porous material based on the material parameters of the fluid-saturated porous material; S2. Based on the stress-strain relationship of fluid-saturated porous materials, derive the wave equation of fluid-saturated porous materials; S3. Based on the surface treatment of fluid-saturated porous materials, derive the expressions for open-pore boundary conditions and closed-pore boundary conditions; S4. Establish a finite element model based on the semi-analytical finite element method, discretize the waveguide into a finite element network in the cross section, and use an analytical solution in the wave propagation direction to obtain the wave equation and boundary conditions of the fluid-saturated porous material in the form of an eigenvalue problem; S5. Convert the wave equation and boundary conditions for fluid-saturated porous materials in the form of eigenvalue problems into the standard form of coefficient-type partial differential equations in commercial finite element software; S6. Solve the eigenvalue problem to obtain the guided wave acoustic field characteristics of the fluid-saturated porous material; The method for obtaining the wave equation and boundary conditions for a fluid-saturated porous material in the form of an eigenvalue problem is: The waveguide is discretized into a finite element mesh in the cross section using the semi-analytical finite element method, and an analytical solution is applied in the propagation direction. Assuming that the simple harmonic wave propagates along the x3 axis, the displacement in the waveguide is expressed as: Among them, x1, x2 and x3 represent the three directions in the three-dimensional Cartesian coordinate system, I is the imaginary unit, K is the wave number, ω = 2πf is the angular frequency, U i represents the spatial distribution function used to describe the solid displacement at any point in the waveguide medium, W i represents the spatial distribution function used to describe the relative displacement of the fluid and solid at any point in the waveguide medium. The subscripts i = 1, 2, and 3 represent the three directions x1, x2, and x3 in the three-dimensional Cartesian coordinate system. Therefore, the displacement gradient is expressed as: Substituting the above formula into the wave equation, we obtain the wave equation for fluid-saturated porous materials in the form of an eigenvalue problem: Where, the sum index j = 1, 2, 3, and the subscripts k, l = 1, 2; Substitute the following equations into the expressions of the open hole boundary condition and the closed hole boundary condition respectively to obtain the boundary conditions in the form of eigenvalue problem; 2. The method for analyzing the characteristics of the guided wave acoustic field of fluid-saturated porous materials based on the semi-analytical finite element method according to claim 1, characterized in that: The method for deriving the stress-strain relationship of fluid-saturated porous materials is: For a fluid-saturated and statistically isotropic porous material, the solid displacement u is defined as [u1,u2,u3] T , fluid displacement u f =[u f1 ,u f2 ,u f3 ] T , the relative displacement between fluid and solid w=φ(u f -u), where φ is the porosity, u1, u2 and u3 are the solid displacements in the x1, x2 and x3 directions respectively, and u f1 、u f2 and u f3 are the relative displacements of the fluid and the solid in the directions x1, x2, and x3, respectively. x1, x2, and x3 represent the three directions in the Cartesian coordinate system of three-dimensional space; The poroelastic parameters M, H, and C are obtained from the known material parameters of the fluid-saturated porous material: Among them, K f , K d and K m are the bulk moduli of the pore fluid, the solid skeleton, and the particles that make up the skeleton, respectively. μ is the shear modulus of the fluid-saturated porous material. is the Biot-Willis coefficient; The stress-strain relationship of the fluid-saturated porous material is expressed as: s ik =C ikjl e jl -Cxd ik ; p f =-Cε ii +Mξ; Among them, σ ik and ε jl are the stress and strain components of the solid skeleton, represents the increment of fluid content, ε ii =ε 11 +ε 22 +ε 33 is the intermediate parameter, ε 11 , ε 22 and ε 33 are the normal strains in the x1, x2 and x3 directions, p f is the stress of the pore fluid, is the Nabla operator, represents the divergence operation, δ ik is the Kronecker symbol, C ikjl is the generalized elastic constant of the solid skeleton of the fluid-saturated porous material. Due to the symmetry of the stress tensor and the strain tensor, the generalized elastic constant C of the solid skeleton of the fluid-saturated porous material is ikjl Converted into stiffness matrix C nm :

3. The method for analyzing the characteristics of the guided wave acoustic field of fluid-saturated porous materials based on the semi-analytical finite element method according to claim 2, characterized in that: The method for deriving the wave equation of a fluid-saturated porous material is: According to Biot's theory, the propagation of elastic waves in fluid-saturated porous materials can be expressed as: Where ρ = φρ f +(1-φ)ρ m is the density of the porous material, φ is the porosity, ρ m and ρ f are the solid density and the fluid density, respectively. q is the effective density of the fluid during relative motion between the fluid and the solid. The subscripts i=1, 2, and 3 represent the three directions x1, x2, and x3 in the Cartesian coordinate system of three-dimensional space. is the Nabla operator, represents the divergence operation, σ represents the stress, express express t represents time; Consider the simple harmonic oscillation of displacement and assume that the time-dependent term of displacement is e -Iωt , where I is the imaginary unit, ω = 2πf is the angular frequency, and f is the frequency. Therefore, the propagation mechanism of elastic waves in fluid-saturated porous materials is given by the wave equation, which is expressed as: q(ω) is the frequency-dependent effective density of the fluid during relative motion between the fluid and the solid.

4. The method for analyzing the characteristics of the guided wave acoustic field of fluid-saturated porous materials based on the semi-analytical finite element method according to claim 3, characterized in that: The frequency-dependent effective density of the fluid during relative motion between the fluid and the solid in: η is the fluid viscosity, I is the imaginary unit, k0 is the low-frequency permeability, τ is the tortuosity, ω t is the transition frequency, Λ is the ratio of weighted volume to surface area, m is a dimensionless constant, and k(ω) is the dynamic permeability.

5. The method for analyzing the characteristics of the guided wave acoustic field of fluid-saturated porous materials based on the semi-analytical finite element method according to claim 4, characterized in that: The open-pore boundary condition indicates the continuity of normal-direction motion at the boundary, and the pore fluid can flow through the surface; the closed-pore boundary condition corresponds to the closure of the surface pores, and the closure of the surface pores includes wrapping a waterproof layer, filling the surface pores, and polishing the surface.

6. The method for analyzing the characteristics of the guided wave acoustic field of fluid-saturated porous materials based on the semi-analytical finite element method according to claim 5, characterized in that: The method for deriving the expressions of the open hole boundary condition and the closed hole boundary condition is: The stress vector T of the solid skeleton at any point on the boundary i Expressed as: Among them, n k is a unit vector perpendicular to the boundary and pointing outward, n1, n2, n3 are unit vectors in the directions of x1, x2, x3; The stress vector p of the pore fluid at any point on the boundary is i Expressed as: Then the expression of the opening boundary condition is: T i =0,p i =0; The expression of the closed-cell boundary condition is: T i =0,w n =0; Among them, w n It represents the normal component of the relative displacement between fluid and solid on the boundary.

7. The method for analyzing the characteristics of the guided wave acoustic field of fluid-saturated porous materials based on the semi-analytical finite element method according to claim 6, characterized in that: The method for converting the wave equation and boundary conditions of a fluid-saturated porous material in the form of an eigenvalue problem into the standard form of a coefficient-type partial differential equation in commercial finite element software is: In commercial finite element software, the standard form of the coefficient-type partial differential equation module is: where λ is the eigenvalue, is the independent variable required for the eigenvalue problem. For the two-dimensional model established by the semi-analytical method, the Nabla operator e a, d a ,c,α,γ,β,a,F are the coefficients of the coefficient-type partial differential equation. The flux boundary conditions and Dirichlet boundaries in the coefficient-type partial differential equation module are expressed as: Where n is the external unit vector of the solution domain, g is the boundary flux term, q is the boundary absorption term, and r is the specified value of the Dirichlet boundary condition; To convert the wave equation and boundary conditions for a fluid-saturated porous material into standard form, the independent variable for: Among them, U k1 =KU1,U k2 =KU2,U k3 =KU3,W k1 = KW1,W k2 = KW2,W k3 = KW3; The coefficients of the coefficient-type partial differential equation are: Among them, 0 is the zero matrix of the corresponding dimension; The hole boundary condition corresponds to the zero flux boundary condition in the Coefficient Form PDE module: The closed-pore boundary condition involves constraints on stress and displacement, and requires the use of both flux boundary conditions and weak contribution boundary conditions; the pore fluid stress vector p on the boundary i = 0 is affected by the normal component w of the fluid-solid relative motion n =0 condition; According to T i = 0, the boundary absorption term q and the boundary flux term g are: According to the normal component w of the fluid-solid relative displacement n =0, we can get w1n1+w2n2=0; w1n1+w2n2=0 is realized by the weak contribution boundary condition in the coefficient-type partial differential equation module. The weak expression is: Lam_a·[test(w1)n1+test(w2)n2]+test(Lam_a)·(w1n1+w2n2); Among them, Lam_a is the auxiliary dependent variable, and test() is the built-in test function of the coefficient-type partial differential equation module.

8. The method for analyzing the characteristics of the guided wave acoustic field of fluid-saturated porous materials based on the semi-analytical finite element method according to claim 7, characterized in that: The sub-matrices in the coefficients of the coefficient-type partial differential equation are: Among them, C 11 、C 66 is the stiffness matrix C nm The elements at the corresponding positions in .

9. The method for analyzing the characteristics of the guided wave acoustic field of fluid-saturated porous materials based on the semi-analytical finite element method according to claim 1 or 8, characterized in that: The method for solving the eigenvalue problem and obtaining the guided wave acoustic field characteristics of the fluid-saturated porous material is as follows: given an angular frequency ω, solving the eigenvalue equation can obtain the wave number K of all possible modes at the angular frequency ω; and screening the wave number that conforms to the physical law according to the following conditions: real(K)>0 and real(K)>abs(Im(K)); Among them, real() and Im() represent the real part and imaginary part of the complex number respectively. In the required frequency range, repeating this process according to a certain frequency step dω can obtain the corresponding K-ω relationship; Phase velocity C in guided wave acoustic field characteristics p and attenuation Att is given by: The wave structure information in the guided wave acoustic field characteristics is given by the eigenvector corresponding to the studied wave number K at a given angular frequency ω. given.

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