Guided wave sound field characteristic analysis method of fluid saturated porous material based on semi-analytical finite element method

Abstract

The invention provides a guided wave sound field characteristic analysis method of a fluid saturated porous material based on a semi-analytical finite element method. The method comprises the following steps: obtaining a stress-strain relationship according to material parameters of the fluid saturated porous material; obtaining a wave equation of the fluid saturated porous material according to the stress-strain relationship; deriving expressions of open pore boundary conditions and closed pore boundary conditions according to the surface treatment mode of the fluid saturated porous material; according to a semi-analytical finite element method, obtaining a wave equation and boundary conditions of the fluid saturated porous material in a characteristic value problem form; converting the wave equation in the characteristic value problem form and the boundary condition into a standard form of a coefficient type partial differential equation in commercial finite element software; and solving the characteristic value problem to obtain the guided wave sound field characteristics of the fluid saturated porous material. The method can be used for solving the guided wave sound field characteristics of the fluid saturated porous material with any cross section, and is small in calculation amount, high in calculation speed, low in calculation cost and not easy to leak.

Description

Technical Field

[0001] The 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 development of materials science, artificial fluid-saturated porous materials such as nanomaterials, artificial bones, and lithium batteries have been widely used. The complex internal structure and service environment make fluid-saturated porous materials very susceptible to various types of composite damage such as corrosion, fracture, and porosity mutation. The gradual development of these damages will seriously affect the service of fluid-saturated porous materials, causing environmental pollution, mechanical structure failure, medical accidents, and other unpredictable casualties and property losses. Therefore, it is necessary to evaluate the health status of fluid-saturated porous materials and monitor their structural health to ensure their safe and stable service.

[0003] During the propagation process, 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 the attenuation, flight time, reflection and refraction information 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 fluid-saturated porous material detection.

[0004] The research results of body wave propagation in porous materials have been widely used in geophysical fields such as acoustic logging and seismic exploration, but the research on guided waves is still in its infancy. The main difference between guided waves and body wave control equations is that the boundary conditions are introduced when solving the guided wave problem, resulting in countless modes of guided waves in theory. Before using guided waves for detection, it is necessary to obtain the dispersion characteristics of ultrasonic guided waves propagating in fluid-saturated porous materials, and then select appropriate modes and frequencies for detection. The success of ultrasonic guided wave detection depends largely on the selection of guided wave modes, so accurately solving the guided wave control equations is the key to the successful application of ultrasonic guided wave detection technology in fluid-saturated porous materials. The interaction between pore fluid and solid skeleton and the complex boundary conditions make the guided wave control equations of fluid-saturated porous materials complex and difficult to solve. Traditional root search methods such as global matrix method and transfer matrix method have been used to study the propagation of guided waves in fluid-saturated porous materials, but there are problems of large computational complexity and poor stability when solving the guided wave control equations of multi-layer and large-thickness fluid-saturated porous materials. Although the spectral method is fast and simple to model, it is only applicable to simple structures such as layered structures and columnar structures that can be accurately modeled by mathematical equations, and it is difficult to solve waveguides with complex geometric shapes. The wave finite element method usually relies on the porous elastic wave frequency domain module of highly integrated commercial finite element software, which has poor scalability and is difficult to adapt to the complex and changeable internal structure and geometry of fluid-saturated porous materials. Summary of the invention

[0005] In view of the technical problems of large amount of calculation, slow calculation speed, poor stability and small scope of application of the existing waveguide acoustic field characteristic analysis method of fluid-saturated porous materials, the present invention proposes a waveguide acoustic field analysis method of fluid-saturated porous materials based on semi-analytical finite element method, which discretizes the cross section of the waveguide into finite element grids, and uses simple harmonic waves to describe the waveform in the propagation direction of the waveguide. The eigenvalue problem is established by introducing the displacement field into the control equation, which not only reduces the amount of calculation, but also is applicable to fluid-saturated porous materials of arbitrary cross-sections, and ensures the 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, the wave equation of fluid-saturated porous materials is derived;

[0010] S3. Based on the surface treatment method of fluid-saturated porous materials, derive the expressions of 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 fluid-saturated porous material, define the solid displacement u = [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 in 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, 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] Then 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 ikjl Transformed 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 when the fluid and the solid are in relative motion. The subscripts i=1,2,3 represent the three directions x1,x2,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;

[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 derived from 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 the normal direction movement at the boundary, and the porous 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 It is 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 It is 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 hole boundary condition is:

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

[0043] Among them, w n It represents the normal component of the fluid-solid relative displacement 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 finite element grids in the cross section by the semi-analytical finite element method, and the 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 flow-solid phase at any point in the waveguide medium. Therefore, the displacement gradient is expressed as:

[0048]

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

[0050]

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

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

[0053]

[0054] Preferably, the method of 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 variables 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 Open 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 the normal component w of the relative motion of the fluid and the solid 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 get w1n1+w2n2=0;

[0073] w1n1+w2n2=0 is realized by the weak contribution boundary condition in the coefficient-type partial differential equation module. 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 built-in test function of 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. is the stiffness matrix C nm The elements at the corresponding positions in .

[0080] Preferably, the method for solving the eigenvalue problem and obtaining the waveguide acoustic field characteristics of the fluid-saturated porous material is: given an angular frequency ω, solving the eigenvalue equation can obtain the wave number K of all possible modes under the angular frequency ω; and screening out the wave number that conforms to the physical law according to the following conditions:

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

[0082] 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;

[0083] Phase velocity C in the characteristics of guided wave acoustic field p And the attenuation Att is given by the following formula:

[0084]

[0085] The wave structure information in the waveguide acoustic field characteristics is given by the eigenvector corresponding to the studied wave number K at a given angular frequency ω. Compared with the prior art, the beneficial effects of the present invention are given as follows:

[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, a 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 the corresponding eigenvector at a given frequency, and the wave structure information can be obtained without additional work; and the method 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 drawings required for use in the embodiments or the description of the prior art will be briefly introduced below. 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 creative work.

[0088] Figure 1 It is a flow chart of the present invention.

[0089] Figure 2 This is a schematic diagram of semi-analytical finite element 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; 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 where the distorted narrow stop band appears in the S0 mode under the open boundary condition of the plate-like fluid saturated porous material in Example 1 of the present invention; wherein, Figure 5 -(a) Figure 4 - Partial magnification of the distorted 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-like 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; wherein, Figure 8 -(a) is the phase velocity dispersion curve and attenuation of L-shaped columnar fluid saturated porous material under open hole boundary conditions. Figure 8 -(b) Phase velocity dispersion curve and attenuation of L-shaped columnar fluid-saturated porous material under closed-cell boundary conditions. DETAILED DESCRIPTION

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

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

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

[0099] For a fluid-saturated and statistically isotropic fluid-saturated porous material, define the solid displacement u = [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 They are the relative displacements of the fluid and the solid in the directions of x1, x2 and x3 respectively, and 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, 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] Then 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, the wave equation of fluid-saturated porous materials is derived.

[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 when the fluid and the solid are in relative motion. The subscripts i=1,2,3 represent the three directions x1,x2,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.

[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 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 is used to describe 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 (Eq. (8) and Eq. (9)) is suitable for the analysis of acoustic characteristics in the full 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 It is expressed as:

[0120]

[0121] Among them, n k is a unit vector perpendicular to the boundary and pointing outward, and n1, n2, n3 are unit vectors in the directions of x1, x2, x3. The pore fluid stress vector p on the boundary i It is 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 the normal motion 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-cell 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 fluid-solid relative displacement on the boundary.

[0129] S4. According to 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, and 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 are established.

[0130] like Figure 2 As shown in Figure 1, the semi-analytical finite element method discretizes the waveguide into a finite element grid in 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-solid phase 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 It is expressed as:

[0137]

[0138] S5. The wave equation and boundary conditions for fluid-saturated porous materials in the form of eigenvalue problems are converted 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 boundaries 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 variables 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 Open 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 the normal component w of the relative motion of the fluid and the solid n = 0. According to T i = 0, the boundary absorption term q and boundary flux term g in equation (18) are:

[0156]

[0157] Where 0 is the zero matrix of the corresponding dimension.

[0158]

[0159] According to the normal component w of the fluid-solid relative displacement n =0, we can get

[0160] w1n1+w2n2=0 (32)

[0161] Formula (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 built-in test function of the coefficient-type partial differential equation module.

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

[0165] When the angular frequency ω is given, solving the eigenvalue equation can obtain the wave number K of all possible modes at this frequency. The wave number that conforms to the physical law is selected according to the following conditions:

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

[0167] Where real( ) and Im( ) represent the real and imaginary parts of the complex number, respectively. Repeating this process in a certain frequency step dω within the required frequency range can obtain the corresponding K-ω relationship.

[0168] Phase velocity C p And the attenuation Att is given by the following formula:

[0169]

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

[0171] Example 1

[0172] Taking a plate-like structure plate-like fluid-saturated porous material as an example, the waveguide 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 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-like structure, a narrow strip structure with periodic boundary conditions is used to simulate the infinitely wide plate-like 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 shown in 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 waveguide mode propagating in the laminar fluid-saturated porous material, the waveguide mode propagating in the laminar fluid-saturated porous material is 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 quasi-Lamb wave. The interaction between the pore fluid and the solid skeleton leads to the existence of a high attenuation waveguide mode propagating at the Biot slow longitudinal wave velocity in the porous material. The phase velocity dispersion curve of the high attenuation mode tends to infinity at the cutoff frequency and intersects with the Lamb wave-like mode. The interaction between the two modes causes the dispersion curve of the Lamb wave-like mode to be distorted. At the intersection, the Lamb wave-like 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 stopband 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 exactly 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 Figure 6 As shown. By comparison, it is found that for the high attenuation mode, the fluid-solid relative displacement is dominant. 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 lamb-like wave mode. For example, in addition to the obvious missing bands 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] Example 1 takes a plate-like structure as an example to verify the effectiveness of the proposed semi-analytical finite element method in the analysis of the waveguide acoustic field characteristics of fluid-saturated porous materials. Furthermore, this example applies the semi-analytical finite element method to an L-shaped columnar fluid-saturated porous material with an irregular interface and no analytical solution, such as Figure 7 shown. Figure 8 (a) and Figure 8 (b) Phase velocity dispersion curve and attenuation of L-type columnar fluid-saturated porous materials under open and closed boundary conditions, respectively. Under closed boundary conditions, the distortion of high attenuation modes is more serious, there are more guided wave modes, and the acoustic field characteristics are more complex, which is consistent with the phenomenon observed in plate-like structures. L-type columnar fluid-saturated porous materials are a type of waveguide with complex geometric structures and material properties. The semi-analytical finite element method provides an efficient solution for solving 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 calculation amount, fast calculation speed, is not easy to miss solutions, and is suitable for 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 principle of the present invention should be included in the protection scope of the present invention.

Claims

1. A method for analyzing the characteristics of guided wave acoustic fields of fluid-saturated porous materials based on 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, the wave equation of fluid-saturated porous materials is derived; S3. Based on the surface treatment method of fluid-saturated porous materials, derive the expressions of 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 and obtain the guided wave acoustic field characteristics of the fluid-saturated porous material.

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 is characterized in that: The method for deriving the stress-strain relationship of fluid-saturated porous materials is: For a fluid-saturated and statistically isotropic fluid-saturated porous material, define the solid displacement u = [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 in 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, 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; Then 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 ikjl Transformed 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 is 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 when the fluid and the solid are in relative motion. The subscripts i=1,2,3 represent the three directions x1,x2,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 derived from 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 is 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 is characterized in that: The open-pore boundary condition represents the continuity of the 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.

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 is 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 It is 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 It is expressed as: Then the expression of the opening boundary condition is: T i =0,p i =0; The expression of the closed hole boundary condition is: T i =0,w n =0; Among them, w n It represents the normal component of the fluid-solid relative displacement 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 is characterized in that: 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 finite element grids in the cross section by the semi-analytical finite element method, and the 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: 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 flow-solid phase at any point in the waveguide medium. Therefore, the displacement gradient is expressed as: Substituting the above equation into the wave equation, we obtain the wave equation for fluid-saturated porous materials in the form of an eigenvalue problem: Among them, the sum index j=1,2,3, and the subscripts k,l=1,2; Substitute the following formulas into the expressions of open hole boundary conditions and closed hole boundary conditions respectively to obtain the boundary conditions in the form of eigenvalue problem; 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 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 variables 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 Open 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 the normal component w of the relative motion of the fluid and the solid 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.

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 8, characterized in that: The sub-matrices in the coefficients of the coefficient-type partial differential equation are: Among them, C 11 , C 66 etc. is the stiffness matrix C nm The elements at the corresponding positions in .

10. 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 9, characterized in that: The method for solving the eigenvalue problem and obtaining the waveguide acoustic field characteristics of the fluid-saturated porous material is as follows: given an angular frequency ω, the eigenvalue equation is solved to obtain the wave number K of all possible modes under the angular frequency ω; the wave number that conforms to the physical law is screened out 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 the attenuation Att is given by the following formula: The wave structure information in the waveguide 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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