Periodic pile structure band gap calculation method based on domain decomposition and linear representation

Abstract

The application discloses a kind of based on the band gap calculation method of periodic pile arrangement structure of area decomposition and linear expression method, the method adopts area decomposition to divide pile soil coordinate system, establishes by periodic boundary constraint condition matrix and pile soil contact interface constraint condition matrix composition pile soil cell total constraint condition matrix;Pile soil cell total constraint condition matrix is handled by linear expression method, and the basic solution set is obtained;According to the basic solution set, the total energy functional of pile soil cell and the motion equation of pile soil cell are established;According to the motion equation of pile soil cell, the wave number of first Brillouin zone is scanned, and the vibration dispersion curve of periodic pile arrangement structure is obtained;The corresponding relationship between wave vector and the frequency of periodic pile arrangement structure is analyzed, and the band gap characteristics of periodic pile arrangement structure are obtained.The application avoids the problem of pile soil parameter distortion and the boundary dependence problem in shape function construction, and has low implementation complexity, accurate calculation result and significantly improved calculation efficiency.

Description

Technical Field

[0001] This invention relates to the field of civil engineering technology, specifically to a method for calculating the band gap of a periodic pile structure based on domain decomposition and linear expression. Background Technology

[0002] Vibration reduction of periodic pile structures is a free vibration problem involving complex composite structures. Energy methods (such as the Rayleigh-Ritz method) can transform the solution of boundary value problems of differential equations into functional extremum problems, which can provide significant assistance in solving coupled structural systems. When analyzing the band gap of periodic pile structures using traditional energy methods, it is necessary to construct displacement field shape functions that satisfy periodic boundaries based on Bloch's theorem. From a mathematical perspective, the periodic reconstruction of shape functions is quite difficult, and the construction methods of different shape functions are not necessarily the same. Energy methods usually place structural cells in a unified coordinate system for processing, but the material properties between piles and soil differ greatly, making waveform fitting difficult, requiring a large number of cutoff terms, and even leading to non-convergence of calculation results. The reconstructed displacement field shape function contains wavenumbers, which results in the presence of wavenumbers in the mass and stiffness matrices of the structure involving the shape function. When calculating the band gap, the mass and stiffness matrices of the structure need to be repeatedly calculated as the wavenumber changes. As the dimensions of the structural mass and stiffness matrices increase, or the number of wavenumber points scanned increases, the computational cost also increases.

[0003] The finite element modeling and analysis method has powerful computational capabilities and can simulate a variety of complex working conditions. However, when simulating the interaction between piles, soil and building structures, in order to obtain accurate calculation results, the element mesh needs to be very fine, the discrete region of the model needs to be very large, and artificial boundary conditions need to be introduced. This often results in a very large number of model elements, with tens of thousands or even millions of degrees of freedom, and high computational costs. Summary of the Invention

[0004] To address the shortcomings of existing technologies, this invention provides a method for calculating the band gap of periodic pile structures based on domain decomposition and linear expression, which has low implementation complexity, high computational efficiency, and accurate results.

[0005] To address the aforementioned problems, this invention employs the following technical solution: a method for calculating the band gap of a periodic pile structure based on domain decomposition and linear expression. This method selects individual pile-soil cells within the periodic pile structure and constructs pile and soil coordinate systems respectively. It establishes displacement expressions for the pile and soil, as well as kinetic and strain energy expressions, based on the statistical energy method. A periodic boundary constraint matrix is ​​established according to periodic structure theory. The pile and soil coordinate systems are solved comprehensively to establish a pile-soil contact interface constraint matrix. A total constraint matrix for the pile-soil cells is established based on the periodic boundary constraint matrix and the pile-soil contact interface constraint matrix. The total constraint matrix for the pile-soil cells is processed using the linear expression method to obtain a set of fundamental solutions. A total energy functional for the pile-soil cells is established based on the set of fundamental solutions, and the motion equations of the pile-soil cells are obtained by combining the Euler-Lagrange equations. Based on the motion equations of the pile-soil cells, the wavenumber of the first Brillouin zone is scanned to obtain the vibration dispersion curve of the periodic pile structure. Based on the vibration dispersion curve, the correspondence between the wave vector and the frequency of the periodic pile structure is analyzed to obtain the band gap characteristics of the periodic pile structure.

[0006] Further optimization yields the following expression for the pile displacement:

[0007] ,

[0008] The expression for soil displacement is:

[0009] ,

[0010] Where: u s ,v s These represent the displacements of the soil in the x and y directions, respectively; u p ,v p These represent the displacements of the pile in the x and y directions, respectively. Let j represent the j-th displacement shape function in the soil coordinate system. Let j represent the j-th displacement shape function in the pile coordinate system. This represents the matrix vector composed of displacement shape functions in the soil coordinate system. This represents a matrix vector composed of displacement shape functions in the pile coordinate system, where i is the complex unit, t is the time variable, and x... s y s Let x and y represent the x and y coordinates of the soil coordinate system, respectively. p y p These represent the horizontal and vertical coordinates of the pile coordinate system, respectively. Let T represent the column vectors of the coefficient matrices consisting of the 1st, 2nd, 3rd, and 4th unknown coefficients, respectively, where T is the symbol for matrix transpose. These represent the 1st, 2nd, 3rd, and 4th time-related unknown coefficients, respectively.

[0011] Further optimization reveals that the kinetic and strain energies of the soil are expressed as follows:

[0012] ,

[0013] Among them, E k U represents the strain energy of the soil. k This represents the kinetic energy of the soil. Indicates soil stress. ρ(x) represents soil strain. s ,y s ) represents the density function of the soil coordinate system, s s Represents a differential unit for soil area. Represents the stiffness matrix of the soil. The mass matrix of the soil. Represents the column vector of the coefficient matrix. The superscript "·" indicates the derivative with respect to time, and the superscript H indicates the conjugate transpose;

[0014] The kinetic and strain energy of the pile are expressed as follows:

[0015] ,

[0016] Where: E c U represents the strain energy of the pile. c This represents the kinetic energy of the pile. Indicates the stress in the pile body. ρ(x) represents the strain of the pile body. p ,y p ) represents the density function of the pile coordinate system, s p Let represent the differential element representing the pile area, and represent the pile stiffness matrix. This represents the mass matrix of the pile.

[0017] Further preferred, the periodic boundary constraint condition matrix satisfy:

[0018] ,

[0019] Right now: ,in, The coordinates in the soil coordinate system are The displacement shape function, The coordinates in the soil coordinate system are The displacement shape function, The coordinates in the soil coordinate system are The displacement shape function, The coordinates in the soil coordinate system are The displacement shape function, It is 1 / 2 the length of the pile soil cell.

[0020] Further optimization of the pile-soil contact interface constraint condition matrix for:

[0021] ,

[0022] Right now: In the formula, Let be the displacement shape function of the pile-soil contact interface in the soil coordinate system. Let x be the displacement shape function of the pile-soil contact interface in the pile coordinate system. r Let y be the x-coordinate of the pile-soil contact interface. r Let be the ordinate of the pile-soil contact interface, and , The radius of the pile is denoted as .

[0023] Further optimization, the total constraint matrix of pile-soil cell elements The Gaussian elimination method was used to transform the total constraint matrix of the pile-soil cell into a row echelon form matrix. Then, linearly independent column vectors were found, and the linear correlation coefficients among the unknown coefficients were linearly expressed by the linearly independent coefficients. Break it down into two parts and add them together, then rewrite it as follows: In the formula, for A full-rank matrix in; Depend on In the column vector of the coefficient matrix The unknown coefficients at the corresponding positions in the text are composed of; for The matrix consisting of the remaining column vectors; Depend on In the column vector of the coefficient matrix The unknown coefficients at the corresponding positions in the middle are composed of; and Arrange vertically to obtain a new column vector. , Through the column vector of the coefficient matrix The elementary row operations are used to obtain it, that is:

[0024] ,

[0025] In the formula It is an elementary transformation matrix; Represents the identity matrix; the coefficient matrix and column vectors. Depend on Linear representation: ,in, This represents the column vector of the coefficient matrix. Depend on The set of fundamental solution sets for linear representations. .

[0026] Further optimized, the total energy functional of the pile-soil cell is:

[0027] .

[0028] Further optimization yields the following equation of motion for the pile-soil cell element: ,in, The frequency of the periodic pile structure.

[0029] Further optimization involves calculating the gap in the periodic pile structure by adjusting the pile-soil material parameters and the periodic pile structure design parameters, thereby designing the optimal periodic pile structure. The periodic pile structure design parameters include pile radius, pile spacing, and pile-soil cell structure type. The pile-soil material parameters include soil density, elastic modulus, and Poisson's ratio, as well as pile density, elastic modulus, and Poisson's ratio.

[0030] Further optimization involves analyzing the vibration dispersion curve to obtain the band gap initiation frequency, cutoff frequency, and band gap width of the periodic pile structure.

[0031] This invention divides a periodic pile structure into several pile-soil cells. It employs domain decomposition to separate the soil coordinate system and the pile coordinate system. An energy method is used to construct a total energy functional that satisfies the periodic boundary conditions. A linear expression method is used to process the constraints, yielding the motion equations of the pile-soil cells. By scanning the first Brillouin zone wavenumber, the vibration dispersion curve of the periodic pile structure is obtained, thus acquiring the band gap characteristics of the periodic pile structure. Compared to traditional methods that require constructing complex displacement shape functions and suffer from waveform fitting difficulties due to pile-soil parameter distortion during overall modeling, this invention models the piles and soil in separate coordinate systems, avoiding pile-soil parameter distortion and boundary dependence issues in shape function construction. This results in lower implementation complexity, more accurate calculation results, and significantly improved computational efficiency. Attached Figure Description

[0032] Figure 1 This is a schematic diagram of a periodic pile structure;

[0033] Figure 2 This is a schematic diagram of a single pile soil cell;

[0034] Figure 3 It is the first Brillouin zone of a single pile soil cell;

[0035] Figure 4 This is a schematic diagram of the soil coordinate system;

[0036] Figure 5 This is a schematic diagram of the pile coordinate system;

[0037] Figure 6 This is a dispersion curve diagram of the periodic pile structure obtained from the embodiment. Detailed Implementation

[0038] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention.

[0039] This embodiment provides a method for calculating the band gap of a periodic pile structure based on domain decomposition and linear expression, which includes the following steps S1 to S8. The implementation of each step is described in detail below.

[0040] Step S1: Select a single pile soil cell in the periodic pile structure and construct the pile coordinate system and soil coordinate system respectively.

[0041] Figure 1 The periodic pile structure shown is selected by taking a single pile soil cell, such as... Figure 2 As shown, a single pile soil cell consists of pile body 100 and soil body 200. The length of the soil cell is 1 / 2 of the pile length, r is the pile radius, u represents the displacement in the x-direction, v represents the displacement in the y-direction, x represents the horizontal axis in the rectangular coordinate system, and y represents the vertical axis in the rectangular coordinate system. The first Brillouin zone of the single pile soil cell is as follows: Figure 3 As shown, the Brillouin zone is controlled by three high-symmetric points, the coordinates of which are Γ. X M In this embodiment, it is assumed that the elastic wave propagates in the xy plane perpendicular to the pile, and that the pile is infinite in the z-direction. Using the domain decomposition method, such as... Figure 4 , Figure 5 As shown, the computational domain is decomposed into two subdomains: the pile body and the soil body, and a soil coordinate system and a pile body coordinate system are constructed respectively.

[0042] Step S2: Establish the displacement expressions for the pile and soil based on the statistical energy method, and establish the kinetic energy and strain energy expressions for the pile and soil;

[0043] Both the soil and the pile have two displacement components, x and y. Based on the energy method calculation principle, the displacement expression for the pile can be written as:

[0044] ,

[0045] The displacement expression for soil can be written as:

[0046] ,

[0047] Where: u s ,v s These represent the displacements of the soil in the x and y directions, respectively; up ,v p These represent the displacements of the pile in the x and y directions, respectively. Let j represent the j-th displacement shape function in the soil coordinate system. Let j represent the j-th displacement shape function in the pile coordinate system. This represents the matrix vector composed of displacement shape functions in the soil coordinate system. This represents a matrix vector composed of displacement shape functions in the pile coordinate system, where i is the complex unit, t is the time variable, and x... s y s Let x and y represent the x and y coordinates of the soil coordinate system, respectively. p y p These represent the horizontal and vertical coordinates of the pile coordinate system, respectively. Let T represent the column vectors of the coefficient matrices consisting of the 1st, 2nd, 3rd, and 4th unknown coefficients, respectively, where T is the symbol for matrix transpose. These represent the 1st, 2nd, 3rd, and 4th time-related unknown coefficients, respectively.

[0048] The kinetic and strain energies of the soil are expressed as follows:

[0049] ,

[0050] Among them, E k U represents the strain energy of the soil. k This represents the kinetic energy of the soil. Indicates soil stress. ρ(x) represents soil strain. s ,y s ) represents the density function of the soil coordinate system, s s Represents a differential unit for soil area. Represents the stiffness matrix of the soil. The mass matrix of the soil. Represents the column vector of the coefficient matrix. The superscript "·" indicates the derivative with respect to time, such as... The superscript H indicates conjugate transpose.

[0051] The kinetic and strain energy of the pile can be expressed as follows:

[0052] ,

[0053] Where: E c U represents the strain energy of the pile. c This represents the kinetic energy of the pile. Indicates the stress in the pile body. ρ(x) represents the strain of the pile body. p ,y p ) represents the density function of the pile coordinate system, sp Let represent the differential element representing the pile area, and represent the pile stiffness matrix. This represents the mass matrix of the pile. Based on the fundamental theory of stress and strain and Hooke's law, the relationships between the stress, strain, displacement, density, elastic modulus, and Poisson's ratio of the soil and the pile can be obtained, thus yielding the stress and strain of the soil and the pile.

[0054] For soil, we have:

[0055] ,

[0056] For the pile body, we have:

[0057] ,

[0058] Where: ρ s ρ is the density of the soil. p For the density of the pile, E(x) s ,y s E(x) represents the elastic modulus function of the soil coordinate system. p ,y p E represents the elastic modulus function of the pile coordinate system. s E represents the elastic modulus of the soil. p This is the elastic modulus of the pile. The radius of the pile is denoted as .

[0059] Step S3: Based on the periodic structure theory, establish the periodic boundary constraint condition matrix.

[0060] The total energy functional of a single period can be expressed as:

[0061]

[0062] at this time, , Because the boundary conditions contain linear correlation coefficients and cannot be variationally determined, it is necessary to handle the periodic boundary conditions by incorporating time-dependent unknown coefficients. According to Bloch theory, the periodic boundary conditions of the periodic pile structure in this embodiment must satisfy the following:

[0063] ,

[0064] in: Let i be the length of 1 / 2 pile soil cell, where i is a complex unit. , k x k y Wavenumbers in the x and y directions, respectively. For unknown coefficients, The coordinates in the soil coordinate system are The displacement of the pile soil cell in the x-direction. The coordinates in the soil coordinate system are The displacement of the pile soil cell in the x-direction. The coordinates in the soil coordinate system are The displacement of the pile soil cell in the y direction. The coordinates in the soil coordinate system are The displacement of the pile soil cell in the y direction. The coordinates in the soil coordinate system are The displacement of the pile soil cell in the x-direction. The coordinates in the soil coordinate system are The displacement of the pile soil cell in the x-direction. The coordinates in the soil coordinate system are The displacement of the pile soil cell in the y direction. The coordinates in the soil coordinate system are The displacement of the pile-soil cell element in the y-direction; the periodic boundary is expressed in matrix form to obtain the periodic boundary constraint matrix. satisfy:

[0065] ,

[0066] Right now: ,in, The coordinates in the soil coordinate system are The displacement shape function, The coordinates in the soil coordinate system are The displacement shape function, The coordinates in the soil coordinate system are The displacement shape function, The coordinates in the soil coordinate system are The displacement shape function.

[0067] Step S4: Solve the coordinate system of the pile body and the coordinate system of the soil body to establish the constraint condition matrix of the pile-soil contact interface.

[0068] For the pile-soil contact interface, in both the pile coordinate system and the soil coordinate system, the x-coordinate of the pile-soil contact interface is x. r The vertical coordinate of the pile-soil contact interface is y. r ,and Assuming the pile and soil are consolidated, that is:

[0069] ,

[0070] Expressing the above equation in matrix form, the pile-soil contact interface constraint condition matrix. for:

[0071] ,

[0072] Right now: In the formula, Let be the displacement shape function of the pile-soil contact interface in the soil coordinate system. Let be the displacement shape function of the pile-soil contact interface in the pile coordinate system.

[0073] Step S5: Based on the periodic boundary constraint matrix and the pile-soil contact interface constraint matrix, establish the total constraint matrix of the pile-soil cell; process the total constraint matrix of the pile-soil cell using the linear expression method to obtain the foundation solution set.

[0074] The total constraint matrix of pile soil cells Where “;” indicates the arrangement of matrix columns, then we have The key to solving this problem lies in finding a set of allowable functions that satisfy all constraints. Gaussian elimination is used to transform the total constraint matrix of the pile-soil cell into a row echelon form matrix. Then, linearly independent column vectors are found, allowing the linear correlation coefficients among the unknown coefficients to be linearly expressed by the linear independence coefficients. Break it down into two parts and add them together, then rewrite it as follows:

[0075] ,

[0076] In the formula, for A full-rank matrix in; Depend on In the column vector of the coefficient matrix The unknown coefficients at the corresponding positions in the text are composed of; for The matrix consisting of the remaining column vectors; Depend on In the column vector of the coefficient matrix The unknown coefficients at the corresponding positions are composed of...

[0077] Will and Vertical arrangement yields a new column vector. ,So It can be obtained through the coefficient matrix and column vectors The elementary row operations are used to obtain it, that is:

[0078] ,

[0079] In the formula It is an elementary transformation matrix; Represents the identity matrix.

[0080] Coefficient matrix column vector can be Linear representation: ,in, This represents the column vector of the coefficient matrix. Depend on The set of fundamental solution sets for linear representations. .

[0081] Step S6: Establish the total energy functional of the pile-soil cell based on the fundamental solution set:

[0082] ,

[0083] Combining the Eular-Lagrange equations The equation of motion for the soil cell element of the pile can be rewritten as: ,in, The frequency of the periodic pile structure.

[0084] Step S7: Scan the first Brillouin zone according to the motion equation of the pile-soil cell. The vibration dispersion curve of the periodic pile structure is obtained by calculating the wave number.

[0085] Step S8: Based on the vibration dispersion curve, analyze the relationship between the wave vector and the frequency of the periodic pile structure. The correspondence is used to obtain the bandgap characteristics of the periodic pile structure, where the wave vector is determined by the wave number k. x k y The vector that makes up the vector.

[0086] In this embodiment, it is assumed that the elastic wave propagates in the xy plane perpendicular to the pile, the soil is single-phase soil, and the pile is infinite in the z-direction. The pile-soil cell length (i.e., period length) is 2a = 4m, the pile radius is r = 0.65m, and the pile-soil material parameters are shown in Table 1. The pile material is concrete, where μ is the Poisson's ratio. To verify the accuracy of this embodiment, the same material parameters are used, and the method proposed in this invention is used to calculate according to steps S1-S7 to obtain the vibration dispersion curve as shown in Table 1. Figure 3 As shown, the results from both methods agree well.

[0087] Table 1. Pile-soil material parameters

[0088]

[0089] By analyzing the vibration dispersion curve, the bandgap initiation frequency, cutoff frequency, and bandgap width of the periodic pile structure are obtained. When a vibration wave passes through the periodic pile structure, the vibration frequencies between the initiation and cutoff frequencies will be attenuated; the bandgap width is the attenuation frequency width. (Refer to...) Figure 6In the vibration dispersion curve of this embodiment, when the frequency falls in region A, there is no real wave number corresponding to the wave vector in any propagation direction. That is, there are no propagation wave modes in this frequency range, which is a complete bandgap. In addition, when the frequency falls in region B, there is no real wave number corresponding to the wave vector in the ΓX direction, which is a directional bandgap. By analyzing the starting frequency, cutoff frequency, and bandgap width of the complete bandgap or directional bandgap of the periodic pile structure, the vibration reduction performance of the periodic pile structure can be effectively evaluated.

[0090] To illustrate the computational efficiency of this method, the pile-soil cell structure type, pile-soil material parameters, pile filling ratio, and wavenumber were used as variables. Under the same configuration and to maintain consistent computational accuracy, six different sets of calculation examples were performed, and the corresponding computation times were recorded as shown in Table 2. The results of each example showed good agreement. Compared with the finite element method, the domain decomposition method improved the computational efficiency by an average of about 3 to 4 times, demonstrating a significant advantage.

[0091] Table 2 Comparison of Calculation Duration Results

[0092]

[0093] By adjusting the soil-pile material parameters (soil density, elastic modulus, Poisson's ratio, pile density, elastic modulus, Poisson's ratio) and the periodic pile structure design parameters (pile radius, pile spacing, pile-soil cell structure type), the gap of the periodic pile structure is calculated, and then the optimal periodic pile structure is designed.

[0094] The embodiments described above are merely preferred embodiments of the present invention and are not intended to limit the concept and scope of the present invention. Various modifications and improvements made to the technical solutions of the present invention by those skilled in the art without departing from the inventive concept should fall within the protection scope of the present invention. All technical contents for which protection is sought in this invention are fully described in the claims.

Claims

1. A method for calculating the band gap of a periodic pile structure based on domain decomposition and linear expression, characterized in that, A single pile-soil cell element in a periodic pile structure is selected, and pile coordinate systems and soil coordinate systems are constructed respectively. Displacement expressions for the pile and soil are established using the statistical energy method, as are expressions for their kinetic and strain energy. A periodic boundary constraint matrix is ​​established based on periodic structure theory. The pile and soil coordinate systems are solved comprehensively to establish a pile-soil contact interface constraint matrix. Based on the periodic boundary constraint matrix and the pile-soil contact interface constraint matrix, a total constraint matrix for the pile-soil cell element is established. The total constraint matrix for the pile-soil cell element is processed using a linear expression method to obtain a set of foundation solutions. A total energy functional for the pile-soil cell element is established based on this set of foundation solutions, and the equations of motion for the pile-soil cell element are obtained by combining the Euler-Lagrange equations. Based on the equations of motion for the pile-soil cell element, the wavenumber of the first Brillouin zone is scanned to obtain the vibration dispersion curve of the periodic pile structure. Based on the vibration dispersion curve, the correspondence between the wave vector and the frequency of the periodic pile structure is analyzed to obtain the bandgap characteristics of the periodic pile structure.

2. The method for calculating the band gap of a periodic pile structure based on domain decomposition and linear expression as described in claim 1, characterized in that, The displacement expression for the pile is: ， The expression for soil displacement is: ， Where: u s ,v s These represent the displacements of the soil in the x and y directions, respectively; u p ,v p These represent the displacements of the pile in the x and y directions, respectively. Let j represent the j-th displacement shape function in the soil coordinate system. Let j represent the j-th displacement shape function in the pile coordinate system. This represents the matrix vector composed of displacement shape functions in the soil coordinate system. This represents a matrix vector composed of displacement shape functions in the pile coordinate system, where i is the complex unit, t is the time variable, and x... s y s Let x and y represent the x and y coordinates of the soil coordinate system, respectively. p y p These represent the horizontal and vertical coordinates of the pile coordinate system, respectively. Let T represent the column vectors of the coefficient matrices consisting of the 1st, 2nd, 3rd, and 4th unknown coefficients, respectively, where T is the symbol for matrix transpose. These represent the 1st, 2nd, 3rd, and 4th time-related unknown coefficients, respectively.

3. The method for calculating the band gap of a periodic pile structure based on domain decomposition and linear expression as described in claim 2, is characterized in that, The kinetic and strain energies of the soil are expressed as follows: ， Among them, E k U represents the strain energy of the soil. k This represents the kinetic energy of the soil. Indicates soil stress. ρ(x) represents soil strain. s ,y s ) represents the density function of the soil coordinate system, s s Represents a differential unit for soil area. Represents the stiffness matrix of the soil. The mass matrix of the soil. Represents the column vector of the coefficient matrix. The superscript "·" indicates the derivative with respect to time, and the superscript H indicates the conjugate transpose; The kinetic and strain energy of the pile are expressed as follows: ， Among them: E c U represents the strain energy of the pile. c This represents the kinetic energy of the pile. Indicates the stress in the pile body. ρ(x) represents the strain of the pile body. p ,y p ) represents the density function of the pile coordinate system, s p Let represent the differential element representing the pile area, and represent the pile stiffness matrix. This represents the mass matrix of the pile.

4. The method for calculating the band gap of a periodic pile structure based on domain decomposition and linear expression as described in claim 3, is characterized in that, The periodic boundary constraint condition matrix satisfy: ， Right now: ,in, The coordinates in the soil coordinate system are The displacement shape function, The coordinates in the soil coordinate system are The displacement shape function, The coordinates in the soil coordinate system are The displacement shape function, The coordinates in the soil coordinate system are The displacement shape function, It is 1 / 2 the length of the pile soil cell.

5. The method for calculating the band gap of a periodic pile structure based on domain decomposition and linear expression as described in claim 4, characterized in that, Pile-soil contact interface constraint matrix for: ， Right now: In the formula, Let be the displacement shape function of the pile-soil contact interface in the soil coordinate system. Let x be the displacement shape function of the pile-soil contact interface in the pile coordinate system. r Let y be the x-coordinate of the pile-soil contact interface. r Let be the ordinate of the pile-soil contact interface, and , The radius of the pile is [value].

6. The method for calculating the band gap of a periodic pile structure based on domain decomposition and linear expression as described in claim 5, is characterized in that, Total constraint matrix of pile soil cells The Gaussian elimination method was used to transform the total constraint matrix of the pile-soil cell into a row echelon form matrix. Then, linearly independent column vectors were found, and the linear correlation coefficients among the unknown coefficients were linearly expressed by the linearly independent coefficients. Break it down into two parts and add them together, then rewrite it as follows: In the formula, for A full-rank matrix in; Depend on In the column vector of the coefficient matrix The unknown coefficients at the corresponding positions in the text are composed of; for The matrix consisting of the remaining column vectors; Depend on In the column vector of the coefficient matrix The unknown coefficients at the corresponding positions in the middle are composed of; and Arrange vertically to obtain a new column vector. , Through the column vector of the coefficient matrix The elementary row operations are obtained as follows: ， In the formula, It is an elementary transformation matrix; Represents the identity matrix; Coefficient matrix column vector Depend on Linear representation: ,in, This represents the column vector of the coefficient matrix. Depend on The set of fundamental solution sets for linear representations. .

7. The method for calculating the band gap of a periodic pile structure based on domain decomposition and linear expression as described in claim 6, characterized in that, The total energy functional of the pile soil cell is: .

8. The method for calculating the band gap of a periodic pile structure based on domain decomposition and linear expression as described in claim 7, is characterized in that, The equation of motion for the soil cell element of the pile is: ,in, The frequency of the periodic pile structure.

9. The method for calculating the band gap of a periodic pile structure based on domain decomposition and linear expression as described in claim 1, characterized in that, The gap of the periodic pile structure is calculated by adjusting the pile-soil material parameters and the periodic pile structure design parameters, and then the optimal periodic pile structure is designed. The periodic pile structure design parameters include pile radius, pile spacing, and pile-soil cell structure type. The pile-soil material parameters include soil density, elastic modulus, and Poisson's ratio, as well as pile density, elastic modulus, and Poisson's ratio.

10. The method for calculating the band gap of a periodic pile structure based on domain decomposition and linear expression as described in claim 1, characterized in that, By analyzing the vibration dispersion curve, the band gap initiation frequency, cutoff frequency, and band gap width of the periodic pile structure are obtained.

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