High-precision calculation method for field effect of complex medium field

Abstract

The invention discloses a high-precision calculation method for a site effect of a complex medium site in the technical field of seismic oscillation analysis, and the method comprises the following steps: 1), carrying out the spectral unit spatial discretization of a fluid-saturated soil-bedrock site, and carrying out the interaction of interface information through a unified calculation framework theory; 2) introducing an improved drift elimination calculation method at the artificial boundary; (3) according to a TMM analysis method, input data of spectrum element node fluctuation at the boundary are given; 4, broadband high-precision simulation is conducted on the displacement response of the earth surface nodes, and an amplification coefficient is given, the precision and stability of fluctuation simulation of the complex medium site are further improved, and a more reliable algorithm framework is provided for site effect analysis of the complex medium site and further earthquake safety evaluation.

Description

Technical Field

[0001] The present invention relates to the technical field of seismic motion analysis and evaluation, and particularly relates to a high-precision calculation method for site effects of complex medium sites. Background Art

[0002] Natural, artificial reefs and coastal ports in such sea areas are mostly located near the southeast coastal seismic zone and the circum-Pacific seismic zone, where earthquakes occur frequently and the high-frequency components of seismic waves are relatively rich. The water-saturated soil-bedrock medium site is one of the main site types, covering cross-river tunnels, coastal buildings and far-offshore reefs, etc. Since such sea area sites contain three media: fluid, saturated soil and single-phase soil body, the seismic response analysis usually needs to consider the fluid-structure interaction between the media. With the rapid development of offshore wind power projects, coastal nuclear power projects, and civilian and military infrastructure on sea area reefs, the value of seismic safety assessment in sea area sites has become prominent.

[0003] Research shows that the spectral element method can achieve numerical simulation of site seismic wave propagation in a wider frequency band than the finite element method. This is crucial for the seismic design of particularly important facilities such as coastal nuclear power plants, large liquid storage tanks and cross-river bridges. Domestic and foreign scholars have used the spectral element method to simulate the seismic motion propagation process of large complex sites such as the Los Angeles Basin, and analyzed the influence of topographic features and propagation media on site effects. However, spectral element simulation studies involving water-saturated soil-bedrock medium sites are very scarce. Considering the calculation scale of large water-saturated soil-bedrock medium sites, the existing hardware conditions, and the requirements for high-precision simulation of broadband seismic motion, applying the spectral element method to local site seismic motion problems considering fluid-structure interaction effects is an efficient choice of calculation method.

[0004] In addition, the seismic motion simulation of complex medium sites also needs to consider aspects such as the setting of artificial boundaries, information interaction between different media, and the input method of external source fluctuations. Among them, the stability of artificial boundaries is a relatively important link. When using traditional commercial finite element software to calculate the seismic response of this type of site, it is necessary to edit the corresponding subroutines and add various contacts and components, and the process is very cumbersome, which is not conducive to the development and application of research.

[0005] The prior art discloses: a control method for the drift problem of the multiple transmission formula (Acta Mechanica Sinica, November 18, 2021); it only proposes a drift elimination scheme for artificial boundaries based on the finite element model of a single solid medium; at the same time, it also discloses: an explicit spectral element simulation method for fluid-structure interaction seismic wave problems (Acta Mechanica Sinica, September 18, 2022). The content disclosed in this article can perform explicit spectral element simulation on fluid-saturated soil-bedrock medium sites, but the artificial boundary does not deal with the problem of drift instability.

[0006] The TMM (Transfer Matrix Method) analysis method is a physical modeling technique based on matrix operations, mainly used to analyze the propagation characteristics of physical quantities such as electromagnetic waves and light waves in multi-layered dielectric structures. Its core principle is to decompose a complex structure into multiple continuous layers, and obtain the transmission characteristics of the overall system by multiplying the transfer matrices of each layer. It is applicable to fields such as optics, acoustics, and quantum mechanics. Summary of the Invention

[0007] In view of the deficiencies in the prior art, the present invention provides a high-precision calculation method for the site effect of a complex medium site to solve the problems in the background art.

[0008] The object of the present invention is achieved as follows: A high-precision calculation method for the site effect of a complex medium site includes the following steps:

[0009] Step 1) Perform spectral element spatial discretization on the fluid-saturated soil-bedrock site, and use the unified calculation framework theory to interact interface information;

[0010] Step 2) Introduce an improved anti-reflection calculation method at the artificial boundary;

[0011] Step 3) According to the TMM analysis method, give the input data of the spectral element node fluctuations at the boundary;

[0012] Step 4) Perform wide-band high-precision simulation on the displacement response of the surface nodes and give the amplification factor.

[0013] Further, in step 1), the spectral element spatial discretization of the fluid-saturated soil-bedrock site is specifically calculated by the following formula:

[0014] (1)

[0015] In the above formula, and are the interaction forces between units;

[0016] (2a)

[0017] (2b)

[0018] (2c)

[0019] (2d)

[0020] (2e)

[0021] (2f)

[0022] (2g)

[0023] (2h)

[0024] Among them, , , where \(N\) is the shape function, and in any one direction , is the solid-phase mass matrix, is the liquid-phase mass matrix, is the damping matrix representing the viscous resistance between the solid and liquid phases, is the solid-phase stiffness matrix, is the liquid-phase stiffness matrix, is the coupling matrix representing the constitutive force between the solid and liquid phases, is the load matrix acting on the solid phase, is the load matrix acting on the liquid phase, and are the mutual acting forces between the elements acting on the solid and liquid phases respectively; when this node is an internal node of various media, during the element assembly process, due to the equal magnitude and opposite direction of the mutual acting forces between the elements, and cancel each other out and their values are 0.

[0025] Furthermore, when performing spectral element spatial discretization calculations, taking a two-dimensional problem as an example, among them,

[0026] (3a)

[0027] (3b)

[0028] (3c)

[0029] , (3d)

[0030] (3e)

[0031] (3f)

[0032] where \(n_g\) is the number of nodes of the spectral element, is the outer normal vector; is the directional derivative matrix.

[0033] Furthermore, the specific formula for step 2) is:

[0034] (3a)

[0035] (3b)

[0036] (3c)

[0037] (3d)

[0038] In the formula, is the solid-phase displacement of the i-th spectral element grid node from the artificial boundary at time, is the interpolation coefficient of the corresponding node, is the distance between each discrete grid node of the spectral element and the boundary node. See Table 2 for details; the media of the model are all single-phase solids. For the saturated two-phase porous media involved, in addition to applying the MTF drift instability control scheme to the solid-phase displacement of the artificial boundary nodes according to the above formula, the liquid-phase displacement also needs to be applied simultaneously.

[0039] Furthermore, step 3) specifically includes: for the external source problem, the input of the ground motion needs to be realized through the artificial boundary and obtained by the TMM analytical method; subtracting the incident wave field solution from the full wave field solution to obtain the scattered wave field at a previous time, obtaining the displacements of each reference point of the MTF, then obtaining the displacements of the boundary nodes at time p + 1, and combining with the displacements of the remaining internal nodes calculated by the wave equation, the complete full wave field solution at time p + 1 can be obtained.

[0040] Furthermore, the wide-band high-precision simulation of the displacement response of the surface nodes in step 4) is specifically as follows:

[0041] Taking any measurement point of the calculation model as the research object, the time history of the corresponding solid-phase vertical displacement and the spectral ratio between it and the input wave are given. The spectral ratio calculation formula is as follows:

[0042] (4-1a)

[0043] (4-1b)

[0044] where S( f ) is the spectral ratio, U( f ) is the spectral amplitude, and the subscripts FEM, SEM, and 0 correspond to the finite element method, spectral element method, and input wave respectively. is a small positive value taken to avoid the denominator being zero. Here, is taken.

[0045] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0046] The present invention adopts the spectral element method which combines the high-precision characteristics of wave field simulation (derived from spectral interpolation) and the flexibility of complex region modeling (derived from the same calculation mode as the finite element method), and combines an improved spectral element multiple transmission boundary anti-reflection scheme to establish an efficient numerical simulation method for fluid-structure interaction seismic wave problems. This method can effectively simulate ground motions in fluid-structure interaction sites within a much wider frequency band, providing more reliable data for ground motion safety evaluation and earthquake engineering design.

[0047] The present invention not only improves the anti-reflection scheme of the artificial boundary from the finite element model to the spectral element model, but also expands from the original single solid medium to fluid media and saturated soil media on this basis, and applies it to the simulation of wave propagation in complex media sites, further improving the accuracy and stability of wave simulation in complex media sites, and providing a more reliable algorithm framework for site effect analysis and further seismic safety evaluation of complex media sites. BRIEF DESCRIPTION OF THE DRAWINGS

[0048] In order to more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the following will briefly introduce the drawings required for the description of the embodiments or the prior art. Obviously, the drawings in the following description are only the embodiments of the present invention. For those of ordinary skill in the art, other drawings can be obtained according to the provided drawings without creative efforts.

[0049] Figure 1 It is a schematic flowchart of an embodiment of the present invention.

[0050] Figure 2 It is a schematic diagram of the site calculation model after the present invention combines the improved MTF boundary.

[0051] Figure 3 It is a schematic diagram of the artificial boundary condition of the multiple transmission formula and the ground motion input in the spectral element model of the present invention.

[0052] Figure 4 It is the solid-phase wave field displacement nephogram of an embodiment of the present invention.

[0053] Figure 5 It is the liquid-phase wave field displacement nephogram of an embodiment of the present invention.

[0054] Figure 6 It is the peak value diagram of the solid-liquid two-phase displacements at the acquisition points of an embodiment of the present invention.

[0055] Figure 7 It is the spectral ratio diagram at the acquisition points of an embodiment of the present invention and the spectral ratio diagram of the traditional finite element method. DETAILED DESCRIPTION OF THE INVENTION

[0056] Next, in combination with the accompanying drawings in the embodiments of the present invention, the technical solutions in the embodiments of the present invention will be described clearly and completely. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all the embodiments. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without creative efforts belong to the scope of protection of the present invention.

[0057] As Figure 1 shown, a high-precision calculation method for the site effect of a complex medium site includes the following steps:

[0058] Step 1: Collect site topography and geological data, and use general grid generation software to generate files such as node geometric information, two-dimensional four-node element information (or three-dimensional eight-node element information), node information of different medium contact surfaces, and collection point information, and convert its grid elements into spectral element grid information (including field function interpolation node information); the specific calculation formula is:

[0059] (1)

[0060] In the above formula, and are the mutual forces between elements, that is,

[0061] (2a)

[0062] (2b)

[0063] (2c)

[0064] (2d)

[0065] (2e)

[0066] (2f)

[0067] (2g)

[0068] (2h)

[0069] Among them, , , and The definitions of are shown in Table 1. N is the shape function, and in any one direction , is the solid-phase mass matrix, is the liquid-phase mass matrix, is the damping matrix representing the viscous resistance between the solid and liquid phases, is the solid-phase stiffness matrix, is the liquid-phase stiffness matrix, is the coupling matrix representing the constitutive force between the solid and liquid phases, is the load matrix acting on the solid phase, is the load matrix acting on the liquid phase, and are the interaction forces acting on the solid and liquid phases between elements respectively; when this node is an internal node of various media, during the element assembly process, due to the equal magnitude and opposite directions of the interaction forces between elements, and cancel each other out and their values are 0.

[0070] Table 1 Symbol Definitions

[0071]

[0072] where A, G, Q, R are non-negative elastic constants, , , is the porosity of the saturated soil, and are the Lame constants of the solid-phase skeleton under drained conditions; and are the constants characterizing the compressibility of the saturated porous medium, measured and calculated by experiments.

[0073] Taking the two-dimensional problem as an example, then it is

[0074] (3a)

[0075] (3b)

[0076] (3c)

[0077] , (3d)

[0078] (3e)

[0079] (3f)

[0080] where ng is the number of nodes of the spectral element; where, is the outer normal vector; is the directional derivative matrix.

[0081] Step 2: Introduce an improved anti-drift scheme at the artificial boundary of the spectral element model. According to the physical wave velocities of different media, adopt corresponding artificial wave velocities, and select appropriate anti-drift factors and anti-drift orders to improve the calculation process of artificial boundary nodes. The specific formula is as follows:

[0082] (3a)

[0083] (3b)

[0084] (3c)

[0085] (3d)

[0086] In the formula, is the solid-phase displacement of the i-th spectral element grid node from the artificial boundary at time, is the interpolation coefficient of the corresponding node, is the distance between each discrete grid node of the spectral element and the boundary node, as shown in Table 2; the media of the model are all single-phase solids. For the saturated two-phase porous media involved, in addition to applying the MTF drift instability control scheme to the solid-phase displacement of the artificial boundary nodes according to the above formula, the liquid-phase displacement also needs to be applied simultaneously.

[0087] Table 2 Local coordinates of interpolation points in the spectral element discretization format

[0088] .

[0089] Step 3: Establish a one-dimensional equivalent model of seismic waves according to the physical parameters of different media at the artificial boundary, and use the transfer matrix analytical solution method to calculate the time history input data of unequally spaced spectral element nodes at the boundary. The specific steps are as follows: For the external source problem, the input of ground motion needs to be realized through the artificial boundary, which is also an advantage of MTF. It is not convenient to realize the input of ground motion on the artificial boundary for stress-type boundaries or PML boundaries. As a displacement-type artificial boundary, MTF can conveniently separate the incident wave field and the scattered wave field of the full wave field. Figure 3Schematic diagram of the artificial boundary condition and seismic motion input for the multiple transmission formula in the spectral element model. Among them, the black nodes represent the full wavefield solution obtained at a previous time; the green nodes represent the known incident wavefield, which is obtained in this paper by the transfer matrix method (TMM); subtracting the incident wavefield solution from the full wavefield solution gives the scattered wavefield at a previous time, that is, the red solid nodes and blue nodes in the figure, and the red hollow dots represent the spatial interpolation reference points of the MTF, whose values are interpolated from Equation (3c). After obtaining the displacements of each reference point of the MTF, according to Equation (3a), the boundary node displacements at time p + 1 can be obtained, and combined with the displacements of the remaining internal nodes calculated by the wave equation, the complete full wavefield solution at time p + 1 is obtained.

[0090] Step 4: Through the numerical simulation of seismic wave propagation in the wave problem, calculate the displacement response time history of the acquisition point and perform spectral analysis on it to obtain the broadband amplification coefficient of the acquisition point response, and complete the analysis of the site effect and the seismic motion safety evaluation in the complex medium site; compared with the finite element method widely used at present, the method of this application can effectively simulate the seismic motion in the fluid-solid coupling site in a much wider frequency band range and better control the artificial boundary drift instability phenomenon.

[0091] To analyze the simulation effects of the two algorithms in different frequency bands of the input seismic wave, taking any measurement point of the calculation model as the research object, the corresponding solid-phase vertical displacement time history and the spectral ratio between it and the input wave are given, and the spectral ratio calculation formula is as follows:

[0092] (4-1a)

[0093] (4-1b)

[0094] Among them, S( f ) is the spectral ratio, U( f ) is the spectral amplitude, and the subscripts FEM, SEM, and 0 correspond to the finite element method, spectral element method, and input wave respectively. A small positive number taken to avoid the denominator being zero, here take .

[0095] Furthermore, regarding the conversion of the two-dimensional four-node geometric element in Step 1 to the spectral element multi-node interpolation nodes, it should be explained and noted that although the fitting function of the coordinates and the field function of the variable to be solved often use the Lagrange function as the basis function, the two do not necessarily have exactly the same function group and are easily confused. The fitting function of the coordinates itself carries clear geometric coordinate information, and its interpolation object is the geometric coordinate. However, the interpolation object of the field function is the variable to be solved, and its essential meaning is to fit the information of any other node in the field based on the known node information (such as displacement) within the element and the distance relationship in the physical space. Therefore, the interpolation nodes of the field function can also be called calculation nodes.

[0096] In the present invention, the Lagrange function is selected as the basis function for the field function of the spectral element method. For two-dimensional or three-dimensional cases, Lagrange-type elements adopt a similar manner in each coordinate direction as in the one-dimensional case. For the convenience of explanation, taking the one-dimensional problem as an example, the interpolation function of the element displacement field in the global coordinate system is given

[0097] (5)

[0098] where x is the global coordinate (physical coordinate system) of the interpolation node; is the number of nodes for element interpolation; is the global coordinate of the i-th node; is the displacement value of the i-th node; is the basis function of the interpolation function of the i-th node.

[0099] For the basis function in Equation (5), the finite element method and the spectral element method usually select the Lagrange function as the basis function of the interpolation function. Similar to the basis function of coordinate transformation, its form is

[0100] (6)

[0101] As mentioned before when discussing coordinate transformation, the global coordinates of the corresponding nodes of each element are different. If the global coordinates are adopted, the expansion expressions of the field functions will be different, resulting in a large amount of computing memory and a complex computing process. Therefore, similar to coordinate transformation, dimensionless local coordinates are introduced for the field function to simplify, standardize, and unify the expression. The local coordinate ( ) and the global coordinate x are related as

[0102] (7)

[0103] where, is the global coordinate of the first node; is the global coordinate of the last node.

[0104] In the local coordinate system, the basis function of the field function of each element is

[0105] (8)

[0106] Substituting the above formula into Equation (5), the expression of the field function of the element displacement field in the local coordinate system can be obtained

[0107] (9)

[0108] where, is the dimensionless local coordinate corresponding to the interpolation node.

[0109] It should be noted that, generally, the preprocessing process of the finite element method and the spectral element method only gives the coordinate information of geometric nodes. The Jacobian matrix and its determinant involved in the coordinate transformation are generated based on the coordinates of these geometric nodes. For the calculation nodes in the field function, only the position parameters of each node in the local coordinate system are required.

[0110] Furthermore, in the implementation process of the improved artificial boundary anti-drift instability scheme of the spectral element model in step (2), it should be noted that for the saturated two-phase porous medium, in addition to applying the MTF drift instability control scheme to the solid phase displacement of the artificial boundary nodes according to the above formula, the liquid phase displacement also needs to be applied simultaneously, and the specific format is similar to formula (3).

[0111] Furthermore, in step (3), first calculate the wave amplitude coefficients of different medium layers according to the TMM analytical solution, and then, based on the position information of different nodes, find the wave amplitude coefficients of the corresponding nodes. After the FFT transformation, the time history input data of the solid-liquid two-phase displacement of the node is obtained. It should be noted that the spectral element node information is different from the finite element node and belongs to unequally spaced nodes.

[0112] Furthermore, in step (4), to facilitate the error analysis of the two-dimensional wave field, the Frobenius norm of the error wave field is calculated, and the maximum error wave field ratio is introduced, and the calculation formula is as follows:

[0113] (10)

[0114] In the formula, U is the matrix form of the fluctuating displacement at each moment, and i and j represent the numbers and displacement components of all nodes in the calculation domain. The subscript represents the selected area of the calculation node. The superscripts and represent the corresponding half-space model and full-space model in the calculation, respectively. t is the corresponding moment or time range in the calculation. The error wave field is the numerical solution in the calculation domain in the half-space model minus the exact solution (i.e., the numerical solution of a large area not affected by the boundary) in the calculation domain in the full-space model , and can be expressed as .

[0115] The present invention will be further described below with specific examples.

[0116] Embodiment

[0117] As Figure 1 shown, a control method for artificial boundary drift instability includes the following steps:

[0118] Collect geological data and establish a model database;

[0119] Discretize the space using the spectral element method and the unified framework of fluid-structure interaction;

[0120] Introduce an improved absorbing boundary method on the artificial boundary;

[0121] Obtain the boundary wave input data using the analytical solution;

[0122] Calculate the broadband displacement response of the surface nodes and provide seismic safety evaluation suggestions.

[0123] As Figure 2 shown, it is a schematic diagram of a simplified model of a complex medium site. The site medium is ideal fluid - saturated soil - bedrock. The bottom of the model is bedrock, with a length of , a height of on the left, a thickness of on the right, a length of at the top of the slope. The upper part of the basin is covered with saturated soil and ideal fluid. The thickness of the ideal fluid is , the length of the water surface is , the thickness of the saturated soil is , the length of the free surface part of the saturated soil is , and the slope angle is (14.03°, 26.56°, 45°; corresponding slopes are 1:4, 1:2, 1:1). Numerous measuring points (red nodes in the figure) are evenly distributed on the surface of the saturated soil. The first-order MTF artificial boundary is used on the left and right sides and the bottom of the model (the artificial wave speed is taken as the larger value of the solid longitudinal wave speed and the liquid longitudinal wave speed). The incident P-wave enters the site from the lower right corner of the model at an angle of (0°, 5°, 10°, 15°, 20°). The grid size of the spectral element is (mainly 10m), the element order is ( ), and the time step is . Figure 4 and Figure 5 are respectively the snapshots of the solid-phase displacement wave field and the liquid-phase displacement wave field when the incident angle is 20°. Through the wave field snapshots, the propagation process of the plane P-wave in the complex medium site can be observed more intuitively.

[0124] Figure 6The peak diagrams of the solid-liquid two-phase displacements of each measuring point are given under different slope conditions. It can be seen from the diagrams that the closer the measuring point is to the foot of the hill slope, the smaller the vertical displacement of the solid-liquid two-phase; within the range of 50 m from the foot of the bedrock slope, the greater the slope, the smaller the peak value of the vertical displacement of the solid-liquid two-phase; within the range of 100 m to 300 m, the greater the slope, the greater the vertical displacement of the solid-liquid two-phase; for the measuring points close to the shore (the distribution distance of the measuring points is 700 m), the vertical displacement of the solid-liquid two-phase is relatively large; due to the small incident angle and the influence of the reflected wave of the bedrock slope, the peak values of the lateral displacements of the solid-liquid two-phase of each measuring point are generally small, and the correlation with the slope is not obvious.

[0125] Figure 7 It can reflect the differences between the spectral element method and the finite element method when simulating the same model. Especially in the frequency band range of 16 - 20 Hz, the spectral ratio of the spectral element method is higher than that of the finite element method, that is, in the simulation of fluctuations in the high-frequency range, the spectral element method is more accurate than the finite element method.

[0126] The present invention proposes a high-precision calculation method and system for the site effect of a complex medium site. Based on the explicit spectral element method in the fluid-solid coupling unified calculation framework whose applicability has been verified, an improved MTF anti-reflection scheme is introduced at the artificial boundary, further increasing the calculation stability and accuracy. The calculation results show that this method can accurately and efficiently simulate the wave propagation problem of a complex medium site, and compared with the traditional method, this method has higher accuracy and stability in the wide frequency band range.

[0127] The description of the above embodiments is only used to help understand the method and its core idea of the present invention. It should be noted that for those of ordinary skill in the art of this technology, without departing from the principle of the present invention, several improvements and modifications can still be made to the present invention, and these improvements and modifications also fall within the protection scope of the claims of the present invention.

Claims

1. A high-precision calculation method for the site effect of a complex medium site, characterized in that, It includes the following steps: Step 1) Conduct spectral element spatial discretization of the fluid-saturated soil-bedrock site and use the unified computational framework theory to interact interface information; Step 2) Introduce an improved anti-reflection calculation method at the artificial boundary; Step 3) According to the TMM analytical method, give the input data of the spectral element node fluctuations at the boundary; Step 4) Conduct wide-band high-precision simulation of the displacement response of the surface nodes and give the amplification factor.

2. The high-precision calculation method for the site effect of a complex medium site according to claim 1, characterized in that In Step 1), the spectral element spatial discretization of the fluid-saturated soil-bedrock site is carried out, and the specific calculation formula is: (1) In the above formula, and are the interaction forces between units; (2a) (2b) (2c) (2d) (2e) (2f) (2g) (2h) Among them, , , N is the shape function, and in any one direction , is the solid-phase mass matrix, is the liquid-phase mass matrix, is the damping matrix representing the viscous resistance between the solid and liquid phases, is the solid-phase stiffness matrix, is the liquid-phase stiffness matrix, is the coupling matrix representing the constitutive force between the solid and liquid phases, is the load matrix acting on the solid phase, is the load matrix acting on the liquid phase, and are the interaction forces acting on the solid and liquid phases between the units respectively; when this node is an internal node of various media, during the element assembly process, due to the interaction forces between the units being equal in magnitude and opposite in direction, and cancel each other out and their values are 0.

3. A high-precision calculation method for the site effect of a complex medium site according to claim 2, characterized in that, When conducting spectral element spatial discretization calculation, taking the two-dimensional problem as an example, where, (3a) (3b) (3c) , (3d) (3e) (3f) where $n_g$ is the number of nodes of the spectral element, is the outward normal vector; is the directional derivative matrix.

4. A high-precision calculation method for the site effect of a complex medium site according to claim 3, characterized in that, The specific formula for Step 2) is: (3a) (3b) (3c) (3d) In the formula, is the solid-phase displacement of the i-th spectral element grid node from the artificial boundary at time, is the interpolation coefficient of the corresponding node, is the distance between each discrete grid node of the spectral element and the boundary node, as shown in Table 2; the media of the model are all single-phase solids. For the saturated two-phase porous media involved, in addition to applying the MTF drift instability control scheme to the solid-phase displacement of the artificial boundary nodes according to the above formula, the liquid-phase displacement also needs to be applied simultaneously.

5. A high-precision calculation method for the site effect of a complex medium site according to claim 4, characterized in that, Step 3) Specifically includes: for the external source problem, the input of ground motion needs to be realized through the artificial boundary and obtained by the TMM analytical method; subtracting the incident wave field solution from the full wave field solution to obtain the scattered wave field at a previous moment, after obtaining the displacements of each reference point of the MTF, obtaining the boundary node displacements at the (p + 1)-th moment, and then combining with the displacements of the remaining internal nodes calculated by the wave equation, the complete full wave field solution at the (p + 1)-th moment is obtained.

6. A high-precision calculation method for the site effect of a complex medium site according to claim 5, characterized in that The wide-band high-precision simulation of the displacement response of the surface nodes in Step 4) is specifically as follows: Taking any measurement point of the calculation model as the research object, give the corresponding solid-phase vertical displacement time history and the spectral ratio between it and the input wave. The spectral ratio calculation formula is as follows: (4-1a) (4-1b) where \(S(f)\) is the spectral ratio, \(U(f)\) is the spectral amplitude, and the subscripts FEM, SEM, and 0 correspond to the finite element method, spectral element method, and input wave, respectively. A small positive value taken to avoid a zero denominator, here we take .

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