Three-dimensional layered site seismic vibration input method based on infinite element artificial boundary

Abstract

The application relates to the technical field of geotechnical engineering anti-seismic analysis, in particular to a three-dimensional layered ground seismic vibration input method based on an infinite element artificial boundary, which comprises the following steps: solving a three-dimensional layered ground seismic equivalent node force expression considering the infinite element artificial boundary under a free field; establishing a three-dimensional layered ground model mixed with infinite elements and finite elements, obtaining the numbering, spatial coordinates and equivalent node area of each boundary node at the boundary of the finite domain, and calculating the ground stress field node force matched with the initial stress field; calculating the seismic equivalent node force of the grid node at the junction of the infinite and finite grids, batch modifying the layered ground model calculation file, and realizing the initial static force and seismic force input of each boundary node. The seismic vibration equivalent node force obtained by the application has high precision, conforms to the propagation characteristics of the seismic wave in the layered ground, and realizes batch application of the seismic node force of the heterogeneous ground, and has simplicity and operability.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of geotechnical earthquake-resistant analysis, and particularly to a three-dimensional layered ground seismic vibration input method based on an infinite element artificial boundary. BACKGROUND

[0002] The finite element method has been widely used in the field of geotechnical earthquake-resistant analysis. The ground has semi-infinite characteristics, but in most cases, seismic analysis only concerns near-field wave problems, so the finite calculation area of interest needs to be cut from the infinite domain, and artificial boundary conditions are set to make the propagation characteristics of mechanical waves at the cut boundary consistent with the actual infinite domain. As a way to simulate the infinite region, the infinite element has similar element shape functions as the conventional finite element, so the infinite element and the traditional finite element have an inherent "coordination", and compared with other dynamic artificial boundary methods such as viscoelastic boundary and boundary element, the infinite element has more advantages.

[0003] The infinite element can greatly reduce the calculation cost of the model, but the current infinite element dynamic artificial boundary is only suitable for local point source vibration problems in the domain. For external source incident problems such as earthquakes, the infinite element cannot directly handle them. Based on this, many scholars at home and abroad have carried out related research on external source incident of earthquakes. It should be noted that natural soil layers have layering characteristics, and the reflection and transmission of waves at the interfaces of each soil layer need to be considered, which is very different from the homogeneous ground model. At present, most of the layered ground seismic input and seismic dynamic analysis methods apply the equivalent node force of the homogeneous ground model, only considering the influence of different materials on wave velocity, and ignoring the influence of medium interfaces on seismic waves. Although some scholars have proposed seismic input methods considering the layering characteristics of the ground, they are too simplified and differ greatly from the actual situation, and need further research. SUMMARY

[0004] In view of the defects in the prior art, the present application provides a three-dimensional layered ground seismic vibration input method based on an infinite element artificial boundary.

[0005] In order to achieve the above object, the application provides a three-dimensional layered ground seismic vibration input method based on an infinite element artificial boundary, which comprises the following steps: solving a three-dimensional layered ground seismic equivalent node force expression considering an infinite element artificial boundary under a free field; establishing a three-dimensional layered ground model mixed with infinite elements and finite elements, obtaining the numbering, spatial coordinates and equivalent node area of each boundary node at the boundary of the finite domain, and calculating the ground stress field node force matched with the initial stress field; calculating the seismic equivalent node force of the grid node at the junction of the infinite and finite grids, and batch modifying the layered ground model calculation file to realize the initial static force and seismic force input of each boundary node. The seismic vibration equivalent node force obtained by the application has high accuracy, conforms to the propagation characteristics of seismic waves in the layered ground, and realizes batch application of the heterogeneous ground seismic node force, which is simple and operable.

[0006] Optionally, the solving of the three-dimensional layered ground seismic equivalent node force expression considering the infinite element dynamic artificial boundary under the free field comprises the following steps:

[0007] According to the one-dimensional wave equation suitable for a plane wave, the element stress balance condition and the dynamic artificial boundary complete absorption boundary condition, the normal damper damping coefficient and the tangential damper damping coefficient in each layered soil layer are determined;

[0008] Considering the case of vertically incident seismic waves at the bottom of the ground, the reflection coefficient and the transmission coefficient of the waves are determined according to the amplitude value relationship of the incident wave, the reflected wave and the transmitted wave;

[0009] According to the normal damper damping coefficient, the tangential damper damping coefficient, the reflection coefficient, the transmission coefficient and the velocity time history of the input seismic wave, the seismic equivalent node force of the three-dimensional layered ground is calculated.

[0010] Optionally, the normal damper damping coefficient and the tangential damper damping coefficient satisfy the following relationships respectively:

[0011]

[0012]

[0013] wherein, the normal damper damping coefficient of the i-th soil layer is the tangential damper damping coefficient of the i-th soil layer is the soil layer density of the i-th soil layer is the P-wave velocity in the i-th soil layer is the S-wave velocity in the i-th soil layer is.

[0014] Optionally, the reflection coefficient satisfies the following relationship:

[0015]

[0016] wherein, is the reflection coefficient of the wave, is the amplitude of the incident wave, is the amplitude of the transmitted wave, is the soil density of a soil layer far from the ground among two adjacent soil layers, is the wave speed in a soil layer far from the ground among two adjacent soil layers, is the soil density of a soil layer close to the ground among two adjacent soil layers, is the wave speed in a soil layer close to the ground among two adjacent soil layers.

[0017] Optionally, the transmission coefficient satisfies the following relationship:

[0018]

[0019] wherein, is the transmission coefficient of the wave, is the reflection coefficient, is the amplitude of the reflected wave, is the amplitude of the transmitted wave, is the soil density of a soil layer far from the ground among two adjacent soil layers, is the wave speed in a soil layer far from the ground among two adjacent soil layers, is the soil density of a soil layer close to the ground among two adjacent soil layers, is the wave speed in a soil layer close to the ground among two adjacent soil layers.

[0020] Optionally, under the action of P wave, the seismic equivalent node force at the bottom boundary of a three-dimensional layered foundation satisfies the following relationship:

[0021]

[0022] wherein, is the seismic equivalent node force at the bottom boundary of a three-dimensional layered foundation under the action of P wave, the subscript z of indicates the direction of the seismic equivalent node force, the superscript -z of indicates the plane where the face normal is located and the face normal points to the negative direction of the z axis, is the normal damper damping coefficient of the first soil layer, is the soil vibration speed at time t, t is time, is the node control area.

[0023] Optionally, under the action of P wave, the seismic equivalent node force at the front and back boundaries of a three-dimensional layered foundation satisfies the following relationship:

[0024]

[0025]

[0026] wherein, and are the seismic equivalent nodal forces at the back side boundary of the three-dimensional layered foundation, and are the seismic equivalent nodal forces at the front side boundary of the three-dimensional layered foundation, denotes the plane in which the surface normal lies and the surface normal points to the negative direction of the axis, and are the seismic equivalent nodal forces at the back side boundary of the three-dimensional layered foundation, and are the seismic equivalent nodal forces at the front side boundary of the three-dimensional layered foundation, denotes the plane in which the surface normal lies and the surface normal points to the positive direction of the axis, , , and are the directions of the seismic equivalent nodal forces, and z are the directions of the seismic equivalent nodal forces, is the first Lame constant of the i-th soil layer, is the P-wave velocity in the i-th soil layer, is the transmission coefficient of the wave at the interface j, is the soil vibration velocity after time delay t is time, is the reflection coefficient of the wave at the interface i, is the soil vibration velocity after time delay t is time, is an algebraic expression, , is an algebraic expression, , is the reflection coefficient of the wave at the interface n, is the transmission coefficient of the wave in the k-th soil layer, is an algebraic expression, , , is a parameter related to i, is the soil vibration velocity after time delay t is time, is the reflection coefficient of the wave at the interface i-1, is the soil vibration velocity after time delay t is time, , , and are four time delays, is the nodal control area.

[0027] Optionally, under the action of P wave, the seismic equivalent node force at the left and right side boundaries of the three-dimensional layered foundation satisfies the following relationship:

[0028]

[0029]

[0030] wherein, and are the seismic equivalent node forces at the left side boundary of the three-dimensional layered foundation, and the superscripts of and indicate the plane in which the face normal is located and the face normal points to the negative direction of the y axis, and are the seismic equivalent node forces at the front side boundary of the three-dimensional layered foundation, and the superscripts of and indicate the plane in which the face normal is located and the face normal points to the positive direction of the y axis, , , and the subscripts y and z of and is the first Lame constant of the i-th layer of soil, is the P wave velocity in the i-th layer of soil, is the transmission coefficient of the wave at the interface j, is the soil vibration velocity after time delay , is the reflection coefficient of the wave at the interface i, is the soil vibration velocity after time delay , is an algebraic expression, , is an algebraic expression, , is the reflection coefficient of the wave at the interface n, is the transmission coefficient of the wave in the k-th layer of soil, is an algebraic expression, , , is an i-related parameter, is the soil vibration velocity after time delay , is the reflection coefficient of the wave at the interface i-1, is the soil vibration velocity after time delay , , , and is 4 time delays, is the node control area.

[0031] Optionally, the three-dimensional layered foundation model of the mixed infinite element and finite element is established, the numbering, spatial coordinates and equivalent node area of each boundary node at the boundary of the finite domain are obtained, and the node force of the geostress field matching the initial stress field is calculated.

[0032] The finite domain range of the layered foundation model is determined, the three-dimensional modeling of the finite element and the infinite element coupling of the foundation soil is carried out, and the three-dimensional layered foundation model is obtained.

[0033] The three degrees of freedom of the boundary of the finite domain are constrained, and a unit pressure is applied to the boundary of the finite domain, so that the numbering, spatial coordinates and equivalent node area of each boundary node at the boundary of the finite domain are obtained.

[0034] The soil layer number of the boundary node is determined according to the Z coordinate of the spatial coordinates and the soil layer height, and the plane where the boundary node is located is determined according to the spatial coordinates.

[0035] The initial geostress field is applied to the three-dimensional layered foundation model, and the node force of the geostress field of each boundary node is calculated according to the initial geostress distribution.

[0036] Optionally, the calculation of the seismic equivalent node force of the grid node at the junction of the infinite and finite grids, the batch modification of the calculation file of the layered foundation model, and the initial static force and seismic force input of each boundary node include the following steps:

[0037] The input seismic wave at the bottom of the foundation is determined, the corresponding velocity time history curve is obtained, and the time delay of each boundary node is calculated.

[0038] The reflection coefficient and the transmission coefficient are calculated, the vibration velocity of each boundary node at any time is determined, and the seismic equivalent node force of each boundary node at any time is calculated.

[0039] Based on the fopen function and the fprintf function embedded in Matlab, the node set is created, the amplitude function is established, and the concentrated node force is applied according to the ABAQUS calculation model inp file format and the convention, and the seismic equivalent node force and the geostress field node force of each boundary node are applied.

[0040] The present application has at least the following beneficial effects:

[0041] 1. The application is based on the infinite element dynamic artificial boundary, and an equivalent node force input method of ground seismic vibration is provided, which can accurately consider the influence of layered heterogeneous characteristics, and the influence of different materials and medium interfaces on the propagation of seismic waves is considered at the same time, compared with the seismic node force obtained by the conventional homogeneous model and the layered foundation model considering only the first transmission and reflection, the precision of the application is higher, and the propagation characteristics of the seismic wave in the layered foundation are more in line with the propagation characteristics of the seismic wave in the layered foundation.

[0042] 2. The application realizes batch application of the heterogeneous ground seismic node force, has simplicity and operability, and is very suitable for popularization and use in the seismic dynamic response analysis of layered heterogeneous soil and structure interaction. DETAILED DESCRIPTION

[0043] In order to more clearly illustrate the technical solutions of the embodiments of the present application, the drawings needed in the embodiments will be briefly introduced as follows, and it should be understood that the following drawings only show some embodiments of the present application, and therefore should not be regarded as a limitation on the scope, and for those skilled in the art, other related drawings can also be obtained without creative labor on the basis of these drawings.

[0044] Figure 1 A flowchart of a three-dimensional layered ground seismic vibration input method based on the infinite element artificial boundary of the embodiment of the present application;

[0045] Figure 2 A schematic diagram of a seismic equivalent node force calculation model of a three-dimensional layered foundation of the embodiment of the present application;

[0046] Figure 3 A schematic diagram of a three-dimensional layered foundation model example of the embodiment of the present application;

[0047] Figure 4 A velocity time history curve of the input seismic wave at the bottom of the three-dimensional layered foundation model of the embodiment of the present application;

[0048] Figure 5 The power spectral density of the input seismic wave of the embodiment of the present application;

[0049] Figure 6 A schematic diagram for verifying the effectiveness of the method of the embodiment of the present application;

[0050] Figure 7 The soil displacement time history diagram of the embodiment of the present application when the method of the present application, the far boundary model, the layered material characteristic considering model and the layered foundation model considering only the first interlayer reflection and transmission are used respectively. DETAILED DESCRIPTION

[0051] Specific embodiments of the present application will now be described in detail with reference to the drawings, which are by way of example only and shall not be taken as limiting the present application. In the following description, numerous specific details are set forth in order to provide a thorough understanding of the present application. It will be apparent to one skilled in the art, however, that the present application can be practiced without using these specific details. In other instances, well-known circuits, software or methods have not been described in detail in order to avoid obscuring the present application.

[0052] Reference throughout this specification to "one embodiment", "an embodiment", "one example", or "an example" means that a particular feature, structure, or characteristic described in connection with the embodiment or example is included in at least one embodiment of the application. Thus, appearances of the phrases "in one embodiment" or "in an embodiment" or "one example" or "an example" in various places throughout this specification are not necessarily all referring to the same embodiment or example. Furthermore, the particular features, structures, or characteristics can be combined in any suitable

[0053] It is noted that, in an alternative embodiment, the same symbols or letters appearing in all formulas have the same meaning and value except as otherwise specifically noted.

[0054] In an alternative embodiment, referring to Figure 1 The present application provides a three-dimensional layered soil seismic vibration input method based on an infinite-element artificial boundary, which comprises the following steps:

[0055] S1, solving the seismic equivalent node force expression of the three-dimensional layered soil foundation considering the infinite-element artificial boundary under free field.

[0056] In the step S1, the following steps are included:

[0057] S11, determining the normal damper damping coefficient and the tangential damper damping coefficient in each layered soil layer according to the one-dimensional wave equation applicable to plane waves, the element force balance condition, and the dynamic artificial boundary complete absorption boundary condition.

[0058] Specifically, in the present embodiment, in the semi-infinite foundation, the seismic wave can be approximately considered as a plane wave, and the wave motion conforms to the one-dimensional wave equation. The displacement component of the plane wave in the propagation process is only a function of the z coordinate and the time t. For the P-wave action, the wave equation of the soil element of the i-th soil layer can be expressed as the following differential equation:

[0059]

[0060] In the step S11, the normal damper damping coefficient and the tangential damper damping coefficient in each layered soil layer are determined according to the one-dimensional wave equation applicable to plane waves, the element force balance condition, and the dynamic artificial boundary complete absorption boundary condition. is the soil displacement of the i-th layer of soil in the z-axis direction, i = 1, 2, 3,..., n, n is the total number of soil layers, is the P-wave velocity in the i-th layer of soil, , is the soil density of the i-th layer of soil, and are the shear modulus and the first Lame constant of the i-th layer of soil, respectively. The positive direction of the z-axis is the direction perpendicular to the horizontal plane upward, and the absolute value of z represents the distance of a point in the soil layer from the coordinate origin. In the subsequent description, the i-th layer of soil can also be referred to as soil layer i. In addition, in the subsequent description, if no separate description is made for the symbols z, x, and y in the formula, z, x, and y all represent the coordinate values on the corresponding coordinate axes.

[0061] The general solution of the above differential equation is , is the upgoing wave propagating in the positive z-axis direction of the i-th layer of soil, is the downgoing wave propagating in the negative z-axis direction of the i-th layer of soil. According to the theory of elasticity, the P-wave soil stress is:

[0062]

[0063]

[0064] wherein, , and all represent the normal stress of the soil, , and The first symbol from left to right before the comma subscript represents the direction of the soil stress, and the second symbol represents the normal direction of the plane on which the soil stress lies.

[0065] In dynamic analysis, the dynamic artificial boundary of the infinite element is essentially a viscous boundary. The additional damping stress of the soil element at the interface between the infinite element and the finite element under the action of P-wave is which can be expressed as wherein is the normal damper damping coefficient at the artificial boundary, and u is the soil displacement.

[0066] According to the force balance condition of the element and the complete absorption boundary condition, we have , is the normal damper damping coefficient of the i-th layer of soil. Similarly, the tangential damper damping coefficient of each soil layer under the action of S-wave is satisfying , is the tangential damper damping coefficient of the i-th layer of soil, S-wave velocity in the i-th layer of soil, .

[0067] S12, considering the case of vertical incidence of seismic wave at the bottom of foundation, the reflection coefficient and transmission coefficient of wave are determined according to the amplitude relationship of incident wave, reflected wave and transmitted wave.

[0068] Specifically, in the embodiment, considering the case of vertical incidence of seismic wave at the bottom of foundation, according to the Fresnel equation, the mathematical relationship between the amplitudes of incident wave and transmitted wave and the mathematical relationship between the amplitudes of reflected wave and transmitted wave are respectively as follows:

[0069]

[0070]

[0071] wherein, is the reflection coefficient of wave, is the transmission coefficient of wave, is the amplitude of incident wave, is the amplitude of reflected wave, is the amplitude of transmitted wave, is the soil density of the layer of soil far away from the ground in the adjacent two layers of soil, is the wave velocity in the layer of soil far away from the ground in the adjacent two layers of soil, is the soil density of the layer of soil close to the ground in the adjacent two layers of soil, is the wave velocity in the layer of soil close to the ground in the adjacent two layers of soil.

[0072] S13, according to the normal damper damping coefficient, the tangential damper damping coefficient, the reflection coefficient, the transmission coefficient and the velocity time history of input seismic wave, the seismic equivalent node force of three-dimensional layered foundation is calculated.

[0073] Specifically, in the embodiment, please refer to Figure 2 , taking the vertical incidence of P wave at the bottom of foundation as an example, if the vertical incidence of wave at the bottom of foundation is , the vibration velocity can be expressed as , for the i-th layer of soil, the displacement of any soil microelement in it is composed of transmitted wave and reflected wave . Considering the delay of wave and twice incidence and reflection at the interface, and respectively satisfy the following relationships:

[0074]

[0075]

[0076] wherein, is the transmission coefficient of the wave at the interface j, is the time delay is the incident wave after the time delay is the time delay is the incident wave after the time delay is the time delay is the incident wave after the time delay is the time delay is the incident wave after the time delay is an algebraic expression, , is the transmission coefficient of the wave in the kth layer of soil and , is the reflection coefficient of the wave at the interface n, is the reflection coefficient of the wave at the interface i, is the reflection coefficient of the wave at the interface i-1, , , and are four time delays. , , and satisfy the following relationships in turn:

[0077]

[0078]

[0079]

[0080]

[0081] wherein, is the thickness of the jth layer of soil, is the seismic wave wave velocity in the jth layer of soil, is the seismic wave wave velocity in the ith layer of soil, is the distance of the soil mass from the lower adjacent interface, is the thickness of the ith layer of soil, Figure 2 is the distance of the soil mass A from the interface i-1. In the relationship of calculating

[0082] , the transmission effect of all interfaces below the current layer of soil on the incident wave is considered, which belongs to the positive propagation of the wave; the backward re-incident wave after the wave propagates to the ground (the nth layer of soil) and is reflected, which belongs to the negative propagation of the wave. In the relationship of calculating , the transmission effect of all interfaces below the current layer of soil on the incident wave is considered, which belongs to the positive propagation of the wave; the backward re-incident wave after the wave propagates to the ground (the nth layer of soil) and is reflected, which belongs to the negative propagation of the wave. In the relationship of calculating , the transmission effect of all interfaces below the current layer of soil on the incident wave is considered, which belongs to the positive propagation of the wave; represents the reflected wave of the wave at the top interface of the i-th layer of soil, and belongs to the negative direction propagation of the wave, represents the reflected wave of the wave at the bottom interface of the i-th layer of soil, and belongs to the positive direction propagation of the wave. The interface refers to the interface between adjacent soil layers, the top interface of the i-th layer of soil can be denoted as interface i, the bottom interface of the i-th layer of soil can be denoted as interface i-1, the top interface of the i-1-th layer of soil can also be denoted as interface i-1, and so on.

[0083] Therefore, the vibration velocity of the layered foundation soil body under the action of the P wave can be represented as:

[0084]

[0085] wherein, is an algebraic expression, , is the vibration velocity of the soil body after time delay , is the vibration velocity of the soil body after time delay , is the vibration velocity of the soil body after time delay , is the vibration velocity of the soil body after time delay , is a parameter related to i. The following relationship is satisfied:

[0086]

[0087] Through the above analysis, it can be concluded that under the action of the P wave, the seismic equivalent node force at the bottom boundary of the three-dimensional layered foundation, i.e. the seismic equivalent node force in the z-axis direction, satisfies the following relationship:

[0088]

[0089] wherein, is the seismic equivalent node force at the bottom boundary of the three-dimensional layered foundation under the action of the P wave, the subscript z of indicates the direction of the seismic equivalent node force, the superscript -z of indicates the plane in which the face normal is located and the face normal points to the negative direction of the z-axis, is the normal damper damping coefficient of the first layer of soil, is the vibration velocity of the soil body at time t, is the node control area.

[0090] Under the action of the P wave, the seismic equivalent node force at the front and back boundaries of the three-dimensional layered foundation, i.e. the seismic equivalent node force in the x-axis direction, satisfies the following relationship:

[0091]

[0092]

[0093] in, and All are seismic equivalent nodal forces at the rear boundary of a three-dimensional layered foundation. and superscript Indicates the plane containing the surface normal and the direction the surface normal points. In the negative direction of the axis, and All are seismic equivalent nodal forces at the front boundary of a three-dimensional layered foundation. and superscript Indicates the plane containing the surface normal and the direction the surface normal points. Positive axis direction , , and subscript Both z and z represent the direction of the equivalent nodal force in an earthquake. For delay The subsequent soil vibration velocity, For delay The subsequent soil vibration velocity, For algebraic expressions, , It is a parameter related to i. For delay The subsequent soil vibration velocity, For delay The subsequent soil vibration velocity.

[0094] Under the action of P-waves, the seismic equivalent nodal forces at the left and right boundaries of a three-dimensional layered foundation, i.e., the seismic equivalent nodal forces in the y-axis direction, satisfy the following relationship:

[0095]

[0096]

[0097] in, and All are seismic equivalent nodal forces at the left boundary of a three-dimensional layered foundation. and superscript This indicates the plane containing the surface normal, and the surface normal points in the negative y-axis direction. and All are seismic equivalent nodal forces at the front boundary of a three-dimensional layered foundation. and superscript This indicates the plane containing the surface normal, with the surface normal pointing in the positive y-axis direction. , , and The subscripts y and z both indicate the direction of the earthquake equivalent nodal force.

[0098] In addition, it should be noted that, in Figure 2 middle, The transmitted wave is located at the top layer (interface i) of the i-th soil layer. The reflected wave is located at the top layer of the i-th soil layer (interface i). This represents the transmitted wave at the top layer of the (i-1)th soil layer (interface i-1). The reflected wave is located at the top layer of the (i-1)th soil layer (interface i-1).

[0099] Furthermore, for S-waves incident vertically at the bottom of the foundation, the derivation process and expression of the equivalent nodal force of the earthquake are similar to those for P-waves, and will not be elaborated here.

[0100] S2. Establish a three-dimensional layered foundation model that combines infinite and finite element methods, obtain the number, spatial coordinates, and equivalent node area of ​​each boundary node at the boundary of the finite domain, and calculate the nodal forces of the geostress field that match the initial stress field.

[0101] Step S2 specifically includes the following steps:

[0102] S21. Determine the finite domain range of the layered foundation model, perform three-dimensional modeling of the foundation soil using finite element and infinite element coupling, and obtain the three-dimensional layered foundation model.

[0103] Specifically, in this embodiment, such as Figure 3 As shown in (a), this embodiment is based on a three-dimensional layered foundation model using ABAQUS numerical software. The three-dimensional layered foundation model is a cuboid with the z-axis pointing vertically upwards and the x and y axes being horizontal axes. The model dimensions are 105m × 55m × 55m, with the finite field dimensions being 100m × 50m × 50m. A three-dimensional cube is created using the Extrusion command, and the three-dimensional cube is divided into near-field and far-field regions using the Partition Cell command. Then, a three-dimensional model of the foundation soil is performed using coupled finite and infinite element methods to obtain the three-dimensional layered foundation model. The element type of the near-field finite field is C3D8R, and the element type of the far-field infinite field is changed to CIN3D8 by modifying the keyword *Element.

[0104] The soil layers are numbered 5 from bottom to top as 1, 2, 3, 4, and 5, with each layer being 20 m thick. The material parameters of each soil layer from bottom to top are as follows: Poisson's ratio... v s=[0.3, 0.3, 0.3, 0.3, 0.3]; density p = [2.0, 2.0, 2.0, 2.0, 2.0] t / m 3 ; modulus of elasticity E = [500000, 300000, 200000, 150000, 60000] kPa; unit weight y = [6.0, 6.0, 6.0, 6.0, 6.0] kN / m 3 . The wave velocity information of each soil layer is shown in Table 1.

[0105] Table 1 Wave velocity information of each soil layer

[0106] Soil layer number Thickness / m E / kPa c s (m / s)]]> c p (m / s) 1 20 500000 310 580 2 20 300000 240 449 3 20 200000 196 367 4 20 150000 170 318 5 20 60000 107 201

[0107] In Table 1 is the S-wave velocity, is the P-wave velocity.

[0108] Further, the mechanical parameters of natural soil have depth dependence. In order to reduce the calculation cost, a non-uniform grid form is used along the height direction when constructing the three-dimensional layered foundation model. The grid size of each soil layer is determined by the following relationship:

[0109]

[0110] wherein, is the highest frequency component of the ground motion, can be determined according to the power spectral density of the selected ground motion, c is the wave velocity in the soil layer, which can be calculated according to the calculation formula provided in step S11 or the relationship formula .

[0111] S22, constrain the three degrees of freedom of the boundary of the finite field, and apply a unit pressure to the boundary of the finite field to obtain the numbering, spatial coordinates and equivalent node area of each boundary node at the boundary of the finite field.

[0112] Specifically, in the present embodiment, for Figure 3 ​In the three-dimensional layered foundation model shown in (a), in the component module of the ABAQUS pre-processing interface, five surfaces (including four side surfaces and one bottom surface) of the finite domain boundary are uniformly established into a surface set "Surf-1", and a corresponding node set "Set-1" is established; in the analysis step module of ABAQUS, a Static general static analysis step is established for the layered foundation model of the established finite element-infinite element grid coupling; in the load module of ABAQUS, a unit pressure is applied to the surface set "Surf-1" using the pressure command, and the three degrees of freedom of the node set "Set-1" are constrained; in the field output requirement of ABAQUS, only the RF variable of the node set "Set-1" is selected for output, and finally the task is submitted. After the calculation is completed, in the Report Field Output window of the post-processing module of ABAQUS, the last analysis step is selected, the node reaction force RF of the node set "Set-1" at the boundary of the finite domain is output, and is written into the "abaqus.csv" file in the csv format, and finally the number, spatial coordinates and equivalent node area of the boundary node are obtained.

[0113] S23, according to the Z coordinate of the spatial coordinate and the soil layer height, determine the soil layer number where the boundary node is located, and according to the spatial coordinate, determine the plane where the boundary node is located.

[0114] Specifically, in the present embodiment, the "abaqus.csv" file is opened, the first row is deleted and only the 5th-8th columns and the 12th-14th columns are retained, and the retained column data is stored in the "Nodes_info.txt" file. In the matlab software, the data in the "Nodes_info.txt" file is imported into matlab using the load('Nodes_info.txt') command, and then the z-axis coordinate is used as the control index to determine the soil layer number of each boundary node in combination with the soil layer height.

[0115] Further, according to the spatial coordinates of each boundary node, the matlab programming is used to determine the attribution of the plane where the node is located, and the specific idea is as follows: when the x coordinate of the boundary node is equal to the minimum / maximum x coordinate of the nodes in the three-dimensional layered foundation model, the boundary node belongs to the front / rear side boundary (front / rear surface) of the three-dimensional layered foundation model; when the y coordinate of the boundary node is equal to the minimum / maximum y coordinate of the nodes in the three-dimensional layered foundation model, the boundary node belongs to the left / right side boundary (left / right surface) of the three-dimensional layered foundation model; when the z coordinate of the boundary node is equal to the minimum z coordinate of the nodes in the three-dimensional layered foundation model, the boundary node belongs to the bottom boundary (bottom surface) of the three-dimensional layered foundation model.

[0116] S24, applying an initial geo-stress field to the three-dimensional layered foundation model, and calculating the geo-stress field node force of each boundary node according to the initial geo-stress distribution.

[0117] Specifically, in the present embodiment, a body force is selected in the load module of ABAQUS, a body force field is applied to the finite domain, and the value of the body force field is-6. A predefined field is defined in ABAQUS, and an initial geo-stress field is applied to the entire three-dimensional layered foundation model, which is matched with the body force field, wherein the lateral earth pressure coefficient is 0.65.

[0118] Further, the geo-stress field node counterforce of each boundary node matched with the initial geo-stress field is calculated according to the specific gravity, the depth of the boundary node and the equivalent node area, and then the geo-stress field node force with the same size and opposite direction of the geo-stress field node counterforce is obtained. The depth of the boundary node is the z coordinate of the boundary node minus the z coordinate of the top surface of the three-dimensional layered foundation model.

[0119] S3, calculating the seismic equivalent node force of the grid node at the junction of the infinite and finite grids, batch modifying the layered foundation model calculation file, and realizing the initial static force and seismic force input of each boundary node.

[0120] Specifically, step S3 includes the following steps:

[0121] S31, determining the input seismic wave at the bottom of the foundation, obtaining the corresponding velocity time history curve, and calculating the time delay of each boundary node.

[0122] Specifically, in the present embodiment, a uniform incident pulse wave is used as the input seismic wave at the bottom of the three-dimensional layered foundation model, and when the data sampling frequency is 100 Hz, the velocity time history curve at the bottom of the three-dimensional layered foundation model is as shown in Figure 4 The pwelch power spectrum function built in the matlab software is selected to calculate the power spectrum density of the input seismic wave, and the result is as shown in Figure 5 It can be seen that the value of in step S21 is 8 Hz, so for the three-dimensional layered foundation model shown in (a) in Figure 3 , the size of the bottom three layers of grid can be set to 2 m, and the size of the top two layers of grid can be set to 1 m.

[0123] Further, the time delay of each boundary node is calculated according to the relationship shown in step S13 , , and Then, using the Hermite interpolation function embedded in the matlab software, combined with the sampling frequency of the selected ground motion, the velocity time history of each boundary node after considering the time delay is obtained, and finally four kinds of velocity time histories are obtained, which correspond to the acquisition 、 、 and . Note that in the four kinds of velocity time histories, for the time less than the time delay 、 、 and , the soil vibration velocity is 0.

[0124] S32, calculate the reflection coefficient and the transmission coefficient, and determine the vibration velocity of each boundary node at any time, and further calculate the seismic equivalent node force of each boundary node at any time.

[0125] Specifically, in this embodiment, the reflection coefficient and the transmission coefficient of different waveforms in each soil layer are calculated using the relationship shown in step S12, and the calculation results are shown in Table 2.

[0126] Table 2 Basic information of each soil layer interface

[0127] Interface / soil layer number Soil layer top z coordinate / m Soil layer bottom z coordinate / m p ]]> ​ β p ]]> s ]]> ​ β s ]]> 1 25 5 0.13 1.13 0.13 1.13 2 45 25 0.10 1.10 0.10 1.10 3 65 45 0.07 1.07 0.07 1.07 4 85 65 0.22 1.23 0.23 1.23 5 105 85 1 1 1 1

[0128] In Table 2, is the reflection coefficient of P wave, is the reflection coefficient of S wave, is the transmission coefficient of P wave, is the transmission coefficient of S wave.

[0129] Further, after calculating the reflection coefficient and the transmission coefficient, the soil vibration velocity is taken as the vibration velocity of the boundary node, and the seismic equivalent node force of each boundary node at any time can be calculated according to the calculation formula of the seismic equivalent node force under the action of P wave shown in step S13. Correspondingly, the calculation formula of the seismic equivalent node force under the action of S wave can also be used to calculate the seismic equivalent node force of each boundary node at any time. For the three-dimensional layered foundation model shown in (a) of Figure 3 , the seismic equivalent node force at a certain typical boundary node E is shown in (b) of Figure 3 .

[0130] S33, based on the fopen function and the fprintf function embedded in matlab, according to the ABAQUS calculation model inp file format and convention, batch create node set, establish amplitude function and apply concentrated node force, complete the application of seismic equivalent node force and ground stress field node force of each boundary node.

[0131] Specifically, in this embodiment, based on the built-in fopen and fprintf functions of MATLAB, the *Nset keyword is used to batch create node sets for each boundary node and write them to the "Set_Node.txt" file; the *Amplitude keyword is used to batch create amplitude functions for the seismic equivalent nodal forces of each boundary node and write them to the "CLoad-AMP.txt" file; the *Cload keyword is used to apply nodal reaction forces matching the geostress field to each boundary node and write them to the "GeoF.txt" file, where the nodal reaction forces matching the geostress field are: , .in, and The lateral reaction force at the boundary node, and The subscript indicates the load direction, and the load direction always points into the interior of the three-dimensional layered foundation model; This represents the distance from the boundary node to the Earth's surface. For the vertical reaction force at the boundary node, The subscript indicates the direction of the load. The value is always positive. Using the *Cload keyword, apply the seismic equivalent concentrated nodal force F to each boundary node and write it to the "Cload_S.txt" file. The seismic equivalent concentrated nodal force F includes... , , , , , , , and To further illustrate, some relevant MATLAB code is provided here:

[0132] "fid=fopen('Output for inp files\CLoad-AMP.txt','w');

[0133] fclose(fid);

[0134] fid=fopen('Output for inp files\CLoad-AMP.txt','a+');

[0135] fprintf(fid,['**Please locate:','\n','*End Assembly','\n']);

[0136] %---------------------------------Cload--------------------------------

[0137] fid1=fopen('Output for inp files\Cload_S.txt','w');

[0138] fclose(fid1);

[0139] fid1=fopen('Output for inp files\Cload_S.txt','a+');

[0140] fprintf(fid1,['**Place at the seismic analysis step','\n','**Please locate:','\n','**LOADS','\n','**','\n']);

[0141] fid2=fopen('Output for inp files\GeoF.txt','w');

[0142] fclose(fid2);

[0143] fid2=fopen('Output for inp files\GeoF.txt','a+');

[0144] fprintf(fid2,['**Starting from the geostress step, geostress nodal forces must be written in each subsequent analysis step','\n','**Please locate:','\n','**LOADS','\n','**','\n']);”

[0145] Using MATLAB, the corresponding .inp calculation files are generated based on the calculation formula for seismic equivalent nodal forces. All content from the "Set_Node.txt" file is placed at the *End Instance keyword; all content from the "CLoad-AMP.txt" file is placed at the *End Assembly keyword; all content from the "GeoF.txt" file is placed at the **Loads keyword in each analysis step; and all content from the "Cload_S.txt" file is placed at the **Loads keyword in the seismic loading step. At this point, the seismic equivalent nodal forces for a three-dimensional layered foundation based on an infinite element dynamic artificial boundary are input.

[0146] Further, in order to verify the effectiveness of the method, the formula of the equivalent seismic node force of the P-wave is degenerated into a homogeneous case in this embodiment to obtain a three-dimensional homogeneous foundation model. Except that the mechanical parameters of the soil layer 1 of the three-dimensional layered foundation model are consistent with those of the three-dimensional homogeneous foundation model, the remaining parameters of the three-dimensional homogeneous foundation model are consistent with those of the three-dimensional layered foundation model. The displacement time histories of the top point B and the bottom point C of the three-dimensional homogeneous foundation model are shown in Figure 6 It can be seen that the result obtained based on the method is consistent with the analytical solution, that is, the three-dimensional layered foundation equivalent seismic node force of the infinite element dynamic artificial boundary proposed in this embodiment can effectively reflect the wave propagation characteristics of the homogeneous semi-infinite foundation.

[0147] Further, for the three-dimensional layered foundation model shown in (a) of Figure 3 The soil displacement time histories of the D point when the method of the application, the far-field boundary model, the foundation model considering only the material layering characteristics, and the layered foundation model considering only the first reflection transmission are used are shown in Figure 7 It can be seen that the error of the top displacement time history of the soil layer 3 obtained by the method of the application and the far-field boundary result is less than 5%, and the soil displacement time history under the far-field boundary result can be considered as a true solution. Therefore, compared with the seismic equivalent node force calculation method considering only the material layering characteristics and the seismic equivalent node force calculation method considering only the first reflection transmission, the method of the application has higher accuracy.

[0148] It should be noted that in some cases, the actions described in the specification can be performed in different orders and still achieve the desired results, and in this embodiment, the order of the steps given is only to make the embodiment look clearer and more understandable, and is not a limitation.

[0149] In summary, the application based on the infinite element dynamic artificial boundary proposes an equivalent seismic node force input method of the layered inhomogeneous foundation, which can accurately consider the influence of the layered inhomogeneous characteristics. The method simultaneously considers the influence of different materials and medium interfaces on the propagation of seismic waves. Compared with the seismic node force obtained by the conventional homogeneous model and the layered foundation model considering only the first transmission and reflection, the method has higher accuracy and is more consistent with the propagation characteristics of the seismic wave in the layered foundation. In addition, the application also realizes the batch application of the inhomogeneous foundation seismic node force, which is simple and operable, and is very suitable for use in the seismic dynamic response analysis of the layered inhomogeneous soil-structure interaction.

[0150] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present application, and are not intended to limit the present application; although the present application has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that the technical solutions recorded in the foregoing embodiments can still be modified, or some or all of the technical features can be replaced by equivalents; and these modifications or replacements do not make the essence of the corresponding technical solutions deviate from the scope of the technical solutions of the embodiments of the present application, and they should be covered in the scope of the claims and the description of the present application.

Claims

1. A three-dimensional layered site seismic ground motion input method based on an infinite element artificial boundary, characterized by, Comprising the following steps: According to the one-dimensional wave equation applicable to plane waves, the unit force balance condition and the dynamic artificial boundary fully absorbing boundary condition, the normal damper damping coefficient and the tangential damper damping coefficient in each layered soil layer are determined; The normal damper damping coefficient and the tangential damper damping coefficient respectively satisfy the following relationship: ， ， wherein, is the normal dashpot damping coefficient for the i-th layer of soil, is the tangential dashpot damping coefficient for the i-th layer of soil, is the soil density for the i-th layer of soil, is the P-wave velocity in the i-th layer of soil, is the S-wave velocity in the i-th layer of soil; Considering the case of vertically incident seismic waves at the bottom of the foundation, the reflection coefficient and the transmission coefficient are determined according to the amplitude relationship of the incident wave, the reflected wave and the transmitted wave; The reflection coefficient satisfies the following relationship: ， wherein, is the reflection coefficient of the wave, is the amplitude of the incident wave, is the amplitude of the transmitted wave, is the soil density of the layer of soil further away from the ground of the two adjacent layers of soil, is the wave speed in the layer of soil further away from the ground of the two adjacent layers of soil, is the soil density of the layer of soil closer to the ground of the two adjacent layers of soil, is the wave speed in the layer of soil closer to the ground of the two adjacent layers of soil; According to the normal damper damping coefficient, the tangential damper damping coefficient, the reflection coefficient, the transmission coefficient and the velocity time history of the input seismic wave, the seismic equivalent node force of the three-dimensional layered foundation is calculated; An infinite element and finite element mixed three-dimensional layered foundation model is established, the numbering, spatial coordinates and equivalent node area of each boundary node at the boundary of the finite domain are obtained, and the geostress field node force matching the initial stress field is calculated; The seismic equivalent node force of the grid node at the junction of the infinite and finite grids is calculated, the layered foundation model calculation file is batch modified, and the initial static force and seismic force input of each boundary node is realized.

2. The three-dimensional layered site ground motion input method based on the infinite element artificial boundary according to claim 1, characterized by, The transmission coefficient satisfies the following relationship: ， wherein, is the transmission coefficient of the wave, is the reflection coefficient, is the amplitude of the reflected wave, is the amplitude of the transmitted wave, is the soil density of the layer of soil further away from the ground of the two adjacent layers of soil, is the wave speed in the layer of soil further away from the ground of the two adjacent layers of soil, is the soil density of the layer of soil closer to the ground of the two adjacent layers of soil, is the wave speed in the layer of soil closer to the ground of the two adjacent layers of soil.

3. The three-dimensional layered site ground motion input method based on the infinite element artificial boundary according to Claim 2, characterized by Under the action of P wave, the seismic equivalent node force at the bottom boundary of the three-dimensional layered foundation satisfies the following relationship: ， wherein, is the seismic equivalent nodal force at the bottom boundary of a three-dimensional layered foundation under P-wave action, the subscript z of indicates the direction of the seismic equivalent nodal force, the superscript -z of indicates the plane in which the face normal lies and the face normal points to the negative direction of the z-axis, is the normal damper damping coefficient of the first layer of soil, is the vibration velocity of the soil body, and t is time, is the nodal control area.

4. The three-dimensional layered site ground motion input method based on the infinite element artificial boundary according to Claim 3, wherein Under the action of P wave, the seismic equivalent node force at the front and back boundaries of the three-dimensional layered foundation satisfies the following relationship: ， ， wherein and are the seismic equivalent nodal forces at the back side boundary of the three-dimensional layered foundation, and are the seismic equivalent nodal forces at the back side boundary of the three-dimensional layered foundation, denotes the plane in which the surface normal lies and the surface normal points in the negative direction of the axis, and are the seismic equivalent nodal forces at the front side boundary of the three-dimensional layered foundation, and are the seismic equivalent nodal forces at the front side boundary of the three-dimensional layered foundation, denotes the plane in which the surface normal lies and the surface normal points in the positive direction of the axis, , , and are the directions of the seismic equivalent nodal forces, and z are the directions of the seismic equivalent nodal forces, is the first Lame constant of the i-th soil layer, is the P-wave velocity in the i-th soil layer, is the transmission coefficient of the wave at the interface j, is the soil vibration velocity after time delay t is time, is the reflection coefficient of the wave at the interface i, is the soil vibration velocity after time delay t is time, is an algebraic expression, , is an algebraic expression, , is the reflection coefficient of the wave at the interface n, is the transmission coefficient of the wave in the k-th soil layer, is an algebraic expression, , , is a parameter related to i, is the soil vibration velocity after time delay t is time, is the reflection coefficient of the wave at the interface i-1, is the soil vibration velocity after time delay t is time, , , and are four time delays, is the nodal control area.

5. The three-dimensional layered site ground motion input method based on the infinite element artificial boundary according to Claim 4, wherein Under the action of P wave, the seismic equivalent node force at the left and right boundaries of the three-dimensional layered foundation satisfies the following relationship: ， ， wherein and are the seismic equivalent nodal forces at the left side boundary of the three-dimensional layered foundation, and are the superscripts denote the plane in which the face normal lies and the face normal points to the negative direction of the y-axis, and are the seismic equivalent nodal forces at the front side boundary of the three-dimensional layered foundation, and are the superscripts denote the plane in which the face normal lies and the face normal points to the positive direction of the y-axis, , , and are the subscripts y and z, respectively, denoting the direction of the seismic equivalent nodal forces, is the first Lame constant of the i-th soil layer, is the P-wave velocity in the i-th soil layer, is the transmission coefficient of the wave at the interface j, is the soil vibration velocity after time delay , is the reflection coefficient of the wave at the interface i, is the soil vibration velocity after time delay , is an algebraic expression, , is an algebraic expression, , is the reflection coefficient of the wave at the interface n, is the transmission coefficient of the wave in the k-th soil layer, is an algebraic expression, , , is a parameter related to i, is the soil vibration velocity after time delay , is the reflection coefficient of the wave at the interface i-1, is the soil vibration velocity after time delay , , , and are four time delays, is the nodal control area.

6. The three-dimensional layered site ground motion input method based on the infinite element artificial boundary according to Claim 1, wherein The establishment of the infinite element and finite element mixed three-dimensional layered foundation model, the numbering, spatial coordinates and equivalent node area of each boundary node at the boundary of the finite domain, and the calculation of the geostress field node force matching the initial stress field include the following steps: Determine the finite domain range of the layered foundation model, perform three-dimensional modeling of the foundation soil by coupling finite elements and infinite elements, and obtain the three-dimensional layered foundation model; Restrict the three degrees of freedom of the finite domain boundary, and apply a unit pressure to the finite domain boundary to obtain the numbering, spatial coordinates and equivalent node area of each boundary node at the finite domain boundary; Determine the soil layer number of the boundary node according to the Z coordinate of the spatial coordinates and the soil layer height, and determine the plane where the boundary node is located according to the spatial coordinates; Apply an initial geostress field to the three-dimensional layered foundation model, and calculate the geostress field node force of each boundary node according to the initial geostress distribution.

7. The three-dimensional layered site ground motion input method based on the infinite element artificial boundary according to Claim 6, wherein The calculation of the seismic equivalent node force of the grid node at the junction of the infinite and finite grids, the batch modification of the layered foundation model calculation file, and the realization of the initial static force and seismic force input of each boundary node include the following steps: Determine the input seismic wave at the bottom of the foundation, obtain the corresponding velocity time history curve, and calculate the time delay of each boundary node; Calculate the reflection coefficient and the transmission coefficient, and determine the vibration velocity of each boundary node at any time, and then calculate the seismic equivalent node force of each boundary node at any time; Based on the fopen function and fprintf function embedded in Matlab, according to the ABAQUS calculation model inp file format and convention, batch node set creation, amplitude function establishment and concentrated node force application are carried out, and the equivalent node force of each boundary node and the node force of the stress field are applied.

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