Method for calculating equivalent nodal force of oblique incidence p wave, storage medium and equipment
By calculating the displacement spectrum of the obliquely incident P-wave and the incident angle of the reflected wave, and combining it with the fast Fourier transform, the accuracy and adaptability problems of the existing finite element model for ground motion input are solved, and high-precision dynamic response analysis of layered foundations is realized.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- CHINA UNIV OF GEOSCIENCES (WUHAN)
- Filing Date
- 2023-11-14
- Publication Date
- 2026-07-14
AI Technical Summary
Existing technologies suffer from reduced accuracy, high computational difficulty, and inability to adapt to dynamic response analysis of layered foundations when calculating seismic motion input finite element models.
By calculating the displacement spectrum of the obliquely incident P-wave, the incident angle and time delay of the reflected wave, and combining the fast Fourier transform, the displacement, velocity and stress spectra of each node of the artificial viscoelastic boundary are calculated, and finally converted to the time domain to obtain the equivalent nodal force.
It improves calculation accuracy, reduces calculation difficulty, can adapt to dynamic response analysis of layered foundations, and expands the application scope of the calculation.
Smart Images
Figure CN117574026B_ABST
Abstract
Description
Technical Field
[0001] This invention relates to the field of earthquake engineering technology, and in particular to a method, storage medium, and device for calculating the equivalent nodal force of obliquely incident P-waves. Background Technology
[0002] In the study of dynamic interactions between soil structures, how to input seismic motions into the finite element model is a crucial aspect of dynamic research. Currently, most common methods for inputting seismic motions involve calculating the equivalent nodal forces after performing wavefield decomposition on the left boundary or both left and right boundaries of a homogeneous foundation in the time domain, and then inputting these forces into the finite element model.
[0003] The above method has several obvious shortcomings: First, the calculation of equivalent nodal forces requires known displacement and velocity fields. Calculations in the time domain necessitate integration or differentiation of the known wavefield, which current software cannot accurately perform, leading to reduced accuracy. Furthermore, integrating and differentiating the wavefield is technically challenging from a programming perspective. Second, the method only considers the superposition of free and scattered fields at the left and right boundaries or the left boundary, and only considers the free field at the bottom boundary, failing to reproduce the wavefield in the foundation soil, thus reducing calculation accuracy. Third, with the deepening of dynamic research, scholars are no longer limited to the dynamic response of homogeneous foundations but have begun to analyze layered foundations. While the above method can adapt to two-dimensional and three-dimensional cases, it cannot be extended to layered foundations. Summary of the Invention
[0004] To address the aforementioned problems, this invention proposes a method for calculating the equivalent nodal force of an obliquely incident P-wave, comprising the following steps:
[0005] S1. Obtain the displacement time history of the seismic wave to obtain the incident wave;
[0006] S2. Based on the incident wave, calculate the displacement spectra of the incident P wave, reflected P wave, incident SV wave, and reflected SV wave;
[0007] S3. Calculate the angle of incidence of the reflected wave based on the angle of incidence of the incident P wave using Shell's law.
[0008] S4. The incident time delay of the wave at each node of the artificial viscoelastic boundary is calculated based on the incident angle of the reflected wave.
[0009] S5. Based on the displacement spectra of the incident P-wave, reflected P-wave and reflected SV-wave and the incident time delay of the wave at each node of the artificial viscoelastic boundary, the displacement spectra of each node of the artificial viscoelastic boundary are calculated.
[0010] S6. Obtain the velocity spectrum from the displacement spectrum of each node of the artificial viscoelastic boundary, and further obtain the stress spectrum. Then, use the inverse fast Fourier transform to convert the displacement spectrum, velocity spectrum, and stress spectrum of each node of the artificial viscoelastic boundary to the time domain.
[0011] S7. Based on the displacement spectrum, velocity spectrum, and stress spectrum of each node of the artificial viscoelastic boundary in the time domain, obtain the equivalent nodal force of each node of the artificial viscoelastic boundary.
[0012] Furthermore, S2 specifically refers to:
[0013] The displacement spectrum of the incident P-wave was obtained by fast Fourier transform.
[0014] Using the parameter matrix calculated from the reflected wave amplitude, the displacement spectra of the reflected P-wave, the incident SV-wave, and the reflected SV-wave are calculated:
[0015]
[0016]
[0017] D1=4sr-(1-r 2 ) 2
[0018] D2 = 4c sv r(1-r 2 ) / c p
[0019] D3 = -4c p s(1-r 2 ) / c sv
[0020] D4=4sr-(1-r 2 ) 2
[0021] Among them, A p B p A sv B sv These are the displacement spectra of the incident P-wave, the reflected P-wave, the incident SV-wave, and the reflected SV-wave, respectively. sv Let s be a zero vector, where s represents the cotangent of the P-wave incident angle, f represents the cotangent of the SV-wave incident angle, and c is a zero vector. sv c represents the wave velocity of SV waves in the soil layer. p This represents the wave velocity of the P-wave in the soil layer.
[0022] Furthermore, S3 can be expressed mathematically as:
[0023]
[0024] Where θ is the incident angle of the incident P-wave and the reflected P-wave; Let c be the incident angle of the reflected SV wave. p c represents the wave velocity of the P-wave in the soil layer. sv This indicates the wave velocity of SV waves in the soil layer.
[0025] Furthermore, S4 can be expressed mathematically as:
[0026]
[0027]
[0028]
[0029]
[0030] Where Δt1 represents the time delay of the incident P-wave reaching the node on the artificial viscoelastic boundary, Δt2 represents the time delay of the reflected P-wave reaching the node on the artificial viscoelastic boundary, Δt3 represents the time delay of the reflected SV-wave reaching the node on the artificial viscoelastic boundary, and Δt4 represents the time delay during the lateral propagation of the P-wave and SV-wave; d represents the soil layer thickness, (x, y) represents the node coordinates on the artificial viscoelastic boundary, and θ represents the incident angle of the incident P-wave and the reflected P-wave. Let c be the incident angle of the reflected SV wave. p This represents the wave velocity of the P-wave in the soil layer.
[0031] Furthermore, S5 can be expressed mathematically as:
[0032]
[0033]
[0034]
[0035]
[0036]
[0037] in, This represents the displacement spectrum of the incident P-wave at the artificial viscoelastic boundary node. This represents the displacement spectrum of the reflected P-wave at the artificial viscoelastic boundary node. A represents the displacement spectrum of the reflected SV wave at the artificial viscoelastic boundary node. p B p B sv Let be the displacement spectrum of the incident P-wave, the displacement spectrum of the reflected P-wave, and the displacement spectrum of the reflected SV-wave, respectively, and θ be the incident angle of the incident P-wave and the reflected P-wave. Let ω represent the incident angle of the reflected SV wave, i represent the imaginary number, Δt1 represent the time delay of the incident P wave reaching the node on the artificial viscoelastic boundary, Δt2 represent the time delay of the reflected P wave reaching the node on the artificial viscoelastic boundary, Δt3 represent the time delay of the reflected SV wave reaching the node on the artificial viscoelastic boundary, and Δt4 represent the time delay during the lateral propagation of the P wave and SV wave. u(x,y,ω) represents the lateral displacement caused by the incident wave and the reflected wave, and w(x,y,ω) represents the longitudinal displacement caused by the incident wave and the reflected wave.
[0038] Furthermore, S6 can be expressed mathematically as:
[0039]
[0040]
[0041]
[0042]
[0043]
[0044] b = 1 + 2iζ
[0045] in, This represents the velocity spectrum of the incident P-wave at the artificial viscoelastic boundary node. This represents the velocity spectrum of the reflected P-wave at the artificial viscoelastic boundary node. The velocity spectrum of the reflected SV wave at the artificial viscoelastic boundary node is represented by θ, where θ is the incident angle of the incident P wave and the reflected P wave. ω represents the incident angle of the reflected SV wave, i represents the imaginary number, and σ represents the frequency. x σ represents the normal stress in the x-direction generated by the incident and reflected waves on the artificial viscoelastic boundary. y Let represent the normal stress in the y-direction generated by the incident and reflected waves on the artificial viscoelastic boundary, b represent the coefficient introducing the damping ratio, λ represent the Lamé constant, G represent the shear modulus, and c represent the normal stress in the y-direction generated by the incident and reflected waves on the artificial viscoelastic boundary. p c represents the P-wave velocity in the soil layer. s ζ represents the SV wave velocity in the soil layer, and ζ is the damping ratio.
[0046] Furthermore, S7 can be expressed mathematically as follows:
[0047]
[0048]
[0049]
[0050]
[0051]
[0052]
[0053]
[0054] Among them, F lx (t) represents the equivalent nodal force at each node along the x-direction of the artificial viscoelastic left boundary, where t represents time, F ly (t) represents the equivalent nodal force at each node along the y-direction of the artificial viscoelastic left boundary, F. rx (t) represents the equivalent nodal force at each node along the x-direction of the artificial viscoelastic right boundary, F. ry (t) represents the equivalent nodal force at each node along the y-direction of the artificial viscoelastic right boundary, F. bx (t) represents the equivalent nodal force at each node in the x-direction of the artificial viscoelastic bottom boundary, F. by (t) represents the equivalent nodal force at each node in the y-direction of the artificial viscoelastic bottom boundary, K. n C represents the normal spring stiffness of the spring damper. n The normal damping of the spring damper is represented by u(t), and the displacement time history of the seismic wave in the x-direction is represented by σ. x A represents the normal stress in the x-direction generated by the incident and reflected waves on the artificial viscoelastic boundary. n K represents the effective area of influence when the equivalent nodal force acts on the artificial viscoelastic boundary node. t C represents the tangential spring stiffness of the spring damper. t τ represents the tangential damping of the spring damper. xy σ represents the shear stress at the artificial viscoelastic boundary node under the action of incident and reflected waves, w(t) represents the time history of the seismic wave displacement in the y-direction, and σ represents the shear stress at the artificial viscoelastic boundary node. y It represents the normal stress in the y-direction generated by the incident and reflected waves on the artificial viscoelastic boundary.
[0055] The present invention also proposes a computer-readable storage medium storing a computer program that, when executed by a processor, implements the steps of the above-described method for calculating the equivalent nodal force of an obliquely incident P-wave.
[0056] The present invention also proposes an electronic device, including a processor and a memory, wherein the processor and the memory are interconnected, wherein the memory is used to store a computer program, the computer program including computer-readable instructions, and the processor is configured to invoke the computer-readable instructions to execute the above-described method for calculating the equivalent nodal force of an obliquely incident P-wave.
[0057] The beneficial effects of the technical solution provided by this invention are:
[0058] This invention calculates the displacement spectra of the incident P-wave, reflected P-wave, incident SV-wave, and reflected SV-wave, as well as the incident angle of the reflected wave. It also calculates the incident time delay of the wave at each node of the artificial viscoelastic boundary, further calculating the displacement, velocity, and stress spectra of each node. These three spectra are then converted to the time domain, finally yielding the equivalent nodal forces at each node of the artificial viscoelastic boundary. This invention performs all calculations in the frequency domain, reducing computational difficulty and time costs. It can consider the influence of soil damping on the dynamic response of the foundation and improves calculation accuracy. The three-boundary wavefield decomposition method (left boundary, right boundary, and bottom boundary) maximizes the reconstruction of the wavefield under seismic loading, thereby improving the accuracy of the calculation results. The theoretical basis of this invention is the layered soil wavefield calculation method; therefore, this invention can be extended to layered foundations, with broad application prospects. Attached Figure Description
[0059] Figure 1 This is a flowchart of the method for calculating the equivalent nodal force of an obliquely incident P-wave according to an embodiment of the present invention;
[0060] Figure 2 This is a two-dimensional computational grid for ABAQUS analysis in an embodiment of the present invention.
[0061] Figure 3 This is a comparison between numerical simulation results and theoretical calculation results in this embodiment of the invention, wherein, Figure 3 In the middle (a), the numerical simulation result of point A among the four vertices of the soil is compared with the theoretical calculation result. Figure 3 (b) represents a comparison between the numerical simulation result and the theoretical calculation result for point B among the four vertices of the soil. Figure 3 (c) represents the comparison between the numerical simulation result and the theoretical calculation result of point C among the four vertices of the soil. Figure 3 (d) represents the comparison between the numerical simulation result and the theoretical calculation result of point D among the four vertices of the soil.
[0062] Figure 4 This is a schematic diagram of the wavefront according to an embodiment of the present invention. Figure 4 Image (a) is a schematic diagram of the wavefront at t = 0.33s. Figure 4 Image (b) is a schematic diagram of the wavefront at t = 0.83 s. Figure 4 Image (c) is a schematic diagram of the wavefront at t = 1.30 s. Figure 4 The middle (d) diagram is a schematic diagram of the wavefront at t = 1.51s. Detailed Implementation
[0063] To make the objectives, technical solutions, and advantages of the present invention clearer, the embodiments of the present invention will be further described below with reference to the accompanying drawings.
[0064] The flowchart of the method for calculating the equivalent nodal force of an obliquely incident P-wave according to an embodiment of the present invention is as follows: Figure 1 Specifically, it includes the following steps:
[0065] S1. Obtain the displacement time history of the seismic wave to obtain the incident wave.
[0066] In a further embodiment, the incident wave is simulated as a seismic wave using an approximate Dirac pulse, lasting 3 seconds, and expressed as follows:
[0067]
[0068] Among them, A pt Let A = 0.05, T = 0.25, and z(a) = a. 3 H(a) is the Heaviside function (H(a) = 0 when a < 0; H(a) = 1 when a ≥ 0).
[0069] S2. Based on the incident wave, calculate the displacement spectra of the incident P wave, reflected P wave, incident SV wave, and reflected SV wave.
[0070] The displacement spectrum of the incident P-wave was obtained by fast Fourier transform.
[0071] Using the parameter matrix calculated from the amplitude of reflected waves from the free surface of the foundation soil, the displacement spectra of the reflected P-wave, incident SV-wave, and reflected SV-wave are obtained:
[0072]
[0073]
[0074] D1=4sr-(1-r 2 ) 2
[0075] D2 = 4c sv r(1-r 2 ) / c p
[0076] D3 = -4c p s(1-r 2 ) / c sv
[0077] D4=4sr-(1-r 2 ) 2
[0078] Among them, A p B p A sv B svThese are the displacement spectra of the incident P-wave, the reflected P-wave, the incident SV-wave, and the reflected SV-wave, respectively. sv Let s be a zero vector, where s represents the cotangent of the P-wave incident angle, f represents the cotangent of the SV-wave incident angle, and c is a zero vector. sv c represents the wave velocity of SV waves in the soil layer. p This represents the wave velocity of the P-wave in the soil layer.
[0079] S3. Calculate the incident angle of the reflected wave using Shell's law based on the incident angle of the incident P wave.
[0080] Furthermore, S3 can be expressed mathematically as:
[0081]
[0082] Where θ is the incident angle of the incident P-wave and the reflected P-wave; Let c be the incident angle of the reflected SV wave. p c represents the wave velocity of the P-wave in the soil layer. sv This indicates the wave velocity of SV waves in the soil layer.
[0083] S4. The incident time delay of the wave at each node of the artificial viscoelastic boundary is calculated based on the incident angle of the reflected wave.
[0084] Expressed mathematically as follows:
[0085]
[0086]
[0087]
[0088]
[0089] Where Δt1 represents the time delay of the incident P-wave reaching the node on the artificial viscoelastic boundary, Δt2 represents the time delay of the reflected P-wave reaching the node on the artificial viscoelastic boundary, Δt3 represents the time delay of the reflected SV-wave reaching the node on the artificial viscoelastic boundary, and Δt4 represents the time delay during the lateral propagation of the P-wave and SV-wave; d represents the soil layer thickness, (x, y) represents the node coordinates on the artificial viscoelastic boundary, and θ represents the incident angle of the incident P-wave and the reflected P-wave. Let c be the incident angle of the reflected SV wave. p This represents the wave velocity of the P-wave in the soil layer.
[0090] S5. Based on the displacement spectra of the incident P-wave, reflected P-wave, and reflected SV-wave, and the incident time delay of the wave at each node of the artificial viscoelastic boundary, the displacement spectra of each node of the artificial viscoelastic boundary are calculated.
[0091] Expressed mathematically as follows:
[0092]
[0093]
[0094]
[0095]
[0096]
[0097] in, This represents the displacement spectrum of the incident P-wave at the artificial viscoelastic boundary node. This represents the displacement spectrum of the reflected P-wave at the artificial viscoelastic boundary node. A represents the displacement spectrum of the reflected SV wave at the artificial viscoelastic boundary node. p B p B sv Let be the displacement spectrum of the incident P-wave, the displacement spectrum of the reflected P-wave, and the displacement spectrum of the reflected SV-wave, respectively, and θ be the incident angle of the incident P-wave and the reflected P-wave. Let ω represent the incident angle of the reflected SV wave, i represent the imaginary number, Δt1 represent the time delay of the incident P wave reaching the node on the artificial viscoelastic boundary, Δt2 represent the time delay of the reflected P wave reaching the node on the artificial viscoelastic boundary, Δt3 represent the time delay of the reflected SV wave reaching the node on the artificial viscoelastic boundary, and Δt4 represent the time delay during the lateral propagation of the P wave and SV wave. u(x, y, ω) represents the lateral displacement caused by the incident wave and the reflected wave, and w(x, y, ω) represents the longitudinal displacement caused by the incident wave and the reflected wave.
[0098] S6. Obtain the velocity spectrum from the displacement spectrum of each node of the artificial viscoelastic boundary, and further obtain the stress spectrum. Then, use the inverse fast Fourier transform to convert the displacement spectrum, velocity spectrum, and stress spectrum of each node of the artificial viscoelastic boundary to the time domain.
[0099] Expressed mathematically as follows:
[0100]
[0101]
[0102]
[0103]
[0104]
[0105] b = 1 + 2iζ
[0106] in, This represents the velocity spectrum of the incident P-wave at the artificial viscoelastic boundary node. This represents the velocity spectrum of the reflected P-wave at the artificial viscoelastic boundary node. The velocity spectrum of the reflected SV wave at the artificial viscoelastic boundary node is represented by θ, where θ is the incident angle of the incident P wave and the reflected P wave. ω represents the incident angle of the reflected SV wave, i represents the imaginary number, and σ represents the frequency. x σ represents the normal stress in the x-direction generated by the incident and reflected waves on the artificial viscoelastic boundary. y Let represent the normal stress in the y-direction generated by the incident and reflected waves on the artificial viscoelastic boundary, b represent the coefficient introducing the damping ratio, λ represent the Lamé constant, G represent the shear modulus, and c represent the normal stress in the y-direction generated by the incident and reflected waves on the artificial viscoelastic boundary. p c represents the P-wave velocity in the soil layer. sv ζ represents the SV wave velocity in the soil layer, and ζ is the damping ratio.
[0107] S7. Based on the displacement spectrum, velocity spectrum, and stress spectrum of each node of the artificial viscoelastic boundary in the time domain, obtain the equivalent nodal force of each node of the artificial viscoelastic boundary.
[0108] Expressed mathematically as follows:
[0109]
[0110]
[0111]
[0112]
[0113]
[0114]
[0115]
[0116] Among them, F lx (t) represents the equivalent nodal force at each node along the x-direction of the artificial viscoelastic left boundary, where t represents time, F ly (t) represents the equivalent nodal force at each node along the y-direction of the artificial viscoelastic left boundary, F. rx (t) represents the equivalent nodal force at each node along the x-direction of the artificial viscoelastic right boundary, F. ry (t) represents the equivalent nodal force at each node along the y-direction of the artificial viscoelastic right boundary, F. bx (t) represents the equivalent nodal force at each node in the x-direction of the artificial viscoelastic bottom boundary, F. by(t) represents the equivalent nodal force at each node in the y-direction of the artificial viscoelastic bottom boundary, K. n C represents the normal spring stiffness of the spring damper. n The normal damping of the spring damper is represented by u(t), and the displacement time history of the seismic wave in the x-direction is represented by σ. x A represents the normal stress in the x-direction generated by the incident and reflected waves on the artificial viscoelastic boundary. n K represents the effective area of influence when the equivalent nodal force acts on the artificial viscoelastic boundary node. t C represents the tangential spring stiffness of the spring damper. t τ represents the tangential damping of the spring damper. xy σ represents the shear stress at the artificial viscoelastic boundary node under the action of incident and reflected waves, w(t) represents the time history of the seismic wave displacement in the y-direction, and σ represents the shear stress at the artificial viscoelastic boundary node. y It represents the normal stress in the y-direction generated by the incident and reflected waves on the artificial viscoelastic boundary.
[0117] In one exemplary embodiment, a computer-readable storage medium is included, which stores a computer program that, when executed by a processor, implements the steps of the above-described method for calculating the equivalent nodal force of an obliquely incident P-wave.
[0118] In one exemplary embodiment, the device further includes an electronic device including at least one processor, at least one memory, and at least one communication bus.
[0119] The memory stores a computer program, which includes computer-readable instructions. The processor calls the computer-readable instructions stored in the memory through the communication bus to execute the above-mentioned method for calculating the equivalent nodal force of the obliquely incident P-wave.
[0120] To verify the method of this invention, a two-dimensional semi-infinite foundation with obliquely incident P-waves is used as an example. A rectangular region with a horizontal dimension of 1200m and a vertical dimension of 600m is taken as the finite element soil model. Using Python and MATLAB software, artificial viscoelastic boundaries are set for the left, right, and bottom boundaries, equivalent nodal forces are calculated, and equivalent nodal forces are applied. The upper surface is a free surface. Soil parameters include: soil density (ρ), elastic modulus (E), Poisson's ratio, and the velocity in the P-wave layer (c). p ), sv wave velocity in soil layer (c sv The soil parameters are shown in Table 1. Three different Dirac pulses with different application times and pulse amplitudes were used, as shown in Table 2.
[0121] Two-dimensional computational grid for ABAQUS analysis, such as Figure 2 As shown, Figure 2 Select four vertices of the soil mass (A, B, C, D).
[0122] To verify the effectiveness of the present invention, the experimental values of the displacements of points A, B, C, and D obtained by ABAQUS analysis were fitted with the theoretical values calculated by Matlab, as follows: Figure 3 The results were good.
[0123] To verify the versatility and accuracy of the present invention, five observation points were selected: the four vertices of the soil (A, B, C, D) and the midpoint E of the upper surface. The theoretical values of the peak displacements at each observation point under three pulses with different incident angles and unit sizes were compared with the ABAQUS analysis values, as shown in Tables 3, 4, and 5.
[0124] Table 1 Soil Parameters
[0125] <![CDATA[ρ / (kg·m -3 )]]> E / GPa Poisson's ratio <![CDATA[c p / (m·s -1 )]]> <![CDATA[c sv / (m·s -1 )]]> 2000 1.25 0.25 866 500
[0126] Table 2 Three different Dirac pulses
[0127]
[0128]
[0129] Table 3 Comparison of theoretical and ABAQUS analysis values of peak displacement at each observation point under different incident angles of pulse 1.
[0130]
[0131]
[0132] Table 4. Comparison of theoretical and ABAQUS analysis values of peak displacement at each observation point under different incident angles of pulse 2.
[0133]
[0134]
[0135]
[0136] Table 5 compares the theoretical values and ABAQUS analysis values of the peak displacement at each observation point under different incident angles of pulse 3.
[0137]
[0138]
[0139]
[0140] The wavefront schematic diagram of this embodiment is shown below. Figure 4 , Figure 4 Image (a) is a schematic diagram of the wavefront at t = 0.33s. Figure 4Image (b) is a schematic diagram of the wavefront at t = 0.83 s. Figure 4 Image (c) is a schematic diagram of the wavefront at t = 1.30 s. Figure 4 Figure (d) shows a schematic diagram of the wavefront at t = 1.51 s. The wave propagation process in the soil medium can be observed intuitively from the figure.
[0141] It should be noted that, in this document, the terms "comprising," "including," or any other variations thereof are intended to cover non-exclusive inclusion, such that a process, method, article, or system that comprises a list of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such a process, method, article, or system. Unless otherwise specified, an element defined by the phrase "comprising one..." does not exclude the presence of other identical elements in the process, method, article, or system that includes that element.
[0142] The above description of the disclosed embodiments enables those skilled in the art to make or use the invention. Various modifications to these embodiments will be readily apparent to those skilled in the art, and the general principles defined herein may be implemented in other embodiments without departing from the spirit or scope of the invention. Therefore, the invention is not to be limited to the embodiments shown herein, but is to be accorded the widest scope consistent with the principles and novel features disclosed herein.
Claims
1. A method for calculating the equivalent nodal force of an obliquely incident P-wave, characterized in that, Includes the following steps: S1. Obtain the displacement time history of the seismic wave to obtain the incident wave; S2. Based on the incident wave, calculate the displacement spectra of the incident P wave, reflected P wave, incident SV wave, and reflected SV wave; S3. Calculate the angle of incidence of the reflected wave based on the angle of incidence of the incident P wave using Shell's law. S4. The incident time delay of the wave at each node of the artificial viscoelastic boundary is calculated based on the incident angle of the reflected wave. S5. Based on the displacement spectra of the incident P-wave, reflected P-wave and reflected SV-wave and the incident time delay of the wave at each node of the artificial viscoelastic boundary, the displacement spectra of each node of the artificial viscoelastic boundary are calculated. S6. Obtain the velocity spectrum from the displacement spectrum of each node of the artificial viscoelastic boundary, and further obtain the stress spectrum. Then, use the inverse fast Fourier transform to convert the displacement spectrum, velocity spectrum, and stress spectrum of each node of the artificial viscoelastic boundary to the time domain. S7. Based on the displacement spectrum, velocity spectrum, and stress spectrum of each node of the artificial viscoelastic boundary in the time domain, obtain the equivalent nodal force of each node of the artificial viscoelastic boundary. S5 can be expressed mathematically as: in, This represents the displacement spectrum of the incident P-wave at the artificial viscoelastic boundary node. This represents the displacement spectrum of the reflected P-wave at the artificial viscoelastic boundary node. This represents the displacement spectrum of the reflected SV wave at the artificial viscoelastic boundary node. , , These are the displacement spectra of the incident P-wave, the reflected P-wave, and the reflected SV-wave, respectively. The incident angles of the incident P-wave and the reflected P-wave; The incident angle of the reflected SV wave. The frequency is represented by 'i', where 'i' represents the imaginary number. This represents the time delay of the incident P-wave reaching the node on the artificial viscoelastic boundary. The time delay for the reflected P-wave to reach the node on the artificial viscoelastic boundary; The time delay for the reflected SV wave to reach the nodes on the artificial viscoelastic boundary. This indicates the time delay during the lateral propagation of P-waves and SV-waves; This represents the lateral displacement caused by the incident wave and the reflected wave. This represents the longitudinal displacement caused by the incident wave and the reflected wave. S6 can be expressed mathematically as: in, This represents the velocity spectrum of the incident P-wave at the artificial viscoelastic boundary node. This represents the velocity spectrum of the reflected P-wave at the artificial viscoelastic boundary node. This represents the velocity spectrum of the reflected SV wave at the artificial viscoelastic boundary node. The incident angles of the incident P-wave and the reflected P-wave; The incident angle of the reflected SV wave. Indicates frequency, i To represent imaginary numbers, This represents the normal stress in the x-direction generated by the incident and reflected waves on the artificial viscoelastic boundary. represents the normal stress in the y-direction generated by the incident and reflected waves on the artificial viscoelastic boundary, and b represents the coefficient introducing the damping ratio. Represents Lamé constant, Indicates shear modulus, This represents the P-wave velocity in the soil layer. Indicates the SV wave velocity in the soil layer. The damping ratio is denoted as .
2. The method for calculating the equivalent nodal force of an obliquely incident P-wave according to claim 1, characterized in that, S2 specifically refers to: The displacement spectrum of the incident P-wave was obtained by fast Fourier transform. Using the parameter matrix calculated from the reflected wave amplitude, the displacement spectra of the reflected P-wave, the incident SV-wave, and the reflected SV-wave are calculated: in, , , , These are the displacement spectra of the incident P-wave, the reflected P-wave, the incident SV-wave, and the reflected SV-wave, respectively. It is a zero vector. s This represents the cotangent value of the P-wave incident angle. r This represents the cotangent value of the SV wave incident angle. This indicates the wave velocity of SV waves in the soil layer. This represents the wave velocity of the P-wave in the soil layer.
3. The method for calculating the equivalent nodal force of an obliquely incident P-wave according to claim 1, characterized in that, S3 can be expressed mathematically as: in, The incident angles of the incident P-wave and the reflected P-wave; The incident angle of the reflected SV wave. This represents the wave velocity of the P-wave in the soil layer. This indicates the wave velocity of SV waves in the soil layer.
4. The method for calculating the equivalent nodal force of an obliquely incident P-wave according to claim 3, characterized in that, S4 can be expressed mathematically as: in, This represents the time delay of the incident P-wave reaching the node on the artificial viscoelastic boundary. The time delay for the reflected P-wave to reach the node on the artificial viscoelastic boundary; The time delay for the reflected SV wave to reach the nodes on the artificial viscoelastic boundary. This indicates the time delay during the lateral propagation of P-waves and SV-waves; For soil layer thickness, ( ) These are the coordinates of the nodes on the artificial viscoelastic boundary. The incident angles of the incident P-wave and the reflected P-wave; The incident angle of the reflected SV wave. This represents the wave velocity of the P-wave in the soil layer.
5. The method for calculating the equivalent nodal force of an obliquely incident P-wave according to claim 4, characterized in that, S7 can be expressed mathematically as: in, The left boundary of artificial viscoelasticity x The equivalent nodal forces at each node in the direction, t Indicates time, The left boundary of artificial viscoelasticity y The equivalent nodal forces at each node in the direction, The right boundary of artificial viscoelasticity x The equivalent nodal forces at each node in the direction, The right boundary of artificial viscoelasticity y The equivalent nodal forces at each node in the direction. Artificial viscoelastic bottom boundary x The equivalent nodal forces at each node in the direction. Artificial viscoelastic bottom boundary y The equivalent nodal forces at each node in the direction. This indicates the normal spring stiffness of the spring damper. This indicates the normal damping of the spring damper. Indicates seismic waves x Directional displacement time history, This represents the effect produced by incident and reflected waves on an artificial viscoelastic boundary. x Normal stress in the direction, This represents the effective area of influence when the equivalent nodal force acts on the artificial viscoelastic boundary node. This indicates the tangential spring stiffness of the spring damper. This indicates the tangential damping of the spring damper. This represents the shear stress at the artificial viscoelastic boundary node under the action of incident and reflected waves. Indicates seismic waves y Directional displacement time history, This represents the effect produced by incident and reflected waves on an artificial viscoelastic boundary. y Normal stress in the direction.
6. A computer-readable storage medium storing a computer program, characterized in that: When the computer program is executed by a processor, it implements the steps of the method as described in any one of claims 1-5.
7. An electronic device, characterized in that, The device includes a processor and a memory, the processor being interconnected with the memory, wherein the memory is used to store a computer program, the computer program including computer-readable instructions, and the processor is configured to invoke the computer-readable instructions to perform the method as described in any one of claims 1-5.