Proportional boundary perfect matching layer method and system for infinite domain fluctuation numerical simulation

Abstract

The invention discloses a proportion boundary perfect matching layer method and system for infinite domain fluctuation numerical simulation, and belongs to the field of dynamic analysis, and the method comprises the steps: mapping an irregular geometric infinite domain under a Cartesian coordinate system to a regular geometric domain under a proportion boundary local coordinate system based on proportion boundary coordinate transformation; analyzing and extending the radial coordinates of the regular geometric domain to a complex space by using a complex coordinate stretching function, and performing truncation processing according to a preset thickness to obtain a proportional boundary perfect matching layer; based on a Galerkin weighted residual principle, constructing a corresponding equivalent integral formula according to an equilibrium equation and a physical equation of a fluctuation problem in a proportion boundary perfect matching layer; performing finite element discrete processing on auxiliary variables of displacement, stress and stress integration in the constructed equivalent integral formula by adopting a mixed displacement-stress element finite element technology to obtain a mixed displacement-stress finite element equation of a proportional boundary perfect matching layer; the method is simple in boundary implementation and good in boundary and internal domain adaptability.

Description

Technical Field

[0001] The present invention relates to the field of dynamic analysis technology, and more particularly to a scaled boundary perfectly matched layer method and system for infinite domain wave numerical simulation. Background Art

[0002] Numerical simulation of complex, large-area media waves is the cornerstone of seismic safety assessments for major projects. Due to limited computing resources, it is often necessary to truncate the original large region at appropriate locations and apply appropriate artificial boundary conditions at the truncation boundaries to minimize the computational domain size while ensuring simulation accuracy.

[0003] Traditional artificial boundary implementation methods mainly include the following two categories: 1. Absorbing boundary condition technology: Absorbing boundary conditions are further divided into two categories: local methods and non-local methods. Local methods include the transmission boundary method, viscous boundary method, and viscoelastic boundary method. These methods allow for the spatiotemporal decoupling of the scattered field response at the boundary nodes, and have the advantage of high computational efficiency. However, their computational accuracy is relatively low, and they are prone to instability phenomena such as high-frequency pseudo-oscillations and low-frequency drift. Non-local methods include the boundary element method, thin layer method, and scaled boundary finite element method. Although these methods can achieve higher computational accuracy, they involve convolution integrals or series expansion operations, resulting in higher computational complexity.

[0004] 2. Absorption layer technology. Absorption layer technology uses artificial damping materials to attenuate outgoing waves. Representative methods include the Rayleigh damping layer method, the Cauchy damping layer method, and the damping extraction method. In recent years, absorption layer technology, represented by the perfectly matched layer method (PML), has received widespread attention. Originally proposed by Bérenger, the PML is primarily used for electromagnetic wave simulation. It achieves near-perfect outgoing wave absorption through a complex coordinate stretching function. It has been applied to elastic, anisotropic, viscoelastic, and poroelastic media, becoming an important tool for modeling wave propagation problems. The perfectly matched layer method is highly dependent on the selected coordinate system (i.e., the inner and outer surfaces of the perfectly matched layer region and the discrete grid lines within the domain must be contour lines). This results in the following technical problems with this type of method: 1) Poor flexibility: The perfectly matched layer method is difficult to use artificial boundaries of general geometric shapes, which reduces the flexibility of selecting a finite internal computational domain; 2) General applicability: Because the complex coordinate extension function is established based on global coordinates, existing methods can only consider physical interfaces parallel to the coordinate plane and cannot consider multiple parallel and radial physical surfaces and interfaces extending to infinity within an infinite domain.

[0005] In summary, there is an urgent need to develop a scaled boundary perfectly matched layer method and system for infinite domain wave numerical simulation to solve one or more of the above problems. Summary of the Invention

[0006] One object of the present invention is to provide a new technical solution for a scaled boundary perfectly matched layer method and system for infinite domain wave numerical simulation.

[0007] According to a first aspect of the present invention, a scaled boundary perfectly matched layer method for infinite domain wave numerical simulation is provided, the method comprising: Step S1: Based on the scaled boundary coordinate transformation, the irregular geometric infinite domain in the Cartesian coordinate system is mapped to a regular geometric domain in the scaled boundary local coordinate system, wherein the scaled boundary local coordinate system is composed of radial coordinates and annular coordinates; Step S2: Analytically extending the radial coordinates of the regular geometric domain to a complex space using a complex coordinate stretching function to obtain complex radial coordinates and an absorption layer domain with a fluctuation exponential decay characteristic, and truncating the absorption layer domain according to a preset thickness to obtain a scaled boundary perfectly matched layer; Step S3: Based on the Galerkin weighted residual principle, corresponding equivalent integral forms are constructed for the equilibrium equation and physical equation of the wave problem in the scaled boundary perfectly matched layer to obtain an equivalent integral formula; Step S4: using a hybrid displacement-stress unit finite element technique to perform finite element discretization on the displacement, stress, and auxiliary variables of the stress integral in the constructed equivalent integral formula to obtain a hybrid displacement-stress finite element equation for a scaled boundary perfectly matched layer.

[0008] Optionally, step S1 specifically includes: Step S11: dividing the irregular geometric infinite domain into a finite inner subdomain and an infinite outer subdomain; Step S12: using the scaled boundary coordinate transformation to perform coordinate transformation processing on the points on the boundary surface of the infinite outer subdomain in the Cartesian coordinate system, so as to obtain a regular geometric domain in the scaled boundary local coordinate system.

[0009] Optionally, the expression for the scale boundary coordinate transformation is: in, represents the coordinates of any point on the artificial boundary, ( ) represents the coordinates of the corresponding points of the scaled stitching surface, represents the radial coordinate of a regular geometric domain, Represents the coordinates of any point on a regular geometric domain.

[0010] Optionally, after step S12, the following steps are further included: Step S13: using a standard finite element method to discretize the artificial boundary surface and the scaled splicing surface of the infinite outer subdomain into an equal number of regional units, and interpolating the internal coordinates of each of the regional units based on the shape function.

[0011] Optionally, in step S13, the corresponding shape function is selected according to the shape of the finite internal subdomain; The shape function includes two-dimensional shape function, bilinear quadrilateral unit shape function, Shape functions for polygonal elements and shape functions for spectral elements.

[0012] Optionally, in step S2, the radial coordinates of the regular geometric domain are analytically extended to the complex space using a complex coordinate stretching function, and the process of obtaining the complex radial coordinates is specifically expressed as follows: in, represents the radial coordinate of a regular geometric domain, Represents a mapping, represents the complex radial coordinate, Indicates the integral variable, which is 0 to Any position on the integral path of represents the complex coordinate stretching function, represents the circular frequency, represents the scaling function, represents the decay function, and i represents the imaginary unit.

[0013] Optionally, step S4 specifically includes: Using the shape function to interpolate the displacement and stress in the constructed equivalent integral formula to obtain a displacement interpolation expression and a stress interpolation expression; Based on the displacement interpolation expression, the stress interpolation expression and auxiliary variables of stress integration, finite element discretization is performed on the scaled boundary perfectly matched layer to obtain a mixed displacement-stress finite element equation of the scaled boundary perfectly matched layer.

[0014] According to a second aspect of the present invention, a scaled boundary perfectly matched layer system for infinite domain wave numerical simulation is provided, the system comprising: A coordinate transformation module is configured to map the irregular geometric infinite domain in the Cartesian coordinate system to a regular geometric domain in the scaled boundary local coordinate system based on a scaled boundary coordinate transformation, wherein the scaled boundary local coordinate system is composed of radial coordinates and annular coordinates; a coordinate stretching module configured to analytically extend the radial coordinates of the regular geometric domain to a complex space using a complex coordinate stretching function to obtain complex radial coordinates and an absorption layer domain with a fluctuation exponential decay characteristic, and to truncate the absorption layer domain according to a preset thickness to obtain a scaled boundary perfectly matched layer; The construction module is configured to construct corresponding equivalent integral forms for the equilibrium equations and physical equations of the wave problem in the scaled boundary perfectly matched layer based on the Galerkin weighted residual principle to obtain equivalent integral forms; The finite element discretization module is configured to use a hybrid displacement-stress unit finite element technology to perform finite element discretization on the displacement, stress and auxiliary variables of the stress integral in the constructed equivalent integral formula to obtain a hybrid displacement-stress finite element equation of the scaled boundary perfectly matched layer.

[0015] According to a third aspect of the present invention, an electronic device is provided, comprising a memory and a processor, wherein the memory stores a computer program, and when the processor executes the computer program, the steps of the scaled boundary perfectly matched layer method for numerical simulation of infinite domain waves as described in the first aspect of the present invention are implemented.

[0016] According to a fourth aspect of the present invention, a computer-readable storage medium is provided, on which a computer program is stored. When the computer program is executed by a processor, the computer program implements the steps of the scaled boundary perfectly matched layer method for infinite domain wave numerical simulation as described in the first aspect of the present invention.

[0017] According to an embodiment disclosed in the present invention, a scaled boundary perfectly matched layer method and system for infinite domain wave numerical simulation of the present invention has the following beneficial effects: The scaled boundary perfectly matched layer method and system for infinite domain wave numerical simulation of the present invention have the following advantages: 1) Simplicity of boundary implementation: Similar to local boundaries, the proposed method is simple to implement and avoids complex analytical and semi-analytical derivation processes; 2) Adaptability between the boundary and the internal domain: Perfectly matched layer technology based on complex coordinate extension can absorb incident waves across the entire frequency band and angle. It can seamlessly couple with internal domain hexahedral elements, prismatic elements, and elements of arbitrary order. It also supports the use of different order elements in the extension function direction and the internal computational domain, achieving a flexible balance between accuracy and efficiency. 3) Flexibility of boundary shape: The use of modified-scale boundary coordinates to describe infinite domains eliminates the traditional method's dependence on global coordinates, allowing the proposed method to adopt artificial boundaries of general geometric shapes and multiple parallel and radial physical interfaces that extend to infinity.

[0018] Further features and advantages of the present invention will become apparent from the following detailed description of exemplary embodiments of the present invention with reference to the accompanying drawings. BRIEF DESCRIPTION OF THE DRAWINGS

[0019] The accompanying drawings, which are incorporated in and constitute a part of this specification, illustrate embodiments of the invention and, together with the description, serve to explain the principles of the invention.

[0020] Figure 1 A schematic flow chart of a scaled boundary perfectly matched layer method for infinite domain wave numerical simulation provided according to an embodiment; Figure 2 A schematic diagram of a construction process of a scaled boundary perfectly matched layer according to an embodiment; Figure 3 A schematic diagram of a layered half-space proportional boundary perfectly matched layer truncation model for a complex site with valley terrain under point source load provided in an embodiment; Figures 4(a)-4(c) Snapshots of the wave field of the layered half-space model of a complex site with valley terrain provided by the embodiment, wherein FIG4(a) is the SBPML model and the VSABC model in t = 2.71 s, and Figure 4(b) shows the vertical displacement wave field snapshot of the SBPML model and the VSABC model. t =3.79 s, and Figure 4(c) shows the vertical displacement wave field snapshot of the SBPML model and the VSABC model. t = 8.00 s vertical displacement wave field snapshot; Figure 5(a)-Figure 5(c) According to the embodiment, the displacement time history comparison curve of the SBPML model and the VSABC model at the sampling point A is provided, wherein FIG5(a) is a displacement time history comparison curve of the SBPML model and the VSABC model at the sampling point A in the x direction, FIG5(b) is a displacement time history comparison curve of the SBPML model and the VSABC model at the sampling point A in the y direction, and FIG5(c) is a displacement time history comparison curve of the SBPML model and the VSABC model at the sampling point A in the z direction; Figure 6 A structural block diagram of a scaled boundary perfectly matched layer system for infinite domain wave numerical simulation provided according to an embodiment; Figure 7 A schematic diagram of an electronic device. DETAILED DESCRIPTION

[0021] Various exemplary embodiments of the present invention will now be described in detail with reference to the accompanying drawings. It should be noted that unless otherwise specifically stated, the relative arrangement of components and steps, numerical expressions and numerical values ​​set forth in these embodiments do not limit the scope of the present invention.

[0022] The following description of at least one exemplary embodiment is merely illustrative in nature and is in no way intended to limit the invention, its application, or uses.

[0023] Technologies, methods, and equipment known to ordinary technicians in the relevant art may not be discussed in detail, but where appropriate, the technologies, methods, and equipment should be considered part of the specification.

[0024] In all examples shown and discussed herein, any specific values ​​should be interpreted as merely exemplary and not limiting. Therefore, other examples of the exemplary embodiments may have different values.

[0025] Example 1: See also Figure 1 As shown, an embodiment of the present invention provides a scaled boundary perfectly matched layer method for infinite domain wave numerical simulation, the method comprising: Step S1: Based on the scaled boundary coordinate transformation, the irregular geometric infinite domain in the Cartesian coordinate system is mapped to a regular geometric domain in the scaled boundary local coordinate system, wherein the scaled boundary local coordinate system is composed of radial coordinates and annular coordinates.

[0026] In order to geometrically describe the shape of an irregular geometric infinite domain, this embodiment proposes a method for performing scaled boundary coordinate transformation on the irregular geometric infinite domain, using modified scaled boundary coordinates to describe the infinite domain, eliminating the traditional method's dependence on global coordinates, allowing this embodiment to use artificial boundaries of general geometric shapes and multiple parallel and radial physical interfaces that can reflect extensions to infinity.

[0027] Optionally, step S1 in the scaled boundary perfectly matched layer method for infinite domain wave numerical simulation in this example specifically includes: Step S11: dividing the irregular geometric infinite domain into a finite inner subdomain and an infinite outer subdomain; Step S12: using the scaled boundary coordinate transformation to perform coordinate transformation processing on the points on the boundary surface of the infinite outer subdomain in the Cartesian coordinate system, so as to obtain a regular geometric domain in the scaled boundary local coordinate system.

[0028] Alternatively, the expression for scaled boundary coordinate transformation in the scaled boundary perfectly matched layer method for infinite domain wave numerical simulation in this example is: in, represents the coordinates of any point on the artificial boundary, ( ) represents the coordinates of the corresponding points of the scaled stitching surface, represents the radial coordinate of a regular geometric domain, Represents the coordinates of any point on a regular geometric domain.

[0029] Specifically, the elastic wave propagation problem in an infinite domain of a three-dimensional complex site is as follows Figure 2 As shown, the irregular geometric infinite domain (i.e. the original infinite domain) is first divided into finite internal subdomains and infinite external subdomains . Limited internal subdomain quilt Figure 2 The red border is removed, which usually contains sites with complex geometric shapes and can be solved by domain discretization methods such as finite element, spectral element, and scaled boundary element methods. Figure 2 in As shown, it contains parallel and radial straight physical surfaces and interfaces extending to infinity.

[0030] Assuming infinite outer subdomain In a linear elastic state. The equilibrium equation of the wave problem in the infinite outer subdomain can be expressed as: (1a) The geometric equation of the wave problem in the infinite outer subdomain can be expressed as: (1b) The physical equations of the wave problem in the infinite outer subdomain can be expressed as: (1c) in, , , ; (2) (3) In the above formula, the total stress, displacement and strain vectors of the infinite external subdomain are expressed as and , represents the second derivative of displacement with respect to time, [ is a linear differential operator, is the elastic constitutive matrix, and Expressed as the Lamé constant.

[0031] like Figure 2 As shown, for the initial boundary value problem of the wave problem, this embodiment combines the advantages of the scaled boundary finite element method and the perfect matching layer method, and introduces the scaled boundary coordinate transformation technology to solve the initial boundary value problem of the wave problem. Figure 2 The infinite outer subdomain of is geometrically described. and Infinite external subdomains The artificial boundary surface and scaled splicing surface of , also called the scaling factor, from Departure, defined ( and .based on and The coordinates of the points on the surface are scaled to obtain an infinite external subdomain Any point on The coordinates are: (4a) (4b) (4c) in, Represents an artificial boundary The coordinates of the previous point; ( ) indicates scaling the splicing surface The coordinates of the corresponding points.

[0032] Optionally, the scaled boundary perfectly matched layer method for infinite domain wave numerical simulation in this example further includes, after step S12: Step S13: using the standard finite element method to discretize the artificial boundary surface and the scaled splicing surface of the infinite outer subdomain into an equal number of regional units, and interpolating the internal coordinates of each regional unit based on the shape function.

[0033] Preferably, in the scaled boundary perfectly matched layer method of infinite domain wave numerical simulation of this example, in step S13, a corresponding shape function is selected according to the shape of the finite internal subdomain; wherein the shape function includes a two-dimensional shape function, a bilinear quadrilateral element shape function, Shape functions for polygonal elements and shape functions for spectral elements.

[0034] Specifically, to describe the shape of the infinite external subdomain, the standard finite element method can be used to and Discretize into equal number of regional units, and introduce shape function to discretize the boundary surface. The coordinates after discretization are specifically expressed as: (5a) (5b) (5c) in, is a two-dimensional shape function, which is composed of natural node coordinates constructed, express and The coordinates of the nodes on the face.

[0035] Select the appropriate shape function based on the shape of the finite interior subdomain. For example, for a discrete model with hexahedral elements in the interior domain, the following bilinear quadrilateral element shape function can be used: (6a) The reference node coordinates .

[0036] For the discrete model whose inner domain is a polyhedral unit, the following method can be used Shape function of polygonal element: (6b) in , . It is Edge and The angle between the centers of the polygons, express Polygon The natural coordinates of the vertices can be expressed as and .

[0037] For a discrete model whose inner domain is a high-order spectral unit, the spectral unit shape function can be expressed as: (6c) in is the order of the one-dimensional shape function spectrum unit, Indicates the The natural coordinates of the nodes. The coordinates for the Gauss-Lobatto-Legendre quadrature are shown in Table 1.

[0038] Table 1 Coordinates of one-dimensional spectral unit nodes of Gauss-LeBataud-Legendre quadrature method Substituting equations (5a), (5b) and (5c) into equations (4a), (4b) and (4c) respectively, we can obtain and Indicated Any point in the domain is represented by: (7a) (7b) (7c) The differential operator under the scale boundary coordinates in formula (3) It can be expressed as: (8) The area element in an infinite domain can be expressed as: (9) Among them, the coefficient matrix as well as is represented as follows: in, Is the artificial boundary surface The matrix related to the node coordinates, It is the scaled splicing surface Matrix related to node coordinates; Hadamard operator , which represents the scalar product of two matrices; a subscript followed by a comma represents the partial derivative with respect to this variable.

[0039] Step S2: Analytically extend the radial coordinates of the regular geometric domain to the complex space using a complex coordinate stretching function to obtain complex radial coordinates and an absorption layer domain with a fluctuation exponential decay characteristic, and truncate the absorption layer domain according to a preset thickness to obtain a proportional boundary perfectly matched layer.

[0040] In this embodiment, the radial coordinate is transformed into Extended to complex space , the infinite outer subdomain will be truncated Mapping to scale boundary perfectly matched layers Since the outgoing wave entering the proportional boundary perfectly matched layer has an exponential decay characteristic, it decays to zero when it propagates to a certain position, allowing truncation at this location. In this embodiment, the thickness of the proportional boundary perfectly matched layer can be set to 1 / 2 the minimum wavelength of the outgoing wave.

[0041] The method of this embodiment is based on the perfect matching layer technology of complex coordinate extension, which can absorb incident fluctuations in all frequency bands and all angles; it can seamlessly couple with internal domain hexahedral units, prismatic units and arbitrary order spectrum units, and supports the use of different order units in the extension function direction and the internal calculation domain, achieving a flexible balance between accuracy and efficiency; and improving the adaptability of the boundary and the internal domain.

[0042] Optionally, in the scaled boundary perfectly matched layer method for infinite domain wave numerical simulation in this example, the complex coordinate stretching function is used to analytically extend the radial coordinates of the regular geometric domain to the complex space. The process of obtaining the complex radial coordinates is specifically expressed as follows: in, represents the radial coordinate of a regular geometric domain, Represents a mapping, represents the complex radial coordinate, Indicates the integral variable, which is 0 to Any position on the integral path of represents the complex coordinate stretching function, represents the circular frequency, represents the scaling function, represents the decay function, and i represents the imaginary unit.

[0043] Specifically, the above formulas (7a), (7b), (7c), (8) and (9) are Simply replace with radial stretch coordinates , we can get the differential operator and area differential equation as follows: (10) (11) in, Represents mapping, radially stretching coordinates It can be expressed as: (12) in, is the complex coordinate stretching function, In this embodiment, the classical complex stretching function is used: (13) The scaling function and decay function Can be written as Related expressions: (14a) (14b) and attenuation coefficient It can be written as: (15a) (15b) Where, for The outer boundary reflection coefficient, is the reference wave speed, is the polynomial attenuation order; is the radial element size; is the average thickness of the matching layer domain in Cartesian coordinates; is the circular frequency; Is an imaginary unit.

[0044] Substituting equations (15a) and (15b) into equations (14a) and (14b) respectively, we can obtain the first coordinate after extension: Express: (16) in: (17a) (17b) Applying the fundamental theorem of calculus to equation (17), we can get (18) Substituting equations (18) and (17) into equation (10), we can obtain the proportional boundary perfect matching layer Operator: (19) in: (20a) (20b) (20c) Substituting Equations (16) and (18) into Equation (11), we get the scaled boundary perfectly matched layer The infinitesimals are: (twenty one) The matrix The expression is as follows: Step S3: Based on the Galerkin weighted residual principle, corresponding equivalent integral forms are constructed for the equilibrium equation and physical equation of the wave problem in the scaled boundary perfectly matched layer to obtain an equivalent integral formula.

[0045] In this embodiment, based on the Galerkin weighted residual principle, the equivalent integral form of the equilibrium equation of formula (1a) is constructed for the fluctuation problem within the proportional boundary perfectly matched layer and is expressed as: (22a) Among them, the superscript symbol ( ) indicates that the variable is in the frequency domain; is the virtual displacement; is the pulling vector.

[0046] The equivalent integral form of the physical equation of formula (1c) is expressed as: (22b) The boundary implementation of this embodiment is similar to the local boundary, which is simple in implementation and avoids complex analytical and semi-analytical solution derivation processes.

[0047] Step S4: using a hybrid displacement-stress unit finite element technique to perform finite element discretization on the displacement, stress, and auxiliary variables of the stress integral in the constructed equivalent integral formula to obtain a hybrid displacement-stress finite element equation for the scaled boundary perfectly matched layer.

[0048] Optionally, step S4 in the scaled boundary perfectly matched layer method for infinite domain wave numerical simulation in this example specifically includes: The displacement and stress in the constructed equivalent integral formula are interpolated using shape functions to obtain displacement interpolation expression and stress interpolation expression; Based on the displacement interpolation expression, stress interpolation expression and auxiliary variables of stress integral, the finite element discretization of the scaled boundary perfectly matched layer is performed to obtain the mixed displacement-stress finite element equation of the scaled boundary perfectly matched layer.

[0049] The method of this embodiment can be seamlessly coupled with internal domain hexahedral elements, prismatic elements, and arbitrary-order spectral elements, and supports the use of different-order elements in the extension function direction and the internal calculation domain, thereby achieving a flexible balance between accuracy and efficiency and improving the adaptability of the boundary and the internal domain.

[0050] Specifically, the spectral unit shape function used in Equation (6c) is used to perform displacement interpolation on a regional unit (i.e., spectral unit) in the scale boundary perfectly matched layer. Figure 2 The unit shown, its radial take The order spectrum unit is discrete, o Surface measures The order spectrum unit is discrete. It can be expressed as: (twenty three) One-dimensional unit (with The shape function is explained using the order spectrum unit as an example) and displacement vector It can be expressed as: (24a) (24b) in, is a 3rd order unit matrix. The displacement on the annular surface can be expressed as: (25) in, The two-dimensional spectral element shape function is expressed as: (26) In order to obtain the interpolation expression of displacement, substitute equations (24) and (25) into (23): (27) is the shape function describing the displacement. Similarly, the interpolation expression of stress can be obtained: (28) in: is the shape function describing the stress.

[0051] (29) in, is the 6th-order identity matrix.

[0052] Combined with auxiliary variables for stress integration, the hybrid displacement-stress unit finite element technique is further applied for spatiotemporal discretization.

[0053] Substituting the differential operator in equation (19) and the volume element in (20a), (20b), (20c) into (22a), (22b) we have: (30a) (30b) Convert the above frequency domain equations to the time domain and we have: (31a) (31b) Among them, the superscript symbol ( represents the unknown field variable related to time. In the inverse Fourier transform of , we use the following formula to represent the stress integral as an auxiliary variable: (32) Substituting Equations (27) and (28) into Equations (31a) and (31b), and taking advantage of the arbitrariness of imaginary variables, the mixed displacement-stress finite element equations can be obtained as follows: (33a) (33b) in: , , Among them, the superscript express The matrix in , each coefficient matrix is ​​expressed as: (34a) (34b) (34c) (34d) (34e) (34f) (34g) (34h) (34i) (34g) (34k) (34l) (34m) (34n) The calculations of the above coefficient matrices all use the Gauss-LeBataud-Legendre integration rule.

[0054] The scale boundary perfect matching layer in (33) The unknown quantity in Can be written as: (35) The integral over time can be written as: (36) The traction force vector can be defined as: with (37) in, Represents finite element assembly.

[0055] In order to verify the effectiveness of the scaled boundary perfectly matched layer method for infinite domain wave numerical simulation proposed in this embodiment in simulating elastic wave propagation in infinite domains of complex sites, the following example is taken as an example to illustrate the wave problem of a complex layered site with valley terrain under the action of a point source load. Figure 3 As shown, there is a horizontal physical interface, and the four material parameters are listed in Table 2. In order to excite the outer wave, a vertical downward point source load is applied at point S in the dotted box. Taking Ricker wavelet as an example, its time history can be written as: (38) In the formula and Represent the amplitude of the traction force and the moment when the traction force reaches its peak value. In this example, the parameters are selected as s and Pa, from which the maximum frequency is 4.5 Hz and the minimum wavelength is .

[0056] Table 2 Soil material parameters This example constructs a finite domain-scaled boundary perfectly matched layer truncation model of a layered site with a valley terrain for wave analysis. Here, the entire construction process is explained using the spectrum unit as a specific example. Figure 3 As shown in the figure, the finite domain size is 9.05 km × 8.70 km, discretized with second-order spectral elements with a 100 m grid. The thickness of the SBPML is 300 m, and it is also discretized into 100 m spectral elements with second order in both radial and circumferential directions. This allows the scaled boundary perfectly matched layer (SBPML domain) to be seamlessly coupled with the inner domain spectral elements. The node displacements on the four sides and bottom of the SBPML model are constrained. Figure 3 In the area shown in the red dotted box, a vertical downward point source load is applied to calculate the site response and perform result analysis.

[0057] Figure 4 shows snapshots of the vertical displacement wave fields of the viscoelastic boundary model (VSABC) and the SBPML model (Scaled boundary perfectly matched layer) at three specific moments in the example. The VSABC model applies viscoelastic boundary conditions to the four sides and bottom of the finite domain grid model in Figure 4 to simulate the radiation conditions at infinity. As shown in Figure 4(a), the SBPML and VSABC models have the following characteristics: t = 2.71s shows the same initial wave field. t =3.79 s, a significant reflection occurs, which can be seen in the red box of Figure 4(b). As shown in Figure 4(c), at t = 8.00 s, the outward traveling wave in the SBPML model decays rapidly and almost disappears completely, while the VSABC model shows a continuous round-trip reflected wave.

[0058] Figure 5 compares the displacement time histories of the SBPML and VSABC models at sampling point A in the second example. As shown in Figures 5(a)-(c), the two methods agree well within the first 3.0 seconds, followed by a deviation, demonstrating that the SBPML effectively absorbs outward-traveling waves from the inner domain in infinite domain sites with complex terrain. These results demonstrate the superior outward-traveling wave absorption capabilities of the scaled-boundary perfectly matched layer method coupled with the spectral element for infinite domain wave analysis with complex terrain and geology.

[0059] The scaled boundary perfectly matched layer method for infinite domain wave numerical simulation according to the embodiment of the present invention is established by combining two typical high-precision artificial boundary methods, the scaled boundary finite element method and the perfectly matched layer method. It can be used for infinite domain wave numerical simulation and has the following effects: 1) It adopts a hybrid displacement-stress non-split field representation, which can be seamlessly coupled with internal domain hexahedral elements, polyhedral elements, and arbitrary order spectral elements, making it more robust.

[0060] 2) It is possible to use artificial boundaries of general geometric shapes, which can simulate physical interfaces in infinite external subdomains that extend parallel and radially to infinity, which is more flexible.

[0061] 3) Its performance has been verified by the field wave propagation analysis in a complex layered half-space with valley terrain, which shows that the proposed SBPML has excellent absorption characteristics, greater stability and accuracy.

[0062] Therefore, compared with traditional methods, the scaled boundary perfectly matched layer method for infinite domain wave numerical simulation in the embodiment of the present invention has the simplicity of boundary realization, good adaptability of the boundary and the inner domain, and high flexibility of the boundary shape.

[0063] Example 2: The embodiment of the present invention provides a scaled boundary perfectly matched layer system for infinite domain wave numerical simulation, see Figure 6 As shown, the system 100 includes: The coordinate transformation module 10 is configured to map the irregular geometric infinite domain in the Cartesian coordinate system to the regular geometric domain in the scaled boundary local coordinate system based on the scaled boundary coordinate transformation, wherein the scaled boundary local coordinate system is composed of radial coordinates and annular coordinates; The coordinate stretching module 20 is configured to analytically extend the radial coordinates of the regular geometric domain to the complex space using a complex coordinate stretching function to obtain complex radial coordinates and an absorption layer domain with a fluctuation exponential decay characteristic, and to truncate the absorption layer domain according to a preset thickness to obtain a scaled boundary perfectly matched layer; The construction module 30 is configured to construct corresponding equivalent integral forms for the equilibrium equation and the physical equation of the wave problem in the scaled boundary perfectly matched layer based on the Galerkin weighted residual principle to obtain an equivalent integral formula; The finite element discretization module 40 is configured to perform finite element discretization on the displacement, stress, and auxiliary variables of the stress integral in the constructed equivalent integral formula using a hybrid displacement-stress unit finite element technology to obtain a hybrid displacement-stress finite element equation for a scaled boundary perfectly matched layer.

[0064] Optionally, the coordinate transformation module 10 in the scaled boundary perfectly matched layer system of the infinite domain wave numerical simulation of this embodiment specifically includes: A partitioning unit is configured to partition the irregular geometric infinite domain into a finite inner subdomain and an infinite outer subdomain; The transformation unit is configured to perform coordinate transformation processing on points on the boundary surface of the infinite external subdomain in the Cartesian coordinate system using the scaled boundary coordinate transformation to obtain a regular geometric domain in the scaled boundary local coordinate system.

[0065] Optionally, the expression for scaled boundary coordinate transformation in the scaled boundary perfectly matched layer system in the infinite domain wave numerical simulation of this embodiment is: in, represents the artificial boundary surface of the infinite outer subdomain, represents the scaled splicing surface of the infinite external subdomain, Represents an artificial boundary The coordinates of any point on ( ) indicates scaling the splicing surface The coordinates of the corresponding points, represents the radial coordinate of a regular geometric domain, Represents the coordinates of any point on a regular geometric domain.

[0066] Optionally, the coordinate transformation module 10 in the scaled boundary perfectly matched layer system of the infinite domain wave numerical simulation of this embodiment further includes: The regional discretization unit is configured to discretize the artificial boundary surface and the scaled splicing surface of the infinite outer subdomain into an equal number of regional units using a standard finite element method, and interpolate the coordinates of each regional unit based on the shape function.

[0067] Optionally, in the scaled boundary perfectly matched layer system of the infinite domain wave numerical simulation of this embodiment, a corresponding shape function is selected according to the shape of the finite internal subdomain; Among them, shape functions include two-dimensional shape functions, bilinear quadrilateral element shape functions, Shape functions for polygonal elements and shape functions for spectral elements.

[0068] Optionally, the coordinate stretching module 20 in the scaled boundary perfectly matched layer system of the infinite domain wave numerical simulation of this embodiment is specifically configured to analytically extend the radial coordinates of the regular geometric domain to the complex space using the complex coordinate stretching function. The process of obtaining the complex radial coordinates is specifically expressed as follows: in, represents the radial coordinate of a regular geometric domain, Represents a mapping, represents the complex radial coordinate, Indicates the integral variable, which is 0 to Any position on the integral path of represents the complex coordinate stretching function, represents the circular frequency, represents the scaling function, represents the decay function, and i represents the imaginary unit.

[0069] Optionally, the finite element discretization module 40 in the scaled boundary perfectly matched layer system of the infinite domain wave numerical simulation of this embodiment specifically includes: The interpolation unit is configured to use a shape function to interpolate the displacement and stress in the constructed equivalent integral expression to obtain an interpolation expression of displacement and an interpolation expression of stress; The discrete elements are configured to perform finite element discretization on the scaled boundary perfectly matched layer based on the displacement interpolation expression, the stress interpolation expression and the auxiliary variables of the stress integral, so as to obtain the mixed displacement-stress finite element equation of the scaled boundary perfectly matched layer.

[0070] Example 3: The present invention discloses an electronic device. The electronic device includes a memory and a processor. The memory stores a computer program. When the processor executes the computer program, the electronic device implements the steps of the scaled boundary perfectly matched layer method for infinite domain wave numerical simulation described in any one of the first embodiments of the present invention.

[0071] Figure 7 FIG. 1 is a structural diagram of an electronic device according to an embodiment of the present invention. Figure 7As shown, the electronic device includes a processor, a memory, a communication interface, a display screen and an input device connected via a system bus. The processor of the electronic device is used to provide computing and control capabilities. The memory of the electronic device includes a non-volatile storage medium and an internal memory. The non-volatile storage medium stores an operating system and a computer program. The internal memory provides an environment for the operation of the operating system and the computer program in the non-volatile storage medium. The communication interface of the electronic device is used to communicate with an external terminal in a wired or wireless manner, and the wireless manner can be achieved through WIFI, an operator network, near field communication (NFC) or other technologies. The display screen of the electronic device can be a liquid crystal display or an electronic ink display screen, and the input device of the electronic device can be a touch layer covering the display screen, or a button, trackball or touchpad provided on the electronic device housing, or an external keyboard, touchpad or mouse.

[0072] Those skilled in the art will understand that Figure 7 The structure shown in the figure is only a structural diagram of the part related to the technical solution of the present disclosure, and does not constitute a limitation on the electronic device to which the solution of the present application is applied. The specific electronic device may include more or fewer components than shown in the figure, or combine certain components, or have a different component arrangement.

[0073] Example 4: An embodiment of the present invention discloses a computer-readable storage medium having a computer program stored thereon. When executed by a processor, the computer program implements the steps of any one of the scaled boundary perfectly matched layer methods for infinite domain wave numerical simulations described in the first embodiment of the present invention.

[0074] Please note that the technical features of the above embodiments can be combined arbitrarily. In order to make the description concise, not all possible combinations of the technical features in the above embodiments are described. However, as long as there is no contradiction in the combination of these technical features, they should be considered to be within the scope of this specification. The above embodiments only express several implementation methods of the present application. The description is relatively specific and detailed, but it cannot be understood as a limitation on the scope of the invention patent. It should be pointed out that for ordinary technicians in this field, without departing from the concept of this application, several variations and improvements can be made, which all fall within the scope of protection of this application. Therefore, the scope of protection of the patent in this application shall be based on the attached claims.

[0075] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principles of the present invention should be included in the scope of protection of the present invention.

[0076] Although some specific embodiments of the present invention have been described in detail by way of examples, it should be understood by those skilled in the art that the above examples are for illustration only and are not intended to limit the scope of the present invention. It should be understood by those skilled in the art that modifications may be made to the above embodiments without departing from the scope and spirit of the present invention. The scope of the present invention is defined by the appended claims.

Claims

1. A scaled boundary perfectly matched layer method for infinite domain wave numerical simulation, characterized by: The method comprises: Step S1: Based on the scaled boundary coordinate transformation, the irregular geometric infinite domain in the Cartesian coordinate system is mapped to a regular geometric domain in the scaled boundary local coordinate system, wherein the scaled boundary local coordinate system is composed of radial coordinates and annular coordinates; Step S2: Analytically extending the radial coordinates of the regular geometric domain to a complex space using a complex coordinate stretching function to obtain complex radial coordinates and an absorption layer domain with a fluctuation exponential decay characteristic, and truncating the absorption layer domain according to a preset thickness to obtain a scaled boundary perfectly matched layer; Step S3: Based on the Galerkin weighted residual principle, corresponding equivalent integral forms are constructed for the equilibrium equation and physical equation of the wave problem in the scaled boundary perfectly matched layer to obtain an equivalent integral formula; Step S4: using a hybrid displacement-stress unit finite element technique to perform finite element discretization on the displacement, stress, and auxiliary variables of the stress integral in the constructed equivalent integral formula to obtain a hybrid displacement-stress finite element equation for a scaled boundary perfectly matched layer.

2. The scaled boundary perfectly matched layer method for infinite domain wave numerical simulation according to claim 1, characterized in that: The step S1 specifically includes: Step S11: dividing the irregular geometric infinite domain into a finite inner subdomain and an infinite outer subdomain; Step S12: using the scaled boundary coordinate transformation to perform coordinate transformation processing on the points on the boundary surface of the infinite outer subdomain in the Cartesian coordinate system, so as to obtain a regular geometric domain in the scaled boundary local coordinate system.

3. The scaled boundary perfectly matched layer method for infinite domain wave numerical simulation according to claim 2, characterized in that: The expression of the scale boundary coordinate transformation is: ； in, represents the coordinates of any point on the artificial boundary, ( ) represents the coordinates of the corresponding points of the scaled stitching surface, represents the radial coordinate of a regular geometric domain, Represents the coordinates of any point on a regular geometric domain.

4. The scaled boundary perfectly matched layer method for infinite domain wave numerical simulation according to claim 3, characterized in that: After step S12, the following steps are further included: Step S13: using a standard finite element method to discretize the artificial boundary surface and the scaled splicing surface of the infinite outer subdomain into an equal number of regional units, and interpolating the internal coordinates of each of the regional units based on the shape function.

5. The scaled boundary perfectly matched layer method for infinite domain wave numerical simulation according to claim 4, characterized in that: In the step S13, the corresponding shape function is selected according to the shape of the finite inner subdomain; The shape function includes two-dimensional shape function, bilinear quadrilateral unit shape function, Shape functions for polygonal elements and shape functions for spectral elements.

6. The scaled boundary perfectly matched layer method for infinite domain wave numerical simulation according to claim 1, characterized in that: In step S2, the radial coordinates of the regular geometric domain are analytically extended to the complex space using the complex coordinate stretching function. The process of obtaining the complex radial coordinates is specifically expressed as follows: ； in, represents the radial coordinate of a regular geometric domain, Represents a mapping, represents the complex radial coordinate, Indicates the integral variable, which is 0 to Any position on the integral path of represents the complex coordinate stretching function, represents the circular frequency, represents the scaling function, represents the decay function, and i represents the imaginary unit.

7. The scaled boundary perfectly matched layer method for infinite domain wave numerical simulation according to claim 4, characterized in that: The step S4 specifically includes: Using the shape function to interpolate the displacement and stress in the constructed equivalent integral formula to obtain a displacement interpolation expression and a stress interpolation expression; Based on the displacement interpolation expression, the stress interpolation expression and auxiliary variables of stress integration, finite element discretization is performed on the scaled boundary perfectly matched layer to obtain a mixed displacement-stress finite element equation of the scaled boundary perfectly matched layer.

8. A scaled boundary perfectly matched layer system for infinite domain wave numerical simulation, characterized in that: The system comprises: A coordinate transformation module is configured to map the irregular geometric infinite domain in the Cartesian coordinate system to a regular geometric domain in the scaled boundary local coordinate system based on a scaled boundary coordinate transformation, wherein the scaled boundary local coordinate system is composed of radial coordinates and annular coordinates; a coordinate stretching module configured to analytically extend the radial coordinates of the regular geometric domain to a complex space using a complex coordinate stretching function to obtain complex radial coordinates and an absorption layer domain with a fluctuation exponential decay characteristic, and to truncate the absorption layer domain according to a preset thickness to obtain a scaled boundary perfectly matched layer; The construction module is configured to construct corresponding equivalent integral forms for the equilibrium equations and physical equations of the wave problem in the scaled boundary perfectly matched layer based on the Galerkin weighted residual principle to obtain equivalent integral forms; The finite element discretization module is configured to use a hybrid displacement-stress unit finite element technology to perform finite element discretization on the displacement, stress and auxiliary variables of the stress integral in the constructed equivalent integral formula to obtain a hybrid displacement-stress finite element equation of the scaled boundary perfectly matched layer.

9. An electronic device, characterized in that: The electronic device includes a memory and a processor, wherein the memory stores a computer program, and when the processor executes the computer program, the steps of the scaled boundary perfectly matched layer method for infinite domain wave numerical simulation according to any one of claims 1 to 7 are implemented.

10. A computer-readable storage medium, characterized in that The computer-readable storage medium stores a computer program, and when the computer program is executed by a processor, the computer program implements the steps of the scaled boundary perfectly matched layer method for infinite domain wave numerical simulation according to any one of claims 1 to 7.

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