Method and system for simulating wave propagation in an infinite domain using a scaled boundary perfectly matched layer

Abstract

The application discloses a proportional boundary perfect matched layer method and system for numerical simulation of infinite domain wave motion, and belongs to the field of dynamic analysis, which comprises the following steps: irregular geometric infinite domain in a Cartesian coordinate system is mapped to a regular geometric domain in a proportional boundary local coordinate system based on proportional boundary coordinate transformation; the radial coordinate of the regular geometric domain is analytically extended to a complex space by using a complex coordinate stretching function, and a proportional boundary perfect matched layer is obtained by truncation processing according to a preset thickness; based on the Galerkin weighted residual principle, corresponding equivalent integral expressions are constructed for the balance equation and the physical equation of the wave motion problem in the proportional boundary perfect matched layer; mixed displacement-stress finite element technology is adopted to perform finite element discrete processing on the displacement, stress and auxiliary variables of stress integral in the constructed equivalent integral expressions, so that mixed displacement-stress finite element equations of the proportional boundary perfect matched layer are obtained; the method has the advantages of simple boundary implementation and good adaptability of the boundary to the inner domain.

Description

Technical Field

[0001] This invention relates to the field of dynamic analysis technology, and more specifically, to a method and system for proportional boundary perfect matching layer in numerical simulation of infinite domain fluctuations. Background Technology

[0002] Numerical simulation of large-area medium fluctuations is the cornerstone of seismic safety assessment for major engineering projects. Due to limited computational resources, it is usually necessary to truncate the original large area at an appropriate location and apply suitable artificial boundary conditions to its truncation boundary in order to minimize the size of the computational domain while ensuring simulation accuracy.

[0003] Traditional methods for implementing artificial boundaries mainly fall into the following two categories:

[0004] 1. Absorbing Boundary Condition Techniques: Absorbing boundary conditions are further subdivided into local and nonlocal methods. Local methods include the transmission boundary method, viscous boundary method, and viscoelastic boundary method. These methods allow for spatiotemporal decoupling of the scattered field response at boundary nodes, offering high computational efficiency, but their computational accuracy is relatively low and they are prone to instability phenomena such as high-frequency pseudo-oscillations and low-frequency drift. Nonlocal methods include the boundary element method, thin-layer method, and scaled boundary finite element method. While these methods can achieve higher computational accuracy, they involve convolution integrals or series expansion operations, resulting in higher computational complexity.

[0005] 2. Absorbing Layer Technology. Absorbing layer technology attenuates traveling waves using artificial damping materials, represented by Rayleigh damping layer method, Cauchy damping layer method, and damping extraction method. In recent years, absorbing layer technology, represented by the Perfectly Matched Layer (PML) method, has received widespread attention. Originally proposed by Bérenger, the PML method is mainly used for electromagnetic wave simulation. It achieves near-perfect absorption of traveling waves through complex coordinate stretching functions and has been applied to elastic, anisotropic, viscoelastic, and porous elastic media, becoming an important tool for modeling wave propagation problems. The perfect matching layer method is highly dependent on the selected coordinate system (i.e., the inner and outer surfaces of the perfect matching layer region and the discrete grid lines within the domain must be contour lines), which leads to the following technical problems: 1) Poor flexibility: The perfect matching layer method is difficult to use artificial boundaries of general geometry, which reduces the flexibility of selecting the finite internal computational domain; 2) Limited applicability: Since the complex coordinate extension function is based on global coordinates, the existing method can only consider physical interfaces parallel to the coordinate plane, and cannot consider multiple parallel and radial physical surfaces and interfaces extending infinitely in an infinite domain.

[0006] In summary, there is an urgent need to develop a proportional boundary perfect matching layer method and system for numerical simulation of infinite domain fluctuations to solve one or more of the problems mentioned above. Summary of the Invention

[0007] One objective of this invention is to provide a new technical solution for a proportional boundary perfect matching layer method and system for numerical simulation of infinite domain fluctuations.

[0008] According to a first aspect of the present invention, a method for a scaled boundary perfect-matching layer in numerical simulation of infinite-domain fluctuations is provided, the method comprising:

[0009] Step S1: Based on the proportional boundary coordinate transformation, the irregular geometric infinite domain in the Cartesian coordinate system is mapped to the regular geometric domain in the proportional boundary local coordinate system, wherein the proportional boundary local coordinate system is composed of radial coordinates and circumferential surface coordinates;

[0010] Step S2: The radial coordinates of the regular geometric domain are analytically extended to the complex space using the complex coordinate stretching function to obtain the complex radial coordinates and the absorption layer domain with the fluctuation exponential decay characteristic. The absorption layer domain is then truncated according to the preset thickness to obtain a proportional boundary perfectly matched layer.

[0011] Step S3: Based on the Galerkin weighted residual principle, construct the corresponding equivalent integral form for the equilibrium equation and physical equation of the wave problem in the proportional boundary perfectly matched layer, so as to obtain the equivalent integral formula.

[0012] Step S4: The displacement, stress, and auxiliary variables of the stress integral in the constructed equivalent integral formula are discretized using the hybrid displacement-stress element finite element technique to obtain the hybrid displacement-stress finite element equation of the scaled boundary perfectly matched layer.

[0013] Optionally, step S1 specifically includes:

[0014] Step S11: Divide the irregular geometric infinite field into a finite interior subfield and an infinite outer subfield;

[0015] Step S12: Use the proportional boundary coordinate transformation to perform coordinate transformation on the points on the boundary surface of the infinite outer subdomain in the Cartesian coordinate system to obtain the regular geometric domain in the proportional boundary local coordinate system.

[0016] Optionally, the expression for the proportional boundary coordinate transformation is:

[0017]

[0018]

[0019]

[0020] in, Represents the coordinates of any point on the artificial boundary, ( () indicates the coordinates of the corresponding point in the scaled splicing face. Represents the radial coordinates of a regular geometric domain. Represents the coordinates of any point on a regular geometric domain.

[0021] Optionally, the step S12 may be followed by:

[0022] Step S13: The artificial boundary surface and the scaled splicing surface of the infinite outer subdomain are discretized into an equal number of regional elements using the standard finite element method, and the coordinates within each regional element are interpolated based on the shape function.

[0023] Optionally, in step S13, the corresponding shape function is selected according to the shape of the finite internal subdomain;

[0024] The shape functions include two-dimensional shape functions, bilinear quadrilateral unit shape functions, and... Shape functions of polygonal units and shape functions of spectral units.

[0025] Optionally, in step S2, the process of analytically extending the radial coordinates of the regular geometric domain to the complex space using a complex coordinate stretching function to obtain the complex radial coordinates is specifically represented as follows:

[0026]

[0027]

[0028] in, Represents the radial coordinates of a regular geometric domain. Represents a mapping. Represents complex radial coordinates. The variable representing integration is from 0 to... At any position on the integration path, Represents the complex coordinate stretching function. Represents angular frequency. This represents the scaling function. Let i represent the decay function, and let i represent the imaginary unit.

[0029] Optionally, step S4 specifically includes:

[0030] The displacement and stress in the constructed equivalent integral are interpolated using the shape function to obtain the displacement interpolation expression and the stress interpolation expression.

[0031] Based on the displacement interpolation expression, the stress interpolation expression, and the auxiliary variable of the stress integral, the scale boundary perfectly matched layer is discretized by finite element method to obtain the hybrid displacement-stress finite element equation of the scale boundary perfectly matched layer.

[0032] According to a second aspect of the present invention, a scaled boundary perfect-matching layer system for numerical simulation of infinite-domain fluctuations is provided, the system comprising:

[0033] The coordinate transformation module is configured to map an irregular geometric infinite domain in the Cartesian coordinate system to a regular geometric domain in the proportional boundary local coordinate system based on the proportional boundary coordinate transformation, wherein the proportional boundary local coordinate system is composed of radial coordinates and circumferential surface coordinates.

[0034] The coordinate stretching module is configured to use a complex coordinate stretching function to analytically extend the radial coordinates of the regular geometric domain to the complex space to obtain complex radial coordinates and an absorption layer domain with fluctuating exponential decay characteristics, and to truncate the absorption layer domain according to a preset thickness to obtain a proportional boundary perfectly matched layer.

[0035] The module is configured to construct the corresponding equivalent integral form of the equilibrium equation and physical equation for the wave problem in the proportional boundary perfectly matched layer, based on the Galerkin weighted residual principle, so as to obtain the equivalent integral expression.

[0036] The finite element discretization module is configured to use hybrid displacement-stress element finite element technology to discretize the displacement, stress, and auxiliary variables of the stress integral in the constructed equivalent integral formula, so as to obtain the hybrid displacement-stress finite element equation of the scaled boundary perfectly matched layer.

[0037] According to a third aspect of the present invention, an electronic device is provided, the electronic device including a memory and a processor, the memory storing a computer program, and the processor executing the computer program to implement the steps of the proportional boundary perfect matching layer method for numerical simulation of infinite domain fluctuations as described in the first aspect of the present invention.

[0038] According to a fourth aspect of the present invention, a computer-readable storage medium is provided, on which a computer program is stored, which, when executed by a processor, implements the steps of the proportional boundary perfect matching layer method for numerical simulation of infinite domain fluctuations as described in the first aspect of the present invention.

[0039] According to an embodiment of the present invention, the proportional boundary perfect matching layer method and system for infinite domain wave numerical simulation has the following beneficial effects:

[0040] The present invention provides a method and system for a scaled boundary perfect matching layer in numerical simulation of infinite domain fluctuations, which has the following advantages:

[0041] 1) Simplicity of boundary implementation: Similar to local boundaries, the proposed method is simple to implement, avoiding complex analytical and semi-analytical derivation processes;

[0042] 2) Adaptability of boundary and inner domain: Based on the perfect matching layer technology of complex coordinate extension, it can absorb incident waves in the whole frequency band and all angles; it can be seamlessly coupled with the inner domain hexahedral unit, prismatic unit and arbitrary order spectral unit, and supports the use of different order units in the extension function direction and the inner computational domain to achieve a flexible balance between accuracy and efficiency.

[0043] 3) Flexibility of boundary shape: The proposed method uses modified scale boundary coordinates to describe the infinite domain, which eliminates the dependence of traditional methods on global coordinates. This allows the proposed method to use artificial boundaries with general geometric shapes and to reflect multiple parallel and radial physical interfaces that extend to infinity.

[0044] Other features and advantages of the invention will become clear from the following detailed description of exemplary embodiments of the invention with reference to the accompanying drawings. Attached Figure Description

[0045] The accompanying drawings, which are incorporated in and form part of this specification, illustrate embodiments of the invention and, together with their description, serve to explain the principles of the invention.

[0046] Figure 1 This is a flowchart illustrating a scaled boundary perfect matching layer method for numerical simulation of infinite domain fluctuations according to an embodiment.

[0047] Figure 2 This is a schematic diagram illustrating the construction process of the scale boundary perfect matching layer provided in the embodiment;

[0048] Figure 3 This is a schematic diagram of a layered half-space scaled boundary perfectly matched layer truncation model of a complex site with valley topography under point source load according to the embodiment.

[0049] Figures 4(a)-4(c) Figure 4(a) shows the wavefield snapshots of a layered half-space model for a complex site with valley topography provided in the embodiments, where the SBPML model and the VSABC model are in... t A snapshot of the vertical displacement wave field at time 2.71 s is shown in Figure 4(b), which shows the SBPML and VSABC models. t A snapshot of the vertical displacement wave field at time 3.79 s is shown in Figure 4(c), which shows the SBPML and VSABC models. t =8.00 s snapshot of the vertical displacement wave field;

[0050] Figures 5(a)-5(c)The displacement time history comparison curves of the SBPML model and VSABC model at sampling point A provided in the embodiment are shown in Figure 5(a), which is a comparison curve of the displacement time history of the SBPML model and VSABC model at sampling point A in the x direction; Figure 5(b) is a comparison curve of the displacement time history of the SBPML model and VSABC model at sampling point A in the y direction; and Figure 5(c) is a comparison curve of the displacement time history of the SBPML model and VSABC model at sampling point A in the z direction.

[0051] Figure 6 This is a structural block diagram of a scaled boundary perfect matching layer system for numerical simulation of infinite domain fluctuations, provided according to an embodiment.

[0052] Figure 7 This is a schematic diagram of an electronic device. Detailed Implementation

[0053] 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, numerical expressions, and values ​​of the components and steps set forth in these embodiments do not limit the scope of the invention.

[0054] The following description of at least one exemplary embodiment is merely illustrative and is in no way intended to limit the invention or its application or use.

[0055] Techniques, methods, and equipment known to those skilled in the art may not be discussed in detail, but where appropriate, such techniques, methods, and equipment should be considered part of the specification.

[0056] In all the examples shown and discussed herein, any specific values ​​should be interpreted as merely exemplary and not as limitations. Therefore, other examples of exemplary embodiments may have different values.

[0057] Example 1:

[0058] See Figure 1 As shown, this embodiment of the invention provides a method for a scaled boundary perfect matching layer in numerical simulation of infinite domain fluctuations. The method includes:

[0059] Step S1: Based on the proportional boundary coordinate transformation, the irregular geometric infinite domain in the Cartesian coordinate system is mapped to the regular geometric domain in the proportional boundary local coordinate system, wherein the proportional boundary local coordinate system consists of radial coordinates and circumferential surface coordinates.

[0060] To geometrically describe the shape of an irregular geometric infinite domain, this embodiment proposes a method for proportional boundary coordinate transformation of the irregular geometric infinite domain. By using modified proportional boundary coordinates to describe the infinite domain, the dependence of traditional methods on global coordinates is eliminated. This allows this embodiment to use artificial boundaries of general geometric shapes and multiple parallel and radial physical interfaces that can reflect the extension to infinity.

[0061] Optionally, step S1 in the scaled boundary perfect matching layer method for infinite domain fluctuation numerical simulation in this example specifically includes:

[0062] Step S11: Divide the irregular geometric infinite field into a finite interior subfield and an infinite outer subfield;

[0063] Step S12: Use the proportional boundary coordinate transformation to perform coordinate transformation on the points on the boundary surface of the infinite outer subdomain in the Cartesian coordinate system to obtain the regular geometric domain in the proportional boundary local coordinate system.

[0064] Optionally, the expression for the scaled boundary coordinate transformation in the scaled boundary perfect matching layer method for infinite domain wave numerical simulation in this example is:

[0065]

[0066]

[0067]

[0068] in, Represents the coordinates of any point on the artificial boundary, ( () indicates the coordinates of the corresponding point in the scaled splicing face. Represents the radial coordinates of a regular geometric domain. Represents the coordinates of any point on a regular geometric domain.

[0069] Specifically, the problem of elastic wave propagation in an infinite domain of a complex three-dimensional field, such as... Figure 2 As shown, the irregular geometric infinite field (i.e., the original infinite field) is first divided into finite internal subfields. and infinite outer subdomains Finite internal subdomains quilt Figure 2 The area enclosed by the red border typically contains sites with complex geometries, which can be solved using domain discretization methods such as the finite element method, spectral element method, and scaled boundary element method. Infinite outer subdomains, such as... Figure 2 In As shown, it contains physical surfaces and interfaces with parallel and radial straight lines extending to infinity.

[0070] Assume an infinite outer subfield It is in a linearly elastic state. The equilibrium equation for the wave problem in an infinite outer subdomain can be expressed as:

[0071] (1a)

[0072] The geometric equation for the wave problem in an infinite outer subdomain can be expressed as:

[0073] (1b)

[0074] The physical equations for the wave problem in an infinite outer subdomain can be expressed as:

[0075] (1c)

[0076] in, , , ;

[0077] (2)

[0078] (3)

[0079] In the above equation, the total stress, displacement, and strain vectors of the infinite outer subdomain are denoted as follows: and , This represents the second derivative of displacement with respect to time. It is a linear differential operator. It is the elastic constitutive matrix. and It is expressed as Lamé constant.

[0080] like Figure 2 As shown, for the initial boundary value problem of the fluctuation problem, this embodiment combines the advantages of the scaled boundary finite element method and the perfectly matched layer method, and introduces the scaled boundary coordinate transformation technique to address the initial boundary value problem. Figure 2 The infinite outer subfield is geometrically described. and They are respectively infinite outer subdomains The artificial boundary surface and scaled splicing surface, with radial coordinates as Also known as the scaling factor, from Departure, defined ( and .based on and Scaling the coordinates of points on the surface yields an infinite outer subdomain. any point above The coordinates are:

[0081] (4a)

[0082] (4b)

[0083] (4c)

[0084] in, Representing artificial boundaries The coordinates of the previous point; ) indicates scaling the splicing surface The coordinates of the corresponding point.

[0085] Optionally, the method for scaling boundary perfect matching layers in the numerical simulation of infinite domain fluctuations in this example also includes the following after step S12:

[0086] Step S13: The artificial boundary surface and the scaled splicing surface of the infinite outer subdomain are discretized into an equal number of regional elements using the standard finite element method, and the coordinates within each regional element are interpolated based on the shape function.

[0087] Preferably, in step S13 of the proportional boundary perfect matching layer method for infinite domain fluctuation numerical simulation in this example, the appropriate shape function is selected according to the shape of the finite internal subdomain; wherein, the shape function includes two-dimensional shape functions, bilinear quadrilateral element shape functions, etc. Shape functions of polygonal units and shape functions of spectral units.

[0088] Specifically, to describe the shape of an infinite outer subdomain, the standard finite element method can be used. and The boundary surface is discretized into an equal number of regional elements, and shape functions are introduced for this discretization. The specific expression of the discretized coordinates is as follows:

[0089] (5a)

[0090] (5b)

[0091] (5c)

[0092] in, It is a two-dimensional shape function, derived from the coordinates of natural nodes. Constructed express and The coordinates of the nodes on the surface.

[0093] The appropriate shape function is selected based on the shape of the finite interior subdomain. For example, for a discrete model with hexahedral elements as the interior domain, the following bilinear quadrilateral element shape function can be used:

[0094] (6a)

[0095] Wherein, reference node coordinates .

[0096] For discrete models whose interior domains are polyhedral elements, the following approach can be used: Shape function of polygonal unit:

[0097] (6b)

[0098] in , . It is the first strip edge and The included angle at the center of the polygon, express The first polygon The natural coordinates of each vertex can be represented as and .

[0099] For a discrete model whose inner domain is a higher-order spectral unit, the shape function of the spectral unit can be expressed as:

[0100] (6c)

[0101] in Let be the order of the one-dimensional shape function spectral unit. Indicates the first The natural coordinates of the nodes. The coordinate selection for the Gauss-Lobatto-Legendre Quadrature is shown in Table 1.

[0102] Table 1. Coordinates of one-dimensional spectral element nodes in the Gauss-Lebarto-Legend quadrature method

[0103]

[0104] Substituting equations (5a), (5b), and (5c) into equations (4a), (4b), and (4c) respectively, we can obtain the following: and Indicated Representation of any point within the domain:

[0105] (7a)

[0106] (7b)

[0107] (7c)

[0108] Differential operator in proportional boundary coordinates in equation (3) It can be represented as:

[0109] (8)

[0110] The area element in an infinite field can be represented as:

[0111] (9)

[0112] Wherein, the coefficient matrix as well as The representation is as follows:

[0113]

[0114]

[0115]

[0116] in, It is an artificial boundary surface The matrix related to node coordinates This is related to the scaling and splicing surface. Matrix related to node coordinates; Hadamard operator , represents the dot product of two matrices; the subscript followed by a comma indicates the partial derivative with respect to that variable.

[0117] Step S2: The radial coordinates of the regular geometric domain are analytically extended to the complex space using the complex coordinate stretching function to obtain the complex radial coordinates and the absorption layer domain with the fluctuating exponential decay characteristics. The absorption layer domain is then truncated according to the preset thickness to obtain a proportional boundary perfectly matched layer.

[0118] In this embodiment, a complex coordinate stretching function is introduced to stretch the radial coordinates. Extending to the multi-level space The truncated infinite outer subdomain Mapped to a scale boundary perfect matching layer Because 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 for truncation at that point. In this embodiment, the thickness of the proportional boundary perfectly matched layer can be taken as 1 / 2 the minimum wavelength of the outgoing wave.

[0119] The method in this embodiment is based on the perfect matching layer technology of complex coordinate extension, which can absorb incident waves across the entire frequency band and at all angles; it can be seamlessly coupled with hexahedral units, prismatic units and arbitrary order spectral units in the inner domain, and supports the use of different order units in the direction of the extension function and the inner computational domain, achieving a flexible balance between accuracy and efficiency; it improves the adaptability of the boundary and the inner domain.

[0120] Optionally, in the scaled boundary perfect matching layer method for infinite domain wave numerical simulation in this example, the process of analytically extending the radial coordinates of the regular geometric domain to the complex space using the complex coordinate stretching function to obtain the complex radial coordinates is specifically represented as follows:

[0121]

[0122]

[0123] in, Represents the radial coordinates of a regular geometric domain. Represents a mapping. Represents complex radial coordinates. The variable representing integration is from 0 to... At any position on the integration path, Represents the complex coordinate stretching function. Represents angular frequency. This represents the scaling function. Let i represent the decay function, and let i represent the imaginary unit.

[0124] Specifically, the above equations (7a), (7b), (7c), (8) and (9) Simple replacement with radial stretch coordinates The differential operator and the area infinitesimal equation can be obtained as follows:

[0125] (10)

[0126] (11)

[0127] in, Represents mapping, radial stretching coordinates It can be represented as:

[0128] (12)

[0129] in, For complex coordinate stretching functions, This represents the angular frequency. In this embodiment, the classic complex stretching function is used:

[0130] (13)

[0131] Scaling function and decay function Can be written as and Related expressions:

[0132] (14a)

[0133] (14b)

[0134] and attenuation coefficient It can be written as:

[0135] (15a)

[0136] (15b)

[0137] In the formula, for outer boundary reflection coefficient, For reference wave speed, The order of the polynomial decay; For radial unit dimensions; To match the average thickness of the layer domain in Cartesian coordinates; It is the angular frequency; It is the imaginary unit.

[0138] Substituting equations (15a) and (15b) into equations (14a) and (14b) respectively, we can obtain the extended first coordinate. Express:

[0139] (16)

[0140] in:

[0141] (17a)

[0142] (17b)

[0143] Applying the fundamental theorem of calculus to equation (17), we can obtain...

[0144] (18)

[0145] Substituting equations (18) and (17) into equation (10), and after simplification, we can obtain the proportional boundary perfectly matched layer. Operators:

[0146] (19)

[0147] in:

[0148] (20a)

[0149] (20b)

[0150] (20c)

[0151] Substituting equations (16) and (18) into equation (11), the proportional boundary perfectly matches the layer. The infinitesimal elements are:

[0152] (twenty one)

[0153] Where the matrix The expression is as follows:

[0154]

[0155] Step S3: Based on the Galerkin weighted residual principle, construct the corresponding equivalent integral form for the equilibrium equation and physical equation of the wave problem in the proportional boundary perfectly matched layer, so as to obtain the equivalent integral formula.

[0156] In this embodiment, based on the Galerkin weighted residual principle, the equivalent integral form of the equilibrium equation (1a) for the proportional boundary perfectly matched layer fluctuation problem is expressed as:

[0157] (22a)

[0158] Among them, superscript symbol ( ) indicates that the variable is in the frequency domain; This is a virtual displacement; This is the traction vector.

[0159] The equivalent integral form of the physical equation (1c) is expressed as:

[0160] (22b)

[0161] The boundary implementation in this embodiment is similar to that of a local boundary, which is simple in implementation and avoids complex analytical and semi-analytical solution derivation processes.

[0162] Step S4: The displacement, stress, and auxiliary variables of the stress integral in the constructed equivalent integral are discretized using the hybrid displacement-stress element finite element technique to obtain the hybrid displacement-stress finite element equation of the scaled boundary perfectly matched layer.

[0163] Optionally, step S4 in the scaled boundary perfect matching layer method for infinite domain fluctuation numerical simulation in this example specifically includes:

[0164] Shape functions are used to interpolate the displacement and stress in the constructed equivalent integral to obtain displacement interpolation and stress interpolation expressions;

[0165] Based on the auxiliary variables of displacement interpolation, stress interpolation, and stress integral, the scale boundary perfectly matched layer is discretized by finite element method to obtain the hybrid displacement-stress finite element equation of the scale boundary perfectly matched layer.

[0166] The method in this embodiment can be seamlessly coupled with hexahedral elements, prismatic elements and arbitrary order spectral elements in the inner domain, and supports the use of different order elements in the direction of the extension function and the inner computational domain, so as to achieve a flexible balance between accuracy and efficiency; and improves the adaptability of the boundary and the inner domain.

[0167] Specifically, the spectral element shape function used in equation (6c) is used to perform displacement interpolation on a region element (i.e., a spectral element) in the perfectly matched scale boundary layer. For example... Figure 2 The unit shown, in its radial direction take Discretization of the spectral unit. o Take the face The order spectrum unit is discretized. It can be represented as:

[0168] (twenty three)

[0169] One-dimensional unit (with) (Using the spectral unit as an example) Shape function and displacement vector It can be represented as:

[0170] (24a)

[0171] (24b)

[0172] in, It is a 3x3 identity matrix. The displacement on the circumferential surface can be expressed as:

[0173]

[0174] (25)

[0175] in, The shape function of a two-dimensional spectral unit is expressed as:

[0176] (26)

[0177] To obtain the interpolated expression of the displacement, substituting equations (24) and (25) into (23) yields:

[0178] (27)

[0179] This is a shape function describing displacement. Similarly, the interpolated expression for stress can be obtained:

[0180] (28)

[0181] in: It is a shape function that describes stress.

[0182] (29)

[0183] in, It is a 6th-order identity matrix.

[0184] By incorporating auxiliary variables related to stress integrals, the hybrid displacement-stress element finite element technique is further applied for spatiotemporal discretization.

[0185] Substituting the differential operator in equation (19) and the volume infinitesimal elements in (20a), (20b), and (20c) into (22a) and (22b), we get:

[0186] (30a)

[0187] (30b)

[0188] Transforming the above frequency domain equations to the time domain and simplifying them, we get:

[0189] (31a)

[0190] (31b)

[0191] Among them, superscript symbol ( This represents the unknown field variables that are time-dependent. In the inverse Fourier transform, we use the following equation to represent the stress integral as an auxiliary variable:

[0192] (32)

[0193] Substituting equations (27) and (28) into equations (31a) and (31b), and utilizing the arbitrariness of the dummy variables, we can obtain the hybrid displacement-stress finite element equation as follows:

[0194] (33a)

[0195] (33b)

[0196] in:

[0197] , ,

[0198]

[0199] Among them, superscript express The matrix in the image, with each coefficient matrix represented as:

[0200] (34a)

[0201] (34b)

[0202] (34c)

[0203] (34d)

[0204] (34e)

[0205] (34f)

[0206] (34g)

[0207] (34h)

[0208] (34i)

[0209] (34g)

[0210] (34k)

[0211] (34l)

[0212] (34m)

[0213] (34n)

[0214] The coefficient matrices above were calculated using the Gauss-Lebatto-Legendé integral rule.

[0215] Perfectly matched layer of proportional boundary in equation (33) Unknown quantities in Can be written as:

[0216] (35)

[0217] The integral over time can be written as:

[0218] (36)

[0219] The traction force vector can be defined as:

[0220] with (37)

[0221] in, This indicates finite element assembly.

[0222] To verify the effectiveness of the proportional boundary perfectly matched layer method for numerical simulation of infinite-domain wave propagation proposed in this embodiment in simulating elastic wave propagation in an infinite domain of complex sites, an example is given below: a complex layered site wave problem with valley topography under point source load. Figure 3 As shown, a horizontal physical interface exists, and the four material parameters are listed in Table 2. To induce external wavelets, a vertically downward point source load is applied at point S within the dashed box. Taking the Ricker wavelet as an example, its time history can be written as:

[0223] (38)

[0224] In the formula and These represent the amplitude of the traction force and the moment when the traction force reaches its peak, respectively. In this example, the parameters are selected as follows: s and Pa, from which we can obtain the maximum frequency as 4.5 Hz and the minimum wavelength as... .

[0225] Table 2 Soil material parameters

[0226]

[0227] This embodiment constructs a finite domain-scale boundary perfectly matched layer truncation model of a layered site with valley topography for wave analysis. The spectral unit is used as a specific example to illustrate the entire construction process. The site area is as follows: Figure 3 As shown, the finite domain has a size of 9.05 km × 8.70 km and is discretized using second-order spectral elements with a 100 m mesh. The SBPML has a thickness of 300 m and is also discretized using second-order spectral elements with a size of 100 m in both the radial and circumferential directions. This ensures seamless coupling between the scaled boundary perfectly matched layer (SBPML domain) and the inner domain spectral elements. The nodal displacements around the perimeter and bottom of the SBPML model are constrained. (See figure...) Figure 3 The area shown in the red dashed box is used to perform site response calculations by applying a vertically downward point source load, and the results are then analyzed.

[0228] Figure 4 shows snapshots of the vertical displacement wavefield at three specific moments for the viscous-spring artificial boudary condition (VSABC) and SBPML (Scaled boundary perfectly matched layer) models in the example. The VSABC model applies viscoelastic boundary conditions to the perimeter and bottom of the finite domain mesh model in Figure 4 to simulate the radiation conditions at infinity of the site. As shown in Figure 4(a), the SBPML and VSABC models...t The same initial wavefield was observed at 2.71 s. Subsequently, the VSABC model... t Significant reflection occurs at 3.79 s, as shown in the cluttered reflected wave within the red box in Figure 4(b). As shown in Figure 4(c), at... t At 8.00 s, the outgoing wave in the SBPML model decays rapidly and almost disappears completely, while the VSABC model shows a continuous round-trip reflected wave.

[0229] Figure 5 shows the displacement-time history comparison curves of the SBPML model and the VSABC model at sampling point A in the second example. As shown in Figures 5(a)-(c), the two methods agree well in the first 3.0 seconds, but then a deviation appears, indicating that SBPML can effectively absorb outgoing waves from the inner domain in infinite-domain complex terrain sites. The results demonstrate that the scaled boundary perfectly matched layer method coupled with spectral units has excellent outgoing wave absorption capability in infinite-domain wave analysis with complex terrain and geology.

[0230] The proportional boundary perfectly matched layer method for numerical simulation of infinite domain fluctuations in this invention is established by combining two typical high-precision artificial boundary methods: proportional boundary finite element method and perfectly matched layer method. It can be used for numerical simulation of infinite domain fluctuations and has the following effects:

[0231] 1) It adopts a hybrid displacement-stress non-split field form, which can be seamlessly coupled with the inner domain hexahedral element, polyhedral element and arbitrary order spectral element, making it more robust.

[0232] 2) It can use artificial boundaries of general geometry to simulate physical interfaces that extend in parallel and radially to infinity in infinite external subdomains, thus providing greater flexibility.

[0233] 3) Its performance has been verified by the field wave propagation analysis of complex layered half-space with valley topography, which shows that the proposed spectral unit SBPML has excellent absorption characteristics and is more stable and accurate.

[0234] Therefore, the proportional boundary perfect matching layer method for infinite domain wave numerical simulation of the present invention has the advantages of simple boundary implementation, good adaptability between the boundary and the inner domain, and high flexibility in boundary shape compared with traditional methods.

[0235] Example 2:

[0236] This invention provides a scaled boundary perfect matching layer system for numerical simulation of infinite domain fluctuations. See [link to relevant documentation]. Figure 6 As shown, the system 100 includes:

[0237] The coordinate transformation module 10 is configured to map an irregular geometric infinite domain in the Cartesian coordinate system to a regular geometric domain in the proportional boundary local coordinate system based on the proportional boundary coordinate transformation. The proportional boundary local coordinate system consists of radial coordinates and circumferential surface coordinates.

[0238] The coordinate stretching module 20 is configured to use a complex coordinate stretching function to analytically extend the radial coordinates of a regular geometric domain to a complex space, thereby obtaining complex radial coordinates and an absorption layer domain with fluctuating exponential decay characteristics. The absorption layer domain is then truncated according to a preset thickness to obtain a proportional boundary perfectly matched layer.

[0239] Module 30 is configured to construct the corresponding equivalent integral form of the equilibrium equation and physical equation for the wave problem within the scale boundary perfectly matched layer, based on the Galerkin weighted residual principle, so as to obtain the equivalent integral expression.

[0240] The finite element discretization module 40 is configured to use hybrid displacement-stress element finite element technology to discretize the displacement, stress, and auxiliary variables of the stress integral in the constructed equivalent integral formula, so as to obtain the hybrid displacement-stress finite element equation of the scaled boundary perfectly matched layer.

[0241] Optionally, the coordinate transformation module 10 in the scaled boundary perfect matching layer system of the infinite domain fluctuation numerical simulation of this embodiment specifically includes:

[0242] The partitioning unit is configured to divide an irregular geometric infinite field into a finite interior subfield and an infinite exterior subfield;

[0243] The transformation unit is configured to perform coordinate transformation on points on the boundary surface of an infinite outer subdomain in Cartesian coordinates using proportional boundary coordinate transformation, so as to obtain a regular geometric domain in a proportional boundary local coordinate system.

[0244] Optionally, the expression for the scaled boundary coordinate transformation in the scaled boundary perfect-matching layer system of the infinite-domain wave numerical simulation in this embodiment is:

[0245]

[0246]

[0247]

[0248] in, Denotes the artificial boundary surface of an infinite outer subfield. Represents a scaled splice surface of an infinite outer subdomain. Representing artificial boundaries The coordinates of any point on the top, ( ) indicates scaling the splicing surface The coordinates of the corresponding point Represents the radial coordinates of a regular geometric domain. Represents the coordinates of any point on a regular geometric domain.

[0249] Optionally, the coordinate transformation module 10 in the scaled boundary perfect matching layer system of the infinite domain fluctuation numerical simulation of this embodiment further includes:

[0250] The region discrete element is configured to discretize the artificial boundary surface and the scaled splicing surface of the infinite outer subdomain into an equal number of region elements using the standard finite element method, and to interpolate the coordinates within each region element based on shape functions.

[0251] Optionally, in the scaled boundary perfect matching layer system of the infinite domain fluctuation numerical simulation of this embodiment, the corresponding shape function is selected according to the shape of the finite internal subdomain;

[0252] Among them, shape functions include two-dimensional shape functions, bilinear quadrilateral unit shape functions, Shape functions of polygonal units and shape functions of spectral units.

[0253] Optionally, in the scaled boundary perfect matching layer system of the infinite domain fluctuation numerical simulation of this embodiment, the coordinate stretching module 20 is specifically configured to analytically extend the radial coordinates of the regular geometric domain to the complex space using a complex coordinate stretching function, and the process of obtaining the complex radial coordinates is specifically represented as follows:

[0254]

[0255]

[0256] in, Represents the radial coordinates of a regular geometric domain. Represents a mapping. Represents complex radial coordinates. The variable representing integration is from 0 to... At any position on the integration path, Represents the complex coordinate stretching function. Represents angular frequency. This represents the scaling function. Let i represent the decay function, and let i represent the imaginary unit.

[0257] Optionally, the finite element discretization module 40 in the scaled boundary perfect matching layer system for infinite domain wave numerical simulation in this embodiment specifically includes:

[0258] The interpolation unit is configured to interpolate the displacement and stress in the constructed equivalent integral using shape functions to obtain displacement interpolation expressions and stress interpolation expressions.

[0259] Discrete elements are configured to perform finite element discretization on the scaled boundary perfectly matched layer based on auxiliary variables of displacement interpolation expression, stress interpolation expression, and stress integral, so as to obtain the hybrid displacement-stress finite element equation of the scaled boundary perfectly matched layer.

[0260] Example 3:

[0261] This invention discloses an electronic device. The electronic device includes a memory and a processor. The memory stores a computer program, and when the processor executes the computer program, it implements the steps in the proportional boundary perfect matching layer method for numerical simulation of infinite domain fluctuations according to any one of the embodiments disclosed in this invention.

[0262] Figure 7 This is a structural diagram of an electronic device according to an embodiment of the present invention, such as... Figure 7 As shown, the electronic device includes a processor, memory, communication interface, display screen, and input device connected via a system bus. The processor provides computing and control capabilities. The memory includes non-volatile storage media and internal memory. The non-volatile storage media stores the operating system and computer programs. The internal memory provides an environment for the operation of the operating system and computer programs stored in the non-volatile storage media. The communication interface is used for wired or wireless communication with external terminals; wireless communication can be achieved through Wi-Fi, carrier networks, Near Field Communication (NFC), or other technologies. The display screen can be an LCD screen or an e-ink screen. The input device can be a touch layer covering the display screen, buttons, a trackball, or a touchpad mounted on the device's casing, or an external keyboard, touchpad, or mouse.

[0263] Those skilled in the art will understand that Figure 7 The structure shown is merely a structural diagram of the part related to the technical solution of this disclosure and does not constitute a limitation on the electronic device to which the solution of this application is applied. The specific electronic device may include more or fewer components than shown in the figure, or combine certain components, or have different component arrangements.

[0264] Example 4:

[0265] This invention discloses a computer-readable storage medium. The computer-readable storage medium stores a computer program, which, when executed by a processor, implements the steps of a proportional boundary perfect matching layer method for numerical simulation of infinite domain fluctuations according to any one of the embodiments of this invention.

[0266] Please note that the technical features of the above embodiments can be combined arbitrarily. For the sake of brevity, not all possible combinations of the technical features in the above embodiments have been described. However, as long as the combination of these technical features does not contradict each other, it should be considered within the scope of this specification. The above embodiments only illustrate several implementation methods of this application, and their descriptions are relatively specific and detailed, but they should not be construed as limiting the scope of the invention patent. It should be pointed out that for those skilled in the art, several modifications and improvements can be made without departing from the concept of this application, and these all fall within the protection scope of this application. Therefore, the protection scope of this patent application should be determined by the appended claims.

[0267] 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 within the scope of protection of the present invention.

[0268] While specific embodiments of the invention have been described in detail by way of examples, those skilled in the art should understand that the examples are for illustrative purposes only and not intended to limit the scope of the invention. Those skilled in the art should understand that modifications can be made to the above embodiments without departing from the scope and spirit of the invention. The scope of the invention is defined by the appended claims.

Claims

1. A method for proportionally bounded perfect-matching layers in numerical simulation of infinite-domain fluctuations, characterized in that, The method includes: Step S1: Based on the proportional boundary coordinate transformation, the irregular geometric infinite domain in the Cartesian coordinate system is mapped to the regular geometric domain in the proportional boundary local coordinate system, wherein the proportional boundary local coordinate system is composed of radial coordinates and circumferential surface coordinates; Step S2: The radial coordinates of the regular geometric domain are analytically extended to the complex space using the complex coordinate stretching function to obtain the complex radial coordinates and the absorption layer domain with the fluctuation exponential decay characteristic. The absorption layer domain is then truncated according to the preset thickness to obtain a proportional boundary perfectly matched layer. Step S3: Based on the Galerkin weighted residual principle, construct the corresponding equivalent integral form for the equilibrium equation and physical equation of the wave problem in the proportional boundary perfectly matched layer, so as to obtain the equivalent integral formula. Step S4: The displacement, stress, and auxiliary variables of the stress integral in the constructed equivalent integral formula are discretized using the hybrid displacement-stress element finite element technique to obtain the hybrid displacement-stress finite element equation of the scaled boundary perfectly matched layer. Specifically, in step S2, the process of analytically extending the radial coordinates of the regular geometric domain to the complex space using a complex coordinate stretching function to obtain the complex radial coordinates is as follows: in, Represents the radial coordinates of a regular geometric domain. Represents a mapping. Represents complex radial coordinates. The variable representing integration is from 0 to... At any position on the integration path, Represents the complex coordinate stretching function. Represents angular frequency. This represents the scaling function. Let i represent the decay function, and let i represent the imaginary unit.

2. The method for proportionally bounded perfect-matching layers in numerical simulation of infinite-domain fluctuations according to claim 1, characterized in that, Step S1 specifically includes: Step S11: Divide the irregular geometric infinite field into a finite interior subfield and an infinite outer subfield; Step S12: Use the proportional boundary coordinate transformation to perform coordinate transformation on the points on the boundary surface of the infinite outer subdomain in the Cartesian coordinate system to obtain the regular geometric domain in the proportional boundary local coordinate system.

3. The method for proportionally bounded perfect-matching layers in numerical simulation of infinite-domain fluctuations according to claim 2, characterized in that, The expression for the proportional boundary coordinate transformation is: in, Represents the coordinates of any point on the artificial boundary, ( () indicates the coordinates of the corresponding point in the scaled splicing face. Represents the radial coordinates of a regular geometric domain. Represents the coordinates of any point on a regular geometric domain.

4. The method for proportionally bounded perfect-matching layers in numerical simulation of infinite-domain fluctuations according to claim 3, characterized in that, Following step S12, the following is also included: Step S13: The artificial boundary surface and the scaled splicing surface of the infinite outer subdomain are discretized into an equal number of regional elements using the standard finite element method, and the coordinates within each regional element are interpolated based on the shape function.

5. The method for proportionally bounded perfect-matching layers in numerical simulation of infinite-domain fluctuations according to claim 4, characterized in that, In step S13, the corresponding shape function is selected according to the shape of the finite internal subdomain; The shape functions include two-dimensional shape functions, bilinear quadrilateral unit shape functions, and... Shape functions of polygonal units and shape functions of spectral units.

6. The method for proportionally bounded perfect-matching layers in numerical simulation of infinite-domain fluctuations according to claim 4, characterized in that, Step S4 specifically includes: The displacement and stress in the constructed equivalent integral are interpolated using the shape function to obtain the displacement interpolation expression and the stress interpolation expression. Based on the displacement interpolation expression, the stress interpolation expression, and the auxiliary variable of the stress integral, the scale boundary perfectly matched layer is discretized by finite element method to obtain the hybrid displacement-stress finite element equation of the scale boundary perfectly matched layer.

7. A scaled boundary perfect-matching layer system for numerical simulation of infinite-domain fluctuations, characterized in that, The system includes: The coordinate transformation module is configured to map an irregular geometric infinite domain in the Cartesian coordinate system to a regular geometric domain in the proportional boundary local coordinate system based on the proportional boundary coordinate transformation, wherein the proportional boundary local coordinate system is composed of radial coordinates and circumferential surface coordinates. The coordinate stretching module is configured to use a complex coordinate stretching function to analytically extend the radial coordinates of the regular geometric domain to the complex space to obtain complex radial coordinates and an absorption layer domain with fluctuating exponential decay characteristics, and to truncate the absorption layer domain according to a preset thickness to obtain a proportional boundary perfectly matched layer. The module is configured to construct the corresponding equivalent integral form of the equilibrium equation and physical equation for the wave problem in the proportional boundary perfectly matched layer, based on the Galerkin weighted residual principle, so as to obtain the equivalent integral expression. The finite element discretization module is configured to use hybrid displacement-stress element finite element technology to discretize the displacement, stress, and auxiliary variables of the stress integral in the constructed equivalent integral formula, so as to obtain the hybrid displacement-stress finite element equation of the scaled boundary perfectly matched layer. Specifically, the coordinate stretching module is configured to analytically extend the radial coordinates of the regular geometric domain to the complex space using a complex coordinate stretching function, and the process of obtaining the complex radial coordinates is specifically represented as follows: in, Represents the radial coordinates of a regular geometric domain. Represents a mapping. Represents complex radial coordinates. The variable representing integration is from 0 to... At any position on the integration path, Represents the complex coordinate stretching function. Represents angular frequency. This represents the scaling function. Let i represent the decay function, and let i represent the imaginary unit.

8. An electronic device, characterized in that, The electronic device includes a memory and a processor. The memory stores a computer program, and when the processor executes the computer program, it implements the steps in the proportional boundary perfect matching layer method for infinite domain fluctuation numerical simulation as described in any one of claims 1 to 6. [The variable] represents the integral variable.

9. A computer-readable storage medium, characterized in that, The computer-readable storage medium stores a computer program, which, when executed by a processor, implements the steps of the proportional boundary perfect matching layer method for numerical simulation of infinite domain fluctuations according to any one of claims 1 to 6.

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