An evaluation method for ground artificial boundary space coupling based on RGA

Abstract

The application discloses an evaluation method for ground artificial boundary space coupling based on RGA, wherein a frequency domain dynamic stiffness matrix formed on an artificial boundary of a soil-structure dynamic interaction system is obtained according to a substructure method; a mean coupling degree evaluation index based on RGA is established by using a linear system theory, and the size of the mean coupling degree is used as an evaluation standard of the coupling degree of the ground artificial boundary space. On the basis of the traditional coupling degree defined based on the RGA array, the classical coupling degree definition is improved, the total number of elements of the RGA array is divided, and the root is taken, so that the shortcoming that the traditional coupling degree does not consider the uncertainty of the number of degrees of freedom of the system is solved, and a quantitative evaluation method is provided, which fills the gap that the traditional method can only be qualitatively analyzed but cannot be quantitatively studied. The evaluation method provides a reference for effectively evaluating the coupling characteristics of the ground artificial boundary and provides certain technical support for the size selection of the ground artificial boundary in engineering.

Description

Technical Field

[0001] This invention belongs to the field of structure-foundation dynamic interaction analysis and calculation technology, specifically involving an evaluation method for the spatial coupling degree of artificial boundary of foundation based on RGA. It can be applied to obtain the frequency domain dynamic stiffness matrix of foundation at the artificial boundary under a given artificial boundary range, forming the mean coupling degree of the dynamic stiffness matrix based on the relative gain matrix (RGA), and the coupling degree value is used to characterize the spatial coupling characteristics of artificial boundary of foundation. Background Technology

[0002] In soil-structure dynamic interaction systems, artificial boundaries reflect the influence of an infinite domain outside the artificial boundary on the near-field structure and surrounding soil. In a natural physical sense, this infinite domain is a three-dimensional space. Currently, two-dimensional artificial boundary conditions are used to simulate this infinite domain, essentially characterizing a three-dimensional space as a boundary condition. This often exhibits spatiotemporal coupling characteristics. Temporal coupling on the artificial boundary reflects a convolutional computation process, indicating that the response at the current moment is correlated with the responses at all historical moments. Spatial coupling is manifested in the narrow-band characteristics of the dynamic stiffness matrix generated across the finite element mesh; it is a full matrix, making all nodes on the boundary correlated. Among these, spatial coupling at the boundary is more restrictive than temporal coupling in the time-domain calculation of large structure-foundation systems.

[0003] In their paper, "Time-domain analysis of wave propagation in 3-D unbounded domains by the scaled boundary finite element method," Chen X, Birk C, and Song C mention that when the artificial boundary is far from the near-field soil of the structure, the outgoing scattered waves exhibit a one-dimensional outgoing decoupling phenomenon, but this is only a qualitative evaluation result. Although the infinite-domain simulation obtained by maximizing the distance of the artificial boundary from the near-field generalized structure is more realistic, the artificial boundary cannot be expanded indefinitely because calculating the dynamic response of the boundary and the degrees of freedom of the nodes within the boundary would result in unacceptable computational complexity and memory requirements. In the frequency domain, taking an infinite domain of a 40m×40m×50m foundation as an example, with an average grid side length of 2m, approximately 10,000 nodes will be generated. The dynamic stiffness matrix formed by the degrees of freedom in its three translational directions has 30,000×30,000 elements. Assuming that each element occupies 8 bytes of double-precision type full memory, the dynamic stiffness matrix alone will occupy approximately 6.7GB of memory. Furthermore, for each element, thousands of different frequency points will also generate matrices of the same size. Even considering symmetry and storing only the lower triangular elements, the memory consumption for the entire system is still enormous.

[0004] Quantitatively evaluating the spatial coupling state of artificial boundaries is a crucial foundation for their spatial decoupling, enabling research into how to reduce the storage and computational costs imposed on the solution process by complex fully coupled states. Previous methods have relied solely on qualitative evaluations such as the sparsity of the dynamic stiffness matrix formed on artificial boundaries or simple diagonal dominance, lacking a direct and intuitive evaluation metric to reflect the degree of spatial coupling, thus posing challenges to studying decoupling methods. Without a quantitative evaluation metric, the objectives and merits of the decoupling process are difficult to ascertain.

[0005] To define a suitable evaluation index for coupling characteristics, when the soil-structure interaction system is treated as a displacement input-force output system, the system's transfer function is dynamic stiffness; when treated as a force input-displacement output system, the system's transfer function is dynamic flexibility (see Zhao Mi and Du Xiuli. Local Higher-Order Spring-Damping-Mass Model of Time Convolution). Artificial boundaries can also be considered as such systems, where the forces and displacements of their degrees of freedom are also multi-input, multi-output variables.

[0006] According to linear system theory, the Relative Gain Matrix (RGA) method has significant advantages in characterizing the degree of coupling in the coupling evaluation of multi-input multi-output variable systems. This invention applies the physical quantity equivalence method to the spatial coupling analysis and evaluation of soil-structure dynamic interaction systems. The spatial coupling evaluation process for artificial boundaries mainly consists of three stages: first, solving for the frequency domain dynamic stiffness matrix formed by the artificial boundary in the soil-structure dynamic interaction system using the substructure method; second, obtaining the RGA matrix of the dynamic stiffness matrix according to linear system theory; and third, calculating the m1 norm of the difference matrix by subtracting the obtained RGA matrix from the identity matrix, dividing it by the total number of elements in the matrix, and finally taking the square root to obtain the defined mean coupling degree. The smaller the mean coupling degree, the lower the spatial coupling degree of the artificial boundary. Summary of the Invention

[0007] This invention is based on the RGA matrix formed by the frequency-domain dynamic stiffness matrix of the soil-structure dynamic interaction system. The spatial coupling characteristics between discrete degrees of freedom on the artificial boundary should ideally be reflected by the dynamic stiffness matrix, but since this cannot be quantitatively and intuitively reflected, the mean coupling degree is constructed as an index for quantitatively evaluating the spatial coupling characteristics on the boundary. Based on various methods for solving artificial boundaries in soil-structure dynamic interaction, the frequency-domain dynamic stiffness matrix of the infinite domain reflected by the artificial boundary is obtained. Combining this with the RGA coupling evaluation method of linear system theory, the RGA matrix corresponding to the frequency-domain dynamic stiffness matrix is ​​calculated. The defined mean coupling degree index is used as a direct medium to reflect the degree of spatial coupling on the artificial boundary.

[0008] The technical solution of the present invention:

[0009] An evaluation method for artificial boundary space coupling on foundations based on RGA, comprising the following steps:

[0010] The first step is to solve the frequency domain dynamic stiffness matrix formed by the artificial boundary in the soil-structure dynamic interaction system using the substructure method.

[0011] First, for a soil-structure interaction system, such as Figure 2 As shown, the system is divided into a near-field finite domain and a far-field infinite domain by artificial boundaries. The dynamic equations of its scattering problem can be expressed in the frequency domain as follows:

[0012]

[0013] In the formula: S represents the dynamic stiffness matrix, S ii It is the dynamic stiffness matrix formed only by the internal structure and soil nodes within the boundary; S ib S bi It is the dynamic stiffness matrix formed by the coupling between the nodes inside the boundary and the nodes on the boundary; S bb This is the dynamic stiffness matrix corresponding only to the nodes on the boundary; U and P are the nodal displacement spectrum and force spectrum vectors, respectively, and the subscripts i and b are the internal nodes and boundary nodes, respectively. These are the dynamic stiffness matrices of an infinite domain and an infinite domain medium of the same volume as the foundation, constructed using finite element methods, U. fb The value of the free field displacement at the boundary.

[0014] The force-displacement relationship of degrees of freedom on an artificial boundary can be expressed in the frequency domain as follows:

[0015]

[0016] Where F b The interaction forces applied to the infinite field by the structure on the boundary. For the dynamic stiffness matrix related to the infinite domain, it is generally a fully coupled full matrix when using artificial boundaries, X b The displacements on the boundary caused by the interaction forces represent the degrees of freedom. It can be seen that the artificial boundary is equivalent to a soil-structure interaction subsystem.

[0017] For the above-mentioned overall soil-structure dynamic interaction system and its artificial boundary subsystem, it can be uniformly written in the following form:

[0018] P(ω)=S(ω)U(ω)(3)

[0019] Where S(ω) is the frequency domain dynamic stiffness matrix, which serves as the transfer function of the soil-structure dynamic interaction system, and U(ω) and P(ω) are the input or output variables of the soil-structure dynamic interaction system.

[0020] The second step is to derive the RGA matrix of the dynamic stiffness matrix using methods from linear system theory.

[0021] For a multi-input multi-output variable system, which contains multiple control variables u i and multiple controlled quantities y i :

[0022] u = [u1 u2…u] j …u n ] T

[0023] y = [y1 y2…y i …y n ] T

[0024] Then the j-th control quantity u j up to the i-th controlled quantity y i The relative gain is defined as:

[0025]

[0026] In the formula p ij ,q ij These are the system's open-loop gain and closed-loop gain, respectively; where the open-loop gain and closed-loop gain can be expressed as... Figure 3 As shown.

[0027] Open-loop gain represents the sum of the gains of the input u and the open-loop gain. j - Output y i Except for channel u, all other channels are in an open-loop state, i.e., u k With all (k≠j) remaining constant, the effect of the input increment of this maintained channel on the change of each of the other outputs.

[0028] Closed-loop gain represents the difference between the input u and the closed-loop gain. j - Output y i When all other loops outside the channel remain closed and the output remains unchanged, the input u j The change produced at point affects the output y i The resulting impact.

[0029] The matrix composed of relative gain elements is called the relative gain matrix (RGA):

[0030]

[0031] When performing direct calculations, it is simpler to use a calculation method equivalent to the aforementioned definition:

[0032] RGA = [S(w)].*([S(w)] -1 ) T (6)

[0033] In the formula, [S(w)] represents the transfer function matrix, and .* represents the element-wise multiplication of the two matrices, called the Hadamard product.

[0034] Overall, selecting the interaction channel corresponding to a relative gain value close to 1 as the dominant channel is beneficial for decoupling. Since the relative gain matrix of the dynamic stiffness matrix calculated on the artificial boundary is close to 1 only in the diagonal elements, its ideal decoupling channel, which is represented by a diagonal dynamic stiffness matrix through the dynamic stiffness matrix, is the identity matrix.

[0035] The third step involves subtracting the obtained RGA matrix from the identity matrix to calculate the m1 norm of the difference matrix, dividing it by the total number of elements in the matrix, and finally taking the square root to obtain the defined mean coupling degree. Numerical methods for solving soil-structure dynamic interactions can study the discrete degrees of freedom coupling at different boundary ranges, artificially setting decoupling thresholds as approximate artificial boundary positions for decoupling. For each new soil-structure dynamic interaction system, the selection of the decoupling artificial boundary position is achieved through iterative iteration. This provides a theoretical basis for reducing computational costs in engineering applications when neglecting off-diagonal elements in the dynamic stiffness matrix and only taking diagonal elements.

[0036] While the RGA matrix above allows for a relatively intuitive observation of ideal matching channels and coupling levels, it cannot quantify the system's coupling degree. The coupling problem has shifted from direct observation of the dynamic stiffness matrix to the RGA matrix, which can reflect the interrelationships between different degrees of freedom in the system. The following analysis will continue from the RGA matrix of the system.

[0037] The decoupling RGA matrix of the target (coupling degree is 0) is the identity matrix Λ0:

[0038]

[0039] Therefore, the residual matrix between a general RGA matrix and a decoupled RGA matrix serves as a mediating measure. Let D = Λ - Λ0, then the D matrix is ​​a residual matrix. The closer the D matrix is ​​to zero, the lower the coupling between the input and output variables of the soil-structure dynamic interaction system. The m1-norm (summation norm) of the D matrix can precisely measure this difference. This norm can be compared when decoupling the same system, but it cannot be compared between D matrices of different orders in different systems. Therefore, the average m1-norm is used for measurement, denoted as the average-m1-norm; that is, the value of the m1-norm divided by the total number of summation terms. Although taking the average reduces the accuracy of the measurement, it is still better than directly observing the coupling degree of the system. Defining the average-m1-norm as the coupling degree between soil-structure dynamic interaction systems has broad significance and can be used to compare the coupling degree between different soil-structure dynamic interaction systems (with different orders of dynamic stiffness matrices).

[0040] The extent of the artificial boundary in a soil-structure dynamic interaction system can vary. The change in the boundary alters the extent of the soil in the original near field. The influence of this changed soil on the fluctuations of the outward scattering field makes the soil-structure interaction system different as the artificial boundary changes. The general m1-norm can be used to measure the system with an invariant artificial boundary extent, but the average m1-norm is used to measure the system with a changing artificial boundary extent.

[0041] For different soil-structure dynamic interaction systems that arise as the artificial boundary expands outward, the mean coupling degree N between different degrees of freedom of each soil-structure dynamic interaction system is given. d It can be defined as:

[0042]

[0043] In the formula: This indicates that the m1-norm of the matrix is ​​calculated, where n is the order of Λ.

[0044] The beneficial effects of this invention are as follows: Based on the traditional definition of coupling degree using the RGA matrix, this paper improves upon the classical definition of coupling degree by dividing by the total number of elements in the RGA matrix and taking the square root. This overcomes the limitation of traditional coupling degree definitions that do not consider the uncertainty of the number of system degrees of freedom. Simultaneously, it provides a quantitative evaluation method, filling the research gap where traditional methods can only perform qualitative analysis and not quantitative analysis. The frequency domain dynamic stiffness matrix formed by the artificial boundary of the soil-structure dynamic interaction system is obtained using the substructure method. A mean coupling degree evaluation index based on RGA is established using linear system theory, with the magnitude of the mean coupling degree serving as the evaluation standard for the spatial coupling degree of the artificial boundary. This complete set of analytical methods provides a reference for effectively evaluating the coupling characteristics of artificial boundaries. Attached Figure Description

[0045] Figure 1 This is a flowchart of an evaluation method for the coupling of artificial boundary space on the foundation based on RGA.

[0046] Figure 2 This is a schematic diagram of an artificial boundary in a soil-structure dynamic interaction system.

[0047] Figure 3 This is a schematic diagram of the open-loop gain and closed-loop gain in RGA.

[0048] Figure 4 This is a schematic diagram of the scattering field conditions in a semi-circular landform.

[0049] Figure 5 It is the coupling attenuation fitting curve on the artificial boundary of SBFEM. Detailed Implementation

[0050] The specific embodiments of the present invention will now be described in detail with reference to the technical solutions and accompanying drawings.

[0051] Artificial boundaries constructed using the scaled boundary finite element method, such as Figure 4 As shown, a semi-circular foundation pit with radius r = 1 is placed under a cover soil layer with a thickness of 2r. The soil layer material parameters are: shear wave velocity c of the cover soil layer. s1 =1.0, Poisson's ratio ν1 = 0.25, density ρ1 = 1.0; elastic half-space shear wave velocity c s2 =4.0, Poisson's ratio ν² = 0.25, density ρ² = 1.0. The area encompassed by the dashed lines abcd represents the near-field generalized substructure, which expands outward continuously. The dashed lines indicate the location of the artificial boundary.

[0052] ab=bc=cd=αr,(α=3.2,4,5.2,...,12,14,16)

[0053] Subject to the dimensionless frequency β=ωr / πc s1P-waves and SV-waves with a value of 0.5 are incident perpendicularly.

[0054] After the model is established, implement it according to the following steps:

[0055] (1) The dynamic stiffness matrix [S(ω)] on the artificial boundary abcd is solved using the improved proportional boundary finite element method. Taking the artificial boundary with a side length of 3.2 as an example, the mesh degrees of freedom of the structure are divided using the improved proportional boundary finite element method, and the artificial boundary forms a total of 98 degrees of freedom.

[0056] (2) Calculate the relative gain matrix (RGA) formed by the coupling between all degrees of freedom under this working condition according to formula (6). Due to space limitations, only the master subarray formed by the first 3 degrees of freedom is listed, as shown in Table 1. It can be seen that the coupling effect between different degrees of freedom is mainly concentrated between the degrees of freedom on the diagonal. When the relative gain values ​​on these diagonals approach 1, the aforementioned decoupling state can be achieved.

[0057] Table 1. Principal subarrays formed by the first 3 degrees of freedom of the calculated RGA

[0058]

[0059] (3) According to formula (7), the mean coupling degree corresponding to the dynamic stiffness matrix on different artificial boundaries is solved and recorded in Table 2. The curve of mean coupling degree with boundary range is plotted. Figure 5 Its function expression is obtained through curve fitting:

[0060]

[0061] Table 2. Coupling Degree N d Variation with the side length x of the artificial boundary

[0062]

[0063]

[0064] Where x represents the side length of the artificial boundary range, and N d The expression (8) represents the mean coupling degree. It can be seen from the expression (8) that the coupling degree between different degrees of freedom on the artificial boundary decreases in the form of a power function as the boundary range expands. The expression can also be used to roughly give the boundary conditions for approximate decoupling. For example, the coupling degree can be taken as a very small limit as the approximate decoupling condition. The specific limit value can be determined according to the actual needs, which has certain guiding significance.

[0065] Other embodiments of the invention will readily occur to those skilled in the art upon consideration of the specification and practice of the invention disclosed herein. This application is intended to cover any variations, uses, or adaptations of the invention that follow the general principles of the invention and include common knowledge or customary techniques in the art not disclosed herein. The specification and examples are to be considered exemplary only, and the true scope and spirit of the invention are indicated by the appended claims.

Claims

1. An evaluation method for artificial boundary space coupling of foundations based on RGA, characterized in that, The steps are as follows: The first step is to solve the frequency domain dynamic stiffness matrix formed by the artificial boundary in the soil-structure dynamic interaction system using the substructure method. First, by dividing a soil-structure interaction system into a near-field finite domain and a far-field infinite domain using artificial boundaries, the dynamic equations for its scattering problem can be expressed in the frequency domain as follows: In the formula: S represents the dynamic stiffness matrix, S ii It is the dynamic stiffness matrix formed only by the internal structure and soil nodes within the boundary; S ib S bi It is the dynamic stiffness matrix formed by the coupling between the nodes inside the boundary and the nodes on the boundary; S bb It is the dynamic stiffness matrix corresponding only to the nodes on the boundary; U and P are the nodal displacement spectrum and force spectrum vectors, respectively, and the subscripts i and b are the internal nodes and boundary nodes, respectively. These are the dynamic stiffness matrices of an infinite domain and an infinite domain medium of the same volume as the foundation, constructed using finite element methods, U. fb The values ​​of the free field displacement at the boundary; The force-displacement relationship of degrees of freedom on an artificial boundary can be expressed in the frequency domain as follows: Among them, F b The interaction forces applied to the structure on the boundary to the infinite field; For the dynamic stiffness matrix of the infinite domain, it is a fully coupled full matrix when the artificial boundary is used; X b The displacements on the boundary degrees of freedom caused by the interaction forces; The artificial boundary is equivalent to a soil-structure interaction subsystem; for the above soil-structure dynamic interaction system and its artificial boundary subsystem, it is uniformly written in the following form: P(ω)=S(ω)U(ω) (3) Where S(ω) is the frequency domain dynamic stiffness matrix, which serves as the transfer function of the soil-structure dynamic interaction system; U(ω) and P(ω) are the input or output variables of the soil-structure dynamic interaction system. The second step is to derive the RGA matrix of the dynamic stiffness matrix using methods from linear system theory. For a multi-input multi-output variable system, which contains multiple control variables u i and multiple controlled quantities y i : u=[u1 u2…u j …u n ] T y=[y1 y2…y i …y n ] T Then the j-th control quantity u j up to the i-th controlled quantity y i The relative gain is defined as: In the formula, p ij ,q ij These are the system's open-loop gain and closed-loop gain, respectively. Open-loop gain represents the sum of the gains of the input u and the open-loop gain. j - Output y i Except for channel u, all other channels are in an open-loop state, i.e., u k When k≠j remain constant, the effect of the input increment of this maintained channel on the change of each other output; the closed-loop gain represents the change of the input u excluding the input u. j - Output y i When all other loops outside the channel remain closed and the output remains unchanged, the input u j The change produced at point affects the output y i The resulting impact; The matrix composed of relative gain elements is called the relative gain matrix (RGA). RGA=[S(w)].*([S(w)] -1 ) T (6) In the formula, [S(w)] represents the transfer function matrix, and .* represents the element-wise multiplication of two matrices, called the Hadamard product; Selecting the interaction channel corresponding to the relative gain value close to 1 as the dominant channel is beneficial for decoupling; since the relative gain matrix of the dynamic stiffness matrix calculated on the artificial boundary is close to 1 only in the value of the diagonal elements, its ideal decoupling channel is a diagonal dynamic stiffness matrix, and the corresponding relative gain matrix RGA is the identity matrix. The third step is to calculate the m1-norm of the difference matrix by subtracting the RGA matrix from the identity matrix, divide it by the total number of elements in the matrix, and finally take the square root to calculate the defined mean coupling degree. The decoupling RGA matrix for a target with zero coupling is the identity matrix Λ0: The residual matrix between a relative gain matrix (RGA) and a decoupled RGA is the mediator; let D = Λ - Λ0, then the D matrix is ​​a residual matrix. The closer the D matrix is ​​to zero, the lower the coupling between the input and output variables of the soil-structure dynamic interaction system; the m1-norm of the D matrix is ​​chosen, which can just measure this difference. When decoupling a system, the m1-norm can be compared. However, the m1-norm values ​​corresponding to D matrices of different orders in different systems cannot be compared. Therefore, the average m1-norm is used for measurement and is denoted as average-m1-norm. That is, the value of m1-norm is divided by the total number of terms in the summation. The extent of the artificial boundary in a soil-structure dynamic interaction system can vary. The change in the boundary alters the extent of the soil in the original near field. The influence of the partially changed soil on the fluctuations of the outward scattering field makes the soil-structure dynamic interaction system different as the artificial boundary changes. The m1-norm can be used to measure the system with an invariant artificial boundary, but the average m1-norm is used to compare the systems with changes in the artificial boundary. For different soil-structure dynamic interaction systems that arise as the artificial boundary expands outward, the mean coupling degree N between different degrees of freedom of each soil-structure dynamic interaction system is given. d Defined as: In the formula: This indicates that the m1-norm of the matrix is ​​calculated, where n is the order of Λ.

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