A complex lining structure response determination method of complex domain mapping-laplace inversion

Abstract

The application provides a complex lining structure response determination method of complex domain mapping-Laplace inversion, comprising the following steps: S1, establishing a dynamic control equation, and establishing control equations of saturated soil and lining; S2, expressing a displacement field; S3, Laplace transformation and wave function expansion; S4, constructing and solving a lining structure control equation; S5, setting a boundary continuous condition; S6, processing a complex boundary through a conformal transformation; S7, substituting a boundary condition to solve; S8, truncation and precision control; and S9, numerical Laplace inverse transformation. The application uses complex function-conformal mapping technology to uniformly map an arbitrary smooth or segmented smooth lining boundary to a unit circular ring for solving, breaks through the limitation that an existing method can only process a circular shape or a few regular sections, and can directly serve the analysis of complex lining structures such as an elliptical shape, a horseshoe shape, a polygonal shape or even a free curve.

Description

Technical Field

[0001] This invention belongs to the field of soil dynamics, specifically relating to a method for determining the response of complex lining structures using complex domain mapping-Laplace inversion. Background Technology

[0002] Currently, the dynamic response analysis of underground lining structures (such as tunnels and pipelines) mainly relies on the finite element method (FEM) and traditional analytical methods. The FEM, through discretization modeling, can handle complex geometries and material nonlinearities, and partially considers soil-water coupling effects based on Biot theory. However, its computational process is extremely cumbersome, and solving the 3D model is time-consuming, making it difficult to meet the needs of rapid evaluation under multiple scheme comparisons or sudden loads in engineering projects. Traditional analytical methods are mainly limited to regular cross-sections such as circles or ellipses, using complex variable functions or wave function expansions to directly solve in the Laplace domain or frequency domain, resulting in high computational efficiency. However, its core drawback is its inability to effectively handle boundaries of arbitrarily complex shapes (such as horseshoes or polygons), and it often simplifies the physical model (e.g., ignoring pore water pressure or assuming completely permeable / impermeable interfaces), leading to insufficient accuracy in evaluating the risk of pore pressure exceeding limits and the actual coupling effects of linings in saturated soft clay or sand. Therefore, existing technologies face an irreconcilable contradiction between computational efficiency, geometric adaptability, and physical field integrity, lacking an effective method for quickly and accurately solving the response of lining structures with arbitrarily complex shapes. Summary of the Invention

[0003] The main objective of this invention is to provide a method for determining the response of complex lining structures using complex domain mapping-Laplace inversion, addressing the problems mentioned above.

[0004] Therefore, the above-mentioned objective of the present invention is achieved through the following technical solution:

[0005] A method for determining the response of complex lining structures using complex domain mapping-Laplace inversion includes the following steps:

[0006] S1. Establish the dynamic control equations and the control equations for saturated soil and lining;

[0007] S2 represents the displacement field;

[0008] S3, Laplace transform, and wave function expansion;

[0009] S4. Construction and solution of the governing equations for the lining structure;

[0010] S5. Set boundary continuity conditions;

[0011] S6. Conformal mapping for complex boundaries;

[0012] S7. Substitute the boundary conditions to solve;

[0013] S8, Truncation and Precision Control;

[0014] S9, Numerical Laplace Inverse Transform.

[0015] While adopting the above technical solutions, the present invention may also adopt or combine the following technical solutions:

[0016] As a preferred embodiment of the present invention, in step S1, the formula is as follows:

[0017] Stress-strain relationship equation of soil skeleton:

[0018] Seepage continuity equation:

[0019] In the formula, ε ij σ and e represent the strain and volumetric strain of the soil skeleton, respectively; ij σ is the total stress of the soil. f δ represents pore pressure; A and N are the Lamé constants of the soil skeleton; δ ij α is the Kronecker notation; α and M are Biot parameters.

[0020] As a preferred technical solution of the present invention: in step S2, the formula is as follows:

[0021] Introducing potential functions of two soil skeletons ψ1 and two fluid potential functions If ψ2, then the soil skeleton displacement u i and the displacement ω of the fluid relative to the soil skeleton i It can be represented as:

[0022] Stress-strain relationship equation of soil skeleton:

[0023] Soil skeleton displacement representation:

[0024] Relative displacement of pore fluid:

[0025] In the formula, e ijk It is the permutation tensor in Cartesian coordinates.

[0026] As a preferred technical solution of the present invention: In step S3, the dynamic control equations are transformed into modified Helmholtz equations using the Laplace transform, and the formula is as follows according to the method of separation of variables:

[0027]

[0028]

[0029]

[0030] In the formula, K is the modified Bessel function of the second kind, r is the polar radius, θ is the polar angle, and a n b n c n d n e n These are undetermined coefficients, and p is the Laplace transform parameter.

[0031] As a preferred technical solution of the present invention: In step S4, the formula is as follows:

[0032] The governing equations for the lining structure:

[0033] Introducing potential functions of two lining structures After performing the Lapace transformation on ψ, the solution in polar coordinates is:

[0034]

[0035]

[0036] Based on the geometric equations, constitutive relations, and equilibrium equations of the lining structure, the rectangular coordinate representation is obtained.

[0037]

[0038]

[0039]

[0040]

[0041] In the formula, A1 and N1 are the Lamé constants of the lining structure.

[0042] As a preferred technical solution of the present invention: In step S5, the expansion coefficient is determined by the boundary continuity condition, taking into account the impermeability condition of the interface between the lining and the saturated soil, and the formula is as follows:

[0043] The displacement continuity condition is expressed as: ,

[0044] The stress continuity condition is expressed as: ,

[0045] On the inner boundary of the lining structure, there are

[0046] ,

[0047] In the formula, subscripts I and II represent the variables of the lining structure and saturated soil, respectively, σ represents stress, i represents the imaginary unit, P represents the amplitude of the applied load, and p represents the Laplace transformation variable.

[0048] As a preferred technical solution of the present invention: In step S6, the arbitrary shape lining is mapped onto the unit circle in complex space, and the mapping function is:

[0049] This transformation maps an arbitrary-shaped lining structure on the z-plane to a unit annulus on the mapping plane ω, and maps the outer region of the arbitrary-shaped lining structure to the outside of a unit annulus on the w-plane. Clearly, there exists ζ = eiθ on the outer circumference of the unit annulus in the ω-plane. The mapped solution rotates the coordinates by an angle γ.

[0050]

[0051]

[0052]

[0053]

[0054]

[0055]

[0056]

[0057]

[0058]

[0059]

[0060] In the formula, z represents the complex variable of the physical plane, defined as z = x + iy, where x and y are the rectangular coordinate components in the physical plane, and i is the imaginary unit, satisfying i 2 =-1; ζ represents the complex variable of the mapping plane, used to construct the conformal mapping relationship, ω(ζ) is the conformal mapping function, used to map the region outside the unit circle on the ζ plane to the external region of an arbitrary initial structure on the z plane; (ζ) represents the first derivative of the mapping function with respect to ζ, used to describe the local scale transformation relationship in the mapping process; (ζ) and ψ(ζ) are complex potential functions used to characterize the stress and displacement field distribution in plane elastic problems; p is a Laplace transform variable; N1, A, and M are constant coefficients determined by material parameters and boundary conditions; α is a coefficient parameter related to the elastic constant of the material; β is a material characteristic parameter used to characterize the anisotropic properties of the material, and its value is determined by the material constitutive relation.

[0061] As a preferred technical solution of the present invention: In step S7, the boundary conditions between the saturated soil and the lining structure and the continuity conditions on the outer boundary of the lining structure are substituted into the mapped displacement stress, and the formula is as follows:

[0062]

[0063]

[0064] Under impermeable conditions at the interface between saturated soil and the lining structure, the normal displacement of the fluid relative to the soil skeleton is zero, and the displacement changes with time:

[0065]

[0066] Under permeable conditions at the interface between saturated soil and lining structure, with zero pore water pressure at the interface, the time evolution of stress and displacement in the foundation is as follows:

[0067]

[0068] Using the boundary stress conditions within the lining structure, the influence of the structure on soil pore pressure under uniformly distributed dynamic loading is as follows:

[0069]

[0070]

[0071] In the formula, E represents the elastic modulus and P represents the load strength.

[0072] As a preferred technical solution of the present invention: In step S9, the numerical inverse Laplace transform is performed by using the Durbin method to perform a numerical inverse transform on the result in the Laplace domain to obtain the time domain response.

[0073]

[0074] In the formula, i is the imaginary unit, a is the convergence factor and a> the real part of all singularities, T represents the time period, and NSUM is generally taken as 20-50.

[0075] Compared with the prior art, the present invention has the following beneficial effects:

[0076] 1) This invention utilizes complex variable function-conformal mapping technology to uniformly map any smooth or piecewise smooth lining boundary to a unit ring for solution, breaking through the limitation of existing methods that can only handle circular or a few regular cross sections, and can directly serve the analysis of complex lining structures such as elliptical, horseshoe, polygonal and even free curves.

[0077] 2) Based on Biot's saturated soil wave theory, the coupling effect of the soil skeleton, pore water and lining is fully considered. It not only gives the dynamic stress concentration factor of the lining, but also outputs the pore water pressure and the relative displacement of the skeleton and fluid at the same time. It solves the problem that traditional elastic or single-phase soil models cannot evaluate the risk of pore pressure exceeding the limit. The calculation results are more in line with the actual situation of soft clay, saturated sand and other engineering conditions.

[0078] 3) By using the chain process of "wave function expansion - Laplace domain solution - Durbin numerical inverse transform", the high-dimensional loading-unloading loop of time-domain finite element - iterative solution is avoided; the overall computational cost can be reduced by 1-2 orders of magnitude compared with the finite element calculation method, which is suitable for scheme comparison and rapid parameter scanning.

[0079] 4) Since high-confidence design parameters can be obtained without expensive large-scale three-dimensional explicit calculations and repeated on-site physical tests, it can save a lot of design and monitoring costs for subway, tunnel and other projects, while significantly improving the structural safety reserve and disaster prevention level under sudden loads. Attached Figure Description

[0080] Figure 1 This is a model diagram of an anisotropic tunnel lining structure in soil.

[0081] Figure 2 The horseshoe-shaped tunnel lining structure is transformed into a complex planar unit circle using the transformation method of this invention. Detailed Implementation

[0082] The present invention will now be described in further detail with reference to the accompanying drawings and specific embodiments.

[0083] like Figure 1-2 As shown, a method for determining the response of complex lining structures using complex domain mapping-Laplace inversion specifically includes the following steps:

[0084] Taking the horseshoe-shaped lining structure as an example, its long semi-axis is 1400 mm, its short semi-axis is 1800 mm, its lining thickness is h=230 mm, and the internal sudden uniformly distributed load is P=100 kPa.

[0085] Model parameters and geometric description:

[0086] Density ρ of lining material L =200 kg / m³, shear modulus G L =2.0×10 8 Pa, Poisson's ratio μ L =0.3; density ρ of the soil skeleton s =2500 kg / m³, shear modulus G=2.0×10 7 Pa, Poisson's ratio μ=0.25, porosity n=0.3; pore fluid density ρ f=1000 kg / m³, dynamic viscosity coefficient η=1.0×10 -3 Pa·s, dynamic permeability coefficient k d =5×10 -6 m / s, Biot parameter α = 0.999, M = 2.0 × 10 9 Pa.

[0087] S1. Establish the dynamic control equations. The control equations for saturated soil and lining are established as follows:

[0088] Stress-strain relationship equation of soil skeleton:

[0089] Seepage continuity equation:

[0090] In the formula, ε ij σ and e represent the strain and volumetric strain of the soil skeleton, respectively; ij σ is the total stress of the soil. f δ represents pore pressure; A and N are the Lamé constants of the soil skeleton; δ ij α is the Kronecker notation; α and M are Biot parameters.

[0091] S2, representing the displacement field, is shown below:

[0092] Stress-strain relationship equation of soil skeleton:

[0093] ① Soil skeleton displacement representation:

[0094] ② Relative displacement of pore fluid:

[0095] In the formula, e ijk It is the permutation tensor in Cartesian coordinates.

[0096] S3. Laplace Transform and Wave Function Expansion: Using the Laplace transform, the dynamic control equations are transformed into modified Helmholtz equations. According to the method of separation of variables, the general solution can be obtained as follows:

[0097]

[0098]

[0099]

[0100] In the formula, K is the modified Bessel function of the second kind, r is the polar radius, θ is the polar angle, and a n b n c n d n en are undetermined coefficients, and p is the Laplace transform parameter.

[0101] S4. Governing equations and solution process for lining structure

[0102] The governing equations for the lining structure can be expressed as:

[0103] Introducing potential functions of two lining structures After performing a Lapace transformation on ψ, the solution in polar coordinates is:

[0104]

[0105]

[0106] Based on the geometric equations, constitutive relations, and equilibrium equations of the lining structure, a rectangular coordinate representation can be obtained.

[0107]

[0108]

[0109]

[0110]

[0111] In the formula, A1 and N1 are the Lamé constants of the lining structure.

[0112] S5. Determine the development coefficient for boundary continuity conditions, taking into account the impermeability of the interface between the lining and the saturated soil, and determine the boundary conditions.

[0113] The displacement continuity condition is expressed as: ,

[0114] The stress continuity condition is expressed as: ,

[0115] On the inner boundary of the lining structure, there are

[0116] ,

[0117] In the formula, subscripts I and II represent the variables of the lining structure and saturated soil, respectively, σ represents stress, i represents the imaginary unit, P represents the amplitude of the applied load, and p represents the Laplace transformation variable.

[0118] S6. Conformal transformation for complex boundaries maps arbitrarily shaped linings onto the unit circle in complex space. The mapping function is:

[0119] This transformation maps an arbitrary-shaped lining structure on the z-plane to a unit annulus on the mapping plane ω, and maps the outer region of the arbitrary-shaped lining structure to the outside of a unit annulus on the w-plane. Clearly, on the outer circumference of the unit annulus in the ω-plane, ζ = e iθ The mapped solution will have its coordinates rotated by an angle γ:

[0120]

[0121]

[0122]

[0123]

[0124]

[0125]

[0126]

[0127]

[0128]

[0129]

[0130] In the formula, z represents the complex variable of the physical plane, defined as z = x + iy, where x and y are the rectangular coordinate components in the physical plane, and i is the imaginary unit, satisfying i 2 =-1; ζ represents the complex variable of the mapping plane, used to construct the conformal mapping relationship, ω(ζ) is the conformal mapping function, used to map the region outside the unit circle on the ζ plane to the external region of an arbitrary initial structure on the z plane; (ζ) represents the first derivative of the mapping function with respect to ζ, used to describe the local scale transformation relationship in the mapping process; (ζ) and ψ(ζ) are complex potential functions used to characterize the stress and displacement field distribution in plane elastic problems; p is a Laplace transform variable; N1, A, and M are constant coefficients determined by material parameters and boundary conditions; α is a coefficient parameter related to the elastic constant of the material; β is a material characteristic parameter used to characterize the anisotropic properties of the material, and its value is determined by the material constitutive relation.

[0131] S7. Substituting the boundary conditions of the saturated soil and the lining structure, and the continuity condition on the outer boundary of the lining structure, into the mapped displacement-stress expression, we get:

[0132]

[0133]

[0134] Under impermeable conditions at the interface between saturated soil and the lining structure, the normal displacement of the fluid relative to the soil skeleton is zero, and the displacement changes with time:

[0135]

[0136] Under permeable conditions at the interface between saturated soil and lining structure, with zero pore water pressure at the interface, the time evolution of stress and displacement in the foundation is as follows:

[0137]

[0138] Using the boundary stress conditions within the lining structure, the influence of the structure on soil pore pressure under uniformly distributed dynamic loading is as follows:

[0139]

[0140]

[0141] In the formula, E represents the elastic modulus and P represents the load strength.

[0142] S8. The infinite algebraic equation system composed of S7 satisfies the given precision (precision in this paper [ε] = 10). -5 By appropriately truncating the parameters under the given conditions, the solution coefficients can be determined. The expressions for stress, displacement, and pore pressure are then determined accordingly.

[0143] S9. Numerical Inverse Laplace Transform: The Durbin method is used to perform a numerical inverse transform on the Laplace domain results to obtain the time-domain response.

[0144]

[0145] In the formula, i is the imaginary unit, a is the convergence factor and a> the real part of all singularities, T represents the time period, and NSUM is generally taken as 20-50.

[0146] The above steps yielded the dynamic response of the horseshoe-shaped lining structure under a sudden uniformly distributed load, including precise solutions for the dynamic stress concentration factor, radial displacement, and pore water pressure.

[0147] Taking the horseshoe-shaped tunnel lining section as an example, such as Figure 2 As shown in the left figure, the lining structure in the physical plane z=x+iy is formed by the outer boundary Γ. o With inner boundary Γ iA lining annular region is formed, with point O in the figure as the selected origin, x and y as the physical rectangular coordinate axes; a and b are the characteristic geometric dimensions used to characterize the outer contour scale of the horseshoe-shaped cross-section (used to limit the outer bounding range and scale normalization of the mapped object). To uniformly transform arbitrarily complex boundaries into regular boundaries so as to apply boundary continuity conditions and perform series expansion, the conformal mapping z=ω(ζ) of this invention is used to map the horseshoe-shaped lining annular region shown in the left figure to the complex plane ζ=ρ. eiθd The unit annular region (right figure). The outer boundary Γ of the lining. o After mapping, the corresponding outer circumference of the unit circle is |ζ|=1, therefore its boundary points can be represented by ζ=e. iθ Indicates; inner boundary of lining Γ i After mapping, the inner circumference of the corresponding annulus |ζ|=ρ0, thus the horseshoe-shaped double-boundary problem, which was originally difficult to handle on the z-plane, is equivalently transformed into a double-circular boundary problem with a constant radius on the ζ-plane. The variable θ is the polar angle parameter in the mapping plane, used to parameterize the circular boundary; ω′(ζ) is the derivative of the mapping function with respect to ζ, used to characterize the local scale transformation and correct the stress and displacement expressions after mapping; γ is the coordinate rotation angle, used to correspond the boundary direction on the mapping plane to the tangential / normal direction of the physical plane boundary, thereby ensuring that displacement continuity, stress continuity, and inner boundary loading conditions can be applied in a unified form on the unit annulus boundary and the expansion coefficient can be solved. Through the above mapping relationship of "horseshoe boundary → unit annulus boundary", this invention breaks through the limitation of traditional analytical methods that can only handle circular or a few regular cross-sections, and achieves rapid and accurate response solutions for complex lining structures such as horseshoe shapes.

[0148] The technical solution of the present invention has been described in conjunction with the specific experimental procedures shown in the accompanying drawings. However, the scope of protection of the present invention is not limited to these specific embodiments. Without departing from the principles of the present invention, those skilled in the art can make equivalent changes or substitutions to the relevant technical features, and the technical solutions resulting from such changes or substitutions will all fall within the scope of protection of the present invention.

Claims

1. A method for determining the response of complex lining structures using complex domain mapping-Laplace inversion, characterized in that, Includes the following steps: S1. Establish the dynamic control equations and the control equations for saturated soil and lining; S2 represents the displacement field; S3, Laplace transform, and wave function expansion; S4. Construction and solution of the governing equations for the lining structure; S5. Set boundary continuity conditions; S6. Conformal mapping for complex boundaries; S7. Substitute the boundary conditions to solve; S8, Truncation and Precision Control; S9, Numerical Laplace Inverse Transform.

2. The method according to claim 1, characterized in that: In step S1, the formula is as follows: Stress-strain relationship equation of soil skeleton: Seepage continuity equation: In the formula, ε ij σ and e represent the strain and volumetric strain of the soil skeleton, respectively; ij σ is the total stress of the soil. f δ represents pore pressure; A and N are the Lamé constants of the soil skeleton; δ ij α is the Kronecker notation; α and M are Biot parameters.

3. The method according to claim 1, characterized in that: In step S2, the formula is as follows: Introducing potential functions of two soil skeletons ψ1 and two fluid potential functions If ψ2, then the soil skeleton displacement u i and the displacement ω of the fluid relative to the soil skeleton i It can be represented as: Soil skeleton displacement representation: Relative displacement of pore fluid: In the formula, e ijk It is the permutation tensor in Cartesian coordinates.

4. The method according to claim 1, characterized in that: In step S3, the dynamic control equations are transformed into modified Helmholtz equations using the Laplace transform. According to the method of separation of variables, the formulas are as follows: In the formula, K is the modified Bessel function of the second kind, r is the polar radius, θ is the polar angle, and a n b n c n d n e n are undetermined coefficients, and p is the Laplace transform parameter.

5. The method according to claim 1, characterized in that: In step S4, the formula is as follows: The governing equations for the lining structure: Introducing potential functions of two lining structures After performing the Lapace transformation on ψ, the solution in polar coordinates is: Based on the geometric equations, constitutive relations, and equilibrium equations of the lining structure, the rectangular coordinate representation is obtained. In the formula, A1 and N1 are the Lamé constants of the lining structure.

6. The method according to claim 1, characterized in that: In step S5, the boundary continuity condition determines the expansion coefficient, considering the impermeability of the interface between the lining and the saturated soil. The formula is as follows: The displacement continuity condition is expressed as: , The stress continuity condition is expressed as: , On the inner boundary of the lining structure, there are ， In the formula, subscripts I and II represent the variables of the lining structure and saturated soil, respectively, σ represents stress, i represents the imaginary unit, P represents the amplitude of the applied load, and p represents the Laplace transformation variable.

7. The method according to claim 1, characterized in that: In step S6, the arbitrary-shaped lining is mapped onto the unit circle in complex space, and the mapping function is: This transformation maps an arbitrary-shaped lining structure on the z-plane to a unit annulus on the mapping plane ω, and maps the outer region of the arbitrary-shaped lining structure to the outside of a unit annulus on the w-plane. Clearly, there exists ζ = eiθ on the outer circumference of the unit annulus in the ω-plane. The mapped solution rotates the coordinates by an angle γ. In the formula, z represents the complex variable of the physical plane, defined as z = x + iy, where x and y are the rectangular coordinate components in the physical plane, and i is the imaginary unit, satisfying i 2 =-1; ζ represents the complex variable of the mapping plane, used to construct the conformal mapping relationship, ω(ζ) is the conformal mapping function, used to map the region outside the unit circle on the ζ plane to the external region of an arbitrary initial structure on the z plane; (ζ) represents the first derivative of the mapping function with respect to ζ, used to describe the local scale transformation relationship in the mapping process; (ζ) and ψ(ζ) are complex potential functions used to characterize the stress and displacement field distribution in plane elastic problems; p is a Laplace transform variable; N1, A, and M are constant coefficients determined by material parameters and boundary conditions; α is a coefficient parameter related to the elastic constants of the material. β is a material characteristic parameter used to characterize the anisotropic properties of materials, and its value is determined by the material constitutive relation.

8. The method according to claim 1, characterized in that: In step S7, the boundary conditions between the saturated soil and the lining structure, and the continuity conditions on the outer boundary of the lining structure, are substituted into the mapped displacement stress, as shown in the following formula: Under impermeable conditions at the interface between saturated soil and the lining structure, the normal displacement of the fluid relative to the soil skeleton is zero, and the displacement changes with time: Under permeable conditions at the interface between saturated soil and lining structure, with zero pore water pressure at the interface, the time evolution of stress and displacement in the foundation is as follows: Using the boundary stress conditions within the lining structure, the influence of the structure on soil pore pressure under uniformly distributed dynamic loading is as follows: In the formula, E represents the elastic modulus and P represents the load strength.

9. The method according to claim 1, characterized in that: In step S9, the numerical inverse Laplace transform is performed, and the Durbin method is used to perform a numerical inverse transform on the result in the Laplace domain to obtain the time domain response: In the formula, i is the imaginary unit, a is the convergence factor and a> the real part of all singularities, T represents the time period, and NSUM is generally taken as 20-50.

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