A method for constructing and solving a variable load geotechnical fluid-structure coupling model

Abstract

The application discloses a construction and solving method of a variable load geotechnical solid-fluid coupling solving model, and belongs to the technical field of cross of numerical calculation of geotechnical engineering and artificial intelligence. The application constructs a solid-fluid coupling multi-physical field operator pre-training data set based on a Fourier base function, constructs and trains three DeepONet neural operators; a main network is established, and the three operators are connected to the back of the main network to obtain a solving model; the main network takes spatial coordinates of a point to be solved as input and takes a vertical effective stress field as output, and a target load is applied through a loss function by stress balance supervision of a top boundary and Fourier superposition internal supervision, without any internal monitoring data. The application can solve the problems of high cost and low efficiency of traditional finite element calculation and the problem that a DeepM&Mnet framework depends on internal monitoring data, can process loads of any position, shape and amplitude, and quickly gives stress prediction results meeting engineering precision requirements.

Description

Technical Field

[0001] This invention belongs to the field of interdisciplinary technology of numerical calculation in geotechnical engineering and artificial intelligence, and specifically relates to the construction and solution method of a variable load geotechnical solid-fluid coupling solution model. Background Technology

[0002] In geotechnical engineering (such as foundation settlement prediction, roadbed consolidation analysis, and tunnel surrounding rock stability assessment), soil deformation and pore water pressure dissipation constitute a typical solid-fluid coupling process, whose physical and mechanical behavior is usually described by complex partial differential equations such as Biot's consolidation theory. Traditional methods rely on finite element numerical solutions to obtain the stress distribution and deformation within the soil, which are computationally expensive and difficult to meet the rapid assessment needs in engineering design.

[0003] In recent years, methods based on Physical Information Neural Networks (PINN) have made significant progress in solving partial differential equations. However, in solid-fluid coupling problems, PINN methods involve complex constitutive relations and higher-order derivative calculations. Automatic Grad is prone to gradient vanishing or exploding problems when dealing with the residuals of the coupling equations, and the training process is unstable.

[0004] Proxy solution methods based on deep operator networks (such as DeepONet) demonstrate the advantage of rapid inference in solving parameterized partial differential equations by learning the mapping relationships between function spaces, providing a feasible way to replace computationally expensive finite element simulations and meet the needs of rapid engineering evaluation. However, traditional neural operator methods are mostly geared towards learning mappings for single physics fields and single equations, lacking an effective mechanism for transferring and constraining coupling information between multiple physics fields. For multiphysics problems, the DeepM&Mnet framework achieves multi-field collaborative solution by combining multiple pre-trained DeepONet operators. However, this framework relies on monitoring data within the computational domain as the supervision source for the main network. In actual engineering, the acquisition cost of stress and displacement monitoring data within the soil is high, and the data points are sparse, making it difficult to meet the data requirements. Furthermore, this framework has not yet been applied to solid-fluid coupling problems in soil with strong coupling between soil deformation fields and pore water pressure fields. To make intelligent fast solution methods truly adaptable to solid-fluid coupling problems and complex and variable load conditions in engineering, and to achieve the required prediction accuracy without the need for internal monitoring data, the following key technical issues still need to be addressed: 1. Multiphysics coupling solution: In the soil-solid-fluid coupling problem, the soil deformation field and the pore water pressure field interact and are strongly coupled. However, traditional neural operators are usually modeled for a single physics field and lack a mechanism to transfer and constrain coupling information between multiple physics fields. It is difficult to guarantee the physical self-consistency between the prediction results of various physics fields such as stress field and displacement field.

[0005] 2. Fixed load location: The load application area in the pre-training data is hard-coded as a specific local interval of the top boundary (e.g., ...). It cannot handle situations where loads are applied in other locations.

[0006] 3. Limited load types: The pre-training data only includes uniformly distributed loads (constant distribution), which cannot handle non-uniformly distributed load types commonly found in engineering, such as triangular distribution, trapezoidal distribution, sine distribution, stepped distribution, and concentrated forces.

[0007] 4. The contradiction between internal constraints and internal data dependence: The DeepM&Mnet framework mentioned above relies on stress and displacement monitoring data within the computational domain as the source of supervision for the main network. However, such data is costly to acquire and sparsely distributed. If internal supervision is removed directly to get rid of this dependence, the main network is only constrained by two loss constraints: load matching at the top boundary and operator loop consistency. It lacks direct physical supervision from within the computational domain, which limits the prediction accuracy of the internal field distribution.

[0008] 5. Lack of data normalization: There are magnitude differences (up to 100 times or more) between different physical fields (stress, displacement). Directly inputting them into the neural network without normalization will make operator training difficult, especially the mapping accuracy of the horizontal displacement field is extremely low.

[0009] Therefore, there is an urgent need for a high-precision intelligent solution method that can handle multi-physics problems in geotechnical engineering, adapt to arbitrary boundary loads, and provide sufficient internal physical constraints. Summary of the Invention

[0010] The purpose of this invention is to provide a method for constructing and solving a variable-load soil-solid-fluid coupling solution model, so as to get rid of the dependence of the original framework on monitoring data inside the computational domain and to balance the solution speed and accuracy.

[0011] To achieve the above objectives, the first aspect of this invention provides a method for constructing a variable-load soil-rock solid-fluid coupling solution model, comprising the following steps: S1. Establish a steady-state Biot solid-fluid coupled finite element model, and establish several Fourier basis functions as load bases. By expanding the amplitude of the Fourier basis functions and / or randomly combining multiple Fourier basis functions, several load cases are obtained. S2. Substitute each load case into the steady-state Biot solid-fluid coupled finite element model for solution to obtain multi-physics coupled data containing vertical effective stress field data, horizontal displacement field data and vertical displacement field data in the computational domain. Combine each load case and its corresponding multi-physics coupled data to obtain a pre-training dataset. S3. Construct three DeepONet neural operators and train them using the pre-training dataset; wherein the three DeepONet neural operators include: a forward operator that maps the vertical effective stress field to the horizontal displacement field, a forward operator that maps the vertical effective stress field to the vertical displacement field, and an inverse operator that jointly maps the horizontal displacement field and the vertical displacement field back to the vertical effective stress field. S4. Establish a main network, and then connect the three DeepONet neural operators trained in step S3 to the main network to obtain the final variable load soil-solid-fluid coupling solution model. The input of the main network is the normalized spatial coordinates of the point to be determined, and the output is the predicted vertical effective stress field. The output layer of the main network is connected to the input layer of two of the three DeepONet neural operators, and the output layer of each of the two forward operators is connected to the input layer of the inverse operator.

[0012] Preferably, in step S1, the plurality of Fourier basis functions include: defining the entire top boundary of the computational domain as the load application region, and establishing... Fourier basis functions are used as the load basis. ,in L x The length of the top of the computational domain is represented by N, which is the order of the Fourier expansion. The Fourier basis functions are defined on the entire top boundary of the computational domain, so that loads of any location, range, and shape can be represented by a linear combination of the same set of basis functions, thereby achieving generalization in the three dimensions of load location, shape, and magnitude. The magnitude expansion is achieved by multiplying each Fourier basis function by multiple magnitude coefficients covering different load magnitudes. Field-by-field normalization is performed on the multiphysics coupled data, and interval mapping normalization is performed on the spatial coordinates. The three DeepONet neural operators are trained using a Cartesian product deep operator network architecture.

[0013] Furthermore, in step S1, the amplitude expansion is to multiply each Fourier basis function by multiple different amplitude coefficients, which cover different load levels; the load cases also include several uniformly distributed load cases acting on different positions on the top of the soil and rock. Step S2 also includes: normalizing the multiphysics coupling data and normalizing the spatial coordinates by interval mapping; In step S3, the three DeepONet neural operators are trained using a Cartesian product deep operator network architecture.

[0014] The second aspect of the present invention provides a variable load soil-rock solid-fluid coupling solution method, wherein the target load on the soil-rock region is input into the variable load soil-rock solid-fluid coupling solution model obtained by the construction method described above, and the model is trained and solved, and finally the effective stress field and displacement field distribution of the soil-rock region are output.

[0015] The total loss function used in the training solution is shown below:

[0016] in, Indicates the boundary load-driven loss. This indicates the Fourier superposition of internal monitoring losses. This represents the operator's closed-loop physical consistency loss. For network parameters Regularization term, , , , These represent the weights of each item.

[0017] Furthermore, the calculation method for the boundary load driving loss is as follows: select the top loaded boundary of the soil mass as the data constraint point, obtain the theoretical stress value of the boundary data constraint point according to the stress balance of the target load, and obtain the mean square error between the predicted stress value and the theoretical stress value at these boundary data constraint points in the computational domain, which is used as the boundary load driving loss. The Fourier superposition internal supervision loss is calculated as follows: the mean square error between the predicted stress value at 1500-2500 sampling points within the computational domain and the effective stress field estimate obtained by Fourier linear superposition is taken as the Fourier superposition internal supervision loss. The operator closed-loop physical consistency loss is calculated as follows: the mean square error between the effective stress field predicted by the main network and the verification stress field obtained by the closed-loop reconstruction of the three DeepONet neural operators is calculated as the operator closed-loop physical consistency loss.

[0018] Furthermore, the Fourier linear superposition specifically includes: Substituting the baseline case with zero top boundary load into the steady-state Biot solid-fluid coupled finite element model for solution, the baseline stress field S220(x,y) is obtained. Based on the target load, Fourier coefficients are calculated across the entire top boundary to obtain the coefficients corresponding to each Fourier basis function. The effective stress field estimate S22 is then obtained by performing a linear superposition of the Fourier coefficients according to the following formula. sup (x,y):

[0019] Among them, S22sup (x,y) represents the target of the synthesized internal stress field monitoring; (x,y) represents the spatial coordinates of any point within the computational domain; S220(x,y) is the zero-load baseline stress field, which contains only the coupling effect of self-weight and pore pressure; c0 is the Fourier coefficient of the constant term of the target load, c i (i=1,…,20) are the Fourier coefficients of the cosine / sine terms of each order; ΔS22 i (x,y) represents the unit magnitude increment response of the i-th basis function condition relative to the baseline, i=0,1,…,20, where i=0 corresponds to the constant term basis function.

[0020] Furthermore, the training solution employs a two-stage training strategy. In the first stage, only... and Two loss functions, using the Adam optimizer, with an initial learning rate. Cosine annealing to In the second stage, all four loss terms are used, employing the Adam optimizer with an initial learning rate of [missing information]. Cosine annealing to .

[0021] A third aspect of the present invention provides a computer-readable storage medium having a computer program stored thereon, wherein the computer program, when executed by a processor, implements a method for constructing a variable-load soil-rock-solid-fluid coupling solution model as described in any of the preceding claims, or a variable-load soil-rock-solid-fluid coupling solution method as described in any of the preceding claims.

[0022] A fourth aspect of the present invention provides a variable-load soil-solid-fluid coupling solution device, comprising a computer-readable storage medium and a processor; The computer-readable storage medium is used to store executable instructions; The processor is used to read executable instructions stored in the computer-readable storage medium and execute the method for constructing a variable-load soil-rock-solid-fluid coupling solution model as described in any of the above claims, or the variable-load soil-rock-solid-fluid coupling solution method as described in any of the above claims.

[0023] In summary, compared with the prior art, the above-described technical solutions conceived by this invention mainly possess the following technical advantages: 1. Rapid solution of multiphysics fields without internal monitoring data: The DeepM&Mnet multi-operator serial closed-loop framework is introduced into the geotechnical solid-fluid coupling problem. The coupling relationship between the deformation field and the pore pressure field is modeled through the closed-loop path of "stress prediction - displacement deduction - stress reconstruction". The vertical stress field and displacement field of the whole domain can be quickly inverted based solely on the top load boundary conditions. This eliminates the dependence of the original framework on monitoring data inside the computational domain, and balances the solution speed and accuracy.

[0024] 2. Full generalization solution for arbitrary distributed loads: Based on full boundary Fourier basis function decomposition, boundary loads of arbitrary location, shape, and magnitude are uniformly represented as a linear combination of finite basis functions. The internal field supervision target is synthesized by linear superposition from the finite element solutions of existing basis functions. Combined with the three-loss joint training and two-stage training strategy, high-precision solution under arbitrary distributed loads can be achieved without having to recalculate the finite element for each new load.

[0025] 3. Accuracy Improvement and Nonlinear Extension: Rich internal physical constraints are provided through Fourier superposition internal supervision, and a Cartesian product-based deep operator network architecture and a dimensionality reduction sensor strategy are employed to improve operator mapping accuracy, extending the applicability of the method from linear elastic constitutive models to nonlinear elastoplastic constitutive models. In the linear elastic rectangular domain, the relative error of the effective stress field for three typical test cases is less than 0.09 (best 0.0184, average 0.0611); in the nonlinear elastoplastic (Mohr-Coulomb) rectangular domain, the relative errors of the stress field, horizontal displacement field, and vertical displacement field are 1.3%, 1.7%, and 3.2%, respectively, verifying the extended applicability to nonlinear constitutive models. Attached Figure Description

[0026] Figure 1 This is a flowchart of the construction and solution method of a variable-load rock-soil solid-fluid coupling solution model according to the present invention.

[0027] Figure 2 This is a diagram of the overall architecture of the serial DeepM&Mnet network of the present invention, showing the serial connection relationship between the main network (MainNet) and the three frozen DeepONet operators, the closed-loop information flow, and the input methods of the three supervision signals.

[0028] Figure 3 This is a schematic diagram of the DeepONet neural operator in an embodiment of the present invention.

[0029] Figure 4 This is a schematic diagram of the Fourier basis function load decomposition principle, showing the original distribution, Fourier reconstruction distribution, and corresponding Fourier coefficients of three typical loads (newly distributed uniformly, triangular, and half-wave sine).

[0030] Figure 5 This is a schematic diagram of the internal supervision principle of Fourier superposition, showing the process of zero-load baseline solution, basis function incremental response calculation and linear weighted superposition.

[0031] Figure 6 This is a flowchart of the Biot coupling training data generation and DeepONet training process.

[0032] Figure 7 This is a schematic diagram of the three-loss joint training and two-stage training strategy.

[0033] Figure 8 The figure shows a comparison between the finite element solutions of the effective stress field and displacement field of the semi-sinusoidal distributed load (S22 RelL2=0.0184) in the rectangular domain embodiment and the prediction results of the method of the present invention, verifying the shape generalization ability.

[0034] Figure 9 This is a comparison diagram of the new position x∈[30,40] (S22 RelL2=0.0763) of the triangular load in the rectangular domain embodiment, verifying the joint generalization ability of position and shape.

[0035] Figure 10 This is a comparison chart of the dual-region load (S22 RelL2=0.0886) in the rectangular domain embodiment, verifying the multi-region generalization ability.

[0036] Figure 11 This is a comparison diagram of the vertical effective stress field (S22) and vertical displacement field (V) of the method of the present invention and the true solution of the ABAQUS finite element method when a uniformly distributed load of 100 kPa is applied to the region x∈[0,25] in the nonlinear elastoplastic (Mohr-Coulomb constitutive) rectangular domain embodiment.

[0037] Figure 12 This is a comparison diagram of the vertical effective stress field (S22) and vertical displacement field (V) of the method of the present invention and the true solution of the ABAQUS finite element method when a uniformly distributed load of 150 kPa is applied to the region x∈[40,50] (high amplitude load at the edge of the region) in the embodiment of nonlinear elastoplastic rectangular domain. Detailed Implementation

[0038] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention. Furthermore, the technical features involved in the various embodiments of this invention described below can be combined with each other as long as they do not conflict with each other.

[0039] The first aspect of this invention provides a method for constructing a variable-load, soil-solid-fluid coupling solution model, comprising the following steps: S1. Establish a steady-state Biot solid-fluid coupled finite element model, and establish several Fourier basis functions as load bases. By expanding the amplitude of the Fourier basis functions and / or randomly combining multiple Fourier basis functions, several load cases are obtained. S2. Substitute each load case into the steady-state Biot solid-fluid coupled finite element model for solution to obtain multi-physics coupled data containing vertical effective stress field data, horizontal displacement field data and vertical displacement field data in the computational domain. Combine each load case and its corresponding multi-physics coupled data to obtain a pre-training dataset. S3. Construct three DeepONet neural operators and train them using the pre-training dataset; wherein the three DeepONet neural operators include: a forward operator that maps the vertical effective stress field to the horizontal displacement field, a forward operator that maps the vertical effective stress field to the vertical displacement field, and an inverse operator that jointly maps the horizontal displacement field and the vertical displacement field back to the vertical effective stress field. S4. Establish a main network, and then connect the three DeepONet neural operators trained in step S3 to the main network to obtain the final variable load soil-solid-fluid coupling solution model. The input of the main network is the normalized spatial coordinates of the point to be determined, and the output is the predicted vertical effective stress field. The output layer of the main network is connected to the input layer of two of the three DeepONet neural operators, and the output layer of each of the two forward operators is connected to the input layer of the inverse operator.

[0040] The core improvements of this invention are reflected in two aspects: First, the DeepM&Mnet multi-operator serial closed-loop framework is introduced into the soil-solid-fluid coupling problem, realizing effective modeling and collaborative solution of the coupling relationship between the soil deformation field and the pore water pressure field and other multi-physics fields. Furthermore, the stress equilibrium condition at the top boundary is used to construct supervisory constraints, eliminating the original framework's dependence on monitoring data within the computational domain. No internal monitoring points are required; only the top load is needed for rapid simulation of the entire multi-physics field. Second, a Fourier superposition internal supervision is proposed. Through full-boundary Fourier basis function decomposition, the load location, shape, and amplitude are fully generalized. The internal field supervision target is directly synthesized from existing basis function finite element solutions, providing near-accurate internal constraints for linear problems and necessary approximate anchor points for nonlinear problems, enabling the method to solve both linear and nonlinear problems well. In addition, the Cartesian product deep operator network architecture and dimensionality reduction sensor strategy significantly improve operator mapping accuracy, extending the applicability of the method from linear elastic constitutive models to nonlinear elastoplastic constitutive models.

[0041] The present invention will be further described below through specific embodiments.

[0042] Example 1: Verification of a Linear Elastic Rectangular Domain like Figure 1 As shown in the figure, this embodiment provides a method for constructing and solving a variable-load soil-solid-fluid coupling solution model, including the following steps: S1. Construct a pre-training dataset for geotechnical solid-fluid coupled multiphysics operators based on Fourier basis functions: The computational domain is A two-dimensional rectangular region. Material parameters are set as follows: elastic modulus... (Force line), Poisson's ratio (Dimensionless), Biot coefficient (Dimensionless) Specific gravity of water This embodiment uses uniform dimensions after normalization of parameters, and the values ​​reflect the load-stiffness ratio and pore pressure-load ratio relationship of typical geotechnical engineering. If the stress unit is 0.2 MPa and the length unit is 1 m, then the above parameters correspond to actual engineering quantities such as an elastic modulus of 20 MPa, a calculation domain of 50 m × 20 m, a water unit weight of 10 kN / m³, a bottom hydrostatic pressure of 200 kPa, and a load amplitude of 0.2~3 MPa. Since the governing equations are scale-independent, the above values ​​can be converted to any actual unit system, and the conclusions such as relative errors do not change with unit conversion. Embodiment 2 directly uses actual engineering units (MPa, kPa, kN / m³).

[0043] Hydraulic boundary conditions: top boundary ( ) set as drainage boundary ( ), bottom boundary ( ) is set to far-field hydrostatic pressure ( (The stress dimension should be the same as the elastic modulus and load amplitude, using the same stress unit.) Both sides are set as impermeable boundaries. This setting ensures a bottom-to-top steady-state pore pressure gradient distribution within the computational domain (pore pressure range). The pore pressure gradient couples with the displacement field through the effective stress balance equation, and the contribution of pore pressure to the stress field accounts for about 7%-10% of the applied load.

[0044] Mechanical boundary conditions: Bottom boundary (y=0) is fixed. ), horizontal rolling constraints on both sides of the boundary (x=0 and x=50 m) The top boundary (y=20 m) is the boundary where the load is applied.

[0045] A steady-state Biot solid-fluid coupled finite element model was established using the open-source finite element software FEniCSx (dolfinx 0.10). The model employed a two-step solution strategy: first, solving for the Laplace equations. The steady-state pore water pressure field is controlled by hydraulic boundary conditions; the second step is to convert the pore pressure field into a Biot solid-fluid coupling term. (In the formula, α is the Biot consolidation coefficient, a dimensionless parameter with 0 ≤ α ≤ 1, representing the proportion of pore water pressure transferred to the effective stress of the solid skeleton; p is the steady-state pore water pressure field obtained in the first step, with stress dimensions; v is the displacement test function (virtual displacement vector) in the weak form, and div(v) is the divergence of v, representing the virtual volume strain; integration is performed over the entire computational domain. This coupling term corresponds to the solid-fluid coupling integral term in the weak form, and after integration by parts, it is equivalent to the equivalent body force acting on the solid skeleton.) α Substituting p (the product of the pore water pressure gradient and the Biot coefficient) into the effective stress equilibrium equation to solve for the displacement field, and then... The effective stress field is calculated using projection, ensuring that the displacement and stress fields truly reflect the solid-fluid coupling effect. The mesh is then divided into... The triangular element outputs a mesh. Each node.

[0046] The entire top boundary of the computation domain Define the load application area and establish Fourier basis functions As a load base; for any load Perform a standard Fourier series expansion:

[0047] In the formula, a0, a n b n The Fourier coefficients are determined by performing a Fourier series expansion on the target load p(x); L x The length of the boundary applied to the top load; N is the Fourier expansion order; n is the harmonic order of the summation.

[0048] Because the basis functions cover the entire boundary, loads of any location, width, and shape can be accurately represented by a linear combination of the same set of basis functions. There are 21 basis functions in total, with the shortest half-wavelength being... It can accurately represent the width. Load distribution.

[0049] Training data generation: Specifically defining 21 Fourier basis functions ( Each basis function is multiplied by eight amplitude coefficients (1, 2, 3, 5, 7, 8, 10, 12), resulting in 168 basis function load cases. Additionally, 15 uniformly distributed load verification load cases are generated (load amplitudes 1~15, applied to...). ); 13 groups of uniformly distributed loads at different locations (covering) to The dataset consists of 52 location-diverse load cases, each with four amplitude values ​​(3, 6, 9, 12) and 150 augmented load cases with random combinations of Fourier coefficients. A total of 385 load cases are generated, each outputting four fields at 4141 grid points: effective stress field data (S22), horizontal displacement field data (U), vertical displacement field data (V), and pore water pressure field data (P).

[0050] Simultaneously solve the zero-load baseline case (top boundary load is zero) to obtain the baseline field containing only self-weight and pore pressure coupling effects. , , , serving as a reference benchmark for internal supervision of Fourier superposition.

[0051] Data normalization: Field-by-field statistical normalization (zero mean, unit variance) is performed on the physical field data (stress, displacement). Specifically, in... The field value distribution for all 385 operating conditions was statistically analyzed across a grid of sensor points, and the global mean and standard deviation for each field were calculated. The mean of field S22 is approximately... Standard deviation approximately U field average value is approximately Standard deviation approximately V average value is approximately Standard deviation approximately It can be seen that there are significant differences in magnitude between different physical fields (the stress field and the horizontal displacement field differ by about 100 times). Without normalization, this will severely affect the operator training effect. All field values ​​are normalized to a standardized space with zero mean and unit variance, while the spatial coordinates are... , Linear mapping to The interval eliminates the magnitude difference between different physical fields, significantly improving the training effect of DeepONet operators and the main network. The number of sensor grid points is determined according to the Nyquist sampling theorem: the grid density must not be lower than the sampling rate corresponding to half the wavelength of the highest-order Fourier basis function (2Nth order) to represent all basis function loads without aliasing, so 51×21 grid points are used; the mean and standard deviation of each field are values ​​under the normalized parameter system (stress type is stress dimension, displacement type is length dimension), where the mean of field S22 is about -2.0, which means that the average value of the vertical effective stress of all working conditions is about -2.0 stress units, and the negative sign indicates compression.

[0052] S2, Pre-trained multiphysics mapping neural operator: like Figure 3As shown, the DeepONet network consists of a branch network and a trunk network. Three DeepONet operators are trained using GPU acceleration, with inputs and outputs all in the normalized space. The specific network parameter settings for each operator are as follows: (1) Positive displacement operator and This is used to map the effective vertical stress field to the horizontal and vertical displacement fields, respectively. The input dimension of the branch network is the number of stress field sampling points (in this embodiment, 1071 points correspond to...). (mesh), network layer structure set to The backbone network has an input dimension of 2 spatial coordinates, and the network layer structure is set to... The activation function is GELU, and the initialization method is Glorot normal.

[0053] (2) Inverse stress operator This is used to jointly map the horizontal and vertical displacement fields back to the effective vertical stress field. The input dimension of the branch network is the sum of the number of sampling points of the two displacement fields (2142 points), and the network layer structure is set to... The backbone network has an input dimension of 2 spatial coordinates, and the network layer structure is set to... .

[0054] Training uses the Adam optimizer with an initial learning rate of A reverse-time decay strategy was employed, with a batch size of 8192, sampling 1000 coordinate points per load case, and a total training step count of 30000. After training, the network parameters of all operators were frozen, and normalized statistics and sensor information were saved. The reverse-time decay strategy refers to gradually reducing the learning rate as the number of training steps increases, in order to achieve rapid convergence in the early stages of training and fine approximation in the later stages. The purpose of this pre-training step is to enable the three neural operators to accurately learn the bidirectional physical mapping relationship between the stress field and displacement field in the solid-fluid coupled system. After freezing the parameters, the operators are used as fixed differentiable physical constraint modules in the subsequent main network training (S5)—providing a stable physical mapping benchmark for closed-loop consistency loss and avoiding repeated training of operators for each new load solution.

[0055] S3. Constructing a serial DeepM&Mnet multiphysics solution framework: like Figure 2 As shown, a fully connected deep neural network is established as the main network (MainNet) to directly predict the vertical effective stress field of the target region.

[0056] The main network (MainNet) has the following structure: the input layer contains 2 neurons (corresponding to normalized coordinates). The hidden layer contains 4 layers, each with 256 neurons, and the output layer contains 1 neuron (corresponding to S22). The activation function used is SiLU (Swish), i.e. .

[0057] The trained operators in S2 are sequentially connected after the main network: the output of the main network. As a positive operator and The branch input is used to deduce the displacement field. and The displacement field then passes through the inverse operator. The reconstructed stress field was verified. This forms a closed-loop path of "stress prediction – displacement deduction – stress reconstruction". This closed-loop reconstruction provides a physical consistency check for the solution, and can constrain the coordinated convergence of each field without the need for internal monitoring data, thereby improving the consistency and reliability of the solution.

[0058] S4. Construct the Fourier superposition internal supervision objective: This invention utilizes the linear superposition principle of Biot's consolidation theory under the small deformation assumption to perform a weighted linear superposition of the finite element solutions of each Fourier basis function. This constructs a monitoring target for the internal stress field of any target load without requiring additional finite element calculations for the target load. First, the zero-load baseline (containing only self-weight and pore pressure coupling effects) is solved. Then, the incremental response of each basis function case relative to the baseline is calculated. Finally, the results are weighted and superimposed according to the Fourier coefficients of the target load to obtain an estimate of the internal field, which serves as the internal monitoring signal for training the main network.

[0059] Specifically, for a given target load First, along the entire top boundary Fourier coefficients were calculated, resulting in 21 coefficients. Then, using the existing unit-amplitude finite element solution of the 21 basis functions and the zero-load baseline solution, the incremental response is calculated. And perform linear superposition according to the following formula:

[0060] The negative sign originates from the convention of the load direction applied in the finite element basis function case (applied load direction). ), randomly sample 2000 supervision points within the computational domain (excluding the top boundary), and then The values ​​at these points serve as internal monitoring targets. The key to this step is that internal monitoring can be constructed for any new load using existing basis function finite element solutions, without the need to run additional finite element calculations for the target load.

[0061] S5. Construct a joint training framework for the three losses and perform a two-stage training solution: Define the total loss function:

[0062] in: (Boundary load driven loss): Only the top loaded boundary of the soil mass is selected. , Using 101 sampling points as data constraint points, the theoretical stress values ​​of these boundary data constraint points are obtained based on the stress balance of the target load. The mean square error between the predicted stress values ​​and the known theoretical stress values ​​of the main network at these boundary points is calculated. This term's weight... Set it to 50.0.

[0063] (Fourier superposition of internal supervision loss): The field estimate obtained by superimposing the predicted values ​​at 2000 sampling points within the constraints with the Fourier linear superposition. The mean square error between them. This weight. Set it to 30.0.

[0064] (Operator closed-loop physical consistency loss): Calculate the stress field predicted by the main network. The verification stress field obtained by closed-loop reconstruction using serial operators The mean square error between them. Its calculation formula is:

[0065] in This term represents the number of sampling points within the computational domain. It implicitly constrains the prediction results of the main network to satisfy the physical equations and constitutive relations of solid-fluid coupling in soil and rock using pre-trained operators. The weight of this term... Set it to 1.0.

[0066] : for network parameters Regularization term, weight Set as .

[0067] The weighted design of the three physical losses reflects the constraint hierarchy: the boundary conditions are the strongest constraints (directly observable), the superimposed internal supervision is the intermediate constraint (constructed based on physical principles), and the operator loop is the refined constraint (ensuring consistency across multiple fields).

[0068] The main network described in this invention does not use the load as the network input, but applies the target load information to the network through a loss function during the solution stage: the stress equilibrium condition of the target load at the top loading boundary is used as the boundary supervision, and the internal stress field of the computational domain synthesized by the Fourier linear superposition of the target load is used as the internal supervision, so that the solution of arbitrary distributed loads can be achieved without any internal monitoring data.

[0069] Two-stage training strategy: Phase 1 (Warm-up Training, 10,000 Steps): Using only and Two loss functions, using the Adam optimizer, with an initial learning rate. Cosine annealing to In this stage, boundary conditions and superimposed internal supervision are used to quickly establish a global distribution estimate of the stress field, without enabling operator closed-loop constraints to avoid gradient conflicts caused by inaccurate predictions in the initial main network.

[0070] Phase 2 (Refined Training, 40,000 steps): All four loss terms are enabled, using the Adam optimizer, with an initial learning rate of... Cosine annealing to This stage utilizes operator closed-loop constraints to further improve the physical consistency of the predictions, ensuring that the predicted stress and displacement fields satisfy the physical mapping relationship encoded by the deep learning operators. Gradient pruning (maximum norm 1.0) is implemented during training, and the optimal model parameters (selected based on the minimum total loss) are continuously recorded.

[0071] The following provides a detailed explanation of the solution results for three typical test cases. All test loads were not present in the training data, thus verifying the method's generalization ability for load shape, load location, and multi-region loads.

[0072] Test Case 1 – Half-sine distributed load (shape generalization): at the load location Applying sinusoidal distributed load .like Figure 8 As shown, the model predictions are in high agreement with the finite element solution, and the relative effective stress field S22 is... The error is 0.0184, and the relative vertical displacement field V... The error is 0.0721. Sinusoidal loads have good smoothness and their Fourier decomposition accuracy is extremely high. Therefore, the model can make full use of the superimposed internal supervision information to achieve high-precision prediction.

[0073] Test Case 2 – New Location of Triangular Load (Location + Shape Generalization): In Applying triangular distributed load .like Figure 9 As shown, the model simultaneously handles both positional and shape changes, with S22 relative to... The error is 0.0763, V relative The error was 0.0560, which verifies the generalization ability of the method under the combined changes of load position and shape.

[0074] Test Case 3 – Dual-Region Load (Multi-Region Generalization): Simultaneously in Apply ,exist Apply .like Figure 10 As shown, this working condition simultaneously tests the ability to generalize from location, superimpose multiple regions, and vary amplitude, making it the most complex working condition among all test cases. The model successfully predicted the stress concentration effects and their spatial superposition distribution in the two independent loading regions, with S22 being the most complex. The error is 0.0886, V relative The error is 0.1521.

[0075] Table 1 summarizes the accuracy metrics for three typical test cases of the rectangular domain: Table 1 Summary of Accuracy of Rectangular Region Test Cases

[0076] As can be seen from Table 1: (1) The S22 values ​​of the three typical test cases are relatively... The errors are all below 0.09, and the average S22 error is 0.0611, which reaches the high precision level acceptable for engineering; (2) The method shows good generalization ability in three dimensions: shape generalization (sine), position + shape joint generalization (new position triangle) and multi-region generalization (dual-region load); (3) The average relative vertical displacement field V The error was 0.0934. The displacement field was indirectly derived from the stress field through the DeepONet operator. The accuracy also reached a high level, which verified the role of the operator closed-loop constraint in ensuring the consistency of multiple fields. (4) None of the test loads appeared in the training data. The high accuracy of the method comes from the rich physical constraints provided by the Fourier superposition internal supervision and the stress-displacement self-consistency guaranteed by the operator closed loop.

[0077] Example 2: Verification of Nonlinear Elastic-Plastic (Mohr-Coulomb) Rectangular Domain To verify the applicability of this method to nonlinear constitutive models, the linear elastic constitutive model in Example 1 was replaced with the Mohr-Coulomb elastoplastic constitutive model.

[0078] The computational domain size remains unchanged. Material parameters are set as follows: elastic modulus Poisson's ratio Cohesion internal friction angle Shear expansion angle Total unit weight of soil The specific gravity of water The boundary conditions are the same as in Example 1: the bottom is fixed, the two sides are horizontally rolling constraints, and the top is the load application boundary.

[0079] For steady-state drainage conditions, the effective stress method is used to simplify the solution: using buoyancy unit weight... As an equivalent body force, a CPE4 plane strain model was established using ABAQUS finite element software. First, the initial stress state under gravity equilibrium was established using the GEOSTATIC step, and then the top load was applied using the STATIC step. Pore water pressure field. Calculated using analytical methods.

[0080] The training data includes 446 load cases: 21 Fourier basis functions multiplied by 5 amplitude levels (30, 60, 100, 150, 200 kPa), totaling 105 basis function load cases; 3 uniformly distributed loads at 10 locations, each with 3 amplitudes (50, 100, 150 kPa), totaling 30 uniformly distributed load cases; 60 stochastic Fourier combination load cases with a maximum amplitude of 40 kPa; 100 stochastic Fourier combination augmented load cases with a maximum amplitude of 60 kPa; 100 stochastic Fourier combination augmented load cases with a maximum amplitude of 80 kPa; 30 uniformly distributed load augmented load cases with different locations and amplitudes; 20 triangular load augmented load cases with different locations and amplitudes; and 1 zero-load baseline load case.

[0081] The neural operator employs a Cartesian product DeepONet (DeepONetCartesianProd) architecture, reducing the dimensionality of the sensor mesh to [missing information]. There are 10 points. The training data is in Cartesian product format, and the branch input is... The field value matrix, with the main input being The coordinate matrix is ​​calculated, and each working condition is trained using all grid nodes. This architecture significantly improves the mapping accuracy of the three neural operators compared to the standard DeepONet architecture, laying the foundation for high-precision joint solutions of the subsequent main network.

[0082] Test Case 1: In Area application Uniformly distributed load. For example... Figure 11 As shown, the model predictions agree well with the true solutions of the ABAQUS finite element method. Test case 2: In Area application Uniformly distributed loads were used to verify the predictive capability under edge loading conditions. For example... Figure 12 As shown, the quantitative evaluation results are as follows: Table 2. Accuracy of Nonlinear Mohr-Coulomb Rectangular Domain Test Cases

[0083] The method of this invention achieves high-precision prediction results in nonlinear elastoplastic problems: the relative error of the effective stress field in both test cases is less than 0.020, the error of the horizontal displacement field is less than 0.045, and the error of the vertical displacement field is less than 0.033. Specifically, the overall error for the wide-domain uniformly distributed load case is less than 3.2%, and even under challenging conditions of high-amplitude uniformly distributed loads at the edges, with loads deviating from the center of the domain and high load amplitudes, the overall error remains below 4.5%. This indicates that the method, through operator closed-loop consistency constraints and the nonlinear mapping capability of DeepONet, can effectively capture the nonlinear stress-displacement relationship under the Mohr-Coulomb elastoplastic constitutive model, achieving high-precision prediction across the entire field.

[0084] In summary, this invention achieves high-precision intelligent solutions for multiphysics fields involving solid-fluid coupling in geotechnical systems under arbitrary loads by constructing internal supervision targets and a three-loss joint training framework through Fourier basis function load decomposition combined with the principle of linear superposition. The core innovation of this invention is the Fourier superposition internal supervision, which utilizes the physical characteristics of solid-fluid coupling problems to directly construct internal field supervision targets for arbitrary loads from existing finite element basis function solutions, eliminating the need for additional finite element calculations and providing the main network with far richer physical constraint information than pure boundary supervision. By adopting a Cartesian product DeepONet architecture and a dimensionality reduction sensor strategy, operator accuracy and training efficiency are significantly improved. The applicability of the method is extended from linear elastic constitutive models to nonlinear elastoplastic constitutive models (Mohr-Coulomb model). In the linear elastic embodiment, the average S22 error is 0.0611, and in the nonlinear elastoplastic embodiment, the overall field error is less than 4.5%. This method requires no internal monitoring data and achieves full generalization across the three dimensions of load location, shape, and amplitude, providing an efficient, high-precision, and universal intelligent solution tool for complex load analysis in geotechnical engineering.

[0085] The above embodiments are only used to illustrate the technical solutions of the present invention and not to limit them. Although the present invention has been described in detail with reference to preferred embodiments, those skilled in the art should understand that modifications or equivalent substitutions can still be made to the technical solutions of the present invention, and these modifications or equivalent substitutions should not cause the modified technical solutions to deviate from the spirit and scope of the technical solutions of the present invention. For example, the order of the Fourier expansion. The network structure, loss weight coefficients, and training strategy parameters of DeepONet and MainNet can be flexibly adjusted according to the complexity of the load. The finite element model can also be replaced with other open source or commercial finite element software, and the computational domain can be extended to engineering scenarios with complex geometries such as tunnels.

[0086] Those skilled in the art will readily understand that the above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A method for solving soil-rock solid-fluid coupling under variable load, characterized in that, include: The target load on the soil and rock region is input into the variable load soil-solid-fluid coupling solution model for training and solving, and finally the effective stress field and displacement field distribution of the soil and rock region are output. The total loss function used in the training solution is shown below: in, Indicates the boundary load-driven loss. This indicates the Fourier superposition of internal monitoring losses. This represents the operator's closed-loop physical consistency loss. For network parameters Regularization term, , , , These represent the weights of each item; The calculation method for the boundary load driving loss is as follows: select the top loaded boundary of the soil mass as the data constraint point, obtain the theoretical stress value of the boundary data constraint point according to the stress balance of the target load, and obtain the mean square error between the predicted stress value and the theoretical stress value at these boundary data constraint points in the computational domain, which is used as the boundary load driving loss. The Fourier superposition internal supervision loss is calculated as follows: the mean square error between the predicted stress value at 1500-2500 sampling points within the computational domain and the effective stress field estimate obtained by Fourier linear superposition is taken as the Fourier superposition internal supervision loss. The operator closed-loop physical consistency loss is calculated as follows: the mean square error between the effective stress field predicted by the main network and the verification stress field obtained by the closed-loop reconstruction of the three DeepONet neural operators is calculated as the operator closed-loop physical consistency loss. The method for constructing the variable-load soil-solid-fluid coupling solution model includes the following steps: S1. Establish a steady-state Biot solid-fluid coupled finite element model, and establish several Fourier basis functions as load bases. By expanding the amplitude of the Fourier basis functions and / or randomly combining multiple Fourier basis functions, several load cases are obtained. S2. Substitute each load case into the steady-state Biot solid-fluid coupled finite element model for solution to obtain multi-physics coupled data containing vertical effective stress field data, horizontal displacement field data and vertical displacement field data in the computational domain. Combine each load case and its corresponding multi-physics coupled data to obtain a pre-training dataset. S3. Construct three DeepONet neural operators and train them using the pre-training dataset; wherein the three DeepONet neural operators include: a forward operator that maps the vertical effective stress field to the horizontal displacement field, a forward operator that maps the vertical effective stress field to the vertical displacement field, and an inverse operator that jointly maps the horizontal displacement field and the vertical displacement field back to the vertical effective stress field. S4. Establish a main network, and then connect the three DeepONet neural operators trained in step S3 to the main network to obtain the final variable load soil-solid-fluid coupling solution model. The input of the main network is the normalized spatial coordinates of the point to be determined, and the output is the predicted vertical effective stress field. The output layer of the main network is connected to the input layer of two of the three DeepONet neural operators, and the output layer of each of the two forward operators is connected to the input layer of the inverse operator.

2. The variable-load soil-solid-fluid coupling solution method according to claim 1, characterized in that, In step S1, the plurality of Fourier basis functions include: defining the entire top boundary of the computational domain as the load application region, and establishing... Fourier basis functions are used as the load basis. ,in L x This represents the length of the top of the computational domain, where N is the order of the Fourier expansion.

3. The variable-load soil-solid-fluid coupling solution method according to claim 1, characterized in that, In step S1, the amplitude expansion is to multiply each Fourier basis function by multiple different amplitude coefficients, which cover different load levels; the load case also includes several uniformly distributed load cases acting on different positions on the top of the soil and rock. Step S2 also includes: normalizing the multiphysics coupling data and normalizing the spatial coordinates by interval mapping; In step S3, the three DeepONet neural operators are trained using a Cartesian product deep operator network architecture.

4. The variable-load soil-solid-fluid coupling solution method according to claim 1, characterized in that, The Fourier linear superposition specifically includes: Substituting the baseline case with zero top boundary load into the steady-state Biot solid-fluid coupled finite element model for solution, the baseline stress field S220(x,y) is obtained. Based on the target load, Fourier coefficients are calculated across the entire top boundary to obtain the coefficients corresponding to each Fourier basis function. The effective stress field estimate S22 is then obtained by performing a linear superposition of the Fourier coefficients according to the following formula. sup (x,y): Among them, S22 sup (x,y) represents the target of the synthesized internal stress field monitoring; (x,y) represents the spatial coordinates of any point within the computational domain; S220(x,y) is the zero-load baseline stress field, which contains only the coupling effect of self-weight and pore pressure; c0 is the Fourier coefficient of the constant term of the target load, c i (i=1,…,20) are the Fourier coefficients of the cosine / sine terms of each order; ΔS22 i (x,y) represents the unit magnitude increment response of the i-th basis function condition relative to the baseline, i=0,1,…,20, where i=0 corresponds to the constant term basis function.

5. The variable-load soil-solid-fluid coupling solution method according to any one of claims 1-4, characterized in that, The training solution employs a two-stage training strategy. In the first stage, only... and Two loss functions, using the Adam optimizer, with an initial learning rate. Cosine annealing to In the second stage, all four loss terms are used, employing the Adam optimizer with an initial learning rate of [missing information]. Cosine annealing to .

6. A computer-readable storage medium, characterized in that, It stores a computer program, which, when executed by a processor, implements the variable-load soil-solid-fluid coupling solution method as described in any one of claims 1-5.

7. A variable-load soil-solid-fluid coupling solution device, comprising a computer-readable storage medium and a processor; The computer-readable storage medium is used to store executable instructions; The processor is used to read executable instructions stored in the computer-readable storage medium and execute the variable-load soil-solid-fluid coupling solution method as described in any one of claims 1-5.

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