Numerical simulation method for rapidly calculating fault dislocation discontinuous deformation field

Abstract

The invention discloses a numerical simulation method for rapidly calculating a fault dislocation discontinuous deformation field, and relates to the field of tectonic geology and seismology, and the method comprises the following steps: constructing a physical coding finite element network SN-PEFEN fused with a daghet node technology; fault dislocation deformation field forward modeling simulation is carried out based on the SN-PEFEN; and fault dislocation deformation field inversion simulation is carried out based on the SN-PEFEN, and numerical simulation of a fault dislocation discontinuous deformation field is realized. According to the method, the problem of essential conflict between the continuous field prediction characteristic based on the continuous microhypothesis and the inherent displacement discontinuous characteristic of the fault dislocation field in the existing method is solved.

Description

Technical Field

[0001] This invention relates to the fields of structural geology and seismology, specifically to a numerical simulation method for rapidly calculating discontinuous deformation fields of fault dislocations. Background Technology

[0002] Simulating crustal deformation fields caused by fault dislocation is one of the fundamental theoretical problems in structural geology and seismology. It is of great significance for understanding the role of fault activity and earthquakes in regional tectonic evolution and for revealing fault slip processes. Simulating surface deformation caused by fault dislocation not only helps to invert source parameters such as fault geometry and coseismic slip distribution, which is the foundation for constructing earthquake period models and advancing earthquake prediction research, but also provides a scientific basis for the prediction and prevention of secondary disasters such as post-earthquake landslides and ground fissures.

[0003] Due to the combined characteristics of data sparsity and spatial discontinuity, conventional forward and inverse modeling methods suffer from low computational efficiency and insufficient accuracy. Although artificial intelligence-based methods such as PINNs and PEFEN have provided new ideas for solving a class of forward and inverse modeling problems with sparse data and clear physical processes, their continuous field prediction characteristics based on the assumption of continuous differentiability are fundamentally in conflict with the inherent displacement discontinuity of fault dislocation fields, and the problem of predicting discontinuous deformation fields remains unsolved. Summary of the Invention

[0004] To address the aforementioned shortcomings in existing technologies, this invention provides a numerical simulation method for rapidly calculating discontinuous deformation fields of fault dislocations. This method resolves the fundamental conflict between the continuous field prediction characteristics based on the assumption of continuous differentiability and the inherent discontinuous displacement characteristics of fault dislocation fields.

[0005] To achieve the aforementioned objectives, the present invention employs the following technical solution: a numerical simulation method for rapidly calculating discontinuous deformation fields of fault dislocations, comprising the following steps: S1: Constructing the SN-PEFEN physical coding finite element network that integrates node-based technology; S2: Perform forward modeling of fault dislocation deformation field based on the SN-PEFEN; S3: Based on the SN-PEFEN, perform fault dislocation deformation field inversion simulation to realize numerical simulation of discontinuous fault dislocation deformation field.

[0006] Furthermore, the SN-PEFEN includes: introducing a finite element node splitting mechanism into the mesh framework of the physically encoded finite element network PEFEN, processing the fault interface in the preprocessing stage, and explicitly encoding the spatial discontinuity into the node topology to obtain SN-PEFEN. The finite element node splitting mechanism represents the nodes on the fault plane as two sets of nodes with identical position coordinates. The two sets of nodes belong to the elements of the two fault disks respectively, and the relative dislocations occurring on the two fault disks are described by the relative motion of the two sets of corresponding nodes.

[0007] Furthermore, the forward modeling of the fault dislocation deformation field includes the following steps: S21: Determine the governing equations and model boundary conditions for fault dislocations; S22: Discretize the solution domain into isoparametric elements, and approximate the field variables within each element by interpolating nodal variables and shape functions; S23: Using the displacements of all free nodes as optimization variables, the potential energy functional is minimized using the L-BFGS optimizer to obtain the forward modeling results of the fault dislocation deformation field in equilibrium.

[0008] Furthermore, the governing equations and model boundary conditions for the fault dislocations are as follows:

[0009]

[0010]

[0011]

[0012] in, For divergence operators, For stress tensor, For the computational domain, and These are the displacement vector fields on both sides of the fault plane. and These are the slip vectors of the two disks on the fault plane. and These are the boundary regions on both sides of the fault plane. This is the displacement boundary region. For displacement, Describes the domain in which the equation holds true; Based on the linear elastic constitutive and small strain assumptions:

[0013]

[0014] in, For stress components, The first Lamé constant, For strain components, For strain components, For Kronecker notation, when i= j The value is 1 if it is true, and 0 otherwise. This is the second Lamé constant. For strain tensor, superscript It is the transpose symbol; The potential energy functional of the governing equation is:

[0015] in, Let be the potential energy functional of the governing equation.

[0016] Furthermore, the solution domain is discretized into isoparametric elements, and the field variables are approximated within each element through nodal variables and shape function interpolation. For a given domain containing... A point within a cell of nodes ,variable for:

[0017] in, For the first Displacement variables of each node, These are the shape functions for the corresponding nodes, used to interpolate approximate field variables within the element; The potential energy functional is then calculated through numerical integration at Gauss points across all elements:

[0018] in, The determinant is the Gaussian point weights multiplied by the Jacobian matrix. For the first Unit 1 Jacobian matrix at each Gaussian point These are the weighting coefficients for the Gaussian integral. In the first Unit 1 The displacement is obtained by interpolation at Gaussian points.

[0019] Furthermore, the fault dislocation deformation field inversion simulation includes a forward solution stage and a reverse solution stage; the forward solution stage is consistent with the fault dislocation deformation field forward simulation. The inverse solution stage uses the Hessian-vector product and conjugate gradient method to solve for the gradient of the slip pair on the total system energy and the observation residuals, thereby enabling sensitivity analysis and optimization updates of the slip distribution. The optimal slip distribution is determined by minimizing the total error functional; the objective function is... for:

[0020] in, For the displacement of the observation point, To predict displacement for the model, For the slip parameter, For hyperparameter weights, For regularization terms; Since the displacement field needs to satisfy the mechanical equilibrium condition in each inversion iteration step, the displacement field can be regarded as an implicit function of the slip parameter:

[0021] in, It is an implicit function. It is an energy functional; The objective function is then expressed as:

[0022] To optimize slip parameters Calculate its gradient using the chain rule:

[0023]

[0024] in, The objective function is... The glide parameters are obtained using the Hessian-vector product and conjugate gradient methods. The gradient is:

[0025] in, This is the adjoint vector.

[0026] The beneficial effects of this invention are as follows: Based on artificial intelligence technologies such as PEFEN and integrating finite element mesh node splitting technology, this invention proposes the Split-Node Physics-Encoded Finite-Element Network (SN-PEFEN) method to conduct forward and inverse modeling theoretical and numerical experimental studies on two-dimensional and three-dimensional discontinuous deformation fields caused by fault dislocations in complex crustal faults. The SN-PEFEN method breaks through the theoretical limitations of the existing PEFEN framework in handling discontinuous problems. Compared with traditional numerical methods (finite element method), this method greatly improves the computational efficiency for handling sparse and discontinuous data problems while maintaining simulation accuracy. Its advantages are particularly evident for forward and inverse modeling of fault dislocations in meshes with millions of degrees of freedom. This invention provides a new paradigm for future research on multi-fault system analysis and sliding forward and inverse modeling. Attached Figure Description

[0027] Figure 1This is a flowchart of a numerical simulation method for rapidly calculating discontinuous deformation fields of fault dislocations.

[0028] Figure 2 This is a flowchart of the forward modeling of the SN-PEFEN method.

[0029] Figure 3 This is a flowchart of the inversion process for the SN-PEFEN method.

[0030] Figure 4 The results of forward numerical modeling of two-dimensional and three-dimensional fault dislocations based on the SN-PEFEN method are shown, along with a comparison with FEM results using the same grid.

[0031] Figure 5 The results of the numerical inversion of three-dimensional fault dislocations based on the SN-PEFEN method are shown, along with a comparison of their convergence curves and the accuracy of the surface deformation field simulation. Detailed Implementation

[0032] The present invention will be further described below with reference to the accompanying drawings and specific embodiments.

[0033] The theoretical basis involved in this study includes the following two aspects: (1) finite element method node-breaking technology, and (2) physical-encoded finite element networks (PEFEN).

[0034] The Finite Element Method (FEM) transforms complex continuum mechanics problems into a discrete system of algebraic equations through discretization and piecewise approximation. The solution domain is divided into a finite number of non-overlapping elements, each of which is approximated by shape functions to obtain the unknown field variables. These elements are connected by nodes to form a mesh. The ability of the material within an element to resist deformation and its contribution to the overall structure is expressed as the element stiffness matrix. The element stiffness matrix is ​​assembled into a global stiffness matrix according to the nodal degrees of freedom. Boundary conditions are applied, and by satisfying variational principles (e.g., the minimum potential energy principle for elasticity problems), the global continuous field variables (such as displacements) can be solved. For discontinuous motions within the solution space, such as fault dislocations, nodalization techniques can be used. Specifically, nodes on the fault plane are represented by two sets of nodes with identical positional coordinates, belonging to the elements of the two fault disks respectively. The relative dislocations occurring on the two fault disks can be described by the relative motion of the corresponding nodes in the two sets on the fault plane. Thus, the relative motion of discontinuous interfaces in a continuous medium is realized.

[0035] PEFEN is a novel computational paradigm that deeply integrates physical modeling principles with deep learning architecture. On one hand, it borrows the ideas of Physical Information Neural Networks (PINNs), directly encoding physical equations (such as partial differential equations, PDEs), conservation laws, and symmetry constraints into the loss function to ensure that the model output conforms to known physical laws and can be seamlessly integrated into mainstream deep learning frameworks. On the other hand, it combines the mesh discretization and shape function approximation mechanisms of the Finite Element Method (FEM), using discretized meshes to approximate continuous field variables (such as stress and temperature fields), thus possessing excellent spatial representation capabilities. Unlike the black-box structure of traditional deep neural networks, PEFEN directly constructs a function approximator based on mesh generation and node shape functions from the FEM, treating node degrees of freedom as trainable parameters and optimizing their physical distribution through end-to-end training. Furthermore, PEFEN constructs its loss function based on variational principles (such as the minimum potential energy principle), achieving physical consistency modeling and solving for field variables. However, PEFEN's continuous field prediction characteristics, based on the assumption of continuous differentiability, implicitly assume that it can only handle continuous field problems.

[0036] This study proposes that to handle spatially discontinuous problems such as fault dislocation deformation fields, the fault interface should be preprocessed during the mesh generation stage of the PEFEN framework, i.e., by introducing the node-split mechanism of the finite element method. This explicitly encodes the spatial discontinuity into the node topology, while the assumption of continuous differentiability is still naturally satisfied in subsequent neural network processing. Therefore, the Split-Node Physics-Encoded Finite-Element Network (SN-PEFEN) method overcomes the theoretical limitations of the existing PEFEN framework in handling discontinuous problems. It not only theoretically solves the spatial discontinuity problem but also retains PEFEN's efficient modeling ability for sparse data and the physical consistency of mechanical equilibrium conditions and material constitutive relations.

[0037] The following section details the forward and inverse simulations of fault dislocation deformation fields based on the SN-PEFEN method.

[0038] like Figure 1 As shown, a numerical simulation method for rapidly calculating discontinuous deformation fields of fault dislocations includes the following steps: S1: Constructing the SN-PEFEN physical coding finite element network that integrates node-based technology; The SN-PEFEN includes: introducing a finite element node splitting mechanism into the mesh framework of the physically encoded finite element network PEFEN, processing the fault interface in the preprocessing stage, and explicitly encoding the spatial discontinuity into the node topology to obtain SN-PEFEN. The finite element node splitting mechanism represents the nodes on the fault plane as two sets of nodes with identical position coordinates. The two sets of nodes belong to the elements of the two fault disks respectively, and the relative dislocations occurring on the two fault disks are described by the relative motion of the two sets of corresponding nodes.

[0039] The SN-PEFEN network structure generally consists of four parts: an input layer, a physical constraint layer, a finite element encoding layer, and an optimization solution layer. Specifically, it includes: Input Layer: The input consists of the spatial coordinates (x, y, z) of all nodes in the solution domain, material parameters, and fault geometry information. For fault plane nodes, the input also includes the fault plane normal vector and split identifier to distinguish the topological affiliation of nodes on the two sides of the fault.

[0040] The Physics Constraint Layer embeds governing equations and boundary conditions, including linear elastic equilibrium equations, boundary constraint equations, and fault slip conditions. It automatically calculates the gradients of field variables through differentiation, encoding the mechanical equilibrium conditions and stress continuity conditions into loss function terms to ensure that the network output conforms to the laws of mechanics.

[0041] Finite-Element Encoding Layer: This layer uses finite element shape functions. As a basis function of the network, the node degrees of freedom (displacement vectors) The parameters are considered as trainable parameters. The mapping from discrete nodes to continuous fields is achieved through node shape function interpolation. Unlike traditional PEFEN, SN-PEFEN explicitly includes bidirectional constraints of split-node pairs in this layer, expressing the discontinuous displacement field on the fault plane through the node splitting mechanism.

[0042] Optimization Layer: Based on variational principles, a total energy functional E(u,s) is constructed. Using the displacements of all free nodes as optimization variables, the total energy loss is minimized using L-BFGS or Adam optimizers to achieve forward modeling of the fault dislocation deformation field. In the inversion stage, the sensitivity of the slip parameters is efficiently calculated using the Hessian-vector product and conjugate gradient method, achieving differentiable end-to-end inversion optimization.

[0043] Overall, SN-PEFEN combines the interpretability of finite element methods with the differentiability of neural networks. Compared to traditional PEFEN, it achieves an explicit discontinuous representation of fault interfaces through node topological encoding in its network structure, thereby overcoming the limitations of the continuous differentiable field assumption while ensuring physical consistency.

[0044] S2: Perform forward modeling of fault dislocation deformation field based on the SN-PEFEN; The forward modeling of the fault dislocation deformation field includes the following steps: S21: Determine the governing equations and model boundary conditions for fault dislocations; In the forward modeling of fault dislocation deformation fields based on the SN-PEFEN method, such as Figure 2 As shown, the fault dislocation is treated as the relative motion of two sets of nodes located at the same coordinate positions on the two sides of the fault in the finite element mesh. The governing equations and boundary conditions of the fault dislocation are:

[0045]

[0046]

[0047]

[0048] in, For divergence operators, For stress tensor, For the computational domain, and These are the displacement vector fields on both sides of the fault plane. and These are the slip vectors of the two disks on the fault plane. and These are the boundary regions on both sides of the fault plane. This is the displacement boundary region. For displacement, Describes the domain in which the equation holds true; Based on the linear elastic constitutive and small strain assumptions:

[0049]

[0050] in, For stress components, The first Lamé constant, For strain components, For strain components, For Kronecker notation, when i = j The value is 1 if it is true, and 0 otherwise. This is the second Lamé constant. For strain tensor, superscript It is the transpose symbol; The potential energy functional of the governing equation is:

[0051] in, Let be the potential energy functional of the governing equation.

[0052] S22: Discretize the solution domain into isoparametric elements, and approximate the field variables within each element by interpolating nodal variables and shape functions; For a certain inclusion A point within a cell of nodes ,variable for:

[0053] in, For the first Displacement variables of each node, These are the shape functions for the corresponding nodes, used to interpolate approximate field variables within the element; The potential energy functional is then calculated through numerical integration at Gauss points across all elements:

[0054] in, The determinant is the Gaussian point weights multiplied by the Jacobian matrix. For the first Unit 1 Jacobian matrix at each Gaussian point These are the weighting coefficients for the Gaussian integral. In the first Unit 1 The displacement is obtained by interpolation at Gaussian points.

[0055] S23: Using the displacements of all free nodes as optimization variables, the potential energy functional is minimized using the L-BFGS optimizer to obtain the forward modeling results of the fault dislocation deformation field in equilibrium.

[0056] This scheme involves the displacement of all free nodes. As optimization variables, fault dislocations and other Dirichlet boundary conditions are applied using a fixed mask. Based on the principle of energy minimization, this invention employs an L-BFGS optimizer to minimize this functional. Thus, the equilibrium solution is obtained. Because It is a strictly convex function with a unique minimum value. The optimization process is theoretically stable and does not have local optima, so it can converge quickly.

[0057] S3: Based on the SN-PEFEN, perform fault dislocation deformation field inversion simulation to realize numerical simulation of discontinuous fault dislocation deformation field.

[0058] In the inversion simulation of fault dislocation deformation fields based on the SN-PEFEN method, this invention constructs an end-to-end differentially perturbable solution framework, sequentially performing forward solving and backpropagation operations in each optimization iteration. Through multiple iterations, the residual between the predicted and observed surface deformation results is ultimately made smaller than the acceptable error, thus obtaining the optimal fault dislocation distribution results, such as... Figure 3 As shown.

[0059] The fault dislocation deformation field inversion simulation includes a forward solution stage and a backward solution stage; the forward solution stage is consistent with the fault dislocation deformation field forward simulation. In the forward solution phase, the framework constructs discontinuous boundary conditions based on the currently predicted fault slip distribution and solves for the full-field displacement under static equilibrium. This process is consistent with the forward simulation, ensuring physical consistency between fault slip and surface deformation response. Subsequently, the simulated surface displacement is compared with the observed values, and the residuals are used as data fitting terms in the total loss function.

[0060] The inverse solution stage uses the Hessian-vector product and conjugate gradient method to solve for the gradient of the slip pair on the total system energy and the observation residuals, thereby enabling sensitivity analysis and optimization updates of the slip distribution. Specifically, the goal of the inverse problem is to determine the optimal slip distribution by minimizing the total error functional (objective function). for:

[0061] in, For the displacement of the observation point, To predict displacement for the model, For the slip parameter, For hyperparameter weights, For regularization terms; Since the displacement field needs to satisfy the mechanical equilibrium condition in each inversion iteration step, the displacement field can be regarded as an implicit function of the slip parameter:

[0062] in, It is an implicit function. It is an energy functional; At this point, the overall objective function can be expressed as the glide parameter. If the objective function is an explicit function, then the objective function is expressed as:

[0063] To optimize slip parameters Calculate its gradient using the chain rule:

[0064] in, Since it is not easy to calculate directly, we solve it using implicit function differential theory: From the equilibrium conditions, we obtain the stationary point equations:

[0065] right Taking the differential, we get:

[0066] therefore:

[0067] in, The objective function is... In practical calculations, due to the Hessian matrix The scale is enormous, and direct inversion is extremely resource-intensive. Therefore, we adopted the Hessian-vector product (HVP) method, the core of which is: For any vector ,calculate:

[0068] in, For displacement variables The Hessian matrix; The above calculations are efficiently performed using PyTorch's automatic differentiation, avoiding the explicit construction of the Hessian matrix. Furthermore, to solve the linear system in the implicit derivatives above, we employ the conjugate gradient method (CG), which does not require an explicit Hessian matrix. The CG algorithm is used to solve the system of linear equations:

[0069] The CG method only requires Hessian-vector product operations in each iteration without explicitly storing the Hessian matrix, thus greatly improving computational efficiency.

[0070] Finally, the slip parameters The gradient can be represented by the adjoint vector. Represented as:

[0071] in, This is the adjoint vector.

[0072] The above methods enable efficient computation of gradients for complex inverse problems, allowing for optimization of the glide parameters using gradient descent methods (such as Adam or LBFGS) to achieve fast and stable inversion.

[0073] In one embodiment of the present invention, numerical experiments on forward and inverse modeling of two-dimensional and three-dimensional deformation fields caused by fault dislocations were conducted based on the SN-PEFEN method.

[0074] like Figure 4 As shown, the dislocations on the fault are all 1m. Models a and b have a Young's modulus of 80 GPa and a Poisson's ratio of 0.25; model c has a Young's modulus of 1 GPa for depths of 0-5 km, 80 GPa for depths of 5-10 km, and 160 GPa for depths of 10-50 km, with a Poisson's ratio of 0.25 for the entire model. The boundary conditions for all three models are: free surface, tangential free at other boundaries, and zero normal displacement. Finite element simulations with the same mesh, dislocations, and boundary conditions were performed using the commercial finite element software ABAQUS. This embodiment compares the computational accuracy and efficiency of SN-PEFEN with the traditional finite element method (FEM).

[0075] The forward modeling results of the three models using the SN-PEFEN method all showed computational accuracy comparable to that of the FEM method. The deformation field modes simulated by both methods were consistent, and the residuals near the fault dislocations with the cleavage node setting were less than 1%. In terms of computational efficiency, the two-dimensional homogeneous medium dislocation model 'a' contains 4724 two-dimensional quadrilateral elements with a total of 9796 degrees of freedom, of which 524 are fixed. The computation times for the SN-PEFEN and FEM methods were 3s and 4s respectively, which are roughly the same. Figure 4 (a) The three-dimensional homogeneous medium tomographic model b contains 500,000 three-dimensional hexahedral elements with a total degree of freedom of 1,561,446, of which 91,743 are fixed. The SN-PEFEN calculation took 41 seconds, while the finite element method took 1831 seconds. The SN-PEFEN method is nearly 42 times faster than the FEM method. Figure 4 (b) Similarly, the three-dimensional layered medium fault dislocation model c has the same mesh structure as model b. For model c, the SN-PEFEN and FEM methods take 42s and 1835s to calculate the deformation field, respectively. Figure 4 (c) It should be noted that when the differences in material parameters in a non-homogeneous medium model are very significant, the number of iterations required for the prediction model to converge is greater. It should be emphasized that, in principle, because... It is a strictly convex function with a unique local optimum with a minimum value. Therefore, the SN-PEFEN forward modeling problem does not have a local optimum, and its optimization process can theoretically converge stably.

[0076] In summary, in the calculation of mesh models with more than one million degrees of freedom, with comparable simulation accuracy, SN-PEFEN improves the computational efficiency by tens of times compared to the FEM method, demonstrating a significant advantage in computational efficiency.

[0077] like Figure 5 As shown, in the inversion experiment, the analytical solution of the surface deformation field caused by the three-dimensional fault dislocation was first calculated using the Okada dislocation model. The fault in the Okada model was set as a vertical strike-slip fault, with the fault plane center as the center, and the two blocks exhibiting an elliptical relative slip with a maximum slip of 1m, decreasing outwards from the center. Figure 5 (a)). The surface deformation field of 50 x 50 km is obtained from the Okada dislocation model. Figure 5 (b) is used as the observation data for the numerical experiment of the SN-PEFEN method inversion.

[0078] Utilize Figure 4 The mesh structure of the three-dimensional homogeneous medium fault model in (c) is shown. The model contains 500,000 three-dimensional hexahedral elements with a total degree of freedom of 1,561,446, of which 91,743 are fixed. The fault plane cuts through a 10 km thick crust and is 20 km long. Numerical experiments on the inversion of three-dimensional fault dislocations using the SN-PEFEN method were conducted according to the inversion steps. The results show that after approximately 60 iterations, the model loss decreased to a relatively stable value. Figure 5 (c) indicates that the inversion converges at this level of accuracy. The fault slip distribution obtained from the inversion in the 100th iteration shows ( Figure 5 In the middle (d), the maximum slip is 0.92m, located at the center of the fault plane; the slip distribution also decreases from the center outwards, showing good similarity to the slip distribution of the fault plane set in the Okada forward model. From this fault displacement, the surface deformation field is obtained ( Figure 5 In the middle (e), the maximum residual is within 0.007m compared with the observation results.

[0079] This embodiment utilizes the SN-PEFEN inversion framework with approximately 1.5 million degrees of freedom to perform 60 iterations of inversion calculations, taking about 2000 seconds, which is comparable to the time required for a single forward modeling using the FEM method with the same number of degrees of freedom. The results of this embodiment also further verify PEFEN's ability to handle inversion problems.

[0080] Those skilled in the art will recognize that the embodiments described herein are intended to help the reader understand the principles of the invention, and should be understood that the scope of protection of the invention is not limited to such specific statements and embodiments. Those skilled in the art can make various other specific modifications and combinations based on the technical teachings disclosed in this invention without departing from the spirit of the invention, and these modifications and combinations are still within the scope of protection of the invention.

Claims

1. A numerical simulation method for rapidly calculating discontinuous deformation fields of fault dislocations, characterized in that, Includes the following steps: S1: Constructing the SN-PEFEN physical coding finite element network that integrates node-based technology; S2: Perform forward modeling of fault dislocation deformation field based on the SN-PEFEN; S3: Based on the SN-PEFEN, perform fault dislocation deformation field inversion simulation to realize numerical simulation of discontinuous fault dislocation deformation field.

2. The numerical simulation method for rapidly calculating the discontinuous deformation field of fault dislocations according to claim 1, characterized in that, The SN-PEFEN includes: introducing a finite element node splitting mechanism into the mesh framework of the physically encoded finite element network PEFEN, processing the fault interface in the preprocessing stage, and explicitly encoding the spatial discontinuity into the node topology to obtain SN-PEFEN. The finite element node splitting mechanism represents the nodes on the fault plane as two sets of nodes with identical position coordinates. The two sets of nodes belong to the elements of the two fault disks respectively, and the relative dislocations occurring on the two fault disks are described by the relative motion of the two sets of corresponding nodes.

3. The numerical simulation method for rapidly calculating the discontinuous deformation field of fault dislocations according to claim 1, characterized in that, The forward modeling of the fault dislocation deformation field includes the following steps: S21: Determine the governing equations and model boundary conditions for fault dislocations; S22: Discretize the solution domain into isoparametric elements, and approximate the field variables within each element by interpolating nodal variables and shape functions; S23: Using the displacements of all free nodes as optimization variables, the potential energy functional is minimized using the L-BFGS optimizer to obtain the forward modeling results of the fault dislocation deformation field in equilibrium.

4. The numerical simulation method for rapidly calculating the discontinuous deformation field of fault dislocations according to claim 3, characterized in that, The governing equations and model boundary conditions for the fault dislocations are as follows: in, For divergence operators, For stress tensor, For the computational domain, and These are the displacement vector fields on both sides of the fault plane. and These are the slip vectors of the two disks on the fault plane. and These are the boundary regions on both sides of the fault plane. This is the displacement boundary region. For displacement, Describes the domain in which the equation holds true; Based on the linear elastic constitutive and small strain assumptions: in, For stress components, The first Lamé constant, For strain components, For strain components, For Kronecker notation, when i = j The value is 1 if it is true, and 0 otherwise. This is the second Lamé constant. For strain tensor, superscript It is the transpose symbol; The potential energy functional of the governing equation is: in, Let be the potential energy functional of the governing equation.

5. The numerical simulation method for rapidly calculating the discontinuous deformation field of fault dislocations according to claim 4, characterized in that, The solution domain is discretized into isoparametric elements, and the field variables are approximated within each element through nodal variables and shape function interpolation. For a given domain containing... A point within a unit of nodes ,variable for: in, For the first Displacement variables of each node, These are the shape functions for the corresponding nodes, used to interpolate approximate field variables within the element; The potential energy functional is then calculated through numerical integration at Gauss points across all elements: in, The determinant is the Gaussian point weights multiplied by the Jacobian matrix. For the first Unit 1 Jacobian matrix at each Gaussian point These are the weighting coefficients for the Gaussian integral. In the first Unit 1 The displacement is obtained by interpolation at Gaussian points.

6. The numerical simulation method for rapidly calculating the discontinuous deformation field of fault dislocations according to claim 5, characterized in that, The fault dislocation deformation field inversion simulation includes a forward solution stage and a backward solution stage; the forward solution stage is consistent with the fault dislocation deformation field forward simulation. The inverse solution stage uses the Hessian-vector product and conjugate gradient method to solve for the gradient of the slip pair on the total system energy and the observation residuals, thereby enabling sensitivity analysis and optimization updates of the slip distribution. The optimal slip distribution is determined by minimizing the total error functional; the objective function is... for: in, For the displacement of the observation point, To predict displacement for the model, For the slip parameter, For hyperparameter weights, For regularization terms; Since the displacement field needs to satisfy the mechanical equilibrium condition in each inversion iteration step, the displacement field can be regarded as an implicit function of the slip parameter: in, It is an implicit function. It is an energy functional; The objective function is then expressed as: To optimize slip parameters Calculate its gradient using the chain rule: in, The objective function is... The glide parameters are obtained using the Hessian-vector product and conjugate gradient methods. The gradient is: in, This is the adjoint vector.

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