Phase-field fracture modeling method based on physical information neural network

Abstract

The application discloses a physical information neural network-based phase field fracture forward and reverse integrated modeling method, which comprises the following steps: setting initial parameters and performing adaptive grid division to obtain an observed displacement field and phase field data; constructing a physical information neural network of a predicted displacement and a phase field, and designing a loss function containing a total variation energy and an observed residual error; at an initial time step, taking a phase field length scale parameter as a to-be-identified variable, training the network in combination with observed data and physical laws, and reversely identifying an optimal parameter; at subsequent time steps, fixing the parameter, and adopting a time step strategy to forwardly predict a displacement field and phase field evolution in a crack propagation process. The application solves the prediction deviation caused by the dependence of a traditional phase field model on experience preset parameters, realizes the cooperation of parameter inversion and fracture forward prediction, and improves the physical consistency and accuracy of simulation.

Description

Technical Field

[0001] This invention relates to the field of computational solid mechanics and artificial intelligence, specifically to a phase-field fracture integrated modeling method based on physical information neural networks. Background Technology

[0002] Fracture is one of the most common failure modes in engineering structures. Its occurrence can lead to a decrease in structural load-bearing capacity or even sudden failure, posing a severe challenge to the safety and reliability of fields such as aerospace, civil engineering, and energy equipment. Accurately predicting the fracture behavior of structures and assessing their safety status is a core requirement for avoiding catastrophic accidents and ensuring the reliable operation of engineering structures.

[0003] Numerical analysis methods are the primary means of structural fracture research, currently mainly divided into discrete methods (such as the finite element method) and continuous methods (such as the phase field method). Among them, the phase field method, as a typical representative of continuous modeling methods, is based on the variational form of Griffith fracture theory. By introducing damage parameters, it represents sharp cracks as diffuse bands with finite widths, avoiding the difficulty of explicitly tracing complex crack topologies. Its core advantage lies in its ability to naturally describe the initiation, propagation, branching, and merging processes of cracks, and it drives the evolution of phase field variables through the principle of energy minimization. It has been widely applied to modeling brittle fracture, fatigue fracture, and multiphysics coupled fracture problems. However, the numerical solution of traditional phase field models highly depends on empirical parameters (such as length scale parameters). The selection of these parameters directly affects the accuracy of crack path prediction: excessively large parameters lead to overly smooth crack propagation paths, failing to capture local features; excessively small parameters significantly increase computational costs and reduce numerical stability, even leading to non-physical force and displacement curve predictions. Therefore, how to achieve automatic identification of length scale parameters through the integration of data-driven and physical constraints has become a key issue in improving the practicality of phase field models.

[0004] In recent years, Physics-Informed Neural Networks (PINNs) have provided a new approach to solving this problem. By embedding physical laws (such as conservation equations and energy minimization principles) into the neural network training process, PINNs can solve partial differential equations under meshless conditions and support the fusion of experimental data or high-fidelity simulation data. Although PINN-based phase-field fracture models have made progress, existing methods still have the following limitations: (1) The length scale parameter is usually set as a preset constant, lacking a dynamic adjustment mechanism based on observation data, which makes the model prediction results sensitive to the initial values ​​of the parameters; (2) The irreversibility of crack evolution is not fully considered during the training process, which easily leads to non-physical phenomena such as crack healing; (3) Multi-timestep crack propagation simulation requires repeated training of the network, resulting in low computational efficiency. Therefore, developing a unified forward and reverse integrated modeling method that can realize parameter inversion and structural response prediction is of great significance for promoting the practical application of phase-field fracture models in engineering. Summary of the Invention

[0005] To address the aforementioned problems, this invention aims to provide an integrated forward and inverse phase-field fracture modeling method based on Physics-informed Neural Networks (PINNs). More specifically, it constructs a neural network-based crack propagation solution for cracked structures. This method integrates deep learning and fracture mechanics theory to build a unified framework for structural response prediction (forward problem) and model parameter inversion (inverse problem). It focuses on solving the challenge of collaborative modeling of physical constraints and data-driven approaches in predicting the fracture behavior of engineering structures. This method can be widely applied to the fracture safety assessment and life prediction of key components in aerospace, energy equipment, civil engineering, and other fields.

[0006] The technical solution adopted in this invention is: a phase-field fracture integrated forward and reverse modeling method based on physical information neural networks, comprising the following steps: Step 1: Perform adaptive mesh generation based on the geometric characteristics of the structure to be analyzed, and generate Gaussian integral sample points; Step 2: Acquire observation data of a local region of the structure at the initial time step. The observation data includes displacement field data and phase field data. Step 3: Construct a physical information neural network, which takes spatial coordinates as input and outputs displacement field components and phase field variables; Step 4: Construct a total loss function that integrates physical laws and observation data, and set the length scale parameter in the phase-field model as a trainable variable; Step 5: At the initial time step, train the neural network based on the total loss function, while optimizing the network parameters and reverse-identifying the length scale parameters; Step 6: In subsequent time steps, fix the length scale parameters identified in step 5, use the network parameters from the previous time step as initial values, and predict the crack propagation process in a positive direction based on the principle of minimizing physical energy.

[0007] Furthermore, in step 3, the output of the neural network is modified to satisfy the displacement boundary conditions: in For the distance function constructed based on the boundary conditions, the input variables are... It is a vector form of two-dimensional spatial coordinate points. For neural networks NN The parameters, u and v These represent the lateral and longitudinal components of the displacement, respectively. Represents the phase field.

[0008] Furthermore, in step 4, the loss function is a weighted average of physical energy loss and data observation loss. The total loss function is defined as follows: in, For neural network parameters, The length scale parameter to be identified, These are the weighting coefficients.

[0009] The physical energy loss Total Energy Functional Based on Phase Field Fracture Theory Build: in, It is the elastic strain energy. External potential energy Let u represent the fracture surface energy, and u represent the displacement field vector. The fracture surface energy... Explicitly includes length scale parameters Its expression is: in, The critical energy release rate. For phase field variables, The problem domain is defined as the data observation loss. Constructed based on the mean square error between predicted and observed values, used to incorporate observed data to guide parameter identification: in, For the number of observation points, and Observation points The displacement field and phase field observations at the location are used. To accurately simulate the irreversibility of crack propagation (i.e., to prevent crack healing during unloading or compression), a strain energy history function is introduced. , is used to record the maximum tensile strain energy density that a material experiences during its loading history, and is defined as follows: in, This represents the tensile strain energy density. When calculating the total energy or solving for the phase field evolution, the history function is used. Replace the current tensile strain energy density to drive the phase field The monotonic evolution. For structures containing initial cracks, it is necessary to define the initial strain energy history field. To characterize the initial crack state: in, For spatial points To the initial crack line l The closest distance; B Scalar parameters used to control the initial historical field intensity.

[0010] Furthermore, in step 6, at subsequent time steps The optimal length scale parameter identified in step 5 is fixed. It is no longer used as a trainable variable, and the observed data residual term is removed from the loss function (i.e., let it be). ), only minimizing the physical energy functional For the goal.

[0011] Considering the irreversibility and nonlinear characteristics of crack evolution, this invention adopts a time-increment training strategy, which is implemented as follows: Discretize the entire loading process as follows A series of consecutive time steps, each time step applying a constant displacement increment. or load increment .

[0012] In the time steps ( At the start of training, the weights and bias parameters of the neural network Instead of random initialization, it directly inherits the optimal parameters from the previous time step. , This parameter inheritance strategy leverages the continuity of crack evolution, enabling the network to retain the crack topology and displacement field features learned in the previous time step, thereby significantly reducing the number of training iterations at the current time step and accelerating convergence.

[0013] During training at each time step, a hybrid optimization algorithm is employed to balance computational efficiency and accuracy: First, the Adam optimizer is used. The Adam optimizer has excellent global search capabilities, quickly reducing the loss function value and allowing the network parameters to escape local minima. Then, training is switched to the L-BFGS optimizer. The L-BFGS optimizer utilizes second-order derivative information, exhibiting faster convergence and higher computational accuracy near the optimal solution, ensuring that the physical energy residuals meet strict convergence criteria.

[0014] At the end of each time step, update the strain energy history field. Record the maximum tensile strain energy that the material experiences during its loading history to prevent cracks from healing during unloading or compression, ensuring the physical irreversibility of crack evolution: [1] in t For the iteration step, For tensile strain energy density, For strain energy, and The first t -1 iteration step and the first t The strain energy history function for the nth iteration step. The above formula represents the strain energy history function for the nth iteration step. t The strain energy history function of each iteration step is the maximum value of the strain energy history function of the previous iteration step and the tensile strain energy of the current iteration step.

[0015] Furthermore, a phase-field fracture integrated forward and reverse modeling method based on physical information neural networks also includes the following steps: First, set the displacement increment step and training time step, and apply the displacement boundary conditions at time step t. Then, a physical information neural network for the cracked specimen is constructed. The input of the network is spatial coordinates, and the transverse displacement, longitudinal displacement and phase field are predicted through a densely connected network. Next, the error between the observed data and the neural network prediction data is used to embed the error into the loss function to correctly identify the specimen length scale; Finally, a positive phase field fracture model is established based on the identified length scale to predict the displacement field and phase field during crack propagation, thereby representing the crack propagation path.

[0016] This invention proposes a phase-field fracture integrated forward and inverse modeling method based on physical information neural networks. By unifying structural response prediction (forward problem) and parameter inversion (inverse problem) within the PINNs framework, the following technical effects are achieved: This invention sets the length scale parameter as a trainable variable and automatically identifies the optimal parameter using a small amount of observational data (such as displacement field and phase field) by constructing a total loss function that integrates physical energy constraints and data observation constraints. This method overcomes the dependence of traditional phase field models on empirical parameters, ensures that the model conforms to the basic laws of fracture mechanics by minimizing the physical energy loss term, and significantly improves the model's predictive ability for real fracture behavior by introducing experimental or high-fidelity simulation data through the data observation loss term.

[0017] This invention introduces a strain energy history function to record the maximum tensile strain energy density during the material loading history and replaces the current strain energy density in the total energy calculation. This effectively prevents the non-physical phenomenon of crack healing during unloading or compression. Furthermore, for structures containing initial cracks, an initial strain energy history field based on spatial distance is defined to ensure accurate characterization of the initial crack state, further enhancing the model's ability to simulate complex fracture processes.

[0018] This invention employs a time-increment training strategy, discretizing the loading process into multiple time steps. In subsequent time steps, the identified optimal length scale parameter is fixed, while the network weights from the previous time step are inherited as initial values. This strategy fully leverages the continuity of crack evolution, allowing the network to retain the crack topology and displacement field features learned in the previous time step, significantly reducing the number of training iterations (experiments show that the number of iterations in subsequent time steps can be reduced to 13.3% of the initial step), thus accelerating model convergence. Furthermore, the combined use of a hybrid optimization algorithm (Adam + L-BFGS) balances global search capability with local fine-tuning optimization, ensuring that the physical energy residual meets strict convergence criteria, thereby achieving high-precision prediction of crack propagation paths.

[0019] This invention improves the calculation accuracy of stress concentration regions by adaptively dividing the mesh according to the structural geometry and increasing the sampling points near the crack tip and expected propagation path. Simultaneously, by introducing a distance function to hard-constrain the neural network output, it enforces the satisfaction of displacement boundary conditions, avoiding the complexity of weight coefficient adjustment in traditional soft-constraint methods and enhancing the model's applicability under complex boundary conditions. Attached Figure Description

[0020] Figure 1 This is a schematic diagram of the logical principle of the present invention; Figure 2 This is a simplified diagram of the phase field fracture problem in this invention; Figure 3 These are the single-sided crack specimens and training sample points under tensile load in this invention; Figure 4 This is a schematic diagram of the convergence process of the loss function and length scale at each time step in this invention; Figure 5This is a schematic diagram comparing the calculation results and errors of the method proposed in this invention and the traditional PINNs-based phase field fracture method. Detailed Implementation

[0021] The technical solution of the present invention will be described in more detail below with reference to the accompanying drawings. The present invention includes, but is not limited to, the following embodiments: like Figure 1-5 As shown, where Figure 1-2 This paper provides the basic concepts and background of phase-field fracture problems, demonstrating how the phase-field method can be used to simulate crack initiation, propagation, branching, and merging processes. By introducing damage parameters, the phase-field method represents sharp cracks as diffuse bands with finite widths, thus avoiding explicit tracing of complex crack topologies. Figure 3 These are specimens with single-sided cracks under tensile load and training sample points, among which... Figure 3 (a): This illustrates the specific geometric dimensions and boundary conditions of a specimen with a single-sided crack subjected to tensile load. The specimen has a side length of L = 1 mm and a crack length of a = 0.5 mm, illustrating the specific problem scenario addressed by this invention. Figure 3 (b) illustrates the distribution of Gaussian sampling points obtained on the cracked specimen through adaptive sampling, particularly the dense distribution of sampling points near the crack tip to accurately calculate the total potential energy. Meanwhile, the yellow area in the figure represents a local observation area used to randomly select sample points as observation data to guide the identification of length-scale parameters. Figure 4 This represents the convergence process of the loss function and length scale at each time step, where... Figure 4 (a): This illustrates the iterative convergence process of the length scale parameter and the loss function at the initial time step (t=1). As can be seen from the figure, the length scale parameter eventually converges from the initial value of 0.005mm to 0.063mm, while the loss function also gradually decreases to a convergent state.

[0022] Figure 4 (b)- Figure 4 (d): This shows the iterative convergence process of the loss function alone at subsequent time steps (t=2, t=3, t=4). Since the training at subsequent time steps uses the network weights converged at the previous time step as initial values, the loss function exhibits faster convergence during the iteration process. Figure 5 The calculation results and error comparison diagram of the proposed method and the traditional PINNs-based phase-field fracture method are presented, in which... Figure 5 (ab) shows the longitudinal displacement field and phase field predicted by the finite element method (FEM), used as the true solution for comparison. Figure 5 (cf) shows the longitudinal displacement field and phase field predicted by the proposed method and their errors. The comparison demonstrates that the proposed method achieves good consistency with the actual solution in terms of displacement field prediction. Figure 5(gj): This section presents the longitudinal displacement field and phase field predicted by the traditional PINNs phase-field fracture method, along with their errors. A comparison reveals significant deviations in the accuracy of the traditional method, particularly in predicting crack propagation paths. The following detailed examples illustrate these findings. Example 1 This invention proposes an integrated forward and inverse phase-field fracture modeling method based on Physical Information Neural Networks (PINNs). This method unifies structural response prediction (forward problem) and parameter inversion (inverse problem) into a single framework. The specific implementation steps are as follows: Step 1, Parameter Initialization and Adaptive Mesh Generation First, an adaptive mesh is generated based on the geometric characteristics of the structure to be analyzed (such as crack location and boundary shape). The problem domain is then defined. Divided into Each sub-region is arranged with [the following] elements. Several Gaussian integration points are used for subsequent calculations of the variational energy integral. Specifically, sampling points are adaptively refined near the crack tip and the expected propagation path to accurately capture stress concentration characteristics. Simultaneously, the phase field length scale parameter is initialized. (Usually, empirical values ​​are taken, such as 1 / 10 of the structural geometric features), and a neural network is constructed, using the Xavier method to initialize the network parameters.

[0023] Step 2, Obtain observation data In the initial stage of the loading process (initial time step) This involves obtaining observational data of local regions of the structure. Including the spatial coordinates of the observation point and the corresponding displacement field and phase field : These data can be obtained through experimental measurements using digital image correlation (DIC) techniques or extracted through high-fidelity numerical simulations.

[0024] Step 3, Construct a physical information neural network Construct a fully connected deep neural network, with spatial coordinates as the input. The output is the displacement field components. and phase field variables To enforce the satisfaction of displacement boundary conditions, a distance function is introduced. Hard constraint modification of network output: in, For neural networks NN The parameters, This is the original network output.

[0025] Step 4: Construct the integrated positive and negative loss function Construct a total loss function that includes physical energy constraints and data observation constraints. : in, For neural network parameters, The length scale parameter to be identified, Weighting coefficients. Physical energy loss. Total Energy Functional Based on Phase Field Fracture Theory Build: in, It is the elastic strain energy. External potential energy The fracture surface energy. Explicitly includes length scale parameters Its expression is: in, The critical energy release rate. For phase field variables, Let this be the problem domain. It is the Neumann boundary The traction force on the surface is f, where f is the body force. and These are the tensile strain energy density function and the compressive strain energy density function, respectively, which can be obtained by spectral decomposition of the strain energy tensor; This is a monotonically decreasing stress degradation function used to describe the decrease in material stiffness as damage evolves. Common stress degradation functions are as follows: The data observation loss Constructed based on the mean square error between predicted and observed values, used to incorporate observed data to guide parameter identification. in, For the number of observation points, and Observation points The displacement field and phase field observations at the location.

[0026] Step 5, Reverse Parameter Identification (Initial Time Step) At the initial time step , length scale parameter Let them be trainable variables. Using the observed data, minimize the total loss function. At the same time, optimize the neural network parameters and length scale parameters The Adam optimizer is used for pre-training, followed by fine-tuning using the L-BFGS optimizer until the loss function converges, thereby automatically identifying the optimal length scale parameter. .

[0027] Step 6, Forward Crack Evolution Prediction (Subsequent Time Steps) In subsequent time steps The length scale parameters identified in step 5 are fixed. The data remains unchanged, and no further observation data is introduced (i.e.) The network parameters trained in the previous time step are used as the initialization parameters for the current time step (weight inheritance), with the sole aim of minimizing physical energy loss. The network is trained for the target, thereby positively predicting the initiation and propagation paths of cracks.

[0028] like Figure 3 As shown, this embodiment illustrates a specimen with a single-sided crack subjected to tensile load.

[0029] In this embodiment, the side length of the specimen is L =1mm, crack length is a =0.5mm, the Lamer constant of the specimen is , fracture energy Initial length scale l 0 is taken as 0.005 mm. The boundary conditions for the specimen are: no longitudinal displacement at the lower boundary, no lateral displacement at the left boundary, a longitudinal displacement boundary applied to the upper boundary, and the remaining edges are free boundaries, unaffected by external loads. The displacement increment step is set. Set the training time step t =[1,5], then the displacement boundary The origin of the Cartesian coordinate system is as follows: Figure 3 As shown, at time step t The time-displacement boundary conditions are as follows: in u and v The elastic field is respectively in x and y The displacement components in the direction.

[0030] A neural network for constructing the physical information of a cracked specimen is established, with spatial coordinates as the input. x ,y ), predicting lateral displacement through a densely connected network. u Longitudinal displacement v and phase field The network has 8 hidden layers, each with 20 neurons. During training, a length scale is assumed. l 0 is the variable, and the observed data are the displacement field and phase field at the crack tip. The specimen length scale is correctly identified by embedding the errors of the observed data and the neural network prediction data into the loss function. To satisfy the Dirichlet boundary conditions, the solved displacement field can be rewritten as follows: in and The output of the neural network, u and v The modified neural network output is designed to satisfy the boundary conditions. In traditional PINN-based phase-field fracture modeling, the length scale parameter is typically taken as a value much smaller than the specimen size. This example uses an initial length scale... l Let's take 0 = 0.005mm as an example for explanation. Figure 3 As shown in (b), the blue dots represent 58,800 Gaussian sampling points obtained on the cracked specimen through adaptive sampling. These sampling points are densely distributed near the crack tip, which is beneficial for accurately calculating the total potential energy of the cracked specimen. The yellow area represents the area centered at the crack tip with a size of... Within a local observation area, 200 sample points were randomly selected as observation data. t When the phase field is 1, the displacement and phase field at the observed sample point are used as the observation data. The PINN phase field fracture model not only requires Gaussian sampling points to calculate the total energy of the specimen, but also relies on the sample point data of the observation area to identify the length scale parameters. Figure 4 The convergence process of the loss function and length scaling parameters during training at different time steps is demonstrated. Only in [specific time step] is the convergence process shown. t When =1, the length scale parameter was identified, therefore only Figure 4 (a) contains length-scale iteration curves. Figure 4 (b)- Figure 4 (d) only shows the convergence process of the loss function. For example... Figure 4 As shown in (a), when the initial length scale of the phase field model... l When 0 = 0.005mm, the final convergence is 0.063mm, and the total number of training iterations is set to 15000. Since subsequent time steps use the network weights after convergence from the previous time step as initial values, therefore... Figure 4 (b)- Figure 4The loss function in (d) exhibits faster convergence during the iteration process, with the total number of training iterations set to 2000. This is based on the identified length scale. A positive phase-field fracture model is established to predict the displacement and phase fields during crack propagation, representing the crack propagation path. High-precision finite element simulation results are used as the true solution; Figure 5 shows the crack propagation path when the length scale... l The results of traditional PINNs phase-field fracture modeling at 0 = 0.005 mm, and the results when the length scale is... The proposed method yields displacement field prediction results and errors. The results show that the proposed method achieves good agreement with the actual solution in displacement field prediction.

[0031] Taking a single-sided crack specimen as an example, this invention identifies the optimal length scale parameter (converging from an initial value of 0.005 mm to 0.063 mm) using data from 200 observation points in the initial time step, and then predicts the crack propagation path based on this parameter in subsequent time steps. Compared with the traditional PINNs method, the displacement field predicted by this invention shows higher consistency with the high-fidelity finite element results (error reduced to less than 1 / 3 of the traditional method), and the training efficiency is significantly improved (the number of iterations in subsequent time steps is reduced by 86.7%), verifying the effectiveness and practicality of the method.

[0032] This invention is not limited to the specific embodiments described above. Those skilled in the art can implement this invention using various other specific embodiments based on the disclosed content of the embodiments and accompanying drawings. Therefore, any design that adopts the design structure and concept of this invention and makes some simple changes or modifications falls within the protection scope of this invention.

Claims

1. A phase-field fracture integrated forward and reverse modeling method based on physical information neural networks, characterized in that, Includes the following steps: Step 1: Perform adaptive mesh generation based on the geometric characteristics of the structure to be analyzed, and generate Gaussian integral sample points; Step 2: Acquire observation data of a local region of the structure at the initial time step. The observation data includes displacement field data and phase field data. Step 3: Construct a physical information neural network, which takes spatial coordinates as input and outputs displacement field components and phase field variables; Step 4: Construct a total loss function that integrates physical laws and observation data, and set the length scale parameter in the phase-field model as a trainable variable; Step 5: At the initial time step, train the neural network based on the total loss function, while optimizing the network parameters and inversely identifying the length scale parameters; Step 6: In subsequent time steps, fix the length scale parameters identified in step 5, use the network parameters from the previous time step as initial values, and predict the crack propagation process in a positive direction based on the principle of minimizing physical energy.

2. The integrated forward and reverse phase-field fracture modeling method based on physical information neural networks according to claim 1, characterized in that, In constructing a physical information neural network, the output of the neural network is modified to satisfy the displacement boundary conditions. The modification formula is as follows: in For the distance function constructed based on the boundary conditions, the input variables are... It is a vector form of two-dimensional spatial coordinate points. For neural networks NN The parameters. u and v These represent the lateral and longitudinal components of the displacement, respectively. Represents the phase field.

3. The integrated forward and reverse phase-field fracture modeling method based on physical information neural networks according to claim 1, characterized in that, The total loss function is a weighted average of physical energy loss and data observation loss, and is defined as follows: in, For neural network parameters, The length scale parameter to be identified, These are the weighting coefficients; The physical energy loss is constructed based on the total energy functional of phase-field fracture theory, including elastic strain energy, external potential energy, and fracture surface energy, i.e., the physical energy loss. Total Energy Functional Based on Phase Field Fracture Theory Build: in, It is the elastic strain energy. External potential energy denoted as the fracture surface energy; u represents the displacement field vector. The data observation loss is constructed based on the mean square error between the predicted and observed values, and is used to introduce observation data to guide parameter identification.

4. The integrated forward and reverse phase-field fracture modeling method based on physical information neural networks according to claim 3, characterized in that, The fracture surface energy Explicitly includes length scale parameters Its expression is in, The critical energy release rate. For phase field variables, Let this be the problem domain.

5. The integrated forward and reverse phase-field fracture modeling method based on physical information neural networks according to claim 3, characterized in that, The data observation loss Constructed based on the mean square error between predicted and observed values, used to incorporate observed data to guide parameter identification: in, The number of observation points. and Observation points The displacement field and phase field observations at the location.

6. The integrated forward and reverse phase-field fracture modeling method based on physical information neural networks according to claim 1, characterized in that, In the positive prediction of crack propagation, a strain energy history function is introduced to record the maximum tensile strain energy density that the material experiences during the loading history, so as to drive the monotonic evolution of the phase field and prevent cracks from healing during unloading or compression.

7. The integrated forward and reverse phase-field fracture modeling method based on physical information neural networks according to claim 6, characterized in that, The introduction of strain energy history function , is used to record the maximum tensile strain energy density that a material experiences during its loading history, and is defined as follows: in, This represents the tensile strain energy density. When calculating the total energy or solving for the phase field evolution, the history function is used. Replace the current tensile strain energy density to drive the phase field The monotonous evolution.

8. The integrated forward and reverse phase-field fracture modeling method based on physical information neural networks according to claim 1, characterized in that, In subsequent time steps, the identified optimal length scale parameter is fixed and no longer used as a trainable variable. The observation data residual term in the loss function is removed, and the goal is to minimize the physical energy functional.

9. A phase-field fracture integrated forward and reverse modeling method based on a physical information neural network according to any one of claims 1 to 8, characterized in that, The method is applied to specimens with single-sided cracks subjected to tensile loads. For structures containing initial cracks, it is necessary to define the initial strain energy history field. To characterize the initial crack state: in, For spatial points To the initial crack line l The closest distance; B Scalar parameters used to control the initial historical field intensity.

10. The integrated forward and reverse phase-field fracture modeling method based on a physical information neural network according to claim 9, characterized in that, It also includes the following steps: First, set the displacement increment step and training time step, and apply the displacement boundary conditions at time step t. Then, a physical information neural network for the cracked specimen is constructed. The input of the network is spatial coordinates, and the transverse displacement, longitudinal displacement and phase field are predicted through a densely connected network. Next, the error between the observed data and the neural network prediction data is used to embed the error into the loss function to correctly identify the specimen length scale; Finally, a positive phase field fracture model is established based on the identified length scale to predict the displacement field and phase field during crack propagation, thereby representing the crack propagation path.

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