Deep artificial neural network-powered phase field model for predicting damage characteristic in brittle composite under varying configurations

This work introduces a novel artificial neural network (ANN)-powered phase field model, offering rapid and precise predictions of fracture propagation in brittle materials. To improve the capabilities of the ANN model, we incorporate a loop of conditions into its core to regulate the absolute percentage error for each observation point, that filters and consistently selects the most accurate outcome. This algorithm enables our model to better adapt to the highly sensitive validation data arising from varying configurations. The effectiveness of the approach is illustrated through three examples involving changes in the microgeometry and material properties of steel fiber-reinforced high-strength concrete structures. Indeed, the predicted outcomes from the improved ANN phase field model in terms of stress–strain relationship, and crack propagation path demonstrates an outperformance compared with that based on the extreme gradient boosting method, a leading regression machine learning technique for tabular data. Additionally, the introduced model exhibits a remarkable speed advantage, being 180 times faster than traditional phase field simulations, and provides results at nearly any fiber location, demonstrating superiority over the phase field model. This study marks a significant advancement in the application of artificial intelligence for accurately predicting crack propagation paths in composite materials, particularly in cases involving the relative positioning of the fiber and initial crack location.


Introduction
Since the introduction by Griffith [1], the theory of linear elastic fracture mechanics has undergone a long history of development, marked by numerous contributions from diverse research directions.Despite the apparent simplicity of the fundamental concept stating that, 'a crack will propagate when the reduction in potential energy that occurs due to crack growth is greater than or equal to the increase in surface energy due to the creation of new free surfaces' [2], the fracture mechanical problem remains incredibly complex, even with simplified structures.Until now, theoretical work related to solving problems in damage mechanics and fracture primarily relies on numerical methods, with comprehensive discussions of their descriptions, advantages, and disadvantages documented in various review papers (e.g. the work of Cervera et al [3]).More recently, the phase field method, as discussed in recent reviews by Wu et al [4] and Zhuang et al [5], has gained increasing prominence as an elegant and precise approach to modeling brittle fracture.In this method, the solution to the fracture problem is obtained by minimizing a global variational energy formulation that integrates elastic and fracture energies.By employing a scalar variable, known as the phase field variable, to represent the progression of structural damage from an undamaged state to a fully collapsed state within a regularized framework, fracture energy is computed through the integration of a crack surface density function.As a result, this technique handles phenomena such as crack nucleation, interaction, and the formation of arbitrary crack morphologies, even in complex scenarios involving multiple crack fronts and branching.Nevertheless, the effectiveness of this method comes at the cost of increased computational complexity due to the necessity of handling additional smeared fields and performing time-dependent calculations.Achieving optimal efficiency in this method, balancing both accuracy and computation time, remains one of its primary challenges in practice.
The integration of machine learning and multiscale modeling to address physical problems represents an intriguing area of research [6].With the rapid advancement and continual enhancement of computational power over the last few decades, machine learning models, including artificial neural networks (ANNs), one of the most proficient machine learning models, have found applications across numerous innovative and critically important domains [7].The application of machine learning models to assist or enhance the value of the phase field model is a topic that has gained attention in recent years.
Firstly, building upon the pioneering work of Raissi et al [8] in the development of Physics-informed neural networks (PINNs) for solving nonlinear partial differential equations, Goswami et al [9] were the first to apply the PINNs algorithm to address brittle fracture and phase-field problems through the minimization of the system's variational energy (referred to as V-PINNs).This model has introduced an innovative framework for addressing conventional computational models rooted in finite element methods, thereby reducing costs while ensuring fidelity to physical principles.Subsequently, a series of research studies have been conducted [10,11], with a primary focus on evaluating the efficacy of this PINNs method for fundamental phase field problems.Nevertheless, at its current stage, PINNs face certain challenges, such as those highlighted in [12] which indicate that PINNs are most effective in very simple cases or when the expected crack growth path must be predefined [9].Naturally, this computational framework is in its early stage and necessitates further research to present a comprehensive overview of its potential.
However, the most common topic in the combined framework of machine learning and the phase field model is the construction of machine learning surrogate models based on data generated by phase field methods.This approach, once suitable datasets are constructed, allows for alleviating the computational time cost, which is a major drawback of the phase field method.Feng et al proposed a machine learning-aided phase field method that has recently emerged to predict the probability of failure under arbitrary loading conditions [13].In this method, a surrogate model is constructed using X-SVM, a modified machine learning model trained on numerical damage data generated by a non-deterministic phase field model.However, it is important to note that this statistical numerical model is designed to determine the domain of the critical state of a structure rather than provide an exact solution for specific cases.At the structural level, Xie et al [14] conducted a new method which combines the back-propagation ANN and phase field method to predict the residual useful life of mechanical structures.The results show accuracy and promising results for predicting the remaining life of the structure.Within the context of the graph-based image recognition framework, some hybrid models for predicting fracture propagation have been recently developed [15,16].Feng et al [17], for the first time, employed convolutional neural networks trained on datasets of damage images generated by the phase field method.For a specific configuration involving defined geometric and mechanical properties, these graph-based hybrid models demonstrated successful predictions of failure states under varying loading conditions.Recently, Perera et al in (2023) [18,19] developed a graph neural network framework for simulating fracture and stress evolution in brittle materials.This framework, trained on a dataset generated by an XFEM-based fracture simulator, can simulate crack propagation and corresponding stress distribution for a wide range of initial microcrack configurations.However, it is noteworthy that the influence of the geometric and mechanical configuration of the material structure on crack propagation and other damage behaviors has not been comprehensively examined in these studies.Furthermore, in contrast to typical real image recognition problems, all numerical solutions related to damage behavior can be structured in tabular form and subsequently learned by a simpler, cost-effective network, eliminating the need for complex deep learning techniques.This aspect warrants further investigation and validation.
In another direction of research on enhancing the effectiveness of combined models, Luo et al [20] developed a data-driven solver aimed at efficiently selecting degradation function forms for specific failure modes in structures.Conversely, Gao et al [21] increased the accuracy of data by using experimentally-fitted tangent modulus instead of elastic modulus before constructing a surrogate model using the data-driven back propagation neural network.However, these research directions all involve interventions in the structure of the phase field algorithm, contributing to increased computational costs, which is a drawback that needs to be addressed.
In contrast to machine learning and the phase field methods, which are both widely discussed topics in the field of mechanic of material, the combination of these two methods to leverage their respective strengths remains somewhat limited, as indicated in the concise review mentioned earlier.Consequently, this paper focuses on a question that has not yet to be adequately explored in the existing literature: can a machine learning model be effectively constructed using numerical data related to damage behavior extracted through the phase field method for variation of both geometric configuration and mechanical properties of material structure.Therefore, this paper develops along a different direction from Luo et al [20] or Gao et al [21], rather than intervening in the phase field algorithm, we aim to improve the predictive efficiency of the ANN model.It is crucial to note that this objective is applied to the domain of fracture problems, which are highly sensitive to minor fluctuations.Consequently, for a new configuration, the corresponding failure behavior is represented randomly and is not easy to capture.So, despite being a seemingly straightforward question in the domain of machine learning-aided numerical methods, it still presents a significant challenge to verify.To address this challenge, the present study introduces a novel hybrid approach that combines an improved ANN with numerical data extracted from the phase field method to predict the damage behavior of composite material structures.The foundation of the phase field method is outlined initially.Subsequently, to enhance the precision of the proposed framework, a loop with multiple conditions is introduced.This augmentation enables the ANN-powered phase field approach to achieve remarkable accuracy with only a marginal increase in computational load, covering a broad range of mechanical conditions.This approach is notably more computationally efficient compared to conventional numerical prediction methods.The model is then put to the test using a typical short steel fiber-reinforced concrete structure.Three scenarios, involving variations in geometric configuration and material properties, are computed and verified.This constitutes the core content of this research.

Fundamental concept of the phase field method
We consider a cracked volume Ω ⊂ R D , where ∂Ω represents its boundary, and D = 1, 2, 3 denoting the dimension of the space.Let Γ be a curve of dimension D − 1 that represents a crack discontinuity within Ω (see figure 1) Assuming small strains, the potential energy describing the cracked structure is expressed as follows: Here ψ (ε) represents the elastic energy density, ε = 1 2 is the infinitesimal strain tensor compatible with the displacement vector u, and G c describes the critical fracture energy.One of the main aspects of the phase field method is based on the regularization technique, in which the crack geometry Γ is approximated by a smeared representation, which can be defined as a function of a spatial phase field, d ∈ [0, 1].This phase field d takes a unit value on Γ and vanishing away from it.This parameter measures the level of damage where d = 1 represents the fully broken material, and d = 0 represents the intact state of the material.With the introduction of a crack surface density function γ d (d, ∇d), a degradation function g (d), and a technique that splits the elastic energy into a damaged part ψ + 0 (u) and an intact part ψ − 0 (u), the total potential energy can be approximated as follows: Of course, the solution of the phase field method strongly depends on the form of these parameters in equation ( 2).The review paper by Wu et al [4] provides a detailed analysis of these effects.In our paper, we have chosen a quadratic type of degradation function, while others have adopted the formulas presented in Miehe et al [22] where ξ is a small numerical parameter to avoid loss of stability in case of fully damaged elements.It is necessary to note that the formulas (3)-( 5) are best suited for simulating concrete structures, as demonstrated in Nguyen et al [23].In equation ( 4), ℓ represents a regularized length that describes the thickness of the smeared crack and is also an internal parameter affecting the critical stress for crack initiation.In equation ( 5), elastic strain is decomposed into extensive ε + and compressive ε − parts (for further details, refer to the formulas in Nguyen et al [23,24]).Applying the principle of maximum dissipation and energy minimization to equation ( 2) yields the set of coupled equations to be solved in the domain Ω associated with the structure, with boundary ∂Ω and outward normal n, to determine d (x) and u (x), for all positions denoted by the vector x in Ω: In equation ( 6), σ is the second-order Cauchy stress tensor, f are body forces, and u and F denote displacements and forces on the corresponding boundaries ∂Ω u and ∂Ω F , respectively.The history strain energy density function H (x, t) is introduced to describe dependence on history and possible loading-unloading: Equations ( 6) and ( 7) are solved using a standard finite element procedure in a staggered scheme at each time step (load increment).In this paper, a program written in the MATLAB language is used to obtain numerical results for a particular material structure.

Improvement of deep ANN and integration with phase-field model
By tailoring the biological principle and advanced statistics, the ANN approach has been derived to solve real-world problems, such as complex structure behavior, pattern recognition, game play, and so on, which are beyond the capability of conventional mathematics and physics.It is known that the key element of the ANN model is the artificial neuron consisting of activation function, a bias, weight, and an output, linking to another for transferring the signal, implemented using commercial software like MATLAB or the free programming language Python.The structure of the deep ANN model is shown in figure 2(a).
The input data are taken from the postprocess of the damage phase field simulations, including the coordinate of the crack path and/or effective stress-strain distribution relative to changes in geometric and mechanical material properties.Similarly, the outcome predicted by the improved deep ANN model is the coordinate of crack path and/or stress distribution, even at various configurations.In this paper, we use an ANN Leaky ReLU (α = 0.1) as activation function.The ANN model is enhanced during training phase by adaptive momentum estimation algorithm (Adam) [25].Adam optimizer dynamically updates the learning rate for each parameter.The initial learning rate is set as 0.01 with the default hyperparameters suggest by Kingma and Ba [25] for machine learning problem are µ = 0.001, β 1 = 0.9, β 2 = 0.999.
The intrinsic problem of the ANN model is that it runs only one time after it is manually assigned the number of neurons and hidden layers, resulting in a passive way to select the most accurate outcome.To remove the passiveness and attain the best outcome of the deep ANN approach, smart conditions are integrated into ANN code to iterate ANN through changing the neuron number and layer number using syntaxes, including for, if, and else.The key component of the smart condition is the search for a minimal gap between the training data and output data.This gap is characterized by the absolute percentage error per observation (APE) and is subjected to variations in the number of neurons in the hidden layer and increasing values of components in a given matrix.The minimal gap, denoted as G m is evaluated as where the term Max (a) is the searching for maximum values among components of matrix a, |.| is an absolute operator.Additionally, O, I and T denote matrices representing the outcome, the real output data, and threshold constants, respectively.The minimal gap G m is an independently given value that controls the precision of calculations, while the threshold matrix is calculated through an interactive process.For a simplified scenario, we can select G m = 0, and the threshold matrix becomes homogeneous, containing only one threshold value, denoted as T = T × 1.The flowchart illustrating the working mechanism of the ANN-coupled phase field framework is depicted in figure 2(b).Initially, the basic configuration is established, employing a simple ANN architecture and a small threshold matrix T .Subsequently, the improved ANN continues to run, with alterations to the configuration, including the number of neurons, hidden layers, and threshold values, until inequality (9) is met.At this point, the ANN is halted, and the outcomes are recorded, ensuring result consistency.It is important to note that the fundamental loss function condition of the core ANN model must always be satisfied.

Results and discussion
In this section, the validation of the ANN-based framework against numerical data derived from the phase field method is presented.This validation encompasses assessments of crack paths and stress-strain curves for a specific composite structure.Three distinct case studies involving variations in geometric configurations and material properties are investigated.Figure 3 depicts the geometry and boundary conditions of 2D tests involving a short steel fiber-reinforced structure embedded in high-performance concrete.The specified boundary conditions are as follows: the displacements along the x1-axis at the bottom edge is constrained to zero, while a monotonic tensile displacement increments of ∆U = 0.0001 mm are applied to the top edge region, which spans 100 mm.The problem assumes a plane stress condition.The three numerical experiments correspond to 3 different situations of material behavior to be considered.The first experiment (figure 3(a)), the geometric conditions remain unchanged while material parameters change.This example allows testing the effectiveness of the model in evaluating the effective average behavior of materials.The second experiment (figure 3(b)), when the geometric parameters of the fibers change vertically while the material parameters and loading remain unchanged, this example considers the effectiveness of the model in predicting crack paths of structures in the vertical direction (parallel to the applied force direction).The third experiment (figure 3(c)), similar to the second experiment but will consider evaluating crack patterns in the horizontal direction (perpendicular to the direction of the applied force).The primary material parameters are outlined in table 1 [26], along with the phase field numerical model parameters.In all case studies, the training data in terms of crack path and stress-strain curve  obtained from the phase field model with respect to nine different configurations.It is worth noting that the validation data, obtained from the phase field model, are distinct from the training samples.
In the first example, the focus is placed on examining the ability of the proposed model to predict the stress-strain curve when the material parameter is varied.For this purpose, the geometric configuration is fixed, as depicted in figure 3(a), where the single short steel fiber is anchored with the coordinate of the fiber center being located at point (20,50).To reflect the variation of the material parameters, priority is given to the examination of the influence of the length scale.In the phase field theory, the length scale assumes a significant role, representing an intrinsic aspect of the material structure rather than a unique numerical coefficient.To illustrate this point, a test of a linear isotropic solid bar with a null Poisson ratio under uniaxial traction is revisited.With the assumption of a homogeneous solution along the bar, equations (3)-( 7) establish a relationship between the critical stress (σ c ), critical strain ε c , and the parameter models as follows, This demonstrates that the critical stress increases as the length scale (ℓ) decreases, and vice versa.It should be noted that equation (10) applies to a specific case and does not provide a general prediction.Therefore, instead of altering material parameters such as elastic modulus or energy release, a dataset is constructed based on varying the length scale (ℓ).In this study (see figure 4), nine stress-strain relationship curves corresponding to nine different length scale values (0.1, 0.25, 0.5, 0.75, 1.0, 1.25, 1.5, 1.75, and 2.0) were used to generate the training sample datasets for the ANN model.The dataset, called dataset1, includes strain (ε) and length scale (ℓ) values as input data, stress (σ) as output data.The statical analysis of this dataset is presented in table 2 and the correlation matrix is presented in figure 5.
Figure 6 presents a comparison of the effective stress-strain curves between the improved ANN model and the numerical results obtained through the phase field method with length scales of 0.3, 0.6, 0.9, and 1.2.It is important to emphasize that the numerical data for these four length scale values, as shown in figure 6, are not part of the training samples.Additionally, the problem is also compared with predictions from the classic ANN model and the extreme gradient boosting method (XGB) for reference.More details about the XGB method can be found in works of Le et al [27,28].It is worth noting that XGB is considered one of the best regression machine learning methods for tabular data, as reported in recent publications [29,30].On the other hand, the classic ANN is a standard back propagation neural network with Adam optimizer.Figure 7 illustrates the tuning phase of the classic ANN's structure.The structure of the classic ANN, obtained by the grid search method, consists of 2 hidden layers with 25 neurons in each layer with the highest value of R 2 = 0.9995.From the figure, the improved ANN model exhibits a good agreement with the results obtained from the phase field methods in terms of the stress-strain relationship.There is a small discrepancy observed in the case of ℓ = 0.3, which is caused by the strong variation of the critical point for small values of ℓ in the training data (see figure 4).Conversely, it appears that the XGB method and the classic ANN are less effective in predicting numerical results, even when dealing with larger values of ℓ.For example, at strain of 0.055 and ℓ = 0.3, the percent difference between the stresses attained by the improved ANN model and phase field model is 12% Meanwhile, it is 31% between the XGB model and the phase field model, and 26% between the classic ANN and the phase field model.This highlights the efficacy of the proposed method in forecasting this type of problem, where the structure of the material system undergoes significant transformations, and the training set lacks or restricts information from the validation set.
In order to gain a deeper understanding of the mechanism underlying the improved ANN model, we illustrate the evolution of the model's loss functions as the ANN architecture increases in figures 8 and 9, representing the maximum absolute percentage error (Max(APE)) and the coefficient of determination (R 2 ) respectively.Each point corresponds to an interaction run.We begin with a single layer and 5 neurons, and for each run, we add 5 new neurons until we reach a maximum of 100 neurons per layer, after which we add another layer.The condition specified in equation ( 9) is initiated with a threshold of T = 1%, and the program is concluded when the number of neurons reaches 300, which corresponds to three fully populated hidden layers, owing to the stability of the loss functions.Subsequently, the second round is recalculated with a threshold of T = 2%, and the condition ( 9) is satisfied with an ANN architecture of three layers (100 100, 15).Another crucial point to note is that both R 2 and Max (APE) exhibit fairly similar developments.However, the R 2 coefficient converges at the second layer, while the additional condition (9) necessitates a larger neural network space.This highlights the enhanced efficiency of the improved model compared to the traditional ANN model.
In the following examples, we focus on predicting crack paths within the framework of a machine learning-assisted numerical finite element model.To achieve this, we maintain the macroscopic conditions of     the test as presented in table 1 and figure 3 while altering the position of the fiber.Two scenarios are considered: • In the first scenario (figure 3(b)), the fiber is moved along the x 2 axis, with the x 1 coordinate of the center fixed at x 1 = 20 mm.• In the second scenario (figure 3(c)), the fiber is moved along the x 1 axis, with the x 2 coordinate of the center fixed at x 2 = 60 mm.For model training, the input data comprise the position of the fiber center and the corresponding crack path.Specifically, when the fiber moves along the x 2 direction with the fiber center's position ranging from 50 mm to 68 mm, we select 10 locations of the fiber (x v 2 = 50, 52, 54, 56, 58, 60, 62, 64, 66, 68 mm) and calculate the coordinates of the crack path at 25 positions along the x 1 -axis for each position of the fiber.As a result, the dataset of the first scenario (called dataset 2) has 25 × 10 = 250 samples.Each sample has coordinates in the x 1 direction (x 1 ), locations of the fiber (x v 2 ) as inputs and the coordinates of the crack in the x2 direction (x 2 ) are output value.Similarly, in the case where the fiber moves along the x 1 -axis, the input data are sampled at 9 locations of the fiber moving along the x 1 direction (x h 1 = 20, 30, 40, 50, 60, 70, 80, 90, 99.5 mm) and 25 positions along the x 1 -axis for each position of the fiber.Therefor, the dataset of the second          when the location of fibers is changed in x 1 -axis, reflecting via the different curves of crack path.Remarkably, we observe a strong agreement between the predictions of the improved ANN method and a significant deviation of the XGB method and classic ANN method from the results of the phase field method.Indeed, at x 1 = 80 mm and x h 1 = 65mm, the percent difference in terms of x 2 between the improved ANN model and phase field model is 2%.Meanwhile, it is 5% between the XGB model and phase field model and 4% between classic ANN.The threshold value T is calculated as 5%, for the ANN architecture comprising two hidden layers with (100, 30) neurons.
Similarly, we observe a similar behavior for the vertical movement case (figure 3(b)) for different positions of fiber along x 2 -axis (x v 2 = 51 mm, 55 mm, 63.5 mm, 65.5 mm), which is depicted in figures 16-18.In this case, the threshold value T is 2%, and the ANN architecture consists of two hidden layers with (100, 50) neurons.However, it is worth noting that the shape of the loss functions and the number of neurons in the vertical movement case (figures 17 and 18) differ from the corresponding curves in the two previous cases.This underscores that the parameters of the improved conditions in (9) strongly depend on the data structure.Furthermore, in this case, generally the improved ANN model still provides a more accurate prediction, demonstrating via closer crack path to that from phase field model, compared with that of the XGB method and classic ANN.

Conclusion
This study introduces an improved ANN-powered phase field model for accurate predictions of fracture propagation in brittle materials, marking a novel contribution to the field.Compared to the phase field method, this method is both cost-effective and rapid.This was achieved by implementing a control mechanism to minimize the APE for each observation point by incorporating a loop of conditions into the core ANN algorithm.A calculation strategy was devised, wherein, following each iterative run, the ANN architecture was expanded until all conditions were met.
The resulting improved ANN model exhibited remarkable accuracy in three examples of high-fidelity simulations derived from the phase field method for bi-dimensional short steel fiber-reinforced high-strength cement concrete.These examples included the variation of the length scale parameter for predicting the stress-strain curve, as well as changes in the microgeometry of the material structure for predicting crack paths.In each example, the training data in terms of crack path or stress-strain curve obtained from the phase field model with respect to nine different configurations encompassing variations in fiber position and length scale parameter.Notably, the validation data, derived from the phase field model for four distinct, independent configurations, were separate from the training samples.In all specific validation cases, the improved ANN model yielded more accurate predictions, manifesting closer alignment with the crack paths and stress-strain curves derived from the phase field model when compared to the performance of the clasic ANN model and the XGB method.
One of the significant contributions of the paper is the incorporation of a smart condition based on APE to enhance the model's accuracy.In the illustrative examples, we only consider the threshold matrix as homogeneous with a single threshold value.However, with the smart condition, we can fully exploit a more complex level by allowing threshold values to vary according to the structure of the dataset.Extending this condition may be more suitable for more complex structures (such as three-dimensional problems or the complex spatial structures of fiber-reinforced concrete materials considering the actual distribution of fibers and matrix).
Returning to the overarching topic mentioned in the Introduction section, where several milestones in this area are reported, particularly regarding the integration of artificial intelligence algorithms and numerical results from the phase field method for brittle materials such as concrete.In fact, the topic of machine learning-aided fracture mechanics, or more narrowly, machine learning for crack detection, is a broad field, with potentially hundreds of different models for various configurations developed recently (see more in the recent reviews [31,32]).Research topics may include constructing surrogate models from training datasets to predict overall damage properties [33] to predicting fracture parameters [34] or monitoring fault diagnosis processes [35].While our paper naturally fits into the broader picture outlined above, it proposes a more efficient model compared to the basic approach in predicting a mechanical property of material failure, through relatively basic and controllable examples.Certainly, it seems challenging to conclude, by adding a few specific examples that we believe would not add further value to the paper, that some of the techniques developed in the paper would be highly beneficial for such a broad field.However, based on the results obtained, extending the proposed technique to other specific configurations is highly promising.These are tasks that will be planned for the future.

Figure 1 .
Figure 1.Illustration of the regularized scheme: sharp crack (left) and smeared crack (right) configurations.

Figure 2 .
Figure 2. Depiction of deep ANN model (a) and improved ANN model-coupled damage phase field framework (b).

Figure 3 .
Figure 3. Illustration of the tests with three examples corresponding to (a) a fixed geometric configuration and varying material properties; (b) vertical movement of the fiber; (c) horizontal movement of the fiber.

Figure 6 .
Figure 6.Prediction of stress-strain relationships by the phase field method (PFM), the improved ANN model (I.ANN), the Classic ANN model (C.ANN) and the extreme gradient boosting model (XGB) for different values of ℓ (0.3, 0.6, 0.9, 1.2).

Figure 7 .
Figure 7. Results of simulations of classic ANN with 2 hidden layer and the number of neurons from 1 to 60 neurons for each layer.

Figure 8 .
Figure 8. Maximum absolute percentage error per observation versus neuron number obtained from the improved ANN-based approach in the varying material parameter case.

Figure 9 .
Figure 9. Coefficient of determination (R 2 ) versus neuron number obtained from the improved ANN-based approach in the varying material parameter case.

Figures 10 and 11
Figures 10 and 11 respectively depict images of crack propagation patterns for the two scenarios: vertical movement case with x v 2 = 51, 55, 63.5, 65.5 mm (figure 3(b)) and horizontal movement case with x h 1 = 25, 45, 65, 85 mm (figure 3(c)).These data are subsequently extracted for validation purposes.For model training, the input data comprise the position of the fiber center and the corresponding crack path.Specifically, when the fiber moves along the x 2 direction with the fiber center's position ranging from 50 mm to 68 mm, we select 10 locations of the fiber (x v 2 = 50, 52, 54, 56, 58, 60, 62, 64, 66, 68 mm) and calculate the coordinates of the crack path at 25 positions along the x 1 -axis for each position of the fiber.As a result, the dataset of the first scenario (called dataset 2) has 25 × 10 = 250 samples.Each sample has coordinates in the x 1 direction (x 1 ), locations of the fiber (x v 2 ) as inputs and the coordinates of the crack in the x2 direction (x 2 ) are output value.Similarly, in the case where the fiber moves along the x 1 -axis, the input data are sampled at 9 locations of the fiber moving along the x 1 direction (x h 1 = 20, 30, 40, 50, 60, 70, 80, 90, 99.5 mm) and 25 positions along the x 1 -axis for each position of the fiber.Therefor, the dataset of the second

Figure 10 .
Figure 10.Illustration of the crack path in the vertical movement case for fiber center position at x v 2 = 51, 55, 63.5, 65.5 mm.

Figure 11 .
Figure 11.Illustration of the crack path in the horizontal movement case for fiber center position at x h 1 = 25, 45, 65, 85 mm.

Figure 13
presents predictions of crack paths in the horizontal movement case (figure 3(c)) for the validation dataset (x h 1 = 25, 45, 65, 85 mm), while figures 14 and 15 display Max(APE) and R 2 concerning the ANN architecture.Based on the figure, the crack nonlinearly propagates different paths in the composite

Figure 13 .
Figure 13.Prediction of the crack path by the phase field method (PFM), the improved ANN model (I.ANN), the Classic ANN model (C.ANN), and the extreme gradient boosting model (XGB) for different position of fiber along x1-axis (x h 1 = 25, 45, 65, 85 mm).

Figure 14 .
Figure 14.Maximum absolute percentage error versus neuron number obtained from the improved ANN-based approach in the horizontal movement case.

Figure 15 .
Figure 15.Coefficient of determination (R 2 ) versus neuron number obtained from the improved ANN-based approach in the horizontal movement case.

Figure 16 .
Figure 16.Prediction of the crack path by the phase field method (PFM), the improved ANN model (I.ANN), the Classic ANN model (C.ANN), and the extreme gradient boosting model (XGB) for different position of fiber along x2-axis (x v 2 = 51, 55, 63.5, 65.5 mm).

Figure 17 .
Figure 17.Maximum absolute percentage error versus neuron number obtained from the improved ANN-based approach in the vertical movement case.

Figure 18 .
Figure 18.Coefficient of determination (R 2 ) versus neuron number obtained from the improved ANN-based approach in the vertical movement case.

Table 1 .
Main material parameters and phase field numerical model parameters.(The length scale is only applicable to the second and third experiments).Elastic moulus Poisson ratio Fracture energy Length scale Degradation coefficient Mesh size

Table 2 .
Statistical parameters of the dataset 1.

Table 3 .
Statistical parameters of dataset 2 and 3.

Table 4 .
Computational time for each example.Using the surrogate model based on the improved ANN framework, each test took approximately 5 s, which is 180 times faster than the FEM technique.All computations were implemented in MATLAB, and the improved ANN model was validated using Python code with support from available libraries.A comparison of computational time required for all the test is provided in table 4. Offline computation is the time used to create and train the ANN network while Online is the time used to solve the problem by phase field method or improved ANN model.