In previous researchtwenty threeit was determined that the trained BILSTM model has effectively learned to map MSC growth parameters (i.e., local crack extension, δ,aand the Kink Angle, φCriticism) Virtual Crack Front Point (CFP) to an input sequence containing microstructural and micromechanical features extracted from the unbroken local neighbour of the CFPS. Microstructures do not evolve with crack propagation in physics-based simulations, so the key variable driving the accuracy of the BILSTM model is the characteristic of micromechanicals.
Interleaved Physics-Based Deep Learning Framework
Figure 1 illustrates the prediction process for the proposed interleaved PBDL framework. The process begins by evaluating the micromechanical fields using a physically-based framework using the assumed initial cracks explicitly represented in the FE mesh, and then collecting the input sequences in the CFP along the crack front. Trained ensemble in the bilstm model predicts δa and φCriticism Point-by-point in each CFP using these input sequences. In addition to predicting MSC growth parameters, the ensemble model also provides estimates of related uncertainty. If the model's uncertainty estimate is below the user-specified threshold for any model, the prediction is accepted. Accepted prediction of δa and φCriticism Combined to effectively propagate the front of the crack k= 0 to k= 1. A new input sequence is then collected from the front of the virtually updated crack k= 1 (shown as a white dot in Figure 1) Predict the increase in crack growth following: In this way, the cracks continue to propagate virtually using only deep learning predictions. However, when the virtual crack propagated far from the front of the initial crack, it was extracted in areas where the micromechanical fields were not extracted. k=0 simulations are less relevant to predictions of deep learning models. With each reliance on previous steps, model errors and uncertainties accumulate over time.
Once the accumulation of uncertainty reaches a user-specified threshold, the MSC propagation process switches from deep learning prediction to physics-based simulation. At this point, the substantially propagated crack surface is explicitly represented by the FE mesh, and physics-based simulations are performed to update the micromachine field, thereby allowing better representation of the drive mechanism close to cracks. After updating the micromechanical fields, the process resumes virtual crack propagation using a deep learning framework with the ability to provide information from the updated micromechanical fields. This interleaving approach in the PBDL framework leverages the strengths of both deep learning and physics-based simulations to ensure accurate and efficient predictions.
Demonstration of the interleaved PBDL framework
This section shows the application of the interleaved PBDL framework using one of the microstructured instantiations of the test dataset. Evaluate and compare the errors and calculation costs associated with propagating cracks k= 0 to k= Seven frameworks across three frameworks: an interleaved PBDL framework, traditional physics-based simulation, and a deep learning-only approach. Furthermore, decisions to update micromechanical fields refer to cumulative uncertainty of crack extension (see section “Uncertainty Quantification and Calibration Procedures”), and therefore show the results of the PBDL framework under different user-specified thresholds due to cumulative uncertainty.
Figure 2 shows the evolution of the crack surface, in discrete increments, as it propagates. k= 0 to k=7, the micromechanical field used to predict the increase in the next crack growth. Three cases are shown in Figure 2 for comparison. k=0 simulation (i.e. without representing stress redistribution associated with the formation of a new traction-free surface), (2) interleaved PBDL predictions corresponding to the cumulative uncertainty threshold t1 =0.8μm, and (3) interleaved PBDL predictions corresponding to the cumulative uncertainty threshold t2 =0.4μm. threshold t1 and t2 Each of the four and two increments of crack growth is arbitrarily selected to update the physics-based model, respectively. Notably, the threshold is smaller than the 1 μm voxel resolution used in physics-based simulations and therefore remains within the intrinsic resolution error. In Figure 2a–C, the cut plane view shows the predicted crack trajectories from the corresponding reference trajectories (black) from the physics-based simulation (black) and the overlaid PBDL framework (white) for visual comparison. In Figure 2B, predictions from k= 0 to k= dependent on the micromechanical field calculated at 4 k= 0, but when cumulative uncertainty reaches a predefined threshold k= 4, the explicit crack surface is updated and the micromechanical field is reevaluated k= 4, and subsequent predictions k= 5 to k= 7 Rely on micromechanical fields corresponding to crack conditions k=4. Figure 2C illustrates more frequent update strategies. Here, the micromechanical field is updated with every increase in two crack growth. Prediction from k= 0 to k= 2 is based on the initial field of kPrediction from = 0; k= 3 to k= Use the updated field with 4 kPrediction from = 2; k= 5 to k= 6 is based on the updated field in k= 4 etc. As expected, more frequent updates of the micromechanical field result in better alignment between the predicted and reference crack trajectories.
The interleaved PBDL framework diagram exceeds the prediction of microstructurally small crack growth over eight times (k) Within 3D polycrystalline microstructure. Typical micromechanical fields used to make forward predictions are overlaid at each increment. a Fields that are not updated later k= 0 (i.e., deep learning only case), b threshold t1 As a result, micromechanical fields are updated. k= 4 (see Supplementary Movie 1 for the animated version), and c threshold t2 As a result, micromechanical fields are updated. k= 2, k= 4, and k= 6. In each cut-plane view, the predicted crack trajectories from the PBDL framework (white) are covered with corresponding reference trajectories from the physics-based simulation (black) for visual comparison.
It is important to recognize that uncertainties are reset in each physics-based update of the interleaved PBDL framework, but that errors continue to accumulate as crack propagates. Unlike long cracks, MSC evolves as a highly tortuous 3D surface, which employs differences in crack surface area as a primary error metric. This choice is particularly meaningful as the predicted crack surfaces of the deep learning framework are built on both predictions of crack extension and kink angles, which naturally integrates the errors that arise from both components. However, for ease of interpretation, we define equivalent crack radii. rEqBy equating the crack surface area error with the ideal flat semicircular crack error. This scalar measurement can be compared to the average crack size and provides the basis for comparing errors.
Figure 3 shows the absolute error of total crack surface area for the corresponding high fidelity physics-based simulation (ground truth), plotted as a function of mean crack radius. The results are presented for a deep learning only framework and an interleaved PBDL framework with two different uncertainty thresholds. t1 and t2. To assess long-term behavior, the error trend is extrapolated to a crack size of 1650 μm. This corresponds to the smallest and reliably detectable crack size via ultrasound examination.twenty four. Extrapolation is performed using polynomial regression, and the polynomial order is determined via trial and error. The errors in the interleaved PBDL framework continued closely with the second polynomial, while the deep learning only framework followed three polynomials. As shown in the inset of Figure 3, absolute error increases with crack growth in both approaches. However, growth rates are more pronounced in cases where deep learning is used only. Additionally, errors associated with the PBDL framework using tight thresholds t2 Using a more looser threshold, it is consistently smaller than that t1highlights the role of update frequency in overall prediction accuracy.
Compared to the corresponding physics-based simulation (ground truth), absolute errors in the total crack surface area are predicted and plotted as a function of nominal crack size. The results are presented for a deep learning only framework and an interleaved PBDL framework with two different cumulative uncertainty thresholds. t1and t2 . The inset shows the actual data points from a particular microstructure instantiation. This is used to extrapolate to the final crack size that can be detected by ultrasound.
The difference in absolute error between the two methods may seem conservative at first, but the differences become increasingly important with propagation. In a deep learning only framework, errors can ultimately lead to very unreliable predictions (equivalent crack radius errors, rEqconverted to about 318% of the actual average radius). In contrast, through periodic physical-based updates, the interleaved PBDL framework limits the propagation of errors. The growth of the quadratic error observed in the PBDL framework is converted to a more reasonable deviation, with the error being limited to about 41% and about 29% of the average crack radius of the threshold t1 and t2respectively.
Figure 4 shows the predicted computational costs that occurred when predicting the evolution of microstructurally small cracks using an interleaved PBDL framework compared to purely physics-based simulation and deep learning only approaches (according to the initial state, according to the initial state, k= 0). The inset shows data points from a particular microstructure instantiation. This is used to extrapolate to the final crack size that can be detected by ultrasound. The results show that interleaved PBDL frameworks are computationally more expensive than deep learning only approaches, but are significantly cheaper than traditional physics-based frameworks, using thresholds to reduce computational costs by 4 times. t1 3x using threshold t2. As shown in Figure 4, the computational savings for the interleaved PBDL framework depend on user-specified tolerance for cumulative uncertainty.
Using the interleaved PBDL framework, predictive computational costs (using two different cumulative uncertainty thresholds) to predict the evolution of microstructurally small cracks. t1and t2), compared to purely physics-based simulation and deep learning-only approaches. The inset shows the actual data points from a particular microstructure instantiation. This is used to extrapolate to the final crack size that can be detected by ultrasound.
