Generative AI and image based numerical mechanics in wind blade adhesive composites

Numerical modelling of adhesive composites in wind energy is complicated in part due to material heterogeneity. Microstructural CT scan fibre composite patterns or representative elements, which play a major role in defining the mechanical behaviour of these adhesive structures, are both difficult to characterize as well as hard to numerically simulate. With advances in deep learning based generative AI, new ways of predicting the mechanical behaviour of heterogeneous materials is now possible. Here we put forward a data driven method to relate input composite adhesive microstructures with field data using deep learning and generative AI based methods. The prediction of mechanical stress or strain fields or other similar patterns and combining them as a function of boundary conditions, fibre composite microstructure and material models is achieved and the models are trained such that they closely approximate computationally expensive simulations based on numerical FE techniques and would have the ability to generalize. We also create a dataset of wind energy adhesives with their numerical mechanics based FE simulations subject to different boundary conditions and material models for further deep learning based composite studies.


Introduction
Establishing numerical models that capture the mechanical behavior of wind blade adhesives considering the fibres at a microscale is an ardent but important task.Traditionally when structures are solved numerically as a Boundary-value problem, two types of equations are combined, equations based on Conservation laws, which are inherently uncertainty-free and equations derived from physical modelling and observations which are empirical and uncertain.The classical numerical paradigm is based on making use of the observed experimental data and then use the calibrated material model in other calculations.Often times the method is expensive and time consuming while being empirical in nature [1][2][3].A new direction based on data driven techniques using machine learning architectures opened up new methods for computational solid mechanics [4][5][6][7][8][9].Off late generative AI is making news and disrupting and revolutionizing various industries.It is based on generative deep learning methods with the ability to leverage both supervised and unsupervised learning approaches, with the ultimate goal of taking some training inputs from a distribution and then learning a model that characterizes that distribution.One such class of Generative AI methods are the Generative Adversarial Networks (GANs) [10].In this work we will explore on how to use Generative Adversarial Networks for stress predictions in a wind energy fibre composite adhesive microstructure.Segmentation of the fibres and creation of representative microstructures from the adhesive matrix CT Scans are important prerequisites that have to be met to perform subsequent analysis.Most images however due to the reasons of image acquisition and various other operational modalities contain noisy fluctuations of varying pixel values which force us to define a segmentation approach.Since the current emphasis is on the extraction and segmentation of fibres which have reasonably well defined boundaries or edges thus high intensity methods are employed.Small subsets of microstructures are cropped and marker controlled watershed algorithms are applied.In some cases, pixels below a certain threshold are removed and then ellipses around the amorphous regions are fitted.The segmented regions are then replaced with these fitted ellipses to complete the segmentation.Noise removal strategies are also sought since excess noise even after binarization can alter the results for further processing (Figure 2).The approach is inspired by the reconstruction approach used for tomographic data of heterogeneous fibre systems [11].
After the segmentation is done and a collection of diverse binarized microstructures are obtained with scanned fibre directions, we do a set of finite element simulations where we employ the input CT binarized patterns to inform the heterogeneous fibre matrix distribution of the domain and subject it to equibiaxial extension.For achieving this goal, the binary bitmaps are converted to two different materials of meshed regions.The segmented fibre matrix coordinates are translated to a mesh with two distinct subdomains using Python-based pygmsh.Quadratic triangular elements are chosen for the mesh.Once the microstructures are meshed, a compressible Neo Hookean material model is chosen, defined by a free energy function W and mathematically representing the deformation gradient as F and the right Cauchy Green Tensor as C. Then the free energy function could be written as: Where I c = trace(C) = trace(F T F ) = F : F and J = det(F ), and µ and λ are the Lamé's parameters respectively.The values of the material parameters are chosen from previous studies on fiber reinforced adhesives in wind energy [12,13].Specifically, moduli of 79050 MPa for fiber and 2798 MPa for matrix are used respectively.The Poisson's ratio for fiber is taken as 0.22, and for the matrix, the value is set as 0.40.A perfectly bonded interface behavior between the matrix and fiber is assumed, and numerically it is handled by defining subdomains inside the domain.A finite element database with varying boundary conditions, magnifications, and loading conditions is created.For this study, however, only biaxial extensions at 2% and 4% are considered.The Frobenius norm of the Cauchy stress maps is obtained for each of the 90 microstructures for each of these load conditions, creating a small but reasonably diverse dataset.The study would use this as the training-testing dataset with a split ratio of 80% − 20% respectively.

Generative Modeling
Once the training dataset is obtained, a strategy is sought for generating corresponding stresses using an artificial intelligence-based approach.The objective of the proposed neural network will be thus to output a stress map, given a new microstructure.This could be thought of as a paired image-to-image problem (I2I) where an input image x a from the microstructural domain a is being translated to an output image in the stress domain v. Mathematically, we need to train a mapping G (a→v) such that it could generate an image x av ∈ v, which is identical to the target image in the stress domain x v ∈ v, given the input microstructure image x a ∈ a. Mathematically, we can model it as follows: Generative modelling answers questions on how to achieve this mapping between the two image domains.Generally, in generative modelling, it is assumed that the creation of data is related to a distribution and usually that distribution is described by parametric and non- parametric variants, respectively [14,15].The goal in generative modeling is then to approximate this underlying distribution with various algorithms and techniques.
Our work is based on Generative Adversarial Networks (GANs) [10], which consist of a generator (G) and a discriminator (D), two neural networks which are in competition with one another.In the most classic setting, the generator turns random noise (z) into an imitation of the data, trying to trick the discriminator.The discriminator tries to discriminate real data from fake ones which are created by the generator.The generator continuously improves with its imitation of the data and tries to fool the discriminator.The objectives are thus adversarial, hence the name.A stage comes where the generator is able to output data so close to the real ones that the discriminator can no longer differentiate them from the real ones, a state of Nash equilibrium.As shown in Figure 4, the generator G inputs a random noise z sampled from the model's prior distribution to generate a fake image G(z) to fit the distribution of real data as much as possible.Then, the discriminator D randomly takes the real sample x from the dataset and the fake sample G(z) as input to output a probability between 0 and 1, indicating whether the input is a real or fake image.In other words, D wants to discriminate the generated fake sample G(z) while G intends to create samples to confuse D. Consequently, the objective optimization problem is as shown below: The only shortcoming with this configuration though is that the user has no control over the final output because the only input is random noise.It was suggested that additional information could be added with random noise to generate the final output [16].In other words, conditioning the input with labels, text and attributes of the images.Using this conditional variable c, we are able to achieve control over the generated output image.This is something called Conditional GANs where in addition to the random noise we also have another factor or label to condition for [17][18][19], in the present scenario the microstructure image.
Thus the problem that we deal with falls under the category of supervised image to image problem, wherein we have a microstructure and the corresponding stresses in the form of paired data.The problem could be thought of as single modal if there is no time dependency and multimodal if time also becomes a variable i.e. when we want to see the evolution of stresses in the microstructure with time.For the case of simplicity, in this paper we will only focus on single modal problems.We will use the Pix to Pix architecture as illustrated in figure (5) with minor changes to fit our problem of stress predictions from a composite microstructure in wind energy [20].
The generator and discriminator models in Pix2Pix GAN use standard Convolution-Batch Normalization-ReLU blocks of layers.The generator uses a popular encoder-decoder network "U-Net" architecture with skip connections [21].A patchGAN network is used by the discriminator which instead of predicting the whole image as fake or real, takes an image patch and then subsequently makes a prediction for every pixel in that patch for its authenticity.The final objective then becomes as follows: where, Here c represents the conditional variable, x denotes the real FEM data, z characterizes the random noise vector, and G(z|c) are the fake samples generated by the generator G under the control of the conditioning variable c.The quantity D(x|c) specifies the probability that the discriminator's conditioned input is real.The adversarial loss A controls whether the generator in the network has the power to produce images that are plausible in the target FEM domain.The L1 loss B regulates the generator to output images that are a possible translation of the image from the source domain.The parameter λ gives the user control as to how much importance needs to be given to the L1 loss.Thus it is kind of a weighting factor between the adversarial loss and the L1 loss.

Results and Discussions
The deep learning model used and proposed was evaluated on biaxial extension both for the case of 2% and 4% loading to check for its performance in adhesive composites and the related mechanical behaviour.The microstructure image is the input, while the Frobenius norm of the Cauchy stress tensor σ eq is the label target.Error Maps are calculated using Mean Absolute Error as a metric to test for performance and to compare the results between FEM ground truth and the conditional GAN (AI) based predictions.Figures (7), (8) show that the results of using conditional GAN's on data out of the training dataset and it could be seen that even with very limited training data, we have good quality predictions.We have some deviations between the ground FEM and the conditional GAN based predictions near the boundaries of the fibres especially in highly loaded cases but nonetheless the predictions are promising considering how frugal the dataset was and the complexity of the problem.The variation in fibre shapes, small sizes and different volume fractions combined with the stresses they carry which could be very different for each probable case makes it quite a complicated task.For a deep learning algorithm to be able to predict the fibre stresses opens up broad applicability and transferability to other use cases.For more trustworthy results and confidence in our predictions, it's important to have a much more diverse dataset and this is something which is left for future work.If the dataset covers the whole spectrum of microstructures used in wind adhesives then in future, we can immediately know the mechanical behavior of the adhesive composite if we have the microstructure.Further extensions include incorporation of further material properties of composites as this will enlarge the scope of application.Also, the study is reporting results only from Generative Adversarial Networks as the base algorithm.Further research needs to be done to see how other generative models like Auto-encoders and Diffusion models behave in these situations.The latest and more powerful text to image models can be adapted where the conditioning prompt is not the text but rather can be taken as an embedding of the input adhesive microstructure from which the model then using a learning strategy is trained to predict stress values or other field data.This is also left as future work.Another very important direction as part of the whole data driven strategy is the effective, cheap and representative generation of artificial microstructures as obtaining the real computed tomography is usually quite expensive.Generative AI can also be used for this purpose.The work could also be well extended to cracks and studies involving crack propagation in wind turbine blades.Also porosity and other defects can be incorporated.These effects can be employed with something called transfer learning where if a model is repurposed for a similar but repurposed activity.The main idea is to use the knowledge that has been attained in one task to enhance generalization in another.So instead of learning from scratch, we start with useful knowledge attained before is repurposed for a similar but repurposed activity.

Conclusions
In this work, Generative modelling specifically a conditional GAN is used to predict the stress fields from the 2D microstructures slices of a fibre reinforced adhesive used in wind energy blades.The microstructures were obtained from computed tomography.The conditional GAN model was trained on finite element based simulations and a database with varying boundary and load conditions was created.Here it is postulated that a conditional GAN model can predict stresses even with frugal data especially on fibres which encourages the use of Generative AI.However, the results at the edges of the borders need further improvement in accuracy.Further methods and configurations in Generative Deep Learning for example Conditional GANs and Diffusion models need to be investigated.Also more randomness need to be incorporated along with big datasets to make the model more generalizable.

2 .
Fibre Adhesive Microstructure and Numerical FE Modelling Short fibre adhesives are composites used in the bondlines of wind turbine blades.Various orientations and alignments are reported in prior experiments where dependencies are based on type and characteristics of adhesive flow during the joining process.This anisotropy then effects the structural properties of the composite.Several CT scans were conducted at various magnifications and the investigations reveal approximately 8-9% fibre volume fraction.One such CT-Scan of the adhesive at a magnification of 10X conducted with Zeiss Xradia 410 Versa is shown in Figure1.The scans depict white fibres (lighter) embedded in an adhesive (dark) matrix.

Figure 1 .
Figure 1.Left: Original Computed Tomography Scan.Right:(A) Photo after rotation (B) Cut and enlarged image (C) Filtered image with non-local mean de-noising (D) Binarized image with OTSU's method (E) Image after using open and close operation.

Figure 3 .
Figure 3. Creation of a FEA database with diversity.

Figure 4 .
Figure 4. Unconditional GAN where G and D denote the Generator and Discriminator respectively.

Figure 5 .
Figure 5. Conditional GAN for stress predictions using Pix to Pix Architecture.(A) Conditioning Map (B) Segmentation Network (C) Generated AI Stress Map (D) FEA based Stress Map.

Figure 6 .
Figure 6.Training process of the conditional GAN.(A) Composite microstructure under biaxial extension (B) FEA based prediction acting as ground truth (C) The iteration process, notice the predictions become better with advancing iterations, finally reaching to the tolerance value where the discriminator no longer can differentiate between the real and the fake predictions.

Figure 7 .
Figure 7. Equivalent stress field predictions for an adhesive microstructure under biaxial extension of 2%.(A)Microstructure of the adhesive composite, where the white colour represents the fibre and the black colour the adhesive matrix respectively (B) FEM predictions (Ground Truth) (C) Conditional GAN (AI) based predictions (D) Mean Absolute Error between the FEM results and the AI based predictions.Please note that whilst the composite image could be acquired at different magnifications but the overall image resolution is 256 pixels × 256 pixels in each case.

Figure 8 .
Figure 8. Equivalent stress field predictions for an adhesive microstructure under biaxial extension of 4%.(A) Microstructure of the adhesive composite, where the white colour represents the fibre and the black colour the adhesive matrix respectively (B) FEM predictions (Ground Truth) (C) Conditional GAN (AI) based predictions (D) Mean Absolute Error between the FEM results and the AI based predictions.Please note that whilst the composite image could be acquired at different magnifications but the overall image resolution is 256 pixels × 256 pixels in each case.