A large amount of data can partly assure good fitting quality for the trained neural networks. When the quantity of experimental or on-site monitoring data is commonly insufficient and the quality is difficult to control in engineering practice, numerical simulations can provide a large amount of controlled high quality data. Once the neural networks are trained by such data, they can be used for predicting the properties/responses of the engineering objects instantly, saving the further computing efforts of simulation tools. Correspondingly, a strategy for efficiently transferring the input and output data used and obtained in numerical simulations to neural networks is desirable for engineers and programmers. In this work, we proposed a simple image representation strategy of numerical simulations, where the input and output data are all represented by images. The temporal and spatial information is kept and the data are greatly compressed. In addition, the results are readable for not only computers but also human resources. Some examples are given, indicating the effectiveness of the proposed strategy.

With the recent developments of machine learning algorithms, frameworks and systems, numerous Artificial Neural Networks (ANNs) have been proposed, built and adopted rapidly and widely in engineering applications. Neural networks can be driven by mechanisms or data. The first type can be represented by the Physical-Informed Neural Network (PINN) [

Furthermore, engineering structures can be relatively large across space and time and the amount of data from on-site sensors and experiments, especially spatial data, is generally insufficient. In addition, these data can deviate considerably because of errors from monitoring or testing. In contrast, the quantity and quality of data from numerical simulations can be assured. Hence, first validating the numerical model by comparing the results to the experimental and monitored results and then training the neural network with numerically obtained data can be an advantageous procedure.

Basic data driven neural networks are sequential learning models. There are input and output datasets, between which the structures of the neurons can be assembled and built in the platforms associated with TensorFlow [

The method is easy to follow and can be implemented into the pre- and pro-processing parts of numerical tools such as those built in the finite element method (FEM) framework.

The input images naturally take into account the spatial information of the cases, which can be understood by not only neural networks but also human resources.

The sizes of the images can be further compressed/decompressed by other models such as autoencoders.

Some examples will be provided to show the flexibility and effectiveness of the method. Moreover, we want to emphasize here that some procedures we proposed in this work could be very basic and natural for researchers working in computational mechanics, who follow similar rules for pre- and pro-processing during programming and computing for a long time. Nevertheless, we believe the method can be inspiring and helpful for researchers and engineers working in other fields such as computer science, and civil and mechanical engineering, which is the main motivation of this work. In the next section, we will provide basic rules and examples together with which the procedures are clarified.

We focus on 2D images and 2D simulations (planer or 1D transient cases) in this work, but the ideas can be extended to higher dimensional cases by using a series of continuous images/animations. Considering RGB images, every pixel has channels of three colors: red, green and blue

The mechanical responses of matrix-inclusion materials are basic numerical simulations for composites, such as concrete, rocks and polymers. The model and mesh are shown in

When the matrix and inclusion are isotropic and linear elastic, basic material properties include elastic modulus and Poisson's ratio, represented by red and green channels as

Meanwhile, the stress tensor

It can be found that transforming the input/output parameters into RGB figures is a normalization step, which shall be done anyway for neural networks.

Considering the elastic modulus of the matrix and inclusion as 20 and 80 GPa respectively and the Poisson's ratios of the matrix and inclusion as 0.3 and 0.1, respectively, the input and output images are shown in

The second example refers to limit analysis of slope, which provides a factor of safety of a slope for assessing its stability and safety. Considering upper bound limit analysis, the necessary material parameters are cohesion

The output results are represented by slip lines or so called failure pattern images, which are obtained by discontinuity layout optimization [

When concrete members are subjected to fire loadings, explosive spalling may occur, which is the violent fracturing and splitting of concrete pieces from the heated structures. Spalling greatly jeopardizes the integrity and duratbility of structures [

The control equations of the THC model of heated concrete are composed of three strongly coupled heat equations, which need to be solved concurrently, as a computing exhausting numerical procedure. Meanwhile, the fire loadings and concrete properties can be complex. Engineers and designers would always like to quickly assess the spalling risk of specific structures considering different conditions, which is a strong motive for developing data driven neural network models.

In [

Considering the 1D case, the horizontal direction of the output image is taken for space distributions and the vertical direction is taken for time evolutions of

A hybrid neural network composed of an autoencoder (AE) and a fully connected neural network (FNN) was built by the authors for learning the coupled THC example in [

Taking the structure shown in

To avoid over-fitting, the K-fold cross-validation method is used to verify the generalization capability of the model. The 6048 groups of input and output images of the training set were divided into 10 groups of disjoint subsets and trained 10 times. Each time, one group was selected as the verification set and the other 9 groups were selected as the training set. Ten groups of data were trained and evaluated. The average loss and standard deviation obtained from ten-fold cross validation were 0.002295 (

Hyper parameters | Index |
---|---|

Learning rate | |

Epochs | 200 |

Batch size | 64 |

Activation function | ReLU for CNN and Sigmoid for FNN |

Loss function | MSE |

Sample size | 6720 |

The evolution of the MSE loss with the evolution time (epoch) of all designs considering training and testing are shown in

In this work, we present a strategy for representing the input parameters and output results of numerical simulations by images. With several examples we show that this strategy is simple and compatible with the pre/post-processing parts of popular numerical tools. The images account for the spatial and temporal information used and obtained in the numerical simulations. In addition, all images can be reprocessed by other algorithms. For one of the examples, we train a hybrid CNN-FNN neural network with the input/output images, indicating the effectiveness of the proposed strategy.