Documents published in Scipedia

• Fuzzy statistics-aided inference in experimental design

  R. Dwornicka, A. Gądek-Moszczak, R. Ulewicz, N. Radek

  WCCM2024.

  
  

  Abstract

  Conducting research based on active influence on the examined object or process requires distinguishing an explained quantity, measured quantitatively, the possible changes of which will be considered as influencing it through a group of quantities considered as explanatory quantities. This approach implicitly postulates the existence of a cause-and-effect relationship between the explanatory quantities and the explained quantity. In practice, especially industrial practice, explanatory quantities are often called controlled factors. Knowledge of possible cause-and-effect relationships can be graded, from the most comfortable situation of the existence of appropriate binding equations and their exact solutions, through the existence of binding equations but without knowing the exact solutions, to the absence of such equations. While in the first case, experimental research serves to refine the results originally calculated for idealized models, in the second case, it is a necessary stage of identifying the parameters of the postulated model, and in the third case, it is a necessary stage of collecting data for which the simplest possible forecasting model will be constructed

  Abstract
  Conducting research based on active influence on the examined object or process requires distinguishing an explained quantity, measured quantitatively, the possible changes [...]

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• Accurate solution of linear operator approximations using green's functions by a multi-level neural network approach

  Z. Aldirany, C. Bilodeau, R. Cottereau, M. Laforest

  WCCM2024.

  
  

  Abstract

  Lately, the approximation of operators for partial differential equations using deep learning has been extensively investigated. However, these deep learning approaches have limitations in terms of accuracy. In this work, we present a multi-level approach to accurately approximate linear operators using physics-informed Green operator networks. This method allows for the iterative reduction of the approximation errors through a sequence of operators, each targeting errors of increasing complexity at progressively smaller scales. Numerical examples for the one-dimensional Poisson problem will be presented to demonstrate the effectivenessof the proposed multi-level approach.

  Abstract
  Lately, the approximation of operators for partial differential equations using deep learning has been extensively investigated. However, these deep learning approaches have [...]

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• Symbolic regression via neural networks

  J. Moehlis, N. Boddupalli, T. Matchen

  WCCM2024.

  
  

  Abstract

  Mathematical models allow researchers to understand, analyze, and predict the behavior of systems of physical, biological, and technological interest, and are required for many techniques from dynamical systems and control theory to be used. Unfortunately, it is often impossible to derive mathematical models from first principles, and in such cases system identification is a powerful tool which can be used to deduce models from observed data. Many existing system identification techniques require pre-specification of a dictionary of possible terms in a mathematical model, limiting their ability to give models with the nonlinearities which can arise for biological and other complex systems. We present a methodology which overcomes this limitation by dynamically generating the terms in a model with the necessary complexity and nonlinearity to accurately describe a system’s dynamics. This uses a multilayered, operation-based symbolic regression approach, with the capacity to learn combinations and compositions of operations by training artificial neural networks. Our approach provides a powerful alternative to genetic programming strategies for symbolic regression, and can exploit many of the attractive features of artificial neural networks such as a straightforward learning strategy.

  Abstract
  Mathematical models allow researchers to understand, analyze, and predict the behavior of systems of physical, biological, and technological interest, and are required for [...]

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• Structural optimization through generative adversarial networks

  L. Pereira, L. Driemeier

  WCCM2024.

  
  

  Abstract

  The finite element method (FEM) is a well known approach to solve partial differential equations. It has important applications in structural engineering, such as in topology optimization (TO). TO involves, at each iteration, the solution of structural problems via FEM, which can add up to a high computational cost. Therefore, a line of research to accelerate TO emerged over the years focusing on machine learning (ML) approaches. Particularly, Artificial Neural Networks (ANNs) have been proposed to significantly speed-up the process by eliminating the iterative algorithm, which is intrinsic to TO. Since ANN is a supervised ML method, first a dataset is generated, containing finite element analysis (FEA) inputs, volume fraction, postprocessing, and final topologies. Then, with the Wasserstein Generative Adversarial Networks (WGANs) is trained on this dataset to map fields of physical quantities, such as the von Mises stress, to the final optimized structure. The final designs obtained via ML are quantitatively analyzed according to the metrics.

  Abstract
  The finite element method (FEM) is a well known approach to solve partial differential equations. It has important applications in structural engineering, such as in topology [...]

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• Data-driven modeling of complex mechanical components for integration in system-level simulations

  S. Vanpaemel, N. Kutz, S. Brunton

  WCCM2024.

  
  

  Abstract

  This contribution presents a data-driven approach featuring a physics-inspired neural network structure for modeling complex components in mecha(tro)nic systems. In the present approach, gated recurrent units (GRUs) are employed to approximate the ordinary differential equations (ODEs) describing the system’s states over time, followed by a deep feedforward neural network (FFNN) mapping these states to a target variable. The networks are shown to predict a latent space capable of modeling the underlying dynamics, without the need for measuring the full state vector and only relying on input-output measurements. Subsequently it is shown that a nonlinear coordinate transformation exists between the latent space of the network and the states obtained from the reference ODE integration (simulation). To have a verification of the network’s performance, it is applied to simulation-based data of an academic example for which the states and equations are known beforehand. Furthermore, the methodology is also applied to real measurement data from an INSTRON testing system capturing shock damper and bushing dynamic behaviour. Lastly, it is demonstrated that an ODE expression can be extracted from the trained network. This feature allows seamless integration of these networks into variable time-step, system-level simulation software.

  Abstract
  This contribution presents a data-driven approach featuring a physics-inspired neural network structure for modeling complex components in mecha(tro)nic systems. In the present [...]

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• Data-efficient one-step mechanical design of composites using generative AI

  M. Masrouri, Z. Qin

  WCCM2024.

  
  

  Abstract

  The distribution of material phases is crucial to determine the composite's mechanical properties. While the entire structure-mechanics relationship of highly ordered material distributions can be studied with a finite number of cases, this relationship is challenging to reveal for complex irregular distributions, preventing the design of such material structures from meeting specific mechanical requirements. The noticeable developments of artificial intelligence algorithms in material design enable the discovery of hidden structure-mechanics correlations, which is essential for designing composites of complex structures. It is intriguing how these tools can assist composite design. Here, we focus on the rapid generation of complex irregular composite structures and the stress distribution in loading. We find that generative AI, enabled through fine-tuned Low-Rank Adaptation models, can be trained with a few inputs to generate synthetic composite structures and the corresponding von Mises stress distribution. The results show that this technique is convenient in generating massive composite designs with useful mechanical information that dictates stiffness, fracture, and robustness of the material with one model, and such must be done by several different experimental or simulation tests. This research offers valuable insights for improving composite design to expand the design space and automatic screening of composite designs for improved mechanical functions.

  Abstract
  The distribution of material phases is crucial to determine the composite's mechanical properties. While the entire structure-mechanics relationship of highly ordered material [...]

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• A dynamic weighted loss function for enhancing the performance of neural networks

  C. Mang, A. Tahmasebi Moradi, D. Danan, M. Yagoubi

  WCCM2024.

  
  

  Abstract

  In machine learning process, hyper parameters are chosen in a way to decrease the prediction error and improve the convergence. However, the optimized hyper parameters have a limit in terms of enhancing the performance of the neural networks. In this work, the datasets used for the numerical experiments arise from the resolution of partial differential equations (PDE) defined on a spatial domain. We propose a DYNAmic WEIghted Loss (DYNAWEIL) function-based approach for neural networks that are used to learn these PDE’s solutions. This a two-step process: first we train for a few numbers of epochs in a classical way then the dynamic weighted loss function replaces the classical loss function by leveraging the information from past training error histories. To validate this method, we carry out numerical experiments with different neural networks on datasets arising on two different physics: Goldstein equation [1] and radiative transfer equation [2]. Thus, in order to demonstrate the relevance of this approach, we provide a comparison among a neural network model using a classical loss function, with and without hyper parameters optimization, and a dynamic weighted loss function for both versions.

  Abstract
  In machine learning process, hyper parameters are chosen in a way to decrease the prediction error and improve the convergence. However, the optimized hyper parameters have [...]

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• Prediction of unsteady heat transfer of temperature field on a circuit board using sub-voxels input data structure

  T. Tsukiji, Y. Wada, Y. Iwata, M. Irikiin

  WCCM2024.

  
  

  Abstract

  In this study, we propose a sub-voxel learning method based on a Neural Operator and predict the thermal temperature field on a circuit board in unsteady heat conduction. CAE analysis reduces the cost of experiments using prototypes in the design phase. However, the computational cost is high because the number of trials increases due to changes in analysis conditions. Predictions made by machine learning are less accurate than those made by CAE analysis, but they can significantly reduce computational costs. In general, machine learning is difficult to extrapolate. Thus, the concept of a Neural Operator has attracted attention as one machine learning method incorporating physical laws. In this study, we propose a sub-voxel learning method based on the Neural Operator, inspired by the forward Euler method, and predict the thermal temperature field on a circuit board during unsteady heat conduction. The input data is the analysis data of the current cycle step, and the output data is the data of the next cycle step. Dummy temperatures were set according to the heat generation of each IC in the input data of the 0th cycle step, and pseudo-difference values were generated to extract features. The prediction accuracy of the next cycle step was compared with and without dummy temperatures

  Abstract
  In this study, we propose a sub-voxel learning method based on a Neural Operator and predict the thermal temperature field on a circuit board in unsteady heat conduction. [...]

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• Graph neural networks for accelerating the discrete element simulation of granular flow

  P. Zhi, Y. Wu

  WCCM2024.

  
  

  Abstract

  Granular flow is a phenomenon widely presented in both the natural and engineering fields. Here granular materials could be either solid particles, e.g. rocks, soil, and grains, or liquid particles, e.g. mud and fresh concrete mortar. Soil landslides, particle transport, and grain accumulation have been edge-cutting hot research topics. Discrete Element Method (DEM) has been regarded as one of the most important methods to simulate granular flows and to investigate discontinuous and large deformation problems. The basic principle of DEM was to view the simulated object as consisting of discrete particles, to define specific constitutive relationships for the particles, and to study the macroscopic properties of the simulated object from a microscopic perspective based on the interactions between particles. However, DEM simulations usually consume very high computational cost for particle contact searching and detection. To accelerate the computational process of discrete element simulation, the Graph Neural Network (GNN) based deep learning model was proposed in this paper. In GNNs, graph nodes and graph edges represent the particles and their interactions. The training and testing datasets were generated using an open-source software named YADE, while the neural network model was constructed using PyTorch and Deep Graph Library (DGL). Replacing the direct calculation of particle collisions in DEM with the trained neural network model, the state of the particles at the next moment could be predicted based on the current state of the particles. It significantly increased computational speed. The proposed technique was applied in various examples, such as drum rotation and hopper stacking, and its accuracy had been verified. This study established a solid foundation and provided robust support for further research and applications of granular flow simulation based on GNN

  Abstract
  Granular flow is a phenomenon widely presented in both the natural and engineering fields. Here granular materials could be either solid particles, e.g. rocks, soil, and grains, [...]

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• Prediction of plural crack propagation using discovered PDE

  G. Muraoka, Y. Wada

  WCCM2024.

  
  

  Abstract

  This study presents a prediction of plural crack propagation using the discovered partial differential equations. 80% of structures fracture due to fatigue failure. Therefore, the evaluation of fatigue cracks is essential. Numerical analysis is costly, and machine-learning surrogate models have been proposed. Hence, the crack propagation path and remaining life are predicted using machine learning. A dataset is obtained from the results of a crack propagation analysis using a s-version FEM combined with an automatic mesh generation technique. The input parameters are the coordinates of the four crack tips, and the output parameters are the crack propagation vector and the number of cycles of 0.25 mm. Also, physics-informed neural networks (PINNs) have been widely studied in recent years. Thus, we took inspiration from PINNs and added a regularization term of PDE discovered by AI Feynman to the loss. As a result, the loss of a validation dataset for training constrained by PDE was reduced by about 77% compared to the unconstrained loss. The error in crack length decreased from -0.50% to 0.17%.

  Abstract
  This study presents a prediction of plural crack propagation using the discovered partial differential equations. 80% of structures fracture due to fatigue failure. Therefore, [...]

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