HiDeNN-FEM: a seamless machine learning approach to nonlinear finite element analysis

Liu, Yingjian; Park, Chan-Wook; Lu, Ye; Mojumder, Satyajit; Liu, Wing Kam; Qian, Dong

doi:10.1007/s00466-023-02293-z

Cited by 7 publications

(4 citation statements)

References 30 publications

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“…A different approach are the hierarchical deep-learning NNs (HiDeNNs) [506] with extensions in [507][508][509][510][511][512]. Here, shape functions are treated as NNs constructed from basic building blocks.…”

Section: Finite Element Methodsmentioning

confidence: 99%

Deep learning in computational mechanics: a review

Herrmann,

Kollmannsberger

2024

Comput Mech

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The rapid growth of deep learning research, including within the field of computational mechanics, has resulted in an extensive and diverse body of literature. To help researchers identify key concepts and promising methodologies within this field, we provide an overview of deep learning in deterministic computational mechanics. Five main categories are identified and explored: simulation substitution, simulation enhancement, discretizations as neural networks, generative approaches, and deep reinforcement learning. This review focuses on deep learning methods rather than applications for computational mechanics, thereby enabling researchers to explore this field more effectively. As such, the review is not necessarily aimed at researchers with extensive knowledge of deep learning—instead, the primary audience is researchers on the verge of entering this field or those attempting to gain an overview of deep learning in computational mechanics. The discussed concepts are, therefore, explained as simple as possible.

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Section: Finite Element Methodsmentioning

confidence: 99%

Deep learning in computational mechanics: a review

Herrmann,

Kollmannsberger

2024

Comput Mech

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“…Moreover, ReLU NNs have proved particularly adept at function approximation tasks [24] due to their rectifying nonlinearity. While most hybrid methods have demonstrated the possibility of FEM and ReLU NN equivalence numerically [19], [20], [21], [22], [23], [11], our work in this paper focuses on establishing this equivalence theoretically. The need for theoretical results is bound to motivate more mathematically founded work in developing frameworks for these hybrids and perhaps the deep learning approaches as numerical schemes with well established qualitative numerical properties.…”

Section: Introductionmentioning

confidence: 98%

“…al, established a result that guarantees that under certain conditions, rectified linear unit (ReLU) neural networks (NNs) can approximate any continuous piece wise linear (CPwL)functions to considerable accuracy [1]. This work did not pioneer but enhanced and also vindicated the development of hybrid methods between the FEM (with CPwL functions in 1D) with ReLU NNs [18], [19], [20], [21], [22], [23]. Some studies have explored the use of FEM solutions as training data for DNNs, leveraging the interpretability and error guarantees of FEM to enhance the generalizability and accuracy of the resulting NN models [24].…”

Section: Introductionmentioning

confidence: 99%

Characterizing Parameter Equivalence and Approximation in 1D FEM and ReLU NNs

Katende,

Oleynik,

Kasumba

et al. 2024

Preprint

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This paper introduces a novel methodology for characterizing parameter equivalence between Rectified Linear Unit (ReLU) Neural Networks (NNs) and 1D Finite Element Models (FEMs). By expressing the FEM solution equation in terms of neural networks and employing a hybrid scheme involving partially trained nested neural networks, we bridge the gap between these two computational paradigms. Our approach offers a promising avenue to leverage the flexibility of neural networks while retaining the established principles of FEM. Through extensive numerical experiments, we demonstrate the efficacy of our scheme in function interpolation and solving partial differential equations (PDEs). In function interpolation tasks, our hybrid scheme consistently outperforms fully trainable ReLU NNs, achieving significantly lower error factors across various scenarios. Similarly, in solving PDEs, our approach exhibits superior accuracy compared to fully trainable ReLU NNs, as evidenced by lower L2-error factors. We further extend the application of our hybrid scheme to diverse PDEs including the heat, Poisson, and wave equations, showcasing its versatility and effectiveness across different problem domains. This work not only presents a significant advancement in computational methods but also lays the groundwork for redefining the FEM algorithm within an algorithmic framework. These findings have implications for a wide range of scientific and engineering disciplines, and we believe they warrant consideration for publication in high-impact journals.

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“…In addition to conventional structural analysis methods, neural networks can predict structural responses nowadays. The emerging trends can be found in [32,33].…”

Section: Introductionmentioning

confidence: 99%

Fatigue Analysis of a Jacket-Supported Offshore Wind Turbine at Block Island Wind Farm

Partovi-Mehr,

DeFrancisci,

Minaeijavid

et al. 2024

Sensors

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Offshore wind-turbine (OWT) support structures are subjected to cyclic dynamic loads with variations in loadings from wind and waves as well as the rotation of blades throughout their lifetime. The magnitude and extent of the cyclic loading can create a fatigue limit state controlling the design of support structures. In this paper, the remaining fatigue life of the support structure for a GE Haliade 6 MW fixed-bottom jacket offshore wind turbine within the Block Island Wind Farm (BIWF) is assessed. The fatigue damage to the tower and the jacket support structure using stress time histories at instrumented and non-instrumented locations are processed. Two validated finite-element models are utilized for assessing the stress cycles. The modal expansion method and a simplified approach using static calculations of the responses are employed to estimate the stress at the non-instrumented locations—known as virtual sensors. It is found that the hotspots at the base of the tower have longer service lives than the jacket. The fatigue damage to the jacket leg joints is less than 20% and 40% of its fatigue capacity during the 25-year design lifetime of the BIWF OWT, using the modal expansion method and the simplified static approach, respectively.

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HiDeNN-FEM: a seamless machine learning approach to nonlinear finite element analysis

Cited by 7 publications

References 30 publications

Deep learning in computational mechanics: a review

Deep learning in computational mechanics: a review

Characterizing Parameter Equivalence and Approximation in 1D FEM and ReLU NNs

Fatigue Analysis of a Jacket-Supported Offshore Wind Turbine at Block Island Wind Farm

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