Finite Elements Using Neural Networks and a Posteriori Error
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ORIGINAL PAPER
Finite Elements Using Neural Networks and a Posteriori Error Atsuya Oishi1 · Genki Yagawa2,3 Received: 21 March 2020 / Accepted: 30 September 2020 © CIMNE, Barcelona, Spain 2020
Abstract As the finite element method requires many nodes or elements to obtain accurate results, the adaptive finite element method has been developed to obtain better results with fewer nodes, where error information is used to refine the initial mesh adaptively. In contrast to this, we propose in this paper two new methods to directly derive accurate results by artificial neural networks using information about the errors of analysis results. One of the proposed methods employs error information obtained using a posteriori error estimator to predict accurate solutions from a single analysis with a coarse mesh. The other utilizes error information obtained from comparison between two analysis results: analysis results by using a coarse mesh and those by using the corresponding refined mesh. In both methods above, the artificial neural network is employed to predict accurate results at any target point in the analysis domain based on the error information around the point. These methods are successfully tested in two-dimensional stress analyses.
1 Introduction One of the most important issues of the finite element analyses is the development of solution methods to achieve an accurate solution with high speed. Under the above circumstance, several techniques have been studied in the last few decades. One is an attempt to improve the computational efficiency of large-scale analysis using multiple CPUs and accelerators, where various hardware such as supercomputers [2, 4, 40, 164], multi-core CPUs [75, 106] or GPUs (Graphics Processing Units) [23, 121, 129] are employed. The other is an attempt to reduce the number of nodes while maintaining accuracy called the adaptive method [5, 6, 156], which is developed as a method for obtaining a highly accurate solution while suppressing the drastic increase of the number of total nodes. In the method, the analysis is first performed with relatively coarse mesh consisting of a * Atsuya Oishi aoishi@tokushima‑u.ac.jp Genki Yagawa [email protected] 1
Graduate School of Technology, Industrial and Social Sciences, Tokushima University, 2‑1 Minami‑Johsanjima, Tokushima 770‑8506, Japan
2
University of Tokyo, Tokyo, Japan
3
Toyo University, Tokyo, Japan
small number of elements of large sizes and then with partially refined mesh consisting of a larger amount of small elements in areas where a posterior error estimation indicates poor accuracy. Mesh refinement is repeated in areas of poor accuracy until the required accuracy is achieved, and the total number of nodes is much smaller than that would be obtained by dividing the whole domain into small elements. This method has the advantage that a solution with high accuracy can be obtained in the desired area. Many theoretical and numerical studies have been conducted on the method [11, 16, 21, 22, 32, 110, 124, 125, 147], and
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