Thermal Deformation Analysis of Shell Structures Using a Weak-Form Physics-Informed Neural Network

Myeongryun Seong; Anna Lee

doi:10.7734/COSEIK.2026.39.5.345

All Issue

2026 Vol.39, Issue 5 Preview Page

Research Paper

Thermal Deformation Analysis of Shell Structures Using a Weak-Form Physics-Informed Neural Network 약형 물리정보 신경망을 이용한 쉘 구조의 열변형 해석

31 October 2026. pp. 345-358

PDF XML

Abstract

Physics-informed neural networks (PINNs) have emerged as mesh-free solvers for boundary-value problems in computational mechanics. Existing shell PINN formulations have mainly considered mechanical loading, whereas thermoelastic PINN studies have focused on three-dimensional solids or one-dimensional beams. This paper presents a weak-form PINN framework for the thermal deformation analysis of shell structures under prescribed thermal loading, based on the Naghdi five-parameter shell theory. Thermal effects are introduced through the Duhamel-Neumann decomposition of the total strain into elastic and thermal parts. Substitution into the shell energy functional adds thermal membrane and bending load terms to the variational loss without changing the neural network architecture. The network is trained using L-BFGS with quasi-Monte Carlo collocation points, and Dirichlet boundary conditions are imposed through geometry-aware trial functions. Three benchmark problems—a fully pinned plate, cantilever plate, and clamped cylindrical shell—are analyzed and compared with Abaqus finite element solutions. Displacement errors remain below 3% in all cases, whereas rotation errors are mostly 2%-4%, with one cantilever component reaching 9.04%. These results show that the proposed weak-form PINN can reproduce thermoelastic Naghdi shell responses under the prescribed thermal fields, providing a variational basis that can be extended to future inverse and data-assimilation problems in thermo-mechanical shell analysis.

Keywords

physics-informed neural network

weak-form formulation

naghdi shell theory

thermal deformation

thermoelasticity

본 연구에서는 Naghdi 5-parameter 쉘 이론에 기반하여 주어진 열하중을 받는 쉘 구조의 열변형 해석을 위한 약형 물리정보 신경망(PINN)을 제시한다. Duhamel-Neumann 관계식으로 열변형률을 분리하고, 이를 쉘 에너지 범함수에 대입하여 열에 의한 막 및 굽힘 하중 항을 변분 손실함수에 포함하였다. 제안한 방법은 fully pinned 평판, cantilever 평판, 양단 고정 원통형 쉘 문제에서 Abaqus 유한요소해와 비교하여 검증하였다. 변위장의 상대 L2 오차는 모든 예제에서 3% 이내였으며, 회전장 오차도 대부분 2~4% 수준으로 나타났다. 다만, cantilever 평판의 회전 성분에서는 9.04%의 상대 오차가 관찰되었다. 본 정식화는 신경망 구조의 변경없이 열변형을 도입할 수 있으며, 향후 측정된 변형장으로부터 열하중 및 물성을 추정하는 역문제 및 데이터 융합 해석으로 확장될 수 있다.

키워드

물리정보 신경망

Naghdi 쉘 이론

열변형

약형 손실

변분원리

References

Bastek, J.H., Kochmann, D.M. (2023) Physics-Informed Neural Networks for Shell Structures, Eur. J. Mech. A/Solids, 97, 104849.

10.1016/j.euromechsol.2022.104849

Boley, B.A., Weiner, J.H. (2012) Theory of Thermal Stresses, John Wiley &Sons, New York, p.608.

Choi, C.K., Yoo, S.W. (1990) Improved Degenerated Shell Finite Elements for Analysis of Shell Structures, J. Comput. Struct. Eng. Inst. Korea, 3(1), pp.97~107.

E, W., Yu, B. (2018) The Deep Ritz Method: A Deep Learning-Based Numerical Algorithm for Solving Variational Problems, Commun. Math. Stat., 6(1), pp.1~12.

10.1007/s40304-018-0127-z

Harandi, A., Moeineddin, A., Kaliske, M., Reese, S., Rezaei, S. (2024) Mixed Formulation of Physics-Informed Neural Networks for Thermo-Mechanically Coupled Systems and Heterogeneous Domains, Int. J. Numer. Methods Eng., 125, e7388.

10.1002/nme.7388

Hendrycks, D., Gimpel, K. (2016) Gaussian Error Linear Units (GELUs), arXiv:1606.08415.

Heo, J.-J., Kim, H.-K., Kim, J.-S. (2000) The Finite Element Analysis of Shell Structures Using Improved Shell Element, J. Comput. Struct. Eng. Inst. Korea, 13(4), pp.449~459.

Jagtap, A.D., Kawaguchi, K., Karniadakis, G.E. (2020) Adaptive Activation Functions Accelerate Convergence in Deep and Physics-Informed Neural Networks, J. Comput. Phys., 404, 109136.

10.1016/j.jcp.2019.109136

Kim, M.-G., Song, Y., Lee, H., Kim, J. (2024) Computational Analysis of Geometrically Exact Shell Elements Using Multipatch IsoGeometric Analysis, J. Comput. Struct. Eng. Inst. Korea, 37(5), pp.345~352.

10.7734/COSEIK.2024.37.5.345

Koiter, W.T., Simmonds, J.G. (1973) Foundations of Shell Theory, in: Becker, E., Mikhailov, G.K. (Eds.), Theoretical and Applied Mechanics, Springer, Berlin, pp.150~176.

10.1007/978-3-642-65590-6_11

Krätzig, W.B. (1980) Introduction to General Shell Theory, in: Olszak, W. (Ed.), Thin Shell Theory: New Trends and Applications, Springer, Vienna, p.61.

10.1007/978-3-7091-2442-0_1

Lagaris, I.E., Likas, A., Fotiadis, D.I. (1998) Artificial Neural Networks for Solving Ordinary and Partial Differential Equations, IEEE Trans. Neural Netw., 9(5), pp.987~1000.

10.1109/72.712178

Li, W., Bazant, M.Z., Zhu, J. (2021) A Physics-Guided Neural Network Framework for Elastic Plates: Comparison of Governing Equations-Based and Energy-Based Approaches, Comput. Methods Appl. Mech. Eng., 383, 113933.

10.1016/j.cma.2021.113933

Liu, D.C., Nocedal, J. (1989) On the Limited Memory BFGS Method for Large Scale Optimization, Math. Program., 45, pp.503~528.

10.1007/BF01589116

Niordson, F.I. (1985) Shell Theory, Vol. 29, North-Holland Series in Applied Mathematics and Mechanics, Elsevier Science, Amsterdam, p.408.

Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., Desmaison, A., Köpf, A., Yang, E., DeVito, Z., Raison, M., Tejani, A., Chilamkurthy, S., Steiner, B., Fang, L., Bai, J., Chintala, S. (2019) PyTorch: An Imperative Style, High-Performance Deep Learning Library, Adv. Neural Inf. Process. Syst., 32, pp.8024~8035.

Raissi, M., Perdikaris, P., Karniadakis, G.E. (2019) Physics-Informed Neural Networks: A Deep Learning Framework for Solving Forward and Inverse Problems Involving Nonlinear Partial Differential Equations, J. Comput. Phys., 378, pp.686~707.

10.1016/j.jcp.2018.10.045

Timoshenko, S.P., Woinowsky-Krieger, S. (1959) Theory of Plates and Shells, 2nd ed., McGraw-Hill, New York, p.580.

Yan, C.A., Vescovini, R., Dozio, L. (2022) A Framework Based on Physics-Informed Neural Networks and Extreme Learning for the Analysis of Composite Structures, Comput. Struct., 265, 106761.

10.1016/j.compstruc.2022.106761

Zhuang, X., Guo, H., Alajlan, N., Zhu, H., Rabczuk, T. (2021) Deep Autoencoder Based Energy Method for the Bending, Vibration, and Buckling Analysis of Kirchhoff Plates with Transfer Learning, Eur. J. Mech. A/Solids, 87, 104225.

10.1016/j.euromechsol.2021.104225

Information

Publisher :Computational Structural Engineering Institute of Korea
Publisher(Ko) :한국전산구조공학회
Journal Title :Journal of the Computational Structural Engineering Institute of Korea
Journal Title(Ko) :한국전산구조공학회 논문집
Volume : 39
No :5
Pages :345-358
Received Date : 2026-06-29
Revised Date : 2026-07-02
Accepted Date : 2026-07-02
DOI :https://doi.org/10.7734/COSEIK.2026.39.5.345

Journal of the Computational Structural Engineering Institute of Korea ISSN:1229-3059(Print) 2287-2302(Online) 한국전산구조공학회 논문집

All Issue