PSI - Issue 52

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Procedia Structural Integrity 52 (2024) 752–761 Structural Integrity Procedia 00 (2023) 000–000 Structural Integrity Procedia 00 (2023) 000–000

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Fracture, Damage and Structural Health Monitoring FFT-based homogenisation for Thin Plate Structures Fracture, Damage and Structural Health Monitoring FFT-based homogenisation for Thin Plate Structures

HaolinLi a, ∗ , Zahra Sharif Khodaei 1 , M.H. Aliabadi a a Department of Aeronautics, Imperial College London, London SW7 2AZ, UK HaolinLi a, ∗ , Zahra Sharif Khodaei 1 , M.H. Aliabadi a a Department of Aeronautics, Imperial College London, London SW7 2AZ, UK

© 2023 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of Professor Ferri Aliabadi Abstract This paper introduces an FFT-based homogenisation approach designed to mitigate the computational demands associated with conventional numerical methods for thin plate structures. The periodic Lippman-Schwinger is proposed equation as an approach to solve the governing equation of both Classical and First-order plate models in cell problems, achieving an explicit solution by Green’s function obtained from the Fourier space. The paper provides comprehensive details on the developed method, its algo rithmic implementation, and its potential application demonstrated through two case studies focused on complex plate structures. The findings reveal a significant alignment with the results derived from Finite Element Method (FEM), with an added advantage of marked time e ffi ciency observed within the FFT-based approach. © 2023 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http: // creativecommons.org / licenses / by-nc-nd / 4.0 / ) Peer-review under responsibility of Professor Ferri Aliabadi. Keywords: FFT-based Homogenisation; Thin plate models; Micro structures; Abstract This paper introduces an FFT-based homogenisation approach designed to mitigate the computational demands associated with conventional numerical methods for thin plate structures. The periodic Lippman-Schwinger is proposed equation as an approach to solve the governing equation of both Classical and First-order plate models in cell problems, achieving an explicit solution by Green’s function obtained from the Fourier space. The paper provides comprehensive details on the developed method, its algo rithmic implementation, and its potential application demonstrated through two case studies focused on complex plate structures. The findings reveal a significant alignment with the results derived from Finite Element Method (FEM), with an added advantage of marked time e ffi ciency observed within the FFT-based approach. © 2023 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http: // creativecommons.org / licenses / by-nc-nd / 4.0 / ) Peer-review under responsibility of Professor Ferri Aliabadi. Keywords: FFT-based Homogenisation; Thin plate models; Micro structures; In the exploration of the microscopic properties of specific materials, homogenisation techniques are often em ployed. These methods are focusing on periodic unit cells that reflect the microstructure of the materials under con sideration Zuo et al. (2013); Li et al. (2023). This process, which is centered around deriving homogenised macro properties from micro cells, is known as cell problems Schneider (2021). Classical numerical methods, such as the finite element method, have been traditionally used to address these cell problems. This entails the application of periodic boundary conditions on the model, thereby facilitating the solution of the partial di ff erential equations via their weak form Li et al. (2022, 2023). In contrast to these conventional numerical techniques, an innovative approach has been introduced to solve the cell problems, utilizing the periodic characteristics inherent in the problem. This method engages the Green’s functions between a referenced polarization field and the resultant strain distribution as the solution to periodic Lippmann Schwinger equations for certain partial di ff erential equations Moulinec and Suquet (1998, 1994). This technique significantly reduces the computational intensity since the introduction of explicit Green’s functions and the incorpo- In the exploration of the microscopic properties of specific materials, homogenisation techniques are often em ployed. These methods are focusing on periodic unit cells that reflect the microstructure of the materials under con sideration Zuo et al. (2013); Li et al. (2023). This process, which is centered around deriving homogenised macro properties from micro cells, is known as cell problems Schneider (2021). Classical numerical methods, such as the finite element method, have been traditionally used to address these cell problems. This entails the application of periodic boundary conditions on the model, thereby facilitating the solution of the partial di ff erential equations via their weak form Li et al. (2022, 2023). In contrast to these conventional numerical techniques, an innovative approach has been introduced to solve the cell problems, utilizing the periodic characteristics inherent in the problem. This method engages the Green’s functions between a referenced polarization field and the resultant strain distribution as the solution to periodic Lippmann Schwinger equations for certain partial di ff erential equations Moulinec and Suquet (1998, 1994). This technique significantly reduces the computational intensity since the introduction of explicit Green’s functions and the incorpo- 1. Introduction 1. Introduction

∗ Corresponding author. E-mail address: haolin.li20@imperial.ac.uk ∗ Corresponding author. E-mail address: haolin.li20@imperial.ac.uk

2452-3216 © 2023 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of Professor Ferri Aliabadi 10.1016/j.prostr.2023.12.075 2210-7843 © 2023 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http: // creativecommons.org / licenses / by-nc-nd / 4.0 / ) Peer-review under responsibility of Professor Ferri Aliabadi. 2210-7843 © 2023 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http: // creativecommons.org / licenses / by-nc-nd / 4.0 / ) Peer-review under responsibility of Professor Ferri Aliabadi.