  Citation: Morse, L.; Mallardo, V.; Sharif-Khodaei, Z.; Aliabadi, F.M.H. Optimisation of Assembled Plate Structures with the Boundary Element Method. Aerospace 2022, 9, 381. https://doi.org/10.3390/ aerospace9070381 Academic Editor: Yui-Chuin Shiah Received: 30 May 2022 Accepted: 13 July 2022 Published: 15 July 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). aerospace Article Shape Optimisation of Assembled Plate Structures with the Boundary Element Method Llewellyn Morse 1, * , Vincenzo Mallardo 1 , Zahra Sharif-Khodaei 2 and Ferri M.H. Aliabadi 2 1 Department of Architecture, University of Ferrara, Via Quartieri 8, 44121 Ferrara, Italy; mlv@unife.it 2 Department of Aeronautics, Imperial College London, South Kensington Campus, City and Guilds Building, Exhibition Road, London SW7 2AZ, UK; z.sharif-khodaei@imperial.ac.uk (Z.S.-K.); m.h.aliabadi@imperial.ac.uk (F.M.H.A.) * Correspondence: mrslwl@unife.it Abstract: A novel methodology is presented for performing sensitivity analyses of assembled plate structures using the Boundary Element Method (BEM). The main novelty of this work is that the exact implicit derivatives of the BEM formulations for assembled plate structures have been derived for the first time and incorporated into a newly developed Implicit Differentiation Method (IDM), enabling sensitivity analyses to be conducted for more complex and realistic structures in a more accurate, robust, and efficient manner than previous approaches. Three numerical examples are investigated to validate the derived exact implicit derivatives and to demonstrate how they could be used for a potential application involving the shape optimisation of a complex X-core structure from the canard of a Eurofighter Typhoon fighter jet. Results show that the newly developed IDM is more accurate, robust, and efficient when compared to alternative methodologies using derivatives obtained from methods such as the Finite Difference Method (FDM) and the Finite Element Method (FEM). Keywords: Boundary Element Method (BEM); Implicit Differentiation Method (IDM); Finite Differ- ence Method (FDM); plates; assembled structures; shape optimisation 1. Introduction Sensitivity analysis plays a key role in the design of structures. It is very important for engineers to understand the sensitivity of a structure’s performance to changes in geometrical, material, or loading parameters. Sensitivity analysis also enables engineers to optimise the design of their structures and maximise their safety. The Boundary Element Method (BEM) is a very useful tool for conducting sensitivity analyses of structures. When modelling a structure, the BEM only requires the outer boundary of the structure to be discretised into elements. This makes the BEM a very efficient tool for sensitivity analyses, because only a relatively small part of the model needs to be re-meshed during the analyses. Furthermore, the BEM can often achieve a similar level accuracy to the Finite Element Method (FEM) while using a courser mesh with fewer elements, indicating that the BEM would be a very efficient method for optimisation problems involving the calculation of many sensitivities over many iterations [1]. Most prior work on structural sensitivity analysis with the BEM amongst the research community has involved 2D structures [2–11], and 3D structures [12–17] to a smaller degree. There have only been a few prior research works on the topic of structural sen- sitivity analysis with the BEM that have involved plate structures, with some examples being [18,19]. There are many structures in engineering, particularly in aircraft, that are not able to be modelled accurately in 2D. Furthermore, although modelling a structure in 3D can lead to very accurate results, it is often more computationally efficient to model it as a plate structure, without a noticeable loss in modelling accuracy. Previous works on plate structural sensitivity analysis with the BEM [18,19] have only considered simple single- plate structures, which have a limited range of application. Assembled-plate structures, Aerospace 2022, 9, 381. https://doi.org/10.3390/aerospace9070381 https://www.mdpi.com/journal/aerospace