Multiscale optimisation of resonant frequencies for lattice-based additive manufactured structures

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RESEARCH PAPER

Multiscale optimisation of resonant frequencies for lattice-based additive manufactured structures Morgan Nightingale1

· Robert Hewson1 · Matthew Santer1

Received: 2 April 2020 / Revised: 11 September 2020 / Accepted: 22 September 2020 © The Author(s) 2020

Abstract This paper introduces a novel methodology for the optimisation of resonant frequencies in three-dimensional lattice structures. The method uses a multiscale approach in which the homogenised material properties of the lattice unit cell are defined by the spatially varying lattice parameters. Material properties derived from precomputed simulations of the small scale lattice are projected onto response surfaces, thereby describing the large-scale metamaterial properties as polynomial functions of the small-scale parameters. Resonant frequencies and mode shapes are obtained through the eigenvalue analysis of the large-scale finite element model which provides the basis for deriving the frequency sensitivities. Frequency tailoring is achieved by imposing constraints on the resonant frequency for a compliance minimisation optimisation. A sorting method based on the Modal Assurance Criterion allows for specific mode shapes to be optimised whilst simultaneously reducing the impact of localised modes on the optimisation. Three cases of frequency constraints are investigated and compared with an unconstrained optimisation to demonstrate the algorithms applicability. The results show that the optimisation is capable of handling strict frequency constraints and with the use of the modal tracking can even alter the original ordering of the resonant mode shapes. Frequency tailoring allows for improved functionality of compliance-minimised aerospace components by avoiding resonant frequencies and hence dynamic stresses. Keywords Homogenisation · Heterogeneous multiscale methods · Resonant frequency optimisation · Lattice · Additive manufacturing

1 Introduction Lightweight structures that are able to mitigate or avoid dynamic loading are a key area of development within the aerospace industry. Failure caused by resonance is of particular interest in areas such as engine design and other applications involving vibration or shock loading. The design of lightweight structures capable of avoiding resonance is the primary focus of this work and is achieved using a novel, multiscale lattice-based method of structural optimisation.

Responsible Editor: Axel Schumacher  Morgan Nightingale

[email protected] 1

Department of Aeronautics, Imperial College London, London, UK

Previous research into the optimisation of resonant frequencies has resulted in several methods being adopted within the industry, such as topology optimisation based on the Solid Isotropic Material with Penalisation (SIMP) (Bendsøe 1989) and Bi-directional Evolutionary Structural Optimisation (BESO) methodologies (Huang et al. 2009; Sivapuram and Picelli 2018). Resonant frequencies have also been optimised for using a level set shape and topology optimisation method (Xia et al. 2011; Xu