This work combines Moving Least Square mesh morphing methods with classical optimization techniques to optimize Composite Stiffened Structures against buckling constraints. The idea is to use standard optimization methods to handle thicknesses and percentages of the composite laminates and Moving Least Square morphing capabilities to modify the shape of the stringers without generating new meshes from geometry during the optimization process.
Methodology
The considered structural assembly is reported in the figure below. A flat panel with five stringers and 2 rib bays has been considered as an exemplificative aeronautical structure. The stiffened panel is 1750 mm wide and 2300 mm long, with equally spaced stringer with a pitch of approximately 250 mm. The panel is made of a unidirectional CRFP.
A three rib composite stiffened flat panel
The goal of the proposed optimization is to reduce the overall panel weight while maximising the buckling load in the attempt to find the best layup (thicknesses and percentages) for skin and stringers, as well as the optimal shape for the stringers via MLS morphing as available in Shaper. The evaluations of the buckling load have been performed using Abaqus®.
The constrained multi-objective genetic algorithm of Nexus (a proprietary implementation of NSGA-II) has been used to identify the solutions representing the best trade-off between weight and first buckling load under pure assail compression.
MLS morphing: a) undeformed stringer geometry, b) and c) morphed stringer geometries
Design variables can be grouped in two main sets: variables controlling the shape of the stringers, i.e. the position of the handle in the morphing process, and variables controlling the thicknesses and the lay-ups of panel skin and stringers.
For the panel skin a pseudohomogenized symmetric stacking sequence has been assumed. Six constraints have been defined to upperand lower-limit the total thicknesses of panel and stringers. These also guarantee to obtain a feasible shape for the stringers, assuring the width of the foot flange to be greater than the top flange width.
Advantages in using Shaper and Nexus
Main advantages of using Shaper and Nexus are:
- easy definition of the design parameterization directly on FEM and CFD numerical grids;
- accurate 3D morphing using advanced nonlinear techniques that preserve the original quality of the mesh;
- easy integration of external FEM and CFD solvers among which Nastran®, Abaqus®, Radioss®, Fluent®, Ansys® and Adams®;
- parallel and concurrent evaluations to exploit your hardware and software resources in the best possible way. A scalable architecture that you can tune application by application;
- access to all your results via organised tables and SQL extern databases;
- advanced Design of Experiment (DoE) and statistical tools to explore and analyse results;
- state-of-the-art libraries for single- and multi-objective optimisations, Design of Experiments and Response Surfaces modelling.
Results
The optimization has been started with an initial population of 300 members randomly selected in the design space.
convergence history: number of invalid members (red) and Pareto’s Size (blue) per iterations.
The main results of the procedure are summarized in the Convergence History figure and in the one below , showing the total evaluations performed and the Pareto’s set of the problem. The Pareto’s front, which represents the minimum-weight panel configuration at a given buckling load, envelopes the performed FE analyses, identifying solutions which range from 30.25 Kg up to 67.37 Kg.
Identified Pareto’s Front and all performed evaluations after 75 iterations
Case Presented at the 3rd CEAS Air&Space Conference. G. Quaranta, L. Lanzi and M.Sirna “Size and Shape optimization of Composite Stiffened Panelsvia MLS free-mesh morphing”, 2011.
