• René Sørensen
  • Jesper Kann
4. term , Design of Mechanical Systems, Master (Master Programme)

This Master's thesis deals with mathematical optimisation of composite structures. A new method is presented, in which it is attempted to overcome the non-convex nature often associated with optimisation of composite structures modelled with finite elements.

First of all, the underlying theory is presented. The brief overview of shell theory is given, and the formulations needed in order to incorporate it into a finite element analysis application are derived. That is, the geometric representations needed are established, followed by an expression for the strain-displacement matrix. From this the layer-wise thickness integration needed to obtain the stiffness for a laminate structures is studied and used to derive an expression for performing explicit thickness integration instead, which involves slight approximations of the element Jacobian. Further approximations can, however, be made resulting in an expression for doing approximate explicit thickness integration. On top of the stiffness formulations the constitutive relations are addressed. Aided by the lamina invariants, the constitutive relations are rephrased to include the lamination parameters assuming that the laminate under inspection is composed of only a single material through all layers (with different orientations though). The result of this, in combination with the formulation of explicit thickness integration, is that the element stiffness matrix is seen to be linear in the lamination parameters. The above reformulations are at last verified numerically and seen to give accurate results. Furthermore it is seen that the expression of explicit thickness integration is especially efficient when working with laminates consisting of many layers.

Based on the presented theory the newly developed method is presented. First of all, the focus is brought to maximum stiffness optimisation with reference to the standard model, which is done by minimising the compliance. With the expressions of explicit thickness in hand the sensitivities needed for such an optimisation can be determined analytically, which is verified numerically to give accurate results.

The method of optimisation is based on a patch-compatible parameterisation where lamination parameters play an essential role. In order to overcome the non-convex nature of the optimisation problem some characteristics of lamination parameters are studied, namely the problem of feasibility and the question of convexity of the objective function. If the objective function is optimised with lamination parameters as design variables the strain energy is in fact convex. However, this approach would give problems with ensuring feasibility of the final result. Hence the problems are sought solved by keeping the fibre orientations of the laminate as design variables and then overcoming the non-convex nature of the problem by designing the optimisation method as a two-step approach. Thus some of the ideas from the two-step approach presented by [Foldager,1999] are utilised in combination with results from [Kann and Sørensen,2010] in order to develop a new and more robust two-step method. The method developed includes an identification process where an identification function based on a local linearisation is minimised by the use of a genetic algorithm.

The developed method have been implemented in MUST with the ability to switch between different numbers of applied lamination parameters in the identification process as well as both a full and an approximate method.

In order to test the new method numeric experiments have been conducted. Three simple problems of "academic character" are presented - a cantilever beam with a distributed load, a flat plate with a uniform pressure normal to the surface, and a pinched hemisphere with different curvature-to-thickness ratios. The three examples show that successful identification is indeed found in several instances, meaning that a local minimum can be overcome. However, the success is dependent on a several-to-one relationship between the number of design variables and the number of lamination parameters. Furthermore the results indicate that one of the first problems arising, when doing stiffness optimisation of composite structures, seems to be that the design variables get stuck at the bounds of the design space.

Asides from the experiments of "academic character", a more "industrial/practical" experiment has been conducted as well. The geometry under inspection is a generic main spar from a wind turbine blade. The conclusion from this experiment is first and foremost that the method can indeed be used on large industrial structures. Furthermore the results are close to what would be expected, however, there are small differences. These differences can be explained by the patch breakdown and effects of so small magnitude that they cannot be captured numerically. Finally, there is a strong indication that post-processing of the optimised structure is indeed necessary, as the design typically contains features which cannot be realised, or maybe is too complicated or too impractical for the manufacturer to fully realise the optimised structural design. The amount of time needed for post-processing may be reduced by incorporating restrictions associated with the manufacturing process into the numerical optimisation routines.

LanguageEnglish
Publication date1 Jun 2011
Number of pages92
ID: 52669010