Microstructure Interpolation for Macroscopic Optimisation

Abstract

Recent advances in additive manufacturing now allow the physical construction of designs with features on the scale of tens of micrometres. It is impractical to design macroscopic objects with such feature sizes by hand, yet designs exploiting this new manufacturing ability can be produced through computational algorithms, such as with structural optimisation. While the computational ability of structural optimisation techniques is improving, the direct optimisation of a structure spanning two or more length scales is still difficult. Despite this, the new fine- scale manufacturing capability can be exploited using multiscale structural optimisation to find the best high resolution design for a particular application. A multiscale design consists of microstructures which are repeatedly placed to create the design according to a macroscopic description. Previous work in multiscale optimisation has focused either on homogeneous microstructures that do not vary throughout the macroscopic design; analytically defined microstructures whose variation is well-known such as square lattices; or “top-down” multiscale design. In top-down multiscale design the macroscopic requirements at various locations prescribe the microstructural optimisation problems to be solved. We seek to solve some of the issues associated with top-down multiscale designs while allowing a wider space of microstructures than those that are analytically defined. We present a novel “bottom-up” method for multiscale structural optimisation over two length scales, which we call microstructure interpolation for macroscopic optimisation (MIMO). Shape interpolation, or morphing, between optimised microstructures produces a continuous set of microstructures that smoothly varies in both geometry and mechanical properties. The smooth set is used for macroscopic optimisation similar to the material distribution method (see Figure 1). The output of any multiscale optimisation method must be transformed into a single uni- fied description before a physical product can be manufactured. This transformation requires smooth transitions between the various microstructures to be well defined; the smoothness is assumed but not enforced in many existing top-down methods. The MIMO approach trades the generality of existing multiscale methods for a stronger unified interpretation: while the space of allowed microstructures is diminished compared to existing top-down methods, the microstructures vary smoothly throughout the design. This smoothness makes it clear how the microstructures can be transitioned between neighbouring macroscopic elements ensuring connectedness in the unified two-scale design. The MIMO method is also straightforward to apply to problems where a functionally graded material is desired. In this thesis the MIMO method is developed and tested. We firstly perform shape interpolation between a number of elastically isotropic microstructures optimised for bulk modulus. They are parameterised by their volume fraction and vary in stiffness. The interpolated microstructures have smoothly varying effective macroscopic elastic properties close to the Hashin-Shtrikman bounds. These microstructures are then used in a number of 2D and 3D compliance minimisa- tion problems. We find that the interpolated microstructures produce better objective values for 2D compliance problems than single-scale structural optimisation. The MIMO method is then applied to the problem of minimising peak shear stress and bone resorption for femoral implants. Femoral implants are used in hip replacements, attaching the femur to the new hip socket. The intent of minimising the interface shear stress is to minimise the likelihood of mechanical failure of the interface between implant and bone. Applying the method of interpolated microstructures with a range of porous candidate microstructure families shows significant benefits over the use of unvarying microstructures for this optimisation problem. We also show that the choice of microstructure family is important, as is their orientation if the microstructure is anisotropic. A physical model is produced as a proof-of- concept, illustrating the feasibility of the method to produce manufacturable designs in 3D. Finally, we generalise the MIMO method to use microstructures with a multi-dimensional pa- rameterisation. Using multiple parameters allows spatial variation in the stiffness profile: the stiffness in different directions may independently vary where before a sole parameter (the vol- ume fraction) controlled all directions in concert. Here the Young’s modulus in three orthogonal directions is used as the parameterisation. The femoral implant problem is revisited with this wider space of microstructures and the macroscopic optimisation is shown to make use of these new degrees of freedom. MIMO is an efficient two-scale structural optimisation approach that allows tailoring of the microstructures for the problem at hand, and generates designs with a ready physical interpre- tation. It provides a broad search space with which to solve structural optimisation problems without forsaking practicality.