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Multi-Property Topology Optimisation with the Level-Set Method

We present a level-set algorithm for topology optimisation and demonstrate its capabilities and advantages in a variety of settings. The algorithm uses discrete element densities so that interpolation schemes are avoided and the boundary of the design is always well defined. A review of the level-set method for topology optimisation, and a description of the mathematical concepts behind the level-set algorithm are given in the introductory chapters. A compact Matlab implementation of the algorithm provides explicit implementation details for the simple example of compliance minimisation with a volume constraint. The remainder of the thesis presents original results obtained using the level-set algorithm. As a new application, we use topology optimisation to maximise fracture resistance. Fracture resistance is assumed to be related to the elastic energy released by a crack propagating in a normal direction from parts of the boundary that are in tension. We develop a suitable fracture resistance objective functional, derive its shape derivative and apply the level-set algorithm to simple examples. Topology optimisation methods that involve intermediate density elements are not suitable to solve this problem because the boundary of the design is not well defined. Our results indicate that the algorithm correctly optimises for fracture resistance. As the method is computationally intensive, we suggest simpler objective functionals that could be used as a proxy for fracture resistance. For example, a perimeter penalty could be added to the compliance objective functional in conjunction with a non-linear elasticity law where the Young's modulus in tension is lower than in compression. The level-set method has only recently been applied to fluid flow problems. We utilise the level-set algorithm to minimise energy dissipation in Stokes flows in both two and three dimensions. The discrete element densities allow the no-slip boundary condition to be applied directly. The Stokes equations therefore need only be solved in the fluid region of the design: this results in significant computational savings compared to conventional material distribution approaches. In order to quantify the computational savings the optimisation problems are resolved using an interpolation scheme to simulate the no-slip boundary condition. This significant advantage of the level-set method for fluid flow problems has not been noted by other authors. The algorithm produces results consistent with those obtained by other topology optimisation approaches, and solves large-scale three dimensional problems with modest computational cost. The first examples of three dimensional periodic microstructure design with the level-set method are presented in this thesis. The level-set algorithm is extended to deal with multiple constraints. This is needed so that materials can be designed with symmetry requirements imposed on their effective properties. To demonstrate the capabilities of the approach, unit cells are designed separately to maximise conductivity and bulk modulus with an isotropy requirement. The resulting materials have properties very close to the relevant Hashin-Shtrikman bounds. The algorithm is then applied to multifunctional material design: unit cells are designed to give isotropic materials that have maximum bulk modulus and maximum conductivity. Cross-property bounds indicate the near-optimality of the microstructures obtained. The design space of the problem is extensively explored with different coefficients of the conductivity and bulk modulus in the objective and different volume constraints. We hypothesise the existence of theoretically optimal single-scale microstructures with the topologies of the computationally optimised microstructures we have found. Structures derived from the Schwartz primitive (P) and diamond (D) minimal surfaces have previously been presented as good multifunctional composites. These structures are elastically anisotropic. Although they have similar conductivity, they have stiffness properties inferior to several of the isotropic optimised microstructures.