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.

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.

Submitted by Maria Cristina (library@lncc.br) on 2015-11-25T12:00:01Z No. of bitstreams: 1 Tese - Análise de Sensibilidade Topológica.pdf: 447139 bytes, checksum: d7d9c80ad59acb3e3cf12ae2d457887f (MD5) / Approved for entry into archive by Maria Cristina (library@lncc.br) on 2015-11-25T12:00:22Z (GMT) No. of bitstreams: 1 Tese - Análise de Sensibilidade Topológica.pdf: 447139 bytes, checksum: d7d9c80ad59acb3e3cf12ae2d457887f (MD5) / Made available in DSpace on 2015-11-25T12:00:35Z (GMT). No. of bitstreams: 1 Tese - Análise de Sensibilidade Topológica.pdf: 447139 bytes, checksum: d7d9c80ad59acb3e3cf12ae2d457887f (MD5) Previous issue date: 2015-11-03 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (Capes) / The topological derivative concept has been proved to be useful in many relevant applications such as topology optimization, inverse problems, image processing, multi-scale constitutive modeling, fracture mechanics and damage evolution modeling. The topological asymptotic analysis has been fully developed for a wide range of problems modeled by partial di erential equations. On the other hand, the topological derivatives associated with coupled problems have been derived only in their abstract forms. In this paper, therefore, we deal with the Reissner-Mindlin plate bending model, which is written in the form of a coupled system of partial di erential equations. In particular, the topological asymptotic analysis of the associated total potential energy is developed and the topological derivative with respect to the nucleation of a circular inclusion is derived in its closed form.Finally, we provide the estimates for the remainders of the topological asymptotic expansion and perform a complete mathematical justi cation for the derived formulas. / O conceito de derivada topológica tem se mostrado útil em muitas aplicações, tais como otimização topológica, problemas inversos, processamento de imagens, modelagem constitutiva multi-escala, mecânica da fratura e modelagem da evolução de dano. A análise assintótica topológica foi amplamente desenvolvida para uma grande variedade de problemas modelados por equações diferenciais parciais. Por outro lado, a derivada topológica associada a problemas acoplados é conhecida apenas em sua forma abstrata. Neste trabalho, portanto, considera-se o modelo de flexão de placa de Reissner-Mindlin, que é escrito na forma de um sistema acoplado de equações diferenciais parciais. Em particular, a análise assintótica topológica da energia potencial total associada é desenvolvida e a derivada topológica com relação a nucleação de uma inclusão circular é obtida na sua forma fechada. Finalmente, os resíduos da expansão assintótica topológica são estimados e uma justificativa matemática completa para a derivada topológica é apresentada.