[pt] As metodologias de projeto estrutural foram fortemente influenciadas pelo advento da computação. Os avanços nas áreas de análise numérica, como o método dos elementos finitos, e os softwares de Desenho Assistido por Computador, literalmente ajudaram a moldar o mundo como ele é hoje. Implementações computacionais das técnicas de otimização estrutural, como a otimização topológica, permitem a determinação das estruturas base, gerando uma grande quantidade de projetos novos, mais eficientes, com o potencial de mudar drasticamente o futuro das aeronaves, automóveis, edifícios, etc. Introduzir restrições de tensão na otimização topológica tradicional permite a obtenção de soluções mais seguras e confiáveis que se assemelhem mais à estrutura final. Contudo, isto não é uma tarefa trivial, apresentando várias dificuldades conceituais e numéricas. Nesta dissertação, as principais questões deste problema são discutidas e as técnicas presentes hoje na literatura são revisadas e criticadas quanto aos seus desempenhos. A principal contribuição deste trabalho é uma nova técnica baseada no Método do Lagrangiano Aumentado que lida eficientemente com um grande número de restrições. Em contraste com os métodos existentes, que são dependentes do problema e da malha, a abordagem proposta apresenta poucos parâmetros que precisam ser ajustados a cada novo caso. Para avaliar suas potencialidades, desenvolveu-se um código em MATLAB, eficaz e robusto. Diversos exemplos representativos, incluindo problemas de larga escala, são apresentados. Finalmente, as soluções obtidas, incluindo algumas complicações inesperadas, são discutidas detalhadamente e sugestões para trabalhos futuros são propostas. / [en] Structural design methodologies were strongly influenced by the advent of computing. The advances in numerical analyses, such as the finite element method, and Computer Aided Design software have literally helped shape the engineering world as it is today. Structural optimization methods such as topology optimization aim to take the next step by letting the computer guide the design, in order to achieve new and more eficient designs. This approach has the potential to change the future of various industries, including aircraft, automobile, construction, etc. The introduction of stress constraints on traditional topology optimization allows for safer and more reliable solutions that will more closely resemble the final structure. The successful solution of this problem poses several conceptual and numerical dificulties. Thus this dissertation details the main issues of this problem and reviews the current techniques discussed in the literature including some critiques of their performance. The main contribution of this work is a novel technique based on the Augmented Lagrangian method that can eficiently handle a large number of constraints. In contrast to existing methods which are both problem- and mesh-dependent, the presented approach contains only a few parameters which need to be adjusted for each new case. In order to verify the technique s capabilities, a user friendly MATLAB code was developed that is both effective and robust. Several representative examples, including large-scale problems, are presented. Finally, the solutions obtained here, including some unexpected complications, are thoroughly discussed and suggestions for future work are also addressed.

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Les développements récents des méthodes de fabrication additive permettent aujourd'hui d'envisager l'usinage de pièces à la topologie complexe, composées de microstructures. Ceci ranime l'intérêt pour les méthodes d'optimisation topologique par méthode d'homogénéisation, développées dans les années 80 et quelque peu oubliées par manque d'applications industrielles.L'objectif de cette thèse est de fournir des méthodes d'optimisation topologique pour des structures constituées de matériau lattice localement périodique, c'est-à-dire dont la microstructure est modulée au sein de la pièce.Trois phases ont été définies. La première consiste à calculer les propriétés élastiques homogénéisées de microstructures en fonction de paramètres définissant leur géométrie. Dans la seconde étape, on optimise la structure constituée de matériau homogénéisé selon les paramètres géométriques de la microstructure ainsi que son orientation. Une structure homogénéisée n'est pas usinable en l'état. En effet, l'homogénéisation revient à considérer que la taille des cellules la composant converge vers zéro. Dans une troisième étape, on propose donc de déshomogénéiser la structure optimisée, c'est-à-dire de construire une suite de structures convergeant vers elle. Pour cela, on introduit un difféomorphisme déformant une grille régulière de sorte que chaque cellule soit orientée selon l'orientation optimale.Nous présentons dans cette thèse les détails de cette méthode, pour des microstructures élastiques isotropes et orthotropes, en deux et en trois dimensions.Nous proposons également un couplage de cette méthode avec la méthode d'optimisation de forme par les lignes de niveau, ce qui permet notamment d'inclure des contraintes géométriques sur les structures finales. / Thanks to the recent developments of the additive manufacturing processes, structures built with modulated microstructures and featuring a complex topology are now manufacturable. This leads to a resurrection of the homogenization method for shape optimization, an approach developed in the 80’s but which progressively faded away because yielding too complex structures for manufacturing processes at this time.The goal of this thesis is to develop shape optimization methods for structures built with modulated locally periodic lattice microstructures.Three steps have been defined. The first consists in computing the homogenized, or effective, elastic properties of microstructures according to few parameters characterizing their geometry. In the second step, the geometric properties of the microstructure and its orientation are optimized in the working domain, yielding a homogenized optimized structure. Such a structure is nevertheless not straightforwardly manufacturable. Indeed, the homogenization is equivalent to have a structure featuring cells whose size is converging to zero. Hence, in the third and last step, a deshomogenization process is proposed. It consists in building a sequence of genuine structures converging to the homogenized optimal structures. The key point is to respect locally the orientation of the cells, which is performed thanks to a grid diffeomorphism.In this thesis, we present the details of the whole method, for isotropic and orthotropic microstructures, in 2D and in 3D.A coupling of this method with the level-set shape optimization method is also presented, thanks which the set of geometric constraints on the final structures may be enlarged.

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Coil supports are integral load-bearing components employed in generators andmotors. They serve the purpose of preventing excessive deformation and maintaininga stable position of the coils responsible for generating power and magnetic fieldswhen rotating. However, a problem with these coil supports is that they block theairflow aimed to cool the coils. Thus, this master thesis aimed to conduct a topologyoptimization to develop a cooling-air permeable coil support and select a suitablematerial. The new design was required to withstand 30,000 operational cycles andan overspeed test running at 120% speed without plastic deformation or failure. The material selection process was initiated and based on mechanical and physicalproperties requirements. One of these was that the material should be non-magnetic.Utilizing Ansys Granta EduPack, two materials were suggested, the reference materialcurrently used for the coil support, and a titanium alloy, Ti-6Al-4V. The subsequentstep was to create a CAD model of the original design based on technical drawingsprovided by ABB. With the generated design, finite element analysis (FEA) simulationand the topology optimization could be performed. The generated topology optimizedmodel was modified and two new models were created, one with smaller central cutoutsand one with larger central cutouts and a top surface cutout. Furthermore, a thirdmodel was created based on the fundamentals of fluid mechanics, the Rounded originalmodel. Computational Fluid Dynamics (CFD) simulations of the four models wereexecuted. The findings indicate that the design with larger central cutouts exhibited the mostsubstantial increase in airflow through and in between the coil supports, achieving a122 % improvement compared to the original design. The model satisfied the fatiguerequirement and successfully passed the overspeed test. Both the current referencematerial and the Ti-6Al4V alloy are suitable to use for coil support. However, theutilization of a titanium alloy might be deemed excessive in terms of its mechanicalproperties and cost. / Spolstöd är integrerade lastbärande komponenter som används i generatorer ochmotorer. De har till syfte att förhindra extrem deformation och bibehålla positionenför spolarna som ansvarar för att generera kraft och magnetfält när de roteras. Ettproblem med spolstöden är att de blockerar luftflödet avsett att kyla spolarna. Däravär syftet och målet med detta examensarbetet att genomföra en topologioptimeringför att utveckla luftgenomsläppligt spolstöd för förbättrad kylning samt att välja ettlämpligt material. Den nya konstruktionen är tvungen att klara av 30,000 driftcykleroch ett överspänningsprov vid 120% av hastigheten utan plastisk deformation ellerbrott. Materialvalsprocessen initierades och baserades på mekaniska och fysikaliska krav.En av dessa krav var att materialet skulle vara icke-magnetiskt. Användandet avAnsys Granta EduPack resulterade i två material, referensmaterialet som användsför att producera spolstödet i nuläget och en titanlegering, Ti-6Al-4V. Därefterskapades en CAD-modell av den ursprungliga designen baserat på tekniska ritningartillhandahållna från ABB. Med den genererade modellen kunde finita elementanalys(FEA) och topologioptimeringen utföras. Detta genererade i en topologioptimeradmodel som modifierades och lade grund till två nya modeller, en modell med mindrecentrala hål och en med större centrala hål. En tredje modell skapades dessutom,baserad på grundläggande principer inom fluidmekanik. Fluidmekanik (CFD) beräkningar av de fyra modellerna utfördes och resultatenvisade på att den tolpologioptimerade modellen med stora centrala hål hade denmest betydande ökningen i luftgneomströmning genom och mellan spolstödenmed en förbättring på 122 % jämfört med den ursprungliga designen. Modellenuppfyller även kraven på utmattning och maximal statisk spänning vid rusningsprov.Både referensmaterialet och titanlegeringen var lämpad att användas som spolstöd.Däremot kan användningen av titanlegeringen anses vara överdriven med hänsyn tilldess mekaniska egenskaper och kostnad.

This work addresses the end-to-end virtual automation of structural optimization up to the derivation of a parametric geometry model that can be used for application areas such as additive manufacturing or the verification of the structural optimization result with the finite element method. A holistic design in structural optimization can be achieved with the weighted sum method, which can be automatically parameterized with curve skeletonization and cross-section regression to virtually verify the result and control the local size for additive manufacturing. is investigated in general. In this paper, a holistic design is understood as a design that considers various compliances as an objective function. This parameterization uses the automated determination of beam parameters by so-called curve skeletonization with subsequent cross-section shape parameter estimation based on moments of area, especially for multi-objective optimized shapes. An essential contribution is the linking of the parameterization with the results of the structural optimization, e.g., to include properties such as boundary conditions, load conditions, sensitivities or even density variables in the curve skeleton parameterization. The parameterization focuses on guiding the skeletonization based on the information provided by the optimization and the finite element model. In addition, the cross-section detection considers circular, elliptical, and tensor product spline cross-sections that can be applied to various shape descriptors such as convolutional surfaces, subdivision surfaces, or constructive solid geometry. The shape parameters of these cross-sections are estimated using stiffness distributions, moments of area of 2D images, and convolutional neural networks with a tailored loss function to moments of area. Each final geometry is designed by extruding the cross-section along the appropriate curve segment of the beam and joining it to other beams by using only unification operations. The focus of multi-objective structural optimization considering 1D, 2D and 3D elements is on cases that can be modeled using equations by the Poisson equation and linear elasticity. This enables the development of designs in application areas such as thermal conduction, electrostatics, magnetostatics, potential flow, linear elasticity and diffusion, which can be optimized in combination or individually. Due to the simplicity of the cases defined by the Poisson equation, no experts are required, so that many conceptual designs can be generated and reconstructed by ordinary users with little effort. Specifically for 1D elements, a element stiffness matrices for tensor product spline cross-sections are derived, which can be used to optimize a variety of lattice structures and automatically convert them into free-form surfaces. For 2D elements, non-local trigonometric interpolation functions are used, which should significantly increase interpretability of the density distribution. To further improve the optimization, a parameter-free mesh deformation is embedded so that the compliances can be further reduced by locally shifting the node positions. Finally, the proposed end-to-end optimization and parameterization is applied to verify a linear elasto-static optimization result for and to satisfy local size constraint for the manufacturing with selective laser melting of a heat transfer optimization result for a heat sink of a CPU. For the elasto-static case, the parameterization is adjusted until a certain criterion (displacement) is satisfied, while for the heat transfer case, the manufacturing constraints are satisfied by automatically changing the local size with the proposed parameterization. This heat sink is then manufactured without manual adjustment and experimentally validated to limit the temperature of a CPU to a certain level.:TABLE OF CONTENT III I LIST OF ABBREVIATIONS V II LIST OF SYMBOLS V III LIST OF FIGURES XIII IV LIST OF TABLES XVIII 1. INTRODUCTION 1 1.1 RESEARCH DESIGN AND MOTIVATION 6 1.2 RESEARCH THESES AND CHAPTER OVERVIEW 9 2. PRELIMINARIES OF TOPOLOGY OPTIMIZATION 12 2.1 MATERIAL INTERPOLATION 16 2.2 TOPOLOGY OPTIMIZATION WITH PARAMETER-FREE SHAPE OPTIMIZATION 17 2.3 MULTI-OBJECTIVE TOPOLOGY OPTIMIZATION WITH THE WEIGHTED SUM METHOD 18 3. SIMULTANEOUS SIZE, TOPOLOGY AND PARAMETER-FREE SHAPE OPTIMIZATION OF WIREFRAMES WITH B-SPLINE CROSS-SECTIONS 21 3.1 FUNDAMENTALS IN WIREFRAME OPTIMIZATION 22 3.2 SIZE AND TOPOLOGY OPTIMIZATION WITH PERIODIC B-SPLINE CROSS-SECTIONS 27 3.3 PARAMETER-FREE SHAPE OPTIMIZATION EMBEDDED IN SIZE OPTIMIZATION 32 3.4 WEIGHTED SUM SIZE AND TOPOLOGY OPTIMIZATION 36 3.5 CROSS-SECTION COMPARISON 39 4. NON-LOCAL TRIGONOMETRIC INTERPOLATION IN TOPOLOGY OPTIMIZATION 41 4.1 FUNDAMENTALS IN MATERIAL INTERPOLATIONS 43 4.2 NON-LOCAL TRIGONOMETRIC SHAPE FUNCTIONS 45 4.3 NON-LOCAL PARAMETER-FREE SHAPE OPTIMIZATION WITH TRIGONOMETRIC SHAPE FUNCTIONS 49 4.4 NON-LOCAL AND PARAMETER-FREE MULTI-OBJECTIVE TOPOLOGY OPTIMIZATION 54 5. FUNDAMENTALS IN SKELETON GUIDED SHAPE PARAMETRIZATION IN TOPOLOGY OPTIMIZATION 58 5.1 SKELETONIZATION IN TOPOLOGY OPTIMIZATION 61 5.2 CROSS-SECTION RECOGNITION FOR IMAGES 66 5.3 SUBDIVISION SURFACES 67 5.4 CONVOLUTIONAL SURFACES WITH META BALL KERNEL 71 5.5 CONSTRUCTIVE SOLID GEOMETRY 73 6. CURVE SKELETON GUIDED BEAM PARAMETRIZATION OF TOPOLOGY OPTIMIZATION RESULTS 75 6.1 FUNDAMENTALS IN SKELETON SUPPORTED RECONSTRUCTION 76 6.2 SUBDIVISION SURFACE PARAMETRIZATION WITH PERIODIC B-SPLINE CROSS-SECTIONS 78 6.3 CURVE SKELETONIZATION TAILORED TO TOPOLOGY OPTIMIZATION WITH PRE-PROCESSING 82 6.4 SURFACE RECONSTRUCTION USING LOCAL STIFFNESS DISTRIBUTION 86 7. CROSS-SECTION SHAPE PARAMETRIZATION FOR PERIODIC B-SPLINES 96 7.1 PRELIMINARIES IN B-SPLINE CONTROL GRID ESTIMATION 97 7.2 CROSS-SECTION EXTRACTION OF 2D IMAGES 101 7.3 TENSOR SPLINE PARAMETRIZATION WITH MOMENTS OF AREA 105 7.4 B-SPLINE PARAMETRIZATION WITH MOMENTS OF AREA GUIDED CONVOLUTIONAL NEURAL NETWORK 110 8. FULLY AUTOMATED COMPLIANCE OPTIMIZATION AND CURVE-SKELETON PARAMETRIZATION FOR A CPU HEAT SINK WITH SIZE CONTROL FOR SLM 115 8.1 AUTOMATED 1D THERMAL COMPLIANCE MINIMIZATION, CONSTRAINED SURFACE RECONSTRUCTION AND ADDITIVE MANUFACTURING 118 8.2 AUTOMATED 2D THERMAL COMPLIANCE MINIMIZATION, CONSTRAINT SURFACE RECONSTRUCTION AND ADDITIVE MANUFACTURING 120 8.3 USING THE HEAT SINK PROTOTYPES COOLING A CPU 123 9. CONCLUSION 127 10. OUTLOOK 131 LITERATURE 133 APPENDIX 147 A PREVIOUS STUDIES 147 B CROSS-SECTION PROPERTIES 149 C CASE STUDIES FOR THE CROSS-SECTION PARAMETRIZATION 155 D EXPERIMENTAL SETUP 158

Topology-based optimization techniques and lattice structures are powerful ways to accomplish lightweight components with enhanced mechanical performance. Recent developments in additive manufacturing (AM) have led the way to extraordinary opportunities in realizing complex designs that are derived from topology and lattice-based structural optimization. The main aim of this work is to give a contribution, in the integration between structural optimization techniques and AM, by proposing a setup of a proper methodology for rapid development of optimized medical implants addressing oseeointegration and minimization of stress shielding related problems. The validity of the proposed methodology for a proof of concept was demonstrated in two real-world case studies: a tibia intramedullary implant and a shoulder hemi prosthetics for two bone cancer patients. The optimization was achieved using topology optimization and replacement of solid volumes by lattice structures. Samples of three lattice unit cell configurations were designed, fabricated, mechanically tested, and compared to select the most proper configuration for the shoulder hemi prosthesis. Weight reductions of 30% and 15% were achieved from the optimization of the initial design of the tibia intramedullary implant and the shoulder hemiprosthesis respectively compared to initial designs. Prototypes were fabricated using selective laser melting (SLM) and direct light processing (DLP) technologies. Validation analysis was performed using finite element analysis and compressive mechanical testing. Future work recommendations are provided for further development and improvement of the work presented in this thesis.

This thesis focuses on the development and convergence study of finite element methods for eigenvalue analysis of arbitrarily shaped domains with multi-material and material-void interfaces. Such configurations can be found in problems with evolving discontinuities and interfaces as in fluid-structure interaction or topology optimization problems. The differential equations considered in this thesis include the elliptic operator, Timoshenko beam and Mindlin plate. The compatibility conditions at the interface are weakly imposed using either Nitsche's method or Lagrange multipliers. The variational statements are derived for each case. The analysis results are benchmarked using Galerkin finite element discretization with bodyitted grids. Nitsche's method shows a direct dependence on a penalty term and for Lagrange multipliers method, additional degrees of freedom are added to the solution vector. The convergence rate of the discretized forms is computationally determined and is shown to be optimal for both Timoshenko beams and Mindlin plates.

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