The dissertation deals with the topology optimization problems of the truss systems. The main aim of the work is to create a technology and implementing algorithms for topology optimization and for simultaneous topology and shape optimization of truss systems. The genetic algorithms are used as the main tool for optimization. The topology optimization problems are formulated using the so-called ground structure approach. / Disertacijoje nagrinėjamos santvarų globalaus optimizavimo problemos. Pagrindinis darbo tikslas – sukurti technologiją ir ją aprašančius algoritmus santvarų topologijos optimizavimui ir sinchroniniam topologijos ir formos optimizavimui. Optimizavimui naudojami genetiniai algoritmai. Topologijai optimizuoti pasirinkta perdėtai sujungtos struktūros strategija (angl. ground structure approach).

The law of bone remodeling, commonly referred to as Wolff's Law, asserts that the internal trabecular bone adapts to external loadings, reorienting with the principal stress trajectories to maximize mechanical efficiency, thereby creating a naturally optimum structure. The primary objective of the research was to utilize an advanced structural optimization algorithm, called design space optimization (DSO), to create a numerical framework to perform a micro-level three-dimensional finite element bone remodeling simulation on the human proximal femur and analyze the results to determine the validity of Wolff's hypothesis. DSO optimizes the layout of material by iteratively distributing it into the areas of highest loading, while simultaneously changing the design domain to increase computational efficiency. The result is a "fully stressed" structure with minimized compliance and increased stiffness. The large-scale computational simulation utilized a 175µm mesh resolution and the routine daily loading activities of walking and stair climbing. The resulting anisotropic human trabecular architecture was compared to both Wolff's trajectory hypothesis and natural femur data from the literature using a variety of visualization techniques, including radiography and computed tomography (CT). The remodeling predictions qualitatively revealed several anisotropic trabecular regions comparable to the natural human femurs. Quantitatively, the various regional bone volume fractions from the computational results were consistent with CT analyses. The strain energy proceeded to become more uniform during optimization; implying increased mechanical efficiency was achieved. The realistic simulated trabecular geometry suggests that the DSO method can accurately predict three-dimensional bone adaptation due to mechanical loading and that the proximal femur is an optimum structure as Wolff hypothesized. The secondary objective was to revise this computational framework to perform the first in-silico hip replacement considering micro-level bone remodeling. Two different commercially available hip prostheses were quantitatively analyzed using stress, strain energy, and bone mineral density as performance criteria and qualitatively visualized using the techniques above. Several important factors for stable fixation, determined from clinical evaluations, were evident: high levels of proximal bone loss, distal bone growth, and medial densification. The results suggest the DSO method can be utilized for comparative prosthetic implant stem design, uniquely considering post-operation bone remodeling as a design criterion. / Thesis (Master, Mechanical and Materials Engineering) -- Queen's University, 2010-08-16 15:30:55.144

A unique mathematical tool is developed to deal with global optimization of a set of engineering problems. These include image processing, mechanical topology optimization, and optimal path planning in a variational framework, as well as some benchmark problems in parameter optimization. The optimization tool in these applications is based on the level set theory by which an evolving contour converges toward the optimum solution. Depending upon the application, the objective function is defined, and then the level set theory is used for optimization. Level set theory, as a member of active contour methods, is an extension of the steepest descent method in conventional parameter optimization to the variational framework. It intrinsically suffers from trapping in local solutions, a common drawback of gradient based optimization methods. In this thesis, methods are developed to deal with this drawbacks of the level set approach. By investigating the current global optimization methods, one can conclude that these methods usually cannot be extended to the variational framework; or if they can, the computational costs become drastically expensive. To cope with this complexity, a global optimization algorithm is first developed in parameter space and compared with the existing methods. This method is called "Spiral Bacterial Foraging Optimization" (SBFO) method because it is inspired by the aggregation process of a particular bacterium called, Dictyostelium Discoideum. Regardless of the real phenomenon behind the SBFO, it leads to new ideas in developing global optimization methods. According to these ideas, an effective global optimization method should have i) a stochastic operator, and/or ii) a multi-agent structure. These two properties are very common in the existing global optimization methods. To improve the computational time and costs, the algorithm may include gradient-based approaches to increase the convergence speed. This property is particularly available in SBFO and it is the basis on which SBFO can be extended to variational framework. To mitigate the computational costs of the algorithm, use of the gradient based approaches can be helpful. Therefore, SBFO as a multi-agent stochastic gradient based structure can be extended to multi-agent stochastic level set method. In three steps, the variational set up is formulated: i) A single stochastic level set method, called "Active Contours with Stochastic Fronts" (ACSF), ii) Multi-agent stochastic level set method (MSLSM), and iii) Stochastic level set method without gradient such as E-ARC algorithm. For image processing applications, the first two steps have been implemented and show significant improvement in the results. As expected, a multi agent structure is more accurate in terms of ability to find the global solution but it is much more computationally expensive. According to the results, if one uses an initial level set with enough holes in its topology, a single stochastic level set method can achieve almost the same level of accuracy as a multi-agent structure can obtain. Therefore, for a topology optimization problem for which a high level of calculations (at each iteration a finite element model should be solved) is required, only ACSF with initial guess with multiple holes is implemented. In some applications, such as optimal path planning, objective functions are usually very complicated; finding a closed-form equation for the objective function and its gradient is therefore impossible or sometimes very computationally expensive. In these situations, the level set theory and its extensions cannot be directly employed. As a result, the Evolving Arc algorithm that is inspired by "Electric Arc" in nature, is proposed. The results show that it can be a good solution for either unconstrained or constrained problems. Finally, a rigorous convergence analysis for SBFO and ACSF is presented that is new amongst global optimization methods in both parameter and variational framework.

Die rechnergestützte Topologieoptimierung wird zur Gestaltoptimierung, also im Wesentlichen zur Gewichtsreduktion von Bauteilen oder komplexeren Strukturen eingesetzt. Da die Funktionalität im Rahmen von FEM-Programmen zur Verfügung gestellt wird, erfordert sie umfangreiche Kenntnisse zur Bedienung der Software und der festigkeitstechnischen Grundlagen und wird daher überwiegend von Berechnungsexperten im rein technischen Kontext im Maschinenbau oder Luft- und Raumfahrzeugbau angewendet. Allerdings zeigen vereinzelte Arbeiten wie die Sitzmöbel »Bone Furniture«, die aus einer Zusammenarbeit des Studios »Joris Laarman Lab« mit Opel resultieren (Laarman 2006, Abbildung 1), dass derartige Methoden für das Design interessant sein können. [... aus der Einleitung]

abstract: The electric transmission grid is conventionally treated as a fixed asset and is operated around a single topology. Though several instances of switching transmission lines for corrective mechaism, congestion management, and minimization of losses can be found in literature, the idea of co-optimizing transmission with generation dispatch has not been widely investigated. Network topology optimization exploits the redundancies that are an integral part of the network to allow for improvement in dispatch efficiency. Although, the concept of a dispatchable network initially appears counterintuitive questioning the wisdom of switching transmission lines on a more regu-lar basis, results obtained in the previous research on transmission switching with a Direct Current Optimal Power Flow (DCOPF) show significant cost reductions. This thesis on network topology optimization with ACOPF emphasizes the need for additional research in this area. It examines the performance of network topology optimization in an Alternating Current (AC) setting and its impact on various parameters like active power loss and voltages that are ignored in the DC setting. An ACOPF model, with binary variables representing the status of transmission lines incorporated into the formulation, is written in AMPL, a mathematical programming language and this optimization problem is solved using the solver KNITRO. ACOPF is a non-convex, nonlinear optimization problem, making it a very hard problem to solve. The introduction of bi-nary variables makes ACOPF a mixed integer nonlinear programming problem, further increasing the complexity of the optimization problem. An iterative method of opening each transmission line individually before choosing the best solution has been proposed as a purely investigative approach to studying the impact of transmission switching with ACOPF. Economic savings of up to 6% achieved using this approach indicate the potential of this concept. In addition, a heuristic has been proposed to improve the computational efficiency of network topology optimization. This research also makes a comparative analysis between transmission switching in a DC setting and switching in an AC setting. Results presented in this thesis indicate significant economic savings achieved by controlled topology optimization, thereby reconfirming the need for further examination of this idea. / Dissertation/Thesis / M.S. Electrical Engineering 2011

The development in additive manufacturing methods has cleared the path for topology optimizationby making it possible to produce complex geometries, which would not be possible to produce bytraditional manufacturing methods. Topology optimization uses iterative structural computations tond an optimal material distribution given a maximum optimization domain, load cases and/or otherstructural criteria. The relation between retained mass and structural performance of a swing armfor the vehicle BvS10 was examined for two different materials. The first material was an estimate of an additive manufactured material and the other for a high structural steel. Given the extreme load cases, the geometrical limits of the swing arm and by specifying how much mass was to be retained the stiffness was to be maximized. The optimization was performed using an elastic material model in thecommercial software ANSYS. This elastic material models was based on standard material parameters of steel. Three geometries were generated, namely OG100, OG90 and OG80, which corresponded to 101 %, 87 % and 81 % of the mass of the original swing arm, respectively. The optimization procedurewas combined with geometry modications in SpaceClaim to simplify the obtained geometries. All these geometries consisted of a hollow geometry with a greater width compared to the original geometry. The geometries were then evaluated using multilinear plastic material models based on respective material. Using the additive manufactured material model no generated geometry could perform structurally better than the original swing arm. This indicates that greater material properties must be obtainedin order to be able to reduce the weight of the swing arm. By using the material properties of the highstructural steel, it was found that at least 31.3 kg per vehicle could be reduced by using the optimizedgeometry OG80, and still not perform structurally worse than of the original swing arm. / Utvecklingen inom additiv tillverkning har öppnat vägen för topologioptimering genom att kunna producera komplexa geometrier, som inte skulle vara möjliga att tillverka med hjälp av traditionella tillverkningsmetoder. Topologioptimering använder iterativa hållfasthetsberäkningar för att finna den optimala materialfördelning givet en maximal optimeringsdomän, lastfall och/eller andra strukturella kriterier. Relationen mellan bibehållen massa och strukturella prestationer hos en pendelarm till fordonet BvS10 har undersökts för två olika material. Det ena materialet var en uppskattning av ett additivt tillverkat material och det andra materialet var ett höghållfasthetsstål. Givet dem extrema lastfall, geometriska begränsningar hos pendelarmen och genom specficera hur mycket massa som skulle behållas så skulle styvheten maximeras. Optimeringarna utfördes med en elastisk materialmodell i den kommersiellamjukvaran ANSYS. Denna elastiska materialmodell var baserad på klassiska materialparametrarfor stål. Tre geometrier genererades. Optimeringsproceduren användes i kombination med geometriska modikationer i SpaceClaim för att förenkla de optimierade geometrierna. Dessa var OG100, OG90 och OG80, vilka motsvarade 101 %, 87 % och 81 % av pendelarmens originalvikt. Alla geometrier bestod av en ihålig geometri med större bredd än originalarmens. Geometrierna utvärderades sedan med hjälp av multilinjära plastiska materialmodeller baserat på respektive material. Ingen av dessa geometrier kunde prestera bättre än originalarmen när det additivt tillverkade materialet användes. Detta indikerar att bättre materialegenskaper måste uppnås för att kunna reducera vikten hos pendelarmen. Genom attanvända höghållfasthetsstålet upptäcktes att åtminstone 31.3 kg per fordon kunde reduceras genom attanvända OG80, och fortfarande inte prestera sämre än originalarmen.

[pt] Este trabalho apresenta um estudo comparativo de métodos numéricos para solução das equações da elasticidade em problemas de otimização topológica. Um sistema computacional é desenvolvido em MATLAB para solução de problemas de otimização topológica usando malhas poligonais não estruturadas em domínios bidimensionais arbitrários. Dois métodos numéricos são implementados e comparados com o método dos elementos finitos (FEM) em relação à precisão e à eficiência computacional: o recém proposto Método dos Elementos Virtuais (VEM) e o Método dos Elementos Finitos Suavizados (SFEM). A principal característica que distingue estes métodos do FEM é que as funções de base canônicas não são obtidas de forma explícita. A utilização de projetores locais apropriados permite a extração do componente linear das deformações dos elementos e, por conseguinte, o cálculo da matriz de rigidez se reduz a avaliações de quantidades puramente geométricas. Exemplos numéricos representativos, usando malhas convexas e não convexas, para minimização da flexibilidade são apresentados para ilustrar as potencialidades dos métodos estudados. / [en] This work presents a comparative study of numerical methods for solving the elasticity equations in topology optimization problems. A computational framework is developed in MATLAB for solving topology optimization problems using unstructured polygonal meshes in arbitrary two-dimensional domains. Two numerical methods are implemented and compared with the finite element method (FEM) with respect to accuracy and computational efficiency: the recentlyproposed Virtual Element Method (VEM) and the Smoothed Finite Element Method (SFEM). The key characteristic that distinguish these methods from the FEM is that the canonical basis functions are not computed explicitly. The use of appropriate local projection maps allows the extraction of the linear component of the element deformations and, therefore, the computation of the stiffness matrix is reduced to the evaluation of purely geometric quantities. Representative numerical examples, using convex and non-convex meshes, for compliance minimization are presented to illustrate the capabilities of the methods studied.

[pt] Métodos de otimização topológica estrutural visam obter a melhor distribuição de material dentro de um dado domínio, sujeito a carga, condições de contorno e restrições de projeto, de forma a minimizar alguma medida especificada. A otimização topológica estrutural pode ser dividida em dois tipos: contínua e discreta, sendo a forma discreta o foco da pesquisa desta dissertação. O objetivo deste trabalho é a criação de um sistema para realizar todos os passos dessa otimização, visando a resolução de problemas com grandes dimensões. Para realizar esse tipo de otimização, é necessária a criação de uma malha densa de barras, esta definida como conjunto de nós cobrindo todo o domínio, conectados através de barras, além da especificação dos apoios e das forças aplicadas. Este trabalho propõe um novo método para geração da malha densa de barras, utilizando como entrada somente o contorno do domínio que se deseja otimizar, contrapondo com métodos que necessitam de um domínio já discretizado, como uma malha de poliedros. Com a malha gerada, este trabalho implementou a otimização topológica, sendo necessário resolver um problema de programação linear. Toda a parte de otimização foi realizada dentro do framework TopSim, tendo implementado o método dos pontos interiores para a resolução da programação linear. Os resultados apresentados possuem boa qualidade, tanto na geração quanto na otimização, para casos 2D e 3D, tratando casos com mais de 68 milhões de barras. / [en] Structural topology optimization methods are used to find the optimal material distribution within a given domain, subject to loading, boundary conditions and design constraints, in order to minimize some specified measure. Structural topology optimization can be divided into two types: continuum and discrete, with the discrete type being the research focus of this dissertation. The goal of this work is the creation of a system to achieve all the steps of this optimization process, aiming problems with large dimensions. In order to perform the optimization, it is necessary create a ground structure, defined as a set of nodes covering the entire domain, connected by bars, with the supports and the applied loads. This work proposes a new method for the ground structure generation, using as input only the domain boundary, in contrast with methods that require a domain already discretized, such as a polyhedron mesh. With the generated mesh, this work has implemented the topological optimization, needing to solve a linear programming problem. All the optimization part was performed within the TopSim framework, implementing the interior point method for the linear programming resolution. The results presented have good quality, both in generation and optimization, for 2D and 3D cases, considering cases with more than 68 million bars.

Este trabalho apresenta o desenvolvimento e implementação computacional de técnicas de otimização de topologia para problemas governados pela equação de Poisson. O método numérico utilizado para solução numérica das equações foi o método dos elementos de contorno (MEC). Para tanto, três metodologias foram desenvolvidas. A primeira é direcionada à aplicação de algoritmos genéticos (AG) para investigar como um domínio inicialmente preenchido com cavidades aleatórias evolui durante um processo de otimização e verificar a possibilidade de se extrair topologias ótimas a partir da interpretação da solução encontrada. Os contornos externos permanecem fixos enquanto as posições e as dimensões das cavidades são otimizadas com o objetivo extremizar uma função custo especificada. O desempenho do algoritmo proposto é ilustrada com uma série de exemplos e os resultados são discutidos. A segunda metodologia apresenta um algoritmo numérico para otimização topológica baseado na avaliação da derivada topológica (DT), adotando a energia potencial total como função custo. Este procedimento é uma alternativa às tradicionais técnicas de otimização, evitando assim soluções de projeto com densidade de material intermediária. Sólidos com comportamento anisotrópico são estudados sob condições de contorno de Robin, Neumann e Dirichlet. Uma transformação linear de coordenadas é utilizada para mapear o problema original e suas condições de contorno para um novo domínio equivalente isotrópico, onde o procedimento de otimização é aplicado. A solução otimizada é então transformada de volta ao domínio original. A metodologia proposta mostrou-se particularmente atrativa para resolver esta classe de problemas já que o MEC dispensa o uso de malha no domínio, reduzindo significantemente o custo computacional. Na última parte deste trabalho foi implementada uma formulação de sensibilidade topológica para problemas de otimização de transferência de calor e massa simultâneos. Como as sensibilidades para cada equação diferencial são diferentes, utiliza-se um coeficiente de ponderação para compor a sensibilidade do problema acoplado. Isto permite a imposição de distintos fatores para cada problema, de acordo com uma prioridade especificada. Diversos exemplos são apresentados e seus resultados comparados com os da literatura, quando disponíveis, a fim de validar as formulações propostas. / This work presents the computational development and implementation of topology optimization techniques for problems governed by the Poisson equation. The boundary element method was the numerical technique chosen to solve the equations. Three different methodologies were developed aiming this objective. The first methodology is directed to the application of genetic algorithms to investigate how a domain previously populated with randomly placed cavities evolves during the optimization process, and to verify the resemblance of the final solution with a optimal design. The external boundaries remain fixed during the process, while the location and dimension of the cavities are optimized in order to extremize a given cost function. The performance of the proposed algorithm is verified with a number of examples and the results are discussed. The second methodology presents a numerical algorithm for topology optimization based on the evaluation of topological derivatives, using the total potential energy as the cost function. This procedure is an alternative to the traditional optimization techniques, avoiding design solutions containing intermediary material densities. Solids with anisotropic constitutive behavior are studied under Robin, Neumann and Dirichlet boundary conditions. A linear coordinate transformation approach is used to map the original problem into an isotropic one, where the optimization is carried out. The final solution is then mapped back to the original coordinate system. The proposed method was found to be an attractive way to solve this class of problems, since no interior mesh is necessary, which reduces significantly the computational cost of the analysis. In the last part of the present work the topological derivative approach was further developed to deal with the optimization of problems under simultaneous heat and mass transfer. Since the sensitivities for each differential equation are different, a weighting factor was used to evaluate the final sensitivities of the coupled problem. This allows the imposition of different priorities for each problem Several examples are presented and their results are compared with the literature, when available, in order to validate the proposed formulations.

Este trabalho apresenta a formulação de um problema de otimização topológica para o posicionamento ótimo de atuadores baseado na teoria de controle. A estrutura é composta por dois materiais: um material passivo elástico e um material ativo piezelétrico, ambos lineares. Pretende-se obter um sistema de controle no qual todos os estados sejam controláveis. O processo de otimização topológica busca a distribuição de material piezelétrico que maximize o menor autovalor do Gramiano de controlabilidade garantindo, assim, sua não singularidade e, consequentemente, que o sistema seja completamente controlável. Programação linear sequencial (SLP) é utilizada para a solução do problema de otimização e as sensibilidades para o modelo de elementos finitos são deduzidas para a função objetivo e restrições. Análises modais das topologias ótimas são utilizadas para a definição de um controlador LQR e as respostas das estruturas controladas submetidas à uma carga impulsiva são analisadas. / This work presents a topology optimization formulation for the actuator placement based on the control theory. The structure is composed by two materials: a passive elastic material and an active piezoelectric material, both linear. The aim is to obtain a control system which all states are controllable. The topology optimization process searches the piezoelectric material distribution which maximizes the smallest eigenvalue of the controllability Gramian ensuring its non-singularity and, therefore, the system is completely controllable. Sequential linear programming (SLP) is used to solve the optimization problem. The sensitivities for the finite element model were derived for the objective function and constraints. Modal analysis from the optimal topologies were employed in an LQR controller and the responses for the controlled structures submitted to an impulsive load are analyzed.