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[pt] A otimização topológica de estruturas sujeitas a carregamentos que variam ao longo do tempo, costuma ter custos computacionais bastante elevados. Isso acontece devido ao grande número de análises dinâmicas que são necessárias. ALém disso, avaliar os gradientes da resposta, que fazem parte da análise de sensibilidade, também resulta em alto custo de computação e requer um grande espaço de armazenamento. Nesta dissertação, em vez de resolver o problema dinâmico original diretamente, é resolvida uma sequência de problemas de otimização de resposta estática com múltiplos casos de carga. Essa abordagem, chamada de método do carregamento estático equivalente, gera cargas estáticas que produzem uma resposta equivalente à obtida na análise dinâmica. Para avaliar as abordagens presentes na literatura, desenvolveu-se um código MATLAB, e diversos exemplos representativos são apresentados. / [en] Topology optimization in the time domain of structures subjected to time-dependent loads is usually computationally expensive, starting with the large number of time-dependent analyses that are required. Futhermore, the computational cost to evaluate the gradients of the response is significantly high and requires a large storage space. In this paper, instead of solving the original dynamical problem directly, we solve a sequence of static response optimization problems with many load cases. This approach, called the equivalent static loads method, generates static loads that produce a similar response in comparison to the same response that the dynamic analysis does. In order to verify the approaches present in the literature, a MATLAB code was developed, and several representatives examples are presented.

[pt] A otimização topológica tem como objetivo encontrar a distribuição mais eficiente de material (ótima topologia) em uma determinada região, satisfazendo as restrições de projeto estabelecidas pelo usuário. Na abordagem tradicional atribui-se uma variável de projeto, constante, denominada densidade, para cada elemento finito da malha. Dessa forma, a qualidade da representação dos novos contornos da estrutura depende do nível de discretização da malha: quanto maior a quantidade de elementos, mais bem definida será a topologia da estrutura otimizada. No entanto, a utilização de malhas super-refinadas implica em um elevado custo computacional, principalmente na etapa de solução numérica das equações de equilíbrio pelo método dos elementos finitos. Este trabalho propõe uma nova estratégia computacional para o refinamento adaptativo local de malhas utilizando elementos finitos poligonais em domínios bidimensionais arbitrários. A ideia consiste em realizar um refinamento da malha nas regiões de concentração de material, sobretudo nos contornos internos e externos, e um desrefinamento nas regiões de baixa concentração de material, como por exemplo, nos furos internos. Desta forma, é possível obter topologias ótimas, com alta resolução e relativamente baixo custo computacional. Exemplos representativos são apresentados para demonstrar a robustez e a eficiência da metodologia proposta por meio de comparações com resultados obtidos com malhas super-refinadas e mantidas constantes durante todo o processo de otimização topológica. / [en] Topology optimization aims to find the most efficient distribution of material (optimal topology) in a given domain, subjected to design constraints defined by the user. The quality of the new boundary representation depends on the level of mesh refinement: the greater the number of elements in the mesh, the better will be the representation of the optimized structure. However, the use of super refined meshes implies in a high computational cost, especially regarding the numerical solution of the linear systems of equations that arise from the finite element method. This work proposes a new computational strategy for adaptive local mesh refinement using polygonal finite elements in arbitrary two-dimensional domains. The idea is to perform a mesh refinement in regions of material concentration, mostly in inner and outer boundaries, and a mesh derefinement in regions of low material concentration such as the internal holes. Thus, it is possible to obtain optimal topologies with high resolution and relatively low computational cost. Representative examples are presented to demonstrate the robustness and efficiency of the proposed methodology by comparing the results obtained herein with the ones from the literature where super refined meshes are held constant throughout all topology optimization process.

Simulation driven design is a method and process that has been developed over many years, and with today’s advanced software, the possibility to embed simulation into the design process has become a reality. The advantages of using simulation driven design in the product development process is well known and compared to a more traditional design process, the simulation driven design process can give the user the possibility to explore, optimize and design products with reduced lead time.  One of the methods that is applied in simulation driven design is the use of topology optimization (structural optimization). Topology optimization is something that GKN uses in the design process. Due to the complexity of the products GKN design and manufacture, the output from the topology optimization lacks good design interpretability and the design process requires a lot of time and effort.  The purpose of the thesis is to explore different simulation tools used for topology optimization and improve the methodology and process with higher design interpretability for a static topology optimization. This requires a good understanding of the component and the product development process. It is imperative that the topology result must have high design interpretability, and the visualization of the result must show the formation of clear rib structures.  The software’s used for performing topology optimization in this thesis are Inspire, SimLab, HyperMesh, and OptiStruct (HyperWorks suite). Static topology optimization is conducted, and manufacturing constraints for the casting process are considered. The methodology developed is robust for similar gearbox housing structures, and the process is set up to be efficient. The proposed method is verified by implementing it on a housing structure.  The resulting concept from the topology optimization is deemed to have higher design interpretability which improves knowledge transfer in the design process when compared to the current topology results. The weight of the product is reduced, and a more optimum design is reached with a lesser number of iterations. / Simuleringsdriven design är en metod och process som har utvecklats i många år, och med dagens avancerade programvaror har möjligheten att få in simulering direkt i designprocessen blivit verklighet. Fördelarna med att använda simuleringsdriven design i produktutvecklingsprocessen är välkända och jämfört med en mer traditionell designprocess kan den simuleringsdrivna designprocessen ge användaren möjlighet att utforska, optimera och designa produkter med reducerade ledtider som följd.  En av de metoder som tillämpas i simuleringsdriven design är användning av topologioptimering (strukturoptimering). Topologioptimering är något som GKN använder i designprocessen. På grund av komplexiteten hos produkterna GKN designar och tillverkar kräver designprocessen mycket ingenjörsarbete och tid. Produktionen har också problem med att tolka topologioptimeringsresultaten.. Syftet med avhandlingen är att utforska olika simuleringsverktyg som används för topologioptimering och förbättra metodiken och processen för att öka designtolkningen av en statisk topologioptimering. Detta kräver en god förståelse för komponenten och produktutvecklingsprocessen. För att förbättra osäkerheterna i resultaten från optimeringen, är det nödvändigt att dessa resultat är lätta att tolka, och visualiseringen av resultaten ska vara tydliga och visa hur lastvägarna går och därmed vart ribbor ska läggas.  Programvarorna som användes för att utföra topologioptimering i denna avhandling är Inspire, SimLab, HyperMesh och OptiStruct (HyperWorks suite). Statisk topologioptimering är utförd och tillverkningsbegränsningar för gjutningsprocesser har inkluderats.  Den metod som utvecklats är robust för liknande växellådshusstrukturer, och processen som föreslås är mera effektiv. Den föreslagna metoden har verifierats genom att den tillämpats för ett växellådshus.  Det resulterande topologikonceptet antas ha en bättre designtolkningsbarhet, vilket möjliggör en förbättrad kommunikation och kunskapsöverföring i konstruktionsprocessen, jämfört med den nuvarande processen. Produktens vikt minskas, och en mer optimal design nås med färre iterationer.

Increasing application of Additive Manufacturing (AM) in industrial production demands product reimagination (assemblies, subsystems) from an AM standpoint. Simulation driven design tools play an important part in achieving this with design optimization subject to the capabilities of AM technologies. Therefore, the bus frames department (RBRF) in Scania CV AB, Södertälje wanted to examine the synergies between topology optimization and Design for AM (DfAM) in the context of this thesis. In this thesis, a methodology is developed to establish a DfAM framework involving topology optimization and is accompanied by a manufacturability analysis stage. A case study implementation of this developed methodology is performed for validation and further development. The case study replaces an existing load bearing cross member with a new structure optimized with respect to weight and manufacturing process. It resulted in a nearly self supporting AM friendly design with improved stiffness along with a 9.5% weight reduction, thereby proving the benefit of incorporating topology optimization and AM design fundamentals during the early design phase. / Ökad användning av Additive Manufacturing (AM) i industriell produktion kräver ett nytänkade av produkter (enheter, delsystem) ur AM-synvinkel. Simuleringsdrivna designverktyg spelar en viktig roll för att nå detta med designoptimering med hänsyn taget till AM-teknikens möjligheter. Därför ville bussramavdelningen (RBRF) på Scania CV AB, Södertälje undersöka synergierna mellan topologioptimering och Design för AM (DfAM) i detta examensarbete. I examensarbetet utvecklas en metodik för att skapa en DfAM-ramverk som involverar topologioptimering och åtföljs av ett tillverkningsanalyssteg. En fallstudieimplementering av denna utvecklade metodik utförs för validering och fortsatt utveckling. Fallstudien ersätter en befintlig lastbärande tvärbalk med en ny struktur optimerad med avseende på vikt och tillverkningsprocess. Det resulterade i en nästan självbärande AM-vänlig design med förbättrad styvhet tillsammans med en viktminskning på 9,5 %, vilket visar fördelen med att integrera topologioptimering och grundläggande AM-design tidigt i designfasen.

Zweitveröffentlichung, ursprünglich veröffentlicht: Jonas Holley: Stress-Constrained Topology Optimization with Application to the Design of Electrical Machines. München: Verlag Dr. Hut, 2023, 199 Seiten, Dissertation Humboldt-Universität Berlin (2023). ISBN 978-3-8439-5378-8 / Während des Designprozesses physischer Gegenstände stellt die mechanische Stabilität in nahezu jedem Anwendungsbereich eine essentielle Anforderung dar. Stabilität kann mittels geeigneter Kriterien, die auf dem mechanischen Spannungstensor basieren, mathematisch quantifiziert werden. Dies dient dem Ziel der Vermeidung von Schädigung in jedem Punkt innerhalb des Gegenstands. Die vorliegende Arbeit behandelt die Entwicklung einer Methode zur Lösung von Designoptimierungsproblemen mit punktweisen Spannungsrestriktionen. Zunächst wird eine Regularisierung des Optimierungsproblems eingeführt, die einen zentralen Baustein für den Erfolg einer Lösungsmethode darstellt. Nach der Analyse des Problems hinsichtlich der Existenz von Lösungen wird ein Gradientenabstiegsverfahren basierend auf einer impliziten Designdarstellung und dem Konzept des topologischen Gradienten entwickelt. Da der entwickelte Ansatz eine Methode im Funktionenraum darstellt, ist die numerische Realisierung ein entscheidender Schritt in Richtung der praktischen Anwendung. Die Diskretisierung der Zustandsgleichung und der adjungierten Gleichung bildet die Basis für eine endlich-dimensionale Version des Optimierungsverfahrens. Im letzten Teil der Arbeit werden numerische Experimente durchgeführt, um die Leistungsfähigkeit des entwickelten Algorithmus zu bewerten. Zunächst wird das Problem des minimalen Volumens unter punktweisen Spannungsrestriktionen anhand der L-Balken Geometrie untersucht. Ein Schwerpunkt wird hierbei auf die Untersuchung der Regularisierung gelegt. Danach wird das multiphysikalische Design einer elektrischen Maschine adressiert. Zusätzlich zu den punktweisen Restriktionen an die mechanischen Spannungen wird die Maximierung des mittleren Drehmoments berücksichtigt, um das elektromagnetische Verhalten der Maschine zu optimieren. Der Erfolg der numerischen Tests demonstriert das Potential der entwickelten Methode in der Behandlung realistischer industrieller Problemstellungen. / In the process of designing a physical object, the mechanical stability is an essential requirement in nearly every area of application. Stability can be quantified mathematically by suitable criteria based on the stress tensor, aiming at the prevention of damage in each point within the physical object. This thesis deals with the development of a framework for the solution of optimal design problems with pointwise stress constraints. First, a regularization of the optimal design problem is introduced. This perturbation of the original problem represents a central element for the success of a solution method. After analyzing the perturbed problem with respect to the existence of solutions, a line search type gradient descent scheme is developed based on an implicit design representation via a level set function. The core of the optimization method is provided by the topological gradient, which quantifies the effect of an infinitesimal small topological perturbation of a given design on an objective functional. Since the developed approach is a method in function space, the numerical realization is a crucial step towards its practical application. The discretization of the state and adjoint equation provide the basis for developing a finite-dimensional version of the optimization scheme. In the last part of the thesis, numerical experiments are conducted in order to assess the performance of the developed algorithm. First, the stress-constrained minimum volume problem for the L-Beam geometry is addressed. An emphasis is put on examining the effect of the proposed regularization. Afterwards, the multiphysical design of an electrical machine is addressed. In addition to the pointwise constraints on the mechanical stress, the maximization of the mean torque is considered in order to improve the electromagnetic performance of the machine. The success of the numerical tests demonstrate the potential of the developed design method in dealing with real industrial problems.

The main theme of this work is effectiveness and efficiency of Krylov subspace methods and Krylov subspace recycling. While solving long, slowly changing sequences of large linear systems, such as the ones that arise in engineering, there are many issues we need to consider if we want to make the process reliable (converging to a correct solution) and as fast as possible. This thesis is built on three main components. At first, we target bilinear and quadratic form estimation. Bilinear form $c^TA^{-1}b$ is often associated with long sequences of linear systems, especially in optimization problems. Thus, we devise algorithms that adapt cheap bilinear and quadratic form estimates for Krylov subspace recycling. In the second part, we develop a hybrid recycling method that is inspired by a complex CFD application. We aim to make the method robust and cheap at the same time. In the third part of the thesis, we optimize the implementation of Krylov subspace methods on Graphic Processing Units (GPUs). Since preconditioners based on incomplete matrix factorization (ILU, Cholesky) are very slow on the GPUs, we develop a preconditioner that is effective but well suited for GPU implementation. / Ph. D.

[pt] O design otimizado de estruturas considerando não-linearidades tem sido amplamente pesquisado nas décadas recentes. A análise de elementos finitos aplicada à otimização topológica é prejudicada pela deformação excessiva de elementos de baixa densidade sob alta compressão, o que impede o processo de encontrar uma solução ótima. Dois métodos, o esquema Interpolação de Energia e a técnica de Hiperelasticidade Aditiva, são implementados para superar essa dificuldade no problema de minimização da flexibilidade, e modelos de materiais hiperelásticos são usados para investigar suas influências na topologia otimizada. O Método das Assíntotas Móveis é usado para atualizar as variáves de projeto cujas sensibilidades foram calculadas pelo método adjunto. A equação de estado é resolvida através do método de Newton-Raphson com um incremento de carga ajustável para reduzir o custo computacional. Resultados de dois problemas de referência são comparado com aqueles já estabelecidos na literatura. O uso de diferentes modelos hiperelásticos apresentou pouca influência no design final da estrutura. O método de Interpolação de Energia foi capaz de convergir para cargas muito maiores que o método padrão, enquanto a Hiperelasticidade Aditiva apresentou dificuldades de convergência em estado plano de deformação. / [en] The optimized design of structures considering nonlinearities has been widely researched in the recent decades. The finite element analysis applied to topology optimization is jeopardized by the excessive deformation of low-density elements under high compression, which hinders the process of finding an optimal solution. Two methods, the Energy Interpolation scheme and the Additive Hyperelasticity technique, are implemented to overcome this difficulty in the minimum compliance problem, and hyperelastic material models are used to investigate their influence on the optimized topology. The Method of Moving Asymptotes is used to update the design variables whose sensitivities were calculated from the adjoint method. The state equation is solved through the Newton-Raphson method with an adjusting load step to reduce computational cost. Results for two benchmark problems are compared with those already established in the literature. The use of different hyperelastic models presented little influence on the final design of the structure. The Energy Interpolation method was able to converge for much higher loads than the default method, while the Additive Hyperelasticity presented convergence difficulties in plane strain.

I och med en ökande konkurrens och tekniska framsteg har flygbranschen tvingats ta till nya metoder för att utveckla komponenter. I flygbranschen är viktoptimering en viktig faktor och på senare tid har man sett stora möjligheter med att genomföra detta med hjälp av topologioptimering och additiv tillverkning. På Saab finns det ett intresse av att undersöka hur topologioptimering kan användas för att om möjligt förenkla och förbättra den utvecklingsprocess man har i dag och göra de produkterna man tillverkar ännu bättre. Speciellt vill man titta på hur topologioptimeringsmodulen i det befintliga CAD systemet Siemens NX fungerar för att se hur konstruktörer skulle kunna använda sig av verktyget för att minska vikten på komponenterna. Syftet med denna studie är att undersöka hur topologioptimering kan genomföras i Siemens NX för att se hur det kan användas i utvecklingsarbetet på Saab samt för att se vilka möjligheter/utmaningar verktyget för med sig. Arbetet är baserat på en litteraturstudie, en intervjustudie, en workshop och undersökningar av Siemens NX topologioptimeringsmodul. Resultatet från intervjuerna visade att man på Saab ser en vinst i att använda sig av topologioptimering på flera ställen i utvecklingsprocessen. Undersökningen av Siemens topologioptimeringsmodul visar att verktyget kan användas, framförallt under en av de tidpunkterna i processen som föreslagits under intervjuerna, för att ta fram en kvalificerad första gissning på en konstruktion. Vinsten med att använda topologioptimering har visat sig vara att antalet iterationer mellan konstruktör och strukturanalytiker kan minska, vilket kan leda till en tidseffektiviserad utvecklingsprocess. Det har framkommit under intervju med en strukturanalytiker att färdiga komponenter, där topologioptimering använts, tenderar att vara bättre uppbyggda och ha lättare att klara av de ställda kraven. Arbetet har också visat att det finns utmaningar med att använda Siemens topologioptimeringsmodul på Saab. Ett av de stora problemen som framkommit under arbetet är att det är svårt att ta fram exakta lastfall på Saabs komponenter vilket krävs för att topologioptimera i Siemens NX. Ytterligare en försvårande faktor är att Saabs produkter i många fall har en låst förformskonstruktion med förutbestämda designparametrar, som dimensioner och funktioner. Detta medför att det i vissa fall varit svårt att få fram relevanta resultat. Resultaten från undersökning av Siemens topologioptimering har visat sig ge bäst resultat då den förformskonstruktion som optimeras har en stor designfrihet. / As a consequence of great technical progress in today's society and the increasing competition industries in between, the aviation industry has been forced to implement new methods in their product development to stay competitive. Topology optimization together with additive manufacturing is one of them and is used to optimize the weight of a component. Optimization in weight is a crucial factor in the aviation industry and topology optimization is recently shown to be a useful method. At Saab there is an interest in evaluating topology optimization and its capability to both improve and simplify today's development process to make even better products in the end. In particular, they want to look at how the topology optimization module in the existing CAD system Siemens NX works to explore the opportunities for the designersto reduce the weights of the components. This study aims to examine the use of topology optimization in Siemens NX to explore if it is a useful method to implement in the development process at Saab. The purpose is also to see what advantages and what challenges it brings. The work is based on a literature review, an interview study, a workshop and an investigation of the topology optimization module in Siemens NX.  Results from the interviews showed that Saab can benefit from the usage of topology optimization in several parts of the development process. Siemens NX topology optimization module, specifically, was found to fit in one of the suggested places, where it can be used to create a first qualified iteration. One of the advantages topology optimization can bring is fewer design iterations between designers and analysts which can help reduce the development time and improve the process. Furthermore one of the interviewed structural analysts claims that parts, where topology optimization tools has been used, has been found to be better constructed and has easier to pass through the validation process. The study has also revealed some difficulties with the usage of Siemens NX topology optimization module at Saab where the biggest question is whether relevant load cases exists or not. Topology optimization in Siemens NX requires definitive load cases which has been hard or even impossible to find in many of Saab´s products. Another difficulty with the usage of Siemens Topology optimization on Saab´s product is the limited freedom in the design space. The respondents mean that most of the constructed parts has a lot of predetermined parameters and in the topology optimization module this fact has sometimes made it difficult to produce relevant results.