Otimização topológica evolucionária de problemas com interação fluido-estrutura

Orientador: Renato PavanelloTese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia MecânicaResumo: O objetivo desta tese é o desenvolvimento de uma ferramenta computacional para projeto de estruturas considerando interação fluido-estrutura usando otimização topológica. Uma metodologia de otimização estrutural topológica é proposta associada à formulações de elementos finitos em problemas fluido-estrutura acoplados. Nesses tipos de problemas a estrutura sofre carregamentos advindos do meio fluido, ou seja, pressão e/ou forças viscosas. As dificuldades em se projetar estruturas sob carregamentos de fluidos surgem devido à variação da localização, direção e magnitude dos carregamentos quando a forma e topologia da estrutura são alteradas durante a otimização. Isso se torna o principal desafio para os métodos tradicionais baseados na interpolação da densidade do material. Nesses métodos, as superfícies em contato com o fluido não são definidas explicitamente devido à existência de elementos estruturais de densidade intermediária. Neste trabalho é proposta uma metodologia alternativa para esse tipo de carregamento dependente da topologia. Com a extensão do método de otimização estrutural evolucionária bidirecional (BESO) associada à formulações fluido-estrutura acopladas, pressões e forças viscosas podem ser modeladas diretamente para qualquer topologia estrutural devido à natureza discreta dos métodos evolucionários. Assim, o problema é resolvido sem a necessidade de parametrização das superfícies de carregamentos de pressão. A metodologia BESO é estendida considerando os procedimentos de alteração entre elementos fluido-estrutura-vazios, novas análises de sensibilidade e restrições. Problemas em estado estacionário são considerados, incluindo elasticidade linear para a análise estrutural e as equações de Laplace, Helmholtz e escoamento incompressível de Navier-Stokes para a análise do fluido. Carregamentos constantes e não constantes são modelados. Diversos exemplos e aplicações são explorados com a metodologia propostaAbstract: The aim of this thesis is the development of a computational tool for the design of structures considering fluid-structure interaction using topology optimization. A methodology of structural topology optimization is proposed in association with finite element formulations of fluid-structure coupled problems. In this type of problems, the structure undergoes fluid loading, i.e., pressure and/or viscous loads. The difficulties in designing fluid loaded structures arise due to the variation of location, direction and magnitude of the loads when the structural shape and topology change along the optimization procedure. This turns out to be an additional difficulty for the traditional density-based topology optimization methods. In density-based methods, the pressure loaded surfaces are not explicitly defined due to the existence of intermediate density elements. In this thesis, it is suggested an alternative methodology to handle this type of design-dependent loads. With an extended bi-directional evolutionary structural optimization (BESO) method associated with different fluid-structure formulations, pressures and viscous loads can be modelled straightforwardly for any structural topology due to the discrete nature of the method. Thus, the problem is solved without any need for pressure load surfaces parametrization. The BESO methodology is extended considering the procedures of switching fluid-structure-void elements, new sensitivity analyses and constraints. Steady state problems are considered, including linear elasticity for the structural analysis and Laplace, Helmholtz and incompressible Navier-tokes flow equations for the fluid analysis. Constant and non constant loads are modelled. Several examples and applications are explored with the proposed methodologyDoutoradoMecanica dos Sólidos e Projeto MecanicoDoutor em Engenharia Mecânica2011/09730-6FAPES

Structural optimization of plate-like aircraft wings under flutter and divergence constraints

Minimum-weight aircraft wing design, with an emphasis on avoiding aeroelastic instability, has been studied since the 1960s. The majority of works to date were posed as sizing problems; only a handful of researchers have employed a topology optimization approach. The aim of this study is to use the level set method for this purpose. The problem is formulated as one of plate thickness distribution, which takes on one of two prescribed values at every point on the wing planform. This is combined with constraints implemented on the eigenvalues of the flutter equation; such an approach is shown to be robust and versatile. Optimization results for rectangular plate wings at a range of sweep configurations studied previously are included to validate the present methods. Delta, high-aspect-ratio, and typical swept transport wing planforms are then optimized. All solutions demonstrate the ability to significantly reduce wing mass while maintaining flutter and divergence speed above a specified limit, which can be higher than that of the reference, maximum-thickness design. The proposed method can be used to provide insights into optimal aeroelastic wing structures and is particularly useful for developing unconventional aircraft structural configurations

Level set topology optimization for design-dependent pressure loads using the reproducing kernel particle method

This paper presents a level set topology optimization method in combination with the reproducing kernel particle method (RKPM) for the design of structures subjected to design-dependent pressure loads. RKPM allows for arbitrary particle placement in discretization and approximation of unknowns. This attractive property in combination with the implicit boundary representation given by the level set method provides an effective framework to handle the design-dependent loads by moving the particles on the pressure boundary without the need of remeshing or special numerical treatments. Moreover, the reproducing kernel (RK) smooth approximation allows for the Young’s modulus to be interpolated using the RK shape functions. This is another advantage of the proposed method as it leads to a smooth Young’s modulus distribution for smooth boundary sensitivity calculation which yields a better convergence. Numerical results show good agreement with those in the literature

Topology optimization for design-dependent hydrostatic pressure loading via the level-set method

A few level-set topology optimization (LSTO) methods have been proposed to address complex fluid-structure interaction. Most of them did not explore benchmark fluid pressure loading problems and some of their solutions are inconsistent with those obtained via density-based and binary topology optimization methods. This paper presents a LSTO strategy for design-dependent pressure. It employs a fluid field governed by Laplace’s equation to compute hydrostatic fluid pressure fields that are loading linear elastic structures. Compliance minimization of these structures is carried out considering the design-dependency of the pressure load with moving boundaries. The Ersatz material approach with fixed grid is applied together with work equivalent load integration. Shape sensitivities are used. Numerical results show smooth convergence and good agreement with the solutions obtained by other topology optimization methods

Stress and strain control via level set topology optimization

This paper presents a level set topology optimization method for manipulation of stress and strain integral functions in a prescribed region (herein called sub-structure) of a linear elastic domain. The method is able to deviate or concentrate the ux of stress in the sub-structure by optimizing the shape and topologies of the boundaries outside of that region. A general integral objective function is proposed and its shape sensitivities are derived. For stress isolation or maximization, a von Mises stress integral is used and results show that stresses in the sub-structure can be drastically reduced. For strain control, a strain integral combined with a vector able to select the component of the strain is used. A combination of both can be used to minimize deformation of a prescribed direction. Numerical results show that strain can be eciently minimized or maximized for a wide range of directions. The proposed methodology can be applied to stress isolation of highly sensitive non strain-based sensors, design for failure, maximization of mechanical strain and strain direction control for strain-based sensors and microdevices

Stress-based shape and topology optimization with the level set method

This paper proposes a level set method to solve minimum stress and stress-constrained shape and topology optimization problems. The method solves a sub-optimization problem every iteration to obtain optimal boundary velocities. A p-norm stress functional is used to aggregate stresses in a single constraint. The shape sensitivity function is derived and a computational procedure based on a least squares interpolation approach is devised in order to compute sensitivities at the boundaries. Adaptive constraint scaling is used to enforce exact control of stress limits. Numerical results show that the method is able to solve the problem e�ciently for single and multiple load cases obtaining solutions with smooth boundaries

Level set topology optimization with nodally integrated reproducing kernel particle method

A level set topology optimization (LSTO) using the stabilized nodally integrated reproducing kernel particle method (RKPM) to solve the governing equations is introduced in this paper. This methodology allows for an exact geometry description of a structure at each iteration without remeshing and without any interpolation scheme. Moreover, useful characteristics of the RKPM such as the easily controlled order of continuity and the ability to freely place particles in a design domain wherever needed are illustrated through stress based and design-dependent surface loading examples. The numerical results illustrate the effectiveness and robustness of the methodology with good optimization convergence behavior and ability to handle large topological changes. Furthermore, it is shown that different particle distributions can be used to increase efficiency without additional complexity

Evolutionary structural optimization using hexagonal meshes

Orientador: Renato PavanelloDissertação (mestrado) - Universidade Estadual de Campinas, Faculdade de Engenharia MecânicaResumo: A otimização estrutural topológica é uma ferramenta aplicada em muitos campos da engenharia. Diversos métodos de otimização topológica (MOT) têm sido desenvolvidos a partir da consolidação do método dos elementos finitos e da evolução das capacidades computacionais. Dentre os diversos métodos, o método denominado de Otimização Estrutural Evolucionária (ESO - Evolutionary Structural Optimization) tem se consolidado na área através da aplicação de estratégias heurísticas na análise do modelo estrutural. Esse método se baseia na eliminação sucessiva de elementos em uma malha de elementos finitos que cobre o espaço de soluções definido inicialmente. Atualmente, a otimização evolucionária conta com a versão bidirecional (BESO - Bidirectional ESO) em que os elementos não são apenas eliminados, mas também adicionados ao domínio inicial. Um dos problemas comuns aos métodos de otimização topológica é o Tabuleiro de Xadrez, que consiste na obtenção de soluções intermediárias com padrões sólido-vazios alternados que sub ou supervalorizam a rigidez da estrutura e divergem da solução ótima. Como soluções para este problema, são utilizados elementos finitos de alta ordem, filtros e a aplicação de malhas hexagonais. Entretanto, as duas primeiras opções implicam no aumento do custo computacional. Neste trabalho, investiga-se o uso de malhas hexagonais nos métodos ESO/BESO, cuja metodologia ainda não foi explorada nos métodos evolucionários. Além de eliminar o Tabuleiro de Xadrez, as malhas hexagonais possibilitam a obtenção de topologias com contornos mais suaves e livres de conexões por apenas um nó. São implementados os métodos ESO em critérios de tensão e rigidez, ESO em otimização de forma e o método BESO para critério baseado em rigidez. A aplicação de malhas hexagonais é investigada em todos esses casosAbstract: The Structural Topology Optimization technique has been applied in many fields of engineering. Several optimization methods have been developed with the Finite Element Method consolidation and the evolution of computational capabilities. Among many techniques, Evolutionary Structural Optimization (ESO) has been consolidated in the area through the successive application of heuristic strategies in the structural analysis. This method is based on a rule of successive elimination of elements in a finite element mesh that covers the feasible solution space defined initially. The Bidirectional Evolutionary Structural Optimization method (BESO) was recently proposed, in which the elements are not only eliminated but also added to the initial domain. One of the common problems in topology optimization methods is the checkerboard that consists in obtaining intermediate solutions with solid-void alternated patterns that under or overvalue the stiffness of the structure and diverges from the optimum solutionMestradoMecanica dos Sólidos e Projeto MecanicoMestre em Engenharia Mecânic

Topology optimization of binary structures using integer linear programming

This work proposes an improved method for gradient-based topology optimization in a discrete setting of design variables. The method combines the features of BESO developed by Huang and Xie [1] and the discrete topology optimization method of Svanberg and Werme [2] to improve the effectiveness of binary variable optimization. Herein the objective and constraint functions are sequentially linearized using Taylor's first order approximation, similarly as carried out in [2]. Integer Linear Programming (ILP) is used to compute globally optimal solutions for these linear optimization problems, allowing the method to accommodate any type of constraints explicitly, without the need for any Lagrange multipliers or thresholds for sensitivities (like the modern BESO [1]), or heuristics (like the early ESO/BESO methods [3]). In the linearized problems, the constraint targets are relaxed so as to allow only small changes in topology during an update and to ensure the existence of feasible solutions for the ILP. This process of relaxing the constraints and updating the design variables by using ILP is repeated until convergence. The proposed method does not require any gradual refinement of mesh, unlike in [2] and the sensitivities every iteration are smoothened by using the mesh-independent BESO filter. Few examples of compliance minimization are shown to demonstrate that mathematical programming yields similar results as that of BESO for volume-constrained problems. Some examples of volume minimization subject to a compliance constraint are presented to demonstrate the effectiveness of the method in dealing with a non-volume constraint. Volume minimization with a compliance constraint in the case of design-dependent fluid pressure loading is also presented using the proposed method. An example is presented to show the effectiveness of the method in dealing with displacement constraints. The results signify that the method can be used for topology optimization problems involving non-volume constraints without the use of heuristics, Lagrange multipliers and hierarchical mesh refinement