Topology Optimization of continuum structures with epsilon-relaxed stress constraints

Topology optimization of linear elastic continuum structures is a challenging problem when considering local stress constraints. The reasons are the singular behavior of the constraint with the density design variables, combined with the large number of constraints even for small finite element meshes. This work presents an alternative formulation for the s-relaxation technique, which provides an workaround for the singularity of the stress constraint. It also presents a new global stress constraint formulation. Derivation of the sensitivities for the constraint by the adjoint method is shown. Results for single and multiple load cases show the potential of the new formulation.

Topology design of three-dimensional structures using hybrid finite elements

Conventional three-dimensional isoparametric elements are susceptible to problems of locking when used to model plate/shell geometries or when the meshes are distorted etc. Hybrid elements that are based on a two-field variational formulation are immune to most of these problems, and hence can be used to efficiently model both "chunky" three-dimensional and plate/shell type structures. Thus, only one type of element can be used to model "all" types of structures, and also allows us to use a standard dual algorithm for carrying out the topology optimization of the structure. We also address the issue of manufacturability of the designs.

Freeform Skeletal Shape Optimization of Compliant Mechanisms

Compliant mechanisms are elastic continua used to transmit or transform force and motion mechanically. The topology optimization methods developed for compliant mechanisms also give the shape for a chosen parameterization of the design domain with a fixed mesh. However, in these methods, the shapes of the flexible segments in the resulting optimal solutions are restricted either by the type or the resolution of the design parameterization. This limitation is overcome in this paper by focusing on optimizing the skeletal shape of the compliant segments in a given topology. It is accomplished by identifying such segments in the topology and representing them using Bezier curves. The vertices of the Bezier control polygon are used to parameterize the shape-design space. Uniform parameter steps of the Bezier curves naturally enable adaptive finite element discretization of the segments as their shapes change. Practical constraints such as avoiding intersections with other segments, self-intersections, and restrictions on the available space and material, are incorporated into the formulation. A multi-criteria function from our prior work is used as the objective. Analytical sensitivity analysis for the objective and constraints is presented and is used in the numerical optimization. Examples are included to illustrate the shape optimization method.

Comparative analysis of strut-and-tie models using Smooth Evolutionary Structural Optimization

The strut-and-tie models are widely used in certain types of structural elements in reinforced concrete and in regions with complexity of the stress state, called regions D, where the distribution of deformations in the cross section is not linear. This paper introduces a numerical technique to determine the strut-and-tie models using a variant of the classical Evolutionary Structural Optimization, which is called Smooth Evolutionary Structural Optimization. The basic idea of this technique is to identify the numerical flow of stresses generated in the structure, setting out in more technical and rational members of strut-and-tie, and to quantify their value for future structural design. This paper presents an index performance based on the evolutionary topology optimization method for automatically generating optimal strut-and-tie models in reinforced concrete structures with stress constraints. In the proposed approach, the element with the lowest Von Mises stress is calculated for element removal, while a performance index is used to monitor the evolutionary optimization process. Thus, a comparative analysis of the strut-and-tie models for beams is proposed with the presentation of examples from the literature that demonstrates the efficiency of this formulation. © 2013 Elsevier Ltd.

Miniature Compliant Grippers With Vision-Based Force Sensing

This paper is concerned with grasping biological cells in aqueous medium with miniature grippers that can also help estimate forces using vision-based displacement measurement and computation. We present the design, fabrication, and testing of three single-piece, compliant miniature grippers with parallel and angular jaw motions. Two grippers were designed using experience and intuition, while the third one was designed using topology optimization with implicit manufacturing constraints. These grippers were fabricated using different manufacturing techniques using spring steel and polydimethylsiloxane ( PDMS). The grippers also serve the purpose of a force sensor. Toward this, we present a vision-based force-sensing technique by solving Cauchy's problem in elasticity using an improved algorithm. We validated this technique at the macroscale, where there was an independent method to estimate the force. In this study, the gripper was used to hold a yeast ball and a zebrafish egg cell of less than 1 mm in diameter. The forces involved were estimated to be about 30 and 10 mN for the yeast ball and the zebrafish egg cell, respectively.

A Compliant Mechanism Kit with Flexible Beams and Connectors

We present the concept, prototypes, and an optimal design method for a compliant mechanism kit as a parallel to the kits available for rigid-body mechanisms. The kit consists of flexible beams and connectors that can be easily hand-assembled using snap fits. It enables users, using their creativity and mechanics intuition, to quickly realize a compliant mechanism. The mechanisms assembled in this manner accurately capture the essential behavior of the topology, shape, size and material aspects and thereby can lead the way for a real compliant mechanism for practical use. Also described in this paper are the design of the connector to which flexible beams can be added in eight different directions; and prototyping of the spring steel connectors as well as beams using wire-cut electro discharge machining. It is noted in this paper that the concept of the kit also resolves a discrepancy in the finite element (FE) modeling of beam-based compliant mechanisms. The discrepancy arises when two or more beams are joining at one point and thus leading to increased stiffness. After resolving this discrepancy, this work extends the topology optimization to automatically generate designs that can be assembled with the kit. Thus, the kit and the accompanying analysis and optimal synthesis procedures comprise a self-contained educational as well as a research and pragmatic toolset for compliant mechanisms. The paper also illustrates how human creativity finds new ways of using the kit beyond the original intended use and how it is useful even for a novice to design compliant mechanisms.

A compliant mechanism kit with flexible beams and connectors along with analysis and optimal synthesis procedures

We present a compliant mechanism kit as a parallel to the kits available for rigid-body mechanisms. The kit consists of flexible beams and connectors that can be easily hand-assembled using snap fits. The mechanisms assembled using the kit accurately capture the aspects of the topology, shape, and size of joint-free compliant mechanisms. Thus, the kit enables designers to conceive and design new, practicable, single-piece compliant mechanisms that do not require assembly. The concept of the kit also resolves a discrepancy in the finite element (FE) modeling of beam-based compliant mechanisms. The discrepancy arises when two or more beams are joined at one point and thus leading to increased stiffness. After resolving this discrepancy, this work extends the topology optimization to automatically generate designs that can be assembled with the kit for quick and easy validation instead of time-consuming prototyping. Thus, the kit and the accompanying analysis and optimal synthesis procedures comprise a self-contained educational as well as a research and practice toolset for compliant mechanisms. The paper also illustrates how human creativity finds new ways of using the kit beyond the original intended use and how it enables even a novice to design compliant mechanisms. (C) 2011 Elsevier Ltd. All rights reserved.

Light-propagation characteristics of photonic crystal waveguide based on SOI materials at different polarized states

Straight single-line defect optical waveguides in photonic crystal slabs are designed by the plane wave expansion method and fabricated into silicon-on-insulator (SOI) wafer by 248-nm deep UV lithography. We present an efficient way to measure the light transmission spectrum of the photonic crystal waveguide (PhC WG) at given polarization states. By employing the Mueller/Stokes method, we measure and analyse the light propagation properties of the PhC WG at different polarized states. It is shown that experimental results are in agreement with the simulation results of the three-dimensional finite-difference-time-domain method.

Metodologias de optimização topológica em cálculo estrutural

A optimização estrutural é uma temática antiga em engenharia. No entanto, com o crescimento do método dos elementos finitos em décadas recentes, dá origem a um crescente número de aplicações. A optimização topológica, especificamente, surge associada a uma fase de definição de domínio efectivo de um processo global de optimização estrutural. Com base neste tipo de optimização, é possível obter a distribuição óptima de material para diversas aplicações e solicitações. Os materiais compósitos e alguns materiais celulares, em particular, encontram-se entre os materiais mais proeminentes dos nossos dias, em termos das suas aplicações e de investigação e desenvolvimento. No entanto, a sua estrutura potencialmente complexa e natureza heterogénea acarretam grandes complexidades, tanto ao nível da previsão das suas propriedades constitutivas quanto na obtenção das distribuições óptimas de constituintes. Procedimentos de homogeneização podem fornecer algumas respostas em ambos os casos. Em particular, a homogeneização por expansão assimptótica pode ser utilizada para determinar propriedades termomecânicas efectivas e globais a partir de volumes representativos, de forma flexível e independente da distribuição de constituintes. Além disso, integra processos de localização e fornece informação detalhada acerca de sensibilidades locais em metodologias de optimização multiescala. A conjugação destas áreas pode conduzir a metodologias de optimização topológica multiescala, nas quais de procede à obtenção não só de estruturas óptimas mas também das distribuições ideais de materiais constituintes. Os problemas associados a estas abordagens tendem, no entanto, a exigir recursos computacionais assinaláveis, criando muitas vezes sérias limitações à exequibilidade da sua resolução. Neste sentido, técnicas de cálculo paralelo e distribuído apresentam-se como uma potencial solução. Ao dividir os problemas por diferentes unidades memória e de processamento, é possível abordar problemas que, de outra forma, seriam proibitivos. O principal foco deste trabalho centra-se na importância do desenvolvimento de procedimentos computacionais para as aplicações referidas. Adicionalmente, estas conduzem a diversas abordagens alternativas na procura simultânea de estruturas e materiais para responder a aplicações termomecânicas. Face ao exposto, tudo isto é integrado numa plataforma computacional de optimização multiobjectivo multiescala em termoelasticidade, desenvolvida e implementada ao longo deste trabalho. Adicionalmente, o trabalho é complementado com a montagem e configuração de um cluster do tipo Beowulf, assim como com o desenvolvimento do código com vista ao cálculo paralelo e distribuído.

Sellupesurin rakenteellinen kehittäminen

Tämä diplomityö on tehty Andritz Oy:lle Washers & Filters tuoteryhmään. Työ on osa pienten sellupesureiden tuotekehitysprojektia. Tavoitteena on vertailla olemassa olevaa tuotekehitysaineistoa ja tuoda esiin suunnitteluprosessi, jolla DD – sellupesurin osien rakenteita voidaan järjestelmällisesti kehittää. Diplomityössä tutkittuja osia ovat tiiviste–elementti, päätypalkki ja rumpu. Tiiviste–elementtejä vertailtiin olemassa olevan tuotekehitysaineiston osalta, sekä tutkittiin geneettisiin algoritmeihin pohjautuvan topologian optimoinnin soveltuvuutta tiiviste-elementin suunnitteluun. Päätypalkin ja rummun optimaaliset geometriat selvitettiin geneettisiä algoritmejä hyödyntävällä topologisella optimoinnilla. Optimaalisten topologioiden perusteella suunniteltiin valmistettavissa olevat rakenteet joiden ainevahvuudet määrättiin alustavasti vakion variointiin perustuvalla optimoinnilla. Tällä menettelyllä saatiin päätypalkista ja rummusta aikaiseksi aikaisempaa kevyemmät rakenteet. Topologian optimointi huomattiin soveltuvan rakenteisiin, joiden kuormitus- ja kiinnitystiedot ovat yksiselitteisesti määrätyt.

Biomecânica e otimização topológica com H-adaptatividade em implantes dentários nitretados a plasma em cátodo oco

In recent years there has been a significant growth in technologies that modify implant surfaces, reducing healing time and allowing their successful use in areas with low bone density. One of the most widely used techniques is plasma nitration, applied with excellent results in titanium and its alloys, with greater frequency in the manufacture of hip, ankle and shoulder implants. However, its use in dental implants is very limited due to high process temperatures (between 700 C o and 800 C o ), resulting in distortions in these geometrically complex and highly precise components. The aim of the present study is to assess osseointegration and mechanical strength of grade II nitrided titanium samples, through configuration of hollow cathode discharge. Moreover, new formulations are proposed to determine the optimum structural topology of the dental implant under study, in order to perfect its shape, make it efficient, competitive and with high definition. In the nitriding process, the samples were treated at a temperature of 450 C o and pressure of 150 Pa , during 1 hour of treatment. This condition was selected because it obtains the best wettability results in previous studies, where different pressure, temperature and time conditions were systematized. The samples were characterized by X-ray diffraction, scanning electron microscope, roughness, microhardness and wettability. Biomechanical fatigue tests were then conducted. Finally, a formulation using the three dimensional structural topology optimization method was proposed, in conjunction with an hadaptive refinement process. The results showed that plasma nitriding, using the hollow cathode discharge technique, caused changes in the surface texture of test specimens, increases surface roughness, wettability and microhardness when compared to the untreated sample. In the biomechanical fatigue test, the treated implant showed no flaws, after five million cycles, at a maximum fatigue load of 84.46 N. The results of the topological optimization process showed well-defined optimized layouts of the dental implant, with a clear distribution of material and a defined edge

Otimização topológica de estruturas termoelásticas tridimensionais

This work presents an optimization technique based on structural topology optimization methods, TOM, designed to solve problems of thermoelasticity 3D. The presented approach is based on the adjoint method of sensitivity analysis unified design and is intended to loosely coupled thermomechanical problems. The technique makes use of analytical expressions of sensitivities, enabling a reduction in the computational cost through the use of a coupled field adjoint equation, defined in terms the of temperature and displacement fields. The TOM used is based on the material aproach. Thus, to make the domain is composed of a continuous distribution of material, enabling the use of classical models in nonlinear programming optimization problem, the microstructure is considered as a porous medium and its constitutive equation is a function only of the homogenized relative density of the material. In this approach, the actual properties of materials with intermediate densities are penalized based on an artificial microstructure model based on the SIMP (Solid Isotropic Material with Penalty). To circumvent problems chessboard and reduce dependence on layout in relation to the final optimal initial mesh, caused by problems of numerical instability, restrictions on components of the gradient of relative densities were applied. The optimization problem is solved by applying the augmented Lagrangian method, the solution being obtained by applying the finite element method of Galerkin, the process of approximation using the finite element Tetra4. This element has the ability to interpolate both the relative density and the displacement components and temperature. As for the definition of the problem, the heat load is assumed in steady state, i.e., the effects of conduction and convection of heat does not vary with time. The mechanical load is assumed static and distributed

Método de Otimização Topológica em Estruturas Tridimensionais

The topology optimization problem characterize and determine the optimum distribution of material into the domain. In other words, after the definition of the boundary conditions in a pre-established domain, the problem is how to distribute the material to solve the minimization problem. The objective of this work is to propose a competitive formulation for optimum structural topologies determination in 3D problems and able to provide high-resolution layouts. The procedure combines the Galerkin Finite Elements Method with the optimization method, looking for the best material distribution along the fixed domain of project. The layout topology optimization method is based on the material approach, proposed by Bendsoe & Kikuchi (1988), and considers a homogenized constitutive equation that depends only on the relative density of the material. The finite element used for the approach is a four nodes tetrahedron with a selective integration scheme, which interpolate not only the components of the displacement field but also the relative density field. The proposed procedure consists in the solution of a sequence of layout optimization problems applied to compliance minimization problems and mass minimization problems under local stress constraint. The microstructure used in this procedure was the SIMP (Solid Isotropic Material with Penalty). The approach reduces considerably the computational cost, showing to be efficient and robust. The results provided a well defined structural layout, with a sharpness distribution of the material and a boundary condition definition. The layout quality was proporcional to the medium size of the element and a considerable reduction of the project variables was observed due to the tetrahedrycal element

The strut-and-tie models in reinforced concrete structures analysed by a numerical technique

Os modelos de bielas e tirantes são procedimentos de análise apropriados para projetar elementos de concreto armado em casos de regiões onde há alterações geométricas ou concentrações de tensões, denominadas regiões D. Trata-se de bons modelos de representação da estrutura para avaliar melhor o seu comportamento estrutural e seu mecanismo resistente. O presente artigo aplica a técnica da otimização topológica para identificar o fluxo de tensões nas estruturas, definindo a configuração dos membros de bielas e tirantes, e quantifica seus valores para dimensionamento. Utilizam-se o método ESO, e uma variante desse, o SESO (Smoothing ESO) com o método dos elementos finitos em elasticidade plana. A filosofia do SESO baseia-se na observação de que se o elemento não for necessário à estrutura, sua contribuição de rigidez vai diminuindo progressivamente. Isto é, sua remoção é atenuada nos valores da matriz constitutiva, como se este estivesse em processo de danificação. Para validar a presente formulação, apresentam-se alguns exemplos numéricos onde se comparam suas respostas com as advindas de trabalhos científicos pioneiros sobre o assunto.

A simple interferometric method to measure the calibration factor and displacement amplification in piezoelectric flextensional actuators

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)