Search results for: structural topology optimization

Authors: Hyun Kyu Cho, Jun Soo Kim, Young Hoon Lee, Sang Hoon Kang, Young Chul Park

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In the study, structure for one of offshore drilling system APV(Air Pressure Vessle) modules was designed by using topology optimum design and performed structural safety evaluation according to DNV rules. 3D model created base on design area and non-design area separated by using topology optimization for the environmental loads. This model separated 17 types for wind loads and dynamic loads and performed structural analysis evaluation for each model. As a result, the maximum stress occurred 181.25MPa.

Keywords: APV, topology optimum design, DNV, structural analysis, stress

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Authors: Dongkyu Lee, Thanh Banh Thien, Soomi Shin

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In the present study, the isogeometric topology optimization is proposed for cracked structures through using Solid Isotropic Material with Penalization (SIMP) as a design model. Design density variables defined in the variable space are used to approximate the element analysis density by the bivariate B-spline basis functions. The mathematical formulation of topology optimization problem solving minimum structural compliance is an alternating active-phase algorithm with the Gauss-Seidel version as an optimization model of optimality criteria. Stiffness and adjoint sensitivity formulations linked to strain energy of cracked structure are proposed in terms of design density variables. Numerical examples demonstrate interactions of topology optimization to structures design with cracks.

Keywords: topology optimization, isogeometric, NURBS, design

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Authors: D. K. Lee, S. M. Shin, J. H. Lee

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Topology optimization technique utilizes constant element densities as design parameters. Finally, optimal distribution contours of the material densities between voids (0) and solids (1) in design domain represent the determination of topology. It means that regions with element density values become occupied by solids in design domain, while there are only void phases in regions where no density values exist. Therefore the void regions of topology optimization results provide design information to decide appropriate depositions of web-opening in structure. Contrary to the basic objective of the topology optimization technique which is to obtain optimal topology of structures, this present study proposes a new idea that topology optimization results can be also utilized for decision of proper web-opening’s position. Numerical examples of linear elastostatic structures demonstrate efficiency of methodological design processes using topology optimization in order to determinate the proper deposition of web-openings.

Keywords: topology optimization, web-opening, structure, element density, material

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Authors: Mengxiao Li, Johnson Zhang

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Currently, topology optimization technique is widely used to define the layout design of structures that are presented as truss-like topologies. However, due to the difficulty in combining optimization technique with more realistic material models where their nonlinear properties should be considered, the achieved optimized topologies are commonly unable to apply straight towards the practical design problems. This study presented an optimization procedure of composite structures where different elastic stiffness, yield criteria, and hardening models are assumed for the candidate materials. From the results, it can be concluded that a more explicit modeling has the significant influence on the resulting topologies. Also, the isotropic or kinematic hardening is important for elastoplastic structural optimization design. The capability of the proposed optimization procedure is shown through several cases.

Keywords: topology optimization, material composition, nonlinear modeling, hardening rules

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Authors: Mehdi Jalalpour, Mazdak Tootkaboni

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We present a computationally efficient method for reliability-based topology optimization under material properties uncertainty, which is assumed to be lognormally distributed and correlated within the domain. Computational efficiency is achieved through estimating the response statistics with stochastic perturbation of second order, using these statistics to fit an appropriate distribution that follows the empirical distribution of the response, and employing an efficient gradient-based optimizer. The proposed algorithm is utilized for design of new structures and the changes in the optimized topology is discussed for various levels of target reliability and correlation strength. Predictions were verified thorough comparison with results obtained using Monte Carlo simulation.

Keywords: material uncertainty, stochastic perturbation, structural reliability, topology optimization

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Authors: Arya Prakash Padhi, Anupam Chakrabarti, Rajib Chowdhury

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Topology or layout optimization often gives better performing economic structures and is very helpful in the conceptual design phase. But traditionally it is being done in finite element-based optimization schemes which, although gives a good result, is very time-consuming especially in 3D structures. Among other alternatives machine learning, especially deep learning-based methods, have a very good potential in resolving this computational issue. Here convolutional neural network (3D-CNN) based variational auto encoder (VAE) is trained using a dataset generated from commercially available topology optimization code ABAQUS Tosca using solid isotropic material with penalization (SIMP) method for compliance minimization. The encoded data in latent space is then fed to a 3D generative adversarial network (3D-GAN) to generate the outcome in 64x64x64 size. Here the network consists of 3D volumetric CNN with rectified linear unit (ReLU) activation in between and sigmoid activation in the end. The proposed network is seen to provide almost optimal results with significantly reduced computational time, as there is no iteration involved.

Keywords: 3D generative adversarial network, deep learning, structural topology optimization, variational auto encoder

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Authors: Wu Qinglong, Zhou Qicai, Xiong Xiaolei, Zhang Richeng

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In order to achieve the layout and size optimization of the web members of tower crane boom, a truss topology and cross section size optimization method based on continuum is proposed considering three typical working conditions. Firstly, the optimization model is established by replacing web members with web plates. And the web plates are divided into several sub-domains so that periodic soft kill option (SKO) method can be carried out for topology optimization of the slender boom. After getting the optimized topology of web plates, the optimized layout of web members is formed through extracting the principal stress distribution. Finally, using the web member radius as design variable, the boom compliance as objective and the material volume of the boom as constraint, the cross section size optimization mathematical model is established. The size optimization criterion is deduced from the mathematical model by Lagrange multiplier method and Kuhn-Tucker condition. By comparing the original boom with the optimal boom, it is identified that this optimization method can effectively lighten the boom and improve its performance.

Keywords: tower crane boom, topology optimization, size optimization, periodic, SKO, optimization criterion

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Authors: Filippo Ranalli, Forest Flager, Martin Fischer

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This paper presents a ground structure method to optimize the topology and discrete member sizing of steel frame structures in order to minimize total installed cost, including material, fabrication and erection components. The proposed method improves upon existing cost-based ground structure methods by incorporating constructability considerations well as satisfying both strength and serviceability constraints. The architecture for the method is a bi-level Multidisciplinary Feasible (MDF) architecture in which the discrete member sizing optimization is nested within the topology optimization process. For each structural topology generated, the sizing optimization process seek to find a set of discrete member sizes that result in the lowest total installed cost while satisfying strength (member utilization) and serviceability (node deflection and story drift) criteria. To accurately assess cost, the connection details for the structure are generated automatically using accurate site-specific cost information obtained directly from fabricators and erectors. Member continuity rules are also applied to each node in the structure to improve constructability. The proposed optimization method is benchmarked against conventional weight-based ground structure optimization methods resulting in an average cost savings of up to 30% with comparable computational efficiency.

Keywords: cost-based structural optimization, cost-based topology and sizing, optimization, steel frame ground structure optimization, multidisciplinary optimization of steel structures

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Authors: Khodamorad Nabaki, Jianhu Shen, Xiaodong Huang

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This paper develops a method for considering the critical fatigue stress as a constraint in the Bi-directional Evolutionary Structural Optimization (BESO) method. Our aim is to reach an optimal design in which high cycle fatigue failure does not occur for a specific life time. The critical fatigue stress is calculated based on modified Goodman criteria and used as a stress constraint in our topology optimization problem. Since fatigue generally does not occur for compressive stresses, we use the p-norm approach of the stress measurement that considers the highest tensile principal stress in each point as stress measure to calculate the sensitivity numbers. The BESO method has been extended to minimize volume an object subjected to the critical fatigue stress constraint. The optimization results are compared with the results from the compliance minimization problem which shows clearly the merits of our newly developed approach.

Keywords: topology optimization, BESO method, p-norm, fatigue constraint

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Authors: Ersilio Tushaj

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The efficient solution that satisfies the optimal condition is an important issue in the structural engineering design problem. The new codes of structural design consist in design methodology that looks after the exploitation of the total resources of the construction material. In recent years some non-deterministic or meta-heuristic structural optimization algorithms have been developed widely in the research community. These methods search the optimum condition starting from the simulation of a natural phenomenon, such as survival of the fittest, the immune system, swarm intelligence or the cooling process of molten metal through annealing. Among these techniques the most known are: the genetic algorithms, simulated annealing, evolution strategies, particle swarm optimization, tabu search, ant colony optimization, harmony search and big bang crunch optimization. In this study, five of these algorithms are applied for the optimum weight design of a steel truss structure with variable geometry but fixed topology. The design process selects optimum distances and size sections from a set of commercial steel profiles. In the formulation of the design problem are considered deflection limitations, buckling and allowable stress constraints. The approach is repeated starting from different initial populations. The design problem topology is taken from an existing steel structure. The optimization process helps the engineer to achieve good final solutions, avoiding the repetitive evaluation of alternative designs in a time consuming process. The algorithms used for the application, the results of the optimal solutions, the number of iterations and the minimal weight designs, will be reported in the paper. Based on these results, it would be estimated, the amount of the steel that could be saved by applying structural analysis combined with non-deterministic optimization methods.

Keywords: structural optimization, non-deterministic methods, truss structures, steel truss

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Authors: Abolfazl Hosseinpour, Javad Marzbanrad

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In this research, the topology and shape optimization of a Macpherson control arm has been accomplished to achieve lighter weight. Present automotive market demands low cost and light weight component to meet the need of fuel efficient and cost effective vehicle. This in turn gives the rise to more effective use of materials for automotive parts which can reduce the mass of vehicle. Since automotive components are under dynamic loads which cause fatigue damage, considering fatigue criteria seems to be essential in designing automotive components. At first, in order to create severe loading condition for control arm, some rough roads are generated through power spectral density. Then, the most critical loading conditions are obtained through multibody dynamics analysis of a full vehicle model. Then, the topology optimization is performed based on fatigue life criterion using HyperMesh software, which resulted to 50 percent mass reduction. In the next step a CAD model is created using CATIA software and shape optimization is performed to achieve accurate dimensions with less mass.

Keywords: topology optimization, shape optimization, fatigue life, MacPherson control arm

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Authors: Mahdi Mottahedi, Peter Zahn, Armin Lechler, Alexander Verl

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Optimal topology of components leads to the maximum stiffness with the minimum material use. For the generation of these topologies, normally algorithms are employed, which tackle manufacturing limitations, at the cost of the optimal result. The global optimum result with penalty factor one, however, cannot be fabricated with conventional methods. In this article, an additive manufacturing method is introduced, in order to enable the production of global topology optimization results. For a benchmark, topology optimization with higher and lower penalty factors are performed. Different algorithms are employed in order to interpret the results of topology optimization with lower factors in many microstructure layers. These layers are then joined to form the final geometry. The algorithms’ benefits are then compared experimentally and numerically for the best interpretation. The findings demonstrate that by implementation of the selected algorithm, the stiffness of the components produced with this method is higher than what could have been produced by conventional techniques.

Keywords: topology optimization, additive manufacturing, 3D-printer, laminated object manufacturing

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Authors: S. Wakim, M. Nemer, B. Zeghondy, B. Ghannam, C. Bouallou

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Designing the optimal heat exchanger is still an essential objective to be achieved. Parametrical optimization involves the evaluation of the heat exchanger dimensions to find those that best satisfy certain objectives. This method contributes to an enhanced design rather than an optimized one. On the contrary, topology optimization finds the optimal structure that satisfies the design objectives. The huge development in metal additive manufacturing allowed topology optimization to find its way into engineering applications especially in the aerospace field to optimize metal structures. Using topology optimization in 3d heat and mass transfer problems requires huge computational time, therefore coupling it with CFD simulations can reduce this it. However, existed CFD models cannot be coupled with topology optimization. The CFD model must allow creating a uniform mesh despite the initial geometry complexity and also to swap the cells from fluid to solid and vice versa. In this paper, a porous media approach compatible with topology optimization criteria is developed. It consists of modeling the fluid region of the heat exchanger as porous media having high porosity and similarly the solid region is modeled as porous media having low porosity. The switching from fluid to solid cells required by topology optimization is simply done by changing each cell porosity using a user defined function. This model is tested on a plate and fin heat exchanger and validated by comparing its results to experimental data and simulations results. Furthermore, this model is used to perform a material reallocation based on local criteria to optimize a plate and fin heat exchanger under a constant heat duty constraint. The optimized fin uses 20% fewer materials than the first while the pressure drop is reduced by about 13%.

Keywords: computational methods, finite element method, heat exchanger, porous media, topology optimization

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Authors: Hanjong Kim, Changwan Han, Seonghun Park

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Heat exchanger manifolds in aircraft play an important role in evenly distributing the fluid entering through the inlet to the heat transfer unit. In order to achieve this requirement, the manifold should be designed to have a light weight by withstanding high internal pressure. Therefore, this study aims at minimizing the weight of the heat exchanger manifold through topology optimization. For topology optimization, the initial design space was created with the inner surface extracted from the currently used manifold model and with the outer surface having a dimension of 243.42 mm of X 74.09 mm X 65 mm. This design space solid model was transformed into a finite element model with a maximum tetrahedron mesh size of 2 mm using ANSYS Workbench. Then, topology optimization was performed under the boundary conditions of an internal pressure of 5.5 MPa and the fixed support for rectangular inlet boundaries by SIMULIA TOSCA. This topology optimization produced the minimized finial volume of the manifold (i.e., 7.3% of the initial volume) based on the given constraints (i.e., 6% of the initial volume) and the objective function (i.e., maximizing manifold stiffness). Weight of the optimized model was 6.7% lighter than the currently used manifold, but after smoothing the topology optimized model, this difference would be bigger. The current optimized model has uneven thickness and skeleton-shaped outer surface to reduce stress concentration. We are currently simplifying the optimized model shape with spline interpolations by reflecting the design characteristics in thickness and skeletal structures from the optimized model. This simplified model will be validated again by calculating both stress distributions and weight reduction and then the validated model will be manufactured using 3D printing processes.

Keywords: topology optimization, manifold, heat exchanger, 3D printing

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Authors: Stojan Kravanja, Andrej Ivanič, Tomaž Žula

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This contribution focuses on structural optimization in civil engineering using mixed integer non-linear programming (MINLP). MINLP is characterized as a versatile method that can handle both continuous and discrete optimization variables simultaneously. Continuous variables are used to optimize parameters such as dimensions, stresses, masses, or costs, while discrete variables represent binary decisions to determine the presence or absence of structural elements within a structure while also calculating discrete materials and standard sections. The optimization process is divided into three main steps. First, a mechanical superstructure with a variety of different topology-, material- and dimensional alternatives. Next, a MINLP model is formulated to encapsulate the optimization problem. Finally, an optimal solution is searched in the direction of the defined objective function while respecting the structural constraints. The economic or mass objective function of the material and labor costs of a structure is subjected to the constraints known from structural analysis. These constraints include equations for the calculation of internal forces and deflections, as well as equations for the dimensioning of structural components (in accordance with the Eurocode standards). Given the complex, non-convex and highly non-linear nature of optimization problems in civil engineering, the Modified Outer-Approximation/Equality-Relaxation (OA/ER) algorithm is applied. This algorithm alternately solves subproblems of non-linear programming (NLP) and main problems of mixed-integer linear programming (MILP), in this way gradually refines the solution space up to the optimal solution. The NLP corresponds to the continuous optimization of parameters (with fixed topology, discrete materials and standard dimensions, all determined in the previous MILP), while the MILP involves a global approximation to the superstructure of alternatives, where a new topology, materials, standard dimensions are determined. The optimization of a convex problem is stopped when the MILP solution becomes better than the best NLP solution. Otherwise, it is terminated when the NLP solution can no longer be improved. While the OA/ER algorithm, like all other algorithms, does not guarantee global optimality due to the presence of non-convex functions, various modifications, including convexity tests, are implemented in OA/ER to mitigate these difficulties. The effectiveness of the proposed MINLP approach is demonstrated by its application to various structural optimization tasks, such as mass optimization of steel buildings, cost optimization of timber halls, composite floor systems, etc. Special optimization models have been developed for the optimization of these structures. The MINLP optimizations, facilitated by the user-friendly software package MIPSYN, provide insights into a mass or cost-optimal solutions, optimal structural topologies, optimal material and standard cross-section choices, confirming MINLP as a valuable method for the optimization of structures in civil engineering.

Keywords: MINLP, mixed-integer non-linear programming, optimization, structures

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Authors: Zuyong Chen, Jianghao Wu, Yanlai Zhang

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Flapping-wing micro aerial vehicle (FMAV) is a new type of aircraft by mimicking the flying behavior to that of small birds or insects. Comparing to the traditional fixed wing or rotor-type aircraft, FMAV only needs to control the motion of flapping wings, by changing the size and direction of lift to control the flight attitude. Therefore, its transmission system should be designed very compact. Lightweight design can effectively extend its endurance time, while engineering experience alone is difficult to simultaneously meet the requirements of FMAV for structural strength and quality. Current researches still lack the guidance of considering nonlinear factors of 3D printing material when carrying out topology optimization, especially for the tiny FMAV transmission system. The coupling of non-linear material properties and non-linear contact behaviors of FMAV transmission system is a great challenge to the reliability of the topology optimization result. In this paper, topology optimization design based on FEA solver package Altair Optistruct for the transmission system of FMAV manufactured by 3D Printing was carried out. Firstly, the isotropic constitutive behavior of the Ultraviolet (UV) Cureable Resin used to fabricate the structure of FMAV was evaluated and confirmed through tensile test. Secondly, a numerical computation model describing the mechanical behavior of FMAV transmission structure was established and verified by experiments. Then topology optimization modeling method considering non-linear factors were presented, and optimization results were verified by dynamic simulation and experiments. Finally, detail discussions of different load status and constraints were carried out to explore the leading factors affecting the optimization results. The contributions drawn from this article helpful for guiding the lightweight design of FMAV are summarizing as follow; first, a dynamic simulation modeling method used to obtain the load status is presented. Second, verification method of optimized results considering non-linear factors is introduced. Third, based on or can achieve a better weight reduction effect and improve the computational efficiency rather than taking multi-states into account. Fourth, basing on makes for improving the ability to resist bending deformation. Fifth, constraint of displacement helps to improve the structural stiffness of optimized result. Results and engineering guidance in this paper may shed lights on the structural optimization and light-weight design for future advanced FMAV.

Keywords: flapping-wing micro aerial vehicle, 3d printing, topology optimization, finite element analysis, experiment

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Authors: M. Y. Bakeir

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Rough set theory is a recent approach for reasoning about data. It has achieved a large amount of applications in various real-life fields. The main idea of rough sets corresponds to the lower and upper set approximations. These two approximations are exactly the interior and the closure of the set with respect to a certain topology on a collection U of imprecise data acquired from any real-life field. The base of the topology is formed by equivalence classes of an equivalence relation E defined on U using the available information about data. The theory of generalized topology was studied by Cs´asz´ar. It is well known that generalized topology in the sense of Cs´asz´ar is a generalization of the topology on a set. On the other hand, many important collections of sets related with the topology on a set form a generalized topology. The notion of Nano topology was introduced by Lellis Thivagar, which was defined in terms of approximations and boundary region of a subset of an universe using an equivalence relation on it. The purpose of this paper is to introduce a new generalized topology in terms of rough set called nano generalized topology

Keywords: rough sets, topological space, generalized topology, nano topology

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Authors: Rony Tawk, Boutros Ghannam, Maroun Nemer

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Topology optimization technique presents a potential tool for the design and optimization of structures involved in mass and heat transfer. The method starts with an initial intermediate domain and should be able to progressively distribute the solid and the two fluids exchanging heat. The multi-objective function of the problem takes into account minimization of total pressure loss and maximization of heat transfer between solid and fluid subdomains. Existing methods account for the presence of only one fluid, while the actual work extends optimization distribution of solid and two different fluids. This requires to separate the channels of both fluids and to ensure a minimum solid thickness between them. This is done by adding a third objective function to the multi-objective optimization problem. This article uses density approach where each cell holds two local design parameters ranging from 0 to 1, where the combination of their extremums defines the presence of solid, cold fluid or hot fluid in this cell. Finite volume method is used for direct solver coupled with a discrete adjoint approach for sensitivity analysis and method of moving asymptotes for numerical optimization. Several examples are presented to show the ability of the method to find a trade-off between minimization of power dissipation and maximization of heat transfer while ensuring the separation and continuity of the channel of each fluid without crossing or mixing the fluids. The main conclusion is the possibility to find an optimal bi-fluid domain using topology optimization, defining a fluid to fluid heat exchanger device.

Keywords: topology optimization, density approach, bi-fluid domain, laminar steady state regime, fluid-to-fluid heat exchanger

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Authors: Omer Oral, Y. Emre Yilmaz

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Topology optimization is an approach that optimizes material distribution within a given design space for a certain load and boundary conditions by providing performance goals. It uses various restrictions such as boundary conditions, set of loads, and constraints to maximize the performance of the system. It is different than size and shape optimization methods, but it reserves some features of both methods. In this study, interior structures of the parts were optimized by using SIMP (Solid Isotropic Material with Penalization) method. The volume of the part was preassigned parameter and minimum deflection was the objective function. The basic idea behind the theory was considered, and different methods were discussed. Rhinoceros 3D design tool was used with Grasshopper and TopOpt plugins to create and optimize parts. A Grasshopper algorithm was designed and tested for different beams, set of arbitrary located forces and support types such as pinned, fixed, etc. Finally, 2.5D shapes were obtained and verified by observing the changes in density function.

Keywords: Grasshopper, lattice structure, microstructures, Rhinoceros, solid isotropic material with penalization method, TopOpt, topology optimization

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Authors: Malcolm Dinovitzer, Calvin Miller, Adam Hacker, Gabriel Wong, Zach Annen, Padmassun Rajakareyar, Jordan Mulvihill, Mostafa S.A. ElSayed

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The research work presented in this paper is developed by the Blended Wing Body (BWB) Unmanned Aerial Vehicle (UAV) team, a fourth-year capstone project at Carleton University Department of Mechanical and Aerospace Engineering. Here, a clean sheet UAV with BWB configuration is designed and optimized using Multiscale Design Optimization (MSDO) approach employing lattice materials taking into consideration design for additive manufacturing constraints. The BWB-UAV is being developed with a mission profile designed for surveillance purposes with a minimum payload of 1000 grams. To demonstrate the design methodology, a single design loop of a sample rib from the airframe is shown in details. This includes presentation of the conceptual design, materials selection, experimental characterization and residual thermal stress distribution analysis of additively manufactured materials, manufacturing constraint identification, critical loads computations, stress analysis and design optimization. A dynamic turbulent critical load case was identified composed of a 1-g static maneuver with an incremental Power Spectral Density (PSD) gust which was used as a deterministic design load case for the design optimization. 2D flat plate Doublet Lattice Method (DLM) was used to simulate aerodynamics in the aeroelastic analysis. The aerodynamic results were verified versus a 3D CFD analysis applying Spalart-Allmaras and SST k-omega turbulence to the rigid UAV and vortex lattice method applied in the OpenVSP environment. Design optimization of a single rib was conducted using topology optimization as well as MSDO. Compared to a solid rib, weight savings of 36.44% and 59.65% were obtained for the topology optimization and the MSDO, respectively. These results suggest that MSDO is an acceptable alternative to topology optimization in weight critical applications while preserving the functional requirements.

Keywords: blended wing body, multiscale design optimization, additive manufacturing, unmanned aerial vehicle

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Authors: M. A. Islam, P. J. Hazell, J. P. Escobedo, M. Saadatfar

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Quasistatic compression and micro structural characterization of closed cell aluminium foams of different pore size and cell distributions has been carried out. Metallic foams have good potential for lightweight structures for impact and blast mitigation and therefore it is important to find out the optimized foam structure (i.e. cell size, shape, relative density, and distribution) to maximize energy absorption. In this paper, we present results for two different aluminium metal foams of density 0.5 g/cc and 0.7 g/cc respectively that have been tested in quasi-static compression. The influence of cell geometry and cell topology on quasistatic compression behavior has been investigated using computed tomography (micro-CT) analysis. The compression behavior and micro structural characterization will be presented.

Keywords: metal foams, micro-CT, cell topology, quasistatic compression

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Authors: Angga S. Fajar, A. Aminullah, J. Kiyono, R. A. Safitri

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This paper discusses about the application of ANN for optimizing of bridge structure design. ANN has been applied in various field of science concerning prediction and optimization. The structural optimization has several benefit including accelerate structural design process, saving the structural material, and minimize self-weight and mass of structure. In this paper, there are three types of bridge structure that being optimized including PSC I-girder superstructure, composite steel-concrete girder superstructure, and RC bridge pier. The different optimization strategy on each bridge structure implement back propagation method of ANN is conducted in this research. The optimal weight and easier design process of bridge structure with satisfied error are achieved.

Keywords: bridge structures, ANN, optimization, back propagation

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Authors: Darwin Castillo Huamaní, Francisco A. M. Gomes

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The aim of the multi-material topology optimization (MTO) is to obtain the optimal topology of structures composed by many materials, according to a given set of constraints and cost criteria. In this work, we seek the optimal distribution of materials in a domain, such that the flexibility of the structure is minimized, under certain boundary conditions and the intervention of external forces. In the case we have only one material, each point of the discretized domain is represented by two values from a function, where the value of the function is 1 if the element belongs to the structure or 0 if the element is empty. A common way to avoid the high computational cost of solving integer variable optimization problems is to adopt the Solid Isotropic Material with Penalization (SIMP) method. This method relies on the continuous interpolation function, power function, where the base variable represents a pseudo density at each point of domain. For proper exponent values, the SIMP method reduces intermediate densities, since values other than 0 or 1 usually does not have a physical meaning for the problem. Several extension of the SIMP method were proposed for the multi-material case. The one that we explore here is the ordered SIMP method, that has the advantage of not being based on the addition of variables to represent material selection, so the computational cost is independent of the number of materials considered. Although the number of variables is not increased by this algorithm, the optimization subproblems that are generated at each iteration cannot be solved by methods that rely on second derivatives, due to the cost of calculating the second derivatives. To overcome this, we apply a globally convergent version of the sequential linear programming method, which solves a linear approximation sequence of optimization problems.

Keywords: globally convergence, multi-material design ordered simp, sequential linear programming, topology optimization

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Authors: Elham Kiyani

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In this paper, we first introduce the concept of Q-efficient solutions in a real linear space not necessarily endowed with a topology, where Q is some nonempty (not necessarily convex) set. We also used the scalarization technique including the Gerstewitz function generated by a nonconvex set to characterize these Q-efficient solutions. The algebraic concepts of interior and closure are useful to study optimization problems without topology. Studying nonconvex vector optimization is valuable since topological interior is equal to algebraic interior for a convex cone. So, we use the algebraic concepts of interior and closure to define Q-weak efficient solutions and Q-Henig proper efficient solutions of set-valued optimization problems, where Q is not a convex cone. Optimization problems with set-valued maps have a wide range of applications, so it is expected that there will be a useful analytical tool in optimization theory for set-valued maps. These kind of optimization problems are closely related to stochastic programming, control theory, and economic theory. The paper focus on nonconvex problems, the results are obtained by assuming generalized non-convexity assumptions on the data of the problem. In convex problems, main mathematical tools are convex separation theorems, alternative theorems, and algebraic counterparts of some usual topological concepts, while in nonconvex problems, we need a nonconvex separation function. Thus, we consider the Gerstewitz function generated by a general set in a real linear space and re-examine its properties in the more general setting. A useful approach for solving a vector problem is to reduce it to a scalar problem. In general, scalarization means the replacement of a vector optimization problem by a suitable scalar problem which tends to be an optimization problem with a real valued objective function. The Gerstewitz function is well known and widely used in optimization as the basis of the scalarization. The essential properties of the Gerstewitz function, which are well known in the topological framework, are studied by using algebraic counterparts rather than the topological concepts of interior and closure. Therefore, properties of the Gerstewitz function, when it takes values just in a real linear space are studied, and we use it to characterize Q-efficient solutions of vector problems whose image space is not endowed with any particular topology. Therefore, we deal with a constrained vector optimization problem in a real linear space without assuming any topology, and also Q-weak efficient and Q-proper efficient solutions in the senses of Henig are defined. Moreover, by means of the Gerstewitz function, we provide some necessary and sufficient optimality conditions for set-valued vector optimization problems.

Keywords: algebraic interior, Gerstewitz function, vector closure, vector optimization

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Authors: Bita Mohajernia, R. J. Urbanic

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Mold making techniques have focused on meeting the customers’ functional and process requirements; however, today, molds are increasing in size and sophistication, and are difficult to manufacture, transport, and set up due to their size and mass. Presently, mold weight saving techniques focus on pockets to reduce the mass of the mold, but the overall size is still large, which introduces costs related to the stock material purchase, processing time for process planning, machining and validation, and excess waste materials. Reducing the overall size of the mold is desirable for many reasons, but the functional requirements, tool life, and durability cannot be compromised in the process. It is proposed to use Finite Element Analysis simulation tools to model the forces, and pressures to determine where the material can be removed. The potential results of this project will reduce manufacturing costs. In this study, a light weight structure is defined by an optimal distribution of material to carry external loads. The optimization objective of this research is to determine methods to provide the optimum layout for the mold structure. The topology optimization method is utilized to improve structural stiffness while decreasing the weight using the OptiStruct software. The optimized CAD model is compared with the primary geometry of the mold from the NX software. Results of optimization show an 8% weight reduction while the actual performance of the optimized structure, validated by physical testing, is similar to the original structure.

Keywords: finite element analysis, plastic injection molding, topology optimization, weight reduction

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Authors: Elham Kiyani, S. Mansour Vaezpour, Javad Tavakoli

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In this paper, we first introduce a new distance function in a linear space not necessarily endowed with a topology. The algebraic concepts of interior and closure are useful to study optimization problems without topology. So, we define Q-weak efficient solutions generated by the algebraic interior of a set Q, where Q is not necessarily convex. Studying nonconvex vector optimization is valuable since, for a convex cone K in topological spaces, we have int(K)=cor(K), which means that topological interior of a convex cone K is equal to the algebraic interior of K. Moreover, we used the scalarization technique including the distance function generated by the vectorial closure of a set to characterize these Q-weak efficient solutions. Scalarization is a useful approach for solving vector optimization problems. This technique reduces the optimization problem to a scalar problem which tends to be an optimization problem with a real-valued objective function. For instance, Q-weak efficient solutions of vector optimization problems can be characterized and computed as solutions of appropriate scalar optimization problems. In the convex case, linear functionals can be used as objective functionals of the scalar problems. But in the nonconvex case, we should present a suitable objective function. It is the aim of this paper to present a new distance function that be useful to obtain sufficient and necessary conditions for Q-weak efficient solutions of general optimization problems via scalarization.

Keywords: weak efficient, algebraic interior, vector closure, linear space

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Authors: Abdulsalam A. Al-Tamimi, Chris Peach, Paulo Rui Fernandes, Paulo J. Bartolo

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Bone fixation implants currently used to treat traumatic fractured bones and to promote fracture healing are built with biocompatible metallic materials such as stainless steel, cobalt chromium and titanium and its alloys (e.g., CoCrMo and Ti6Al4V). The noticeable stiffness mismatch between current metallic implants and host bone associates with negative outcomes such as stress shielding which causes bone loss and implant loosening leading to deficient fracture treatment. This paper, part of a major research program to design the next generation of bone fixation implants, describes the combined use of three-dimensional (3D) topology optimization (TO) and additive manufacturing powder bed technology (Electron Beam Melting) to redesign and fabricate the plates based on the current standard one (i.e., locking compression plate). Topology optimization is applied with an objective function to maximize the stiffness and constraint by volume reductions (i.e., 25-75%) in order to obtain optimized implant designs with reduced stress shielding phenomenon, under different boundary conditions (i.e., tension, bending, torsion and combined loads). The stiffness of the original and optimised plates are assessed through a finite-element study. The TO results showed actual reduction in the stiffness for most of the plates due to the critical values of volume reduction. Additionally, the optimized plates fabricated using powder bed techniques proved that the integration between the TO and additive manufacturing presents the capability of producing stiff reduced plates with acceptable tolerances.

Keywords: additive manufacturing, locking compression plate, finite element, topology optimization

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Authors: Sara A. Pakvis, Giulia Scalet, Stefania Marconi, Ferdinando Auricchio, Matthijs Langelaar

Abstract:

In recent developments in the field of multi-material additive manufacturing, differences in material properties are exploited to create printed shape-memory structures, which are referred to as 4D-printed structures. New printing techniques allow for the deliberate introduction of prestresses in the specimen during manufacturing, and, in combination with the right design, this enables new functionalities. This research focuses on bi-polymer 4D-printed structures, where the transformation process is based on a heat-induced glass transition in one material lowering its Young’s modulus, combined with an initial prestress in the other material. Upon the decrease in stiffness, the prestress is released, which results in the realization of an essentially pre-programmed deformation. As the design of such functional multi-material structures is crucial but far from trivial, a systematic methodology to find the design of 4D-printed structures is developed, where a finite element model is combined with a density-based topology optimization method to describe the material layout. This modeling approach is verified by a convergence analysis and validated by comparing its numerical results to analytical and published data. Specific aspects that are addressed include the interplay between the definition of the prestress and the material interpolation function used in the density-based topology description, the inclusion of a temperature-dependent stiffness relationship to simulate the glass transition effect, and the importance of the consideration of geometric nonlinearity in the finite element modeling. The efficacy of topology optimization to design 4D-printed structures is explored by applying the methodology to a variety of design problems, both in 2D and 3D settings. Bi-layer designs composed of thermoplastic polymers are printed by means of the fused deposition modeling (FDM) technology. Acrylonitrile butadiene styrene (ABS) polymer undergoes the glass transition transformation, while polyurethane (TPU) polymer is prestressed by means of the 3D-printing process itself. Tests inducing shape transformation in the printed samples through heating are performed to calibrate the prestress and validate the modeling approach by comparing the numerical results to the experimental findings. Using the experimentally obtained prestress values, more complex designs have been generated through topology optimization, and samples have been printed and tested to evaluate their performance. This study demonstrates that by combining topology optimization and 4D-printing concepts, stimuli-responsive structures with specific properties can be designed and realized.

Keywords: 4D-printing, glass transition, shape memory polymer, topology optimization

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Authors: H. Nikzad, S. Yoshitomi

Abstract:

In this paper, an optimization procedure is applied for 3D Reinforced concrete building structure with shear wall.  In the optimization problem, cross sections of beams, columns and shear wall dimensions are considered as design variables and the optimal cross sections can be derived to minimize the total cost of the structure. As for final design application, the most suitable sections are selected to satisfy ACI 318-14 code provision based on static linear analysis. The validity of the method is examined through numerical example of 15 storied 3D RC building with shear wall.  This optimization method is expected to assist in providing a useful reference in design early stage, and to be an effective and powerful tool for structural design of RC shear wall structures.

Keywords: structural optimization, linear static analysis, ETABS, MATLAB, RC moment frame, RC shear wall structures

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Authors: Shai Sarussi

Abstract:

Let S be an integral domain which is not a field, let F be its field of fractions, and let A be an F-algebra. An S-subalgebra R of A is called S-nice if R ∩ F = S and F R = A. A topological space whose set of open sets is closed under arbitrary intersections is called an Alexandroff space. Inspired by the well-known Zariski-Riemann space and the Zariski topology on the set of prime ideals of a commutative ring, we define a topology on the set of all S-nice subalgebras of A. Consequently, we get an interplay between Algebra and topology, that gives us a better understanding of the S-nice subalgebras of A. It is shown that every irreducible subset of S-nice subalgebras of A has a supremum; and a characterization of the irreducible components is given, in terms of maximal S-nice subalgebras of A.

Keywords: Alexandroff topology, integral domains, Zariski-Riemann space, S-nice subalgebras

Procedia PDF Downloads 74