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This paper proposes an effective algorithm based on the level set method (LSM) to solve shape and topology optimization problems. Since the conventional LSM has several limitations, a binary level set method (BLSM) is used instead. In the BLSM, the level set function can only take 1 and -1 values at convergence. Thus, it is related to phase-field methods. We don’t need to solve the Hamilton-Jacobi equation, so it is free of the CFL condition and the reinitialization scheme. This favorable properties lead to a great time advantage in this method. In this paper, the BLSM is implemented with the additive operator splitting (AOS) scheme and several numerical issues of the implementation are discussed. The proposed scheme is much more efficient than the conventional level set method. Several 2D examples are presented which demonstrate the effectiveness and robustness of the proposed method.

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In this article, an efficient methodology is presented to optimize the topology of structural systems under transient loads. Equivalent static loads concept is used to deal with transient loads and to solve an alternate quasi-static optimization problem. The maximum strain energy of the structure under the transient load during the loading interval is used as objective function. The objective function is calculated in each iteration and then the dynamic optimization problem is replaced by a static optimization problem, which is subsequently solved by a convex linearization approach combining linear and reciprocal approximation functions. The optimal layout of a deep beam subjected to transient loads is considered as a case study to verify the effectiveness of the presented methodology. Results indicate that the optimal layout is dependant of the loading interval.

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A two-stage optimization method is presented by employing the evolutionary structural optimization (ESO) and ant colony optimization (ACO), which is called ESO-ACO method. To implement ESO-ACO, size optimization is performed using ESO, first. Then, the outcomes of ESO are employed to enhance ACO. In optimization process, the weight of double layer grid is minimized under various constraints which artificial ground motion is used to calculate the structural responses. The presence or absence of elements in bottom and web grids and also cross-sectional areas are selected as design variables. The numerical results reveal the computational advantages and effectiveness of the proposed method.

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This paper presents the bidirectional evolutionary structural optimization (BESO) method for the design of two-phase composite materials with optimal properties of stiffness and thermal conductivity. The composite material is modelled by microstructures in a periodical base cell (PBC). The homogenization method is used to derive the effective bulk modulus and thermal conductivity. BESO procedures are presented to optimize the two individual properties and their various combinations. Three numerical examples are studied. The results agree well with those of the benchmark microstructures and the Hashin-Shtrikman (HS) bounds.

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For reliability-based topology optimization (RBTO) of double layer grids, a two-stage optimization method is presented by applying “Solid Isotropic Material with Penalization” and “Ant Colony Optimization” (SIMP-ACO method). To achieve this aim, first, the structural stiffness is maximized using SIMP. Then, the characteristics of the obtained topology are used to enhance ACO through six modifications. As numerical examples, reliability-based topology designs of typical double layer grids are obtained by ACO and SIMP-ACO methods. Their numerical results reveal the effectiveness of the proposed SIMPACO method for the RBTO of double layer grids.

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In the present paper, an approach is proposed for structural topology optimization based on combination of Radial Basis Function (RBF) Level Set Method (LSM) with Isogeometric Analysis (IGA). The corresponding combined algorithm is detailed. First, in this approach, the discrete problem is formulated in Isogeometric Analysis framework. The objective function based on compliance of particular locations of materials in the structure is used and find the optimal distribution of material in the domain to minimize the compliance of the system under a volume constraint. The refinement is employed for construction of the physical mesh to be consistent with the mesh is used for level set function. Then a parameterized level set method with radial basis functions (RBFs) is used for structural topology optimization. Finally, several numerical examples are provided to confirm the validity of the method.

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The prevalent strategy in the topology optimization phase is to select a subset of members existing in an excessively connected truss, called Ground Structure, such that the overall weight or cost is minimized. Although finding a good topology significantly reduces the overall cost, excessive growth of the size of topology space combined with existence of varied types of design variables challenges applicability of evolutionary algorithms tailored for simultaneous optimization of topology, shape and size (TSS) in more complicated cases which are of great practical interest. In practice, large-scale truss structures are often modular, formed by joining periodically repeated units. This article organizes a novel simulation approach for this class of truss structures where the main drawbacks of the ground structure-based simulation approach are greatly moderated. The two approaches are independently employed for simultaneous TSS optimization of a modular truss example and the size of topology space as well as the required computation budget to generate an acceptable candidate design is compared. Result comparison reveals by employing the novel approach, problem complexity grows linearly with respect to the number of modules which allows for expanding application of TSS optimizers to complex modular trusses. Use of relative coordinates is also warranted for shape optimization which concludes to a more efficient optimization process.

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The Isogeometric Analysis (IA) method is applied for structural topology optimization instead of the finite element method. For this purpose, the material density is considered as a continuous function throughout the design domain and approximated by the Non-Uniform Rational B-Spline (NURBS) basis functions. The coordinates of control points which are also used for constructing the density function, are considered as design variables of the optimization problem. In order to change the design variables towards optimum, the Method of Moving Asymptotes (MMA) is used. To alleviate the formation of layouts with porous media, the density function is penalized during the optimization process. A few examples are presented to demonstrate the performance of the method.

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In this article, the ant colony method is utilized for topology optimization of space structures. Strain energy of the structure is minimized while the material volume is limited to a certain amount. In other words, the stiffest possible structure is sought when certain given materials are used. In addition, a noise cleaning technique is addressed to prevent undesirable members in optimum topology. The performance of the method for topology optimization of space structures are demonstrated by three numerical examples.

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A new method for structural topology optimization is introduced which employs the Isogeometric Analysis (IA) method. In this approach, an implicit function is constructed over the whole domain by Non-Uniform Rational B-Spline (NURBS) basis functions which are also used for creating the geometry and the surface of solution of the elasticity problem. Inspiration of the level set method zero level of the function describes the boundary of the structure. An optimality criterion is derived to improve the implicit function towards the optimum boundaries. The last section of this paper is devoted to some numerical examples in order to demonstrate the performance of the method as well as the concluding remarks.

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In this paper, for topology optimization of double layer grids, an efficient optimization method is presented by combination of Imperialist Competitive Algorithm (ICA) and Gravitational Search Algorithm (GSA) which is called ICA-GSA method. The present hybrid method is based on ICA but the moving of countries toward their relevant imperialist is done using the law of gravity of GSA. In topology optimization process, the weight of the structure is minimized subjected to displacements of joints, internal stress and slenderness ratio of members constraints. Through numerical example, topology optimization of a typical large-scale double layer grid is obtained by ICA, GSA and ICA-GSA methods. The numerical results indicate that the proposed algorithm, ICA-GSA, executes better than ICA, GSA and the other methods presented in the literatures for topology optimization of largescale skeletal structures.

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In this paper the piecewise level set method is combined with phase field method to solve the shape and topology optimization problem. First, the optimization problem is formed based on piecewise constant level set method then is updated using the energy term of phase field equations. The resulting diffusion equation which updates the level set function and optimization problem is solved through finite element method. The proposed method enhances the convergence rate and solution efficiency. Various two-dimensional examples are solved to verify the performance of proposed method.

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This study has been inspired by the paper "An efficient 3D topology optimization code written in MATLAB” written by Liu and Tovar (2014) demonstrating that SIMP-based three-dimensional (3D) topology optimization of continuum structures can be implemented in 169 lines of MATLAB code. Based on the above paper, we show here that, by simple and easy-to-understand modifications we get a few lines longer code, which is able to solve robust topology optimization problems with uncertain load directions. In the presented worst load direction oriented approach, the varying load directions are handled by quadratic constrains, which describe spherical regions about the nominal loads. The result of the optimization is a robust compliance-minimal volume constrained design, which is invariant to the investigated directional uncertainty. The key element of the robustification is a worstload-direction searching process, which is formulated as a small quadratic programming problem with quadratic constraints. The presented approach is a 3D extension of the robust approach originally developed by Csébfalvi (2014) for 2D continuum structures. In order to demonstrate the viability and efficiency of the extension, we present the model and algorithm with detailed benchmark results for robust topology optimization of 3D continuum structures. It will be demonstrated that the computational cost of the robustification is comparable with its deterministic equivalent because its central element is a standard 3D deterministic multi-load structure optimization problem and the worst-loaddirection searching process is formulated as a significantly smaller quadratically constrained quadratic programming problem, which can be solved efficiently by several different ways.

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The Isogeometric Analysis (IA) is utilized for structural topology optimization  considering minimization of weight and local stress constraints. For this purpose, material density of the structure  is  assumed  as  a  continuous  function  throughout  the  design  domain  and approximated using the Non-Uniform Rational B-Spline (NURBS) basis functions. Control points of the density surface are considered as design variables of the optimization problem that can change the topology during the optimization process. For initial design, weight and stresses of the structure are obtained based on full material density over the design domain. The  Method  of  Moving  Asymptotes  (MMA)  is  employed  for  optimization  algorithm. Derivatives of the objective function and constraints with respect to the design variables are determined  through  a  direct  sensitivity  analysis.  In  order  to  avoid  singularity  a  relaxation technique  is  used  for  calculating  stress  constraints.  A  few  examples  are  presented  to demonstrate the performance of the method. It is shown that using the IA method and an appropriate stress relaxation technique can lead to reasonable optimum layouts.

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This study focuses on the topology optimization of structures using a hybrid of level set method (LSM) incorporating sensitivity analysis and isogeometric analysis (IGA). First, the topology optimization problem is formulated using the LSM based on the shape gradient. The shape gradient easily handles boundary propagation with topological changes. In the LSM, the topological gradient method as sensitivity analysis is also utilized to precisely design new holes in the interior domain. The hybrid of these gradients can yield an efficient algorithm which has more flexibility in changing topology of structure and escape from local optimal in the optimization process. Finally, instead of the conventional finite element method (FEM) a Non–Uniform Rational B–Splines (NURBS)–based IGA is applied to describe the field variables as the geometry of the domain. In IGA approach, control points play the same role with nodes in FEM, and B–Spline functions are utilized as shape functions of FEM for analysis of structure. To demonstrate the performance of the proposed method, three benchmark examples widely used in topology optimization are presented. Numerical results show that the proposed method outperform other LSMs.

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In this paper, a displacement-constrained volume-minimizing topology optimization model is present for two-dimensional continuum problems. The new model is a generalization of the displacement-constrained volume-minimizing model developed by Yi and Sui [1] in which the displacement is constrained in the loading point. In the original model the displacement constraint was formulated as an equality relation, which practically means that the number of “interesting points” may be exactly one. The recent model resolves this weakness replacing the equality constraint with an inequality constraint. From engineering point of view it is a very important result because we can replace the inequality constraint with a set of inequality constraints without any difficulty. The other very important fact, that the modified displacement-oriented model can be extended very easily to handle stress-oriented relations, which will be demonstrated in the forthcoming paper. Naturally, the more general theoretical model needs more sophisticated numerical problem handling method. Therefore, we replaced the original “optimality-criteria-like” solution searching process with a standard nonlinear programming approach which is able to handle linear (nonlinear) objectives with linear (nonlinear) equality (inequality) constrains. The efficiency of the new approach is demonstrated by an example investigated by several authors. The presented example with reproducible numerical results as a benchmark problem may be used for testing the quality of exact and heuristic solution procedures to be developed in the future for displacement-constrained volume-minimization problems.

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Keeping the eigenfrequencies of a structure away from the external excitation frequencies is one of the main goals in the design of vibrating structures in order to avoid risk of resonance. This paper is devoted to the topological design of freely vibrating continuum structures with the aim of maximizing the fundamental eigenfrequency. Since in the process of topology optimization some areas of domain can potentially be removed, it is quite possible to encounter the problem of localized modes. Hence, the modified Solid Isotropic Material with Penalization (SIMP) model is here used to avoid artificial modes in low density areas. As during the optimization process, the first natural frequency increases, it may become close to the second natural frequency. Due to lack of the usual differentiability of the multiple eigenfrequencies, their sensitivity are calculated by the mathematical perturbation analysis. The optimization problem is formulated by a variable bound formulation and it is solved by the Method of Moving Asymptotes (MMA). Two dimensional plane elasticity problems with different sets of boundary conditions and attachment of a concentrated nonstructural mass are considered. Numerical results show the validity and supremacy of this approach.

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The present study sets out to integrate the performance-based seismic design approach with the connection topology optimization method. Performance-based connection topology optimization concept aims to simultaneously optimize the size of members and the type of connections with respect to the framework of performance-based seismic design. This new optimization concept is carried out for unbraced and X-braced steel frames in order to assess its efficiency. The cross-sectional area of components and the type of beam-to-column connections are regarded as design variables. The objective function is formulated in terms of the material costs and the cost of rigid connections. The nonlinear pushover analysis is adopted to acquire the response of the structure at various performance levels. In order to cope with the optimization problem, CBO algorithm is employed. The achieved results demonstrate that incorporating the optimal arrangement of beam-to-column connections into the optimum performance-based design procedure of either unbraced or X-braced steel frame could lead to a design that significantly reduces the overall cost of the structure and offers a predictable and reliable performance for the structure subjected to hazard levels.

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This paper presents the topology optimization of plane structures using a binary level set (BLS) approach and isogeometric analysis (IGA). In the standard level set method, the domain boundary is descripted as an isocountour of a scalar function of a higher dimensionality. The evolution of this boundary is governed by Hamilton–Jacobi equation. In the BLS method, the interfaces of subdomains are implicitly represented by the discontinuities of BLS functions taking two values 1 or −1. The subdomains interfaces are represented by discontinuities of these functions. Using a two–phase approximation and the BLS approach the original structural optimization problem is reformulated as an equivalent constrained optimization problem in terms of this level set function. For solving drawbacks of the conventional finite element method (FEM), IGA based on a Non–Uniform Rational B–Splines (NURBS) is adopted to describe the field variables as the geometry of the domain. For this purpose, the B–Spline functions are utilized as the shape functions of FEM for analysis of structure and the control points are considered the same role with nodes in FEM. Three benchmark examples are presented to investigate the performance the topology optimization based on the proposed method. Numerical results demonstrate that the BLS method with IGA can be utilized in this field.

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Due to the favorable performance of structural topology optimization to create a proper understanding in the early stages of design, this issue is taken into consideration from the standpoint of research or industrial application in recent decades. Over the last three decades, several methods have been proposed for topology optimization. One of the methods that has been effectively used in structural topology optimization is level set method. Since in the level set method, the boundary of design domain is displayed implicitly, this method can easily modify the shape and topology of structure. Topological design with multiple constraints is of great importance in practical engineering design problems. Most recent topology optimization methods have used only the volume constraint; so in this paper, in addition to current volume constraint, the level set method combines with other constraints such as displacement and frequency. To demonstrate the effectiveness of the proposed level set approach, several examples are presented.