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Topology optimization of the heat transfer quality in two-dimensional heat conduction problem at enclosure as one of the typical thermo-physical problems has always been quite important. In this paper a level set-based topological optimization procedure of two-dimensional heat conduction problem include point and speared thermal on computational domain load using finite elements method is developed. In level-set method, all structural boundaries are parameterized by a level of dynamic implicit scalar function of higher order. Changes of this function can easily model the detachment and attachment of dynamic boundaries in topology procedures. The same shape functions of finite elements analysis are employed to approximate the unknown temperatures and geometry modeling of the design domain. The objective function is to minimize thermal power capacity and sensitivity analysis on some heat conduction problems is investigated to deal with the topology optimization using level-set method with the finite elements scheme. Finally, topology optimization results of 3 heat conduction problems under both include point and spread thermal load cases are presented to demonstrate the validity of the proposed method. The proposed method lead to a significant reduction of the computational cost and time and it can be applied to a wide range of topology optimization problems arising from the heat transfer.

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Reliability based Topology optimization (RBTO) is a process of determining of optimal design satisfying uncertainties of design variables. Sometimes frequency optimization might produce a design with low stiffness or stiffness optimization might lead to a design with low frequency. In this paper, the multi-objective optimization for both stiffness and frequencies isare presented. This article presents (RBTO) using bi-directional evolutionary structural optimization (BESO) with an improved filter scheme. A multi-objective topology optimization technique is implemented to simultaneously considering the stiffness and natural frequency. In order to compute reliability index the first order reliability method (FORM) and standard response surface method (SRSM) for generating limit state function is employed. To increase the efficiency of the solution process the reliability estimates areis coupled with the topology optimization process. Topology optimization is formulated as volume minimization problem with probabilistic displacement and frequency constraints. Young’s module, density, and external load are considered as uncertain variables. The topologies are obtained by (RBTO) are compared with that obtained by deterministic topology optimization (DTO). Results show that (RBTO) using (BESO) method is capable of the multi-objective optimization problem for stiffness and frequency effectively.

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This paper presents an improved approach for handling stress constraints in minimum weight topological design. The Finite Element Method (FEM) and the material model of Solid Isotropic Material with Penalization (SIMP) is used to formulate the topology optimization problem. To evaluate the stress values in elements, the von Mises stresses are calculated at the so called super-convergent Gauss quadrature points. To reduce the time and computational cost, a clustering approach is here adopted and the P-norm integrated stress constraints are used. Doing this, a large number of local constraints are replaced with a few global ones and consequently the stress constraint sensitivities are calculated by using the adjoint method. The employed formulation as well as a complete explanation of the sensitivity analysis is provided. Due to the complexity of the topology optimization problem in the presence of stress constraints, the Method of Moving Asymptotes (MMA) is here employed. To demonstrate the performance and capability of the procedure, a couple of plane stress elasticity problems are taken into consideration. The resulted layouts indicate the superiority of the approach in generating acceptable and practical topological designs.

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Large-scale spatial skeletal structures belong to a special kind of 3D structures widely used in exhibition centers, supermarkets, sport stadiums, airports, etc., to cover large surfaces without intermediate columns. Space structures are often categorized as grids, domes and barrel vaults. Double layer grid structures are classical instances of prefabricated space structures and also the most popular forms which are frequently used nowadays.Topology optimization of large-scale skeletal structures has been recognized as one of the most challenging tasks in structural design. In topology optimization of these structures with discrete cross-sectional areas, the performance of meta-heuristic optimization algorithms can be increased if they are combined with continuous-based topology optimization methods. In this article, a hybrid methodology combining evolutionary structural optimization (ESO) and harmony search algorithm (HSA) methods is proposed for topologyoptimization of double layer grid structures subject to vertical load. In the present methodology, which is called ESO-HSA method, the size optimization of double layer grid structures is first performed by the ESO. Then, the outcomes of the ESO are used to improve the HSA. In fact, a sensitivity analysis is carried out using an optimization method (ESO) to determine more important members based on the cross-sectional areas of members. Then, the obtained optimum cross-sectional areas of members are used to enhance the HSA through two modifications. Structural weight is minimized against constraints on the displacements of nodes, internal stresses and element slenderness ratio. In topology optimization of double layer grid structures, the geometry of the structure, support locations and coordinates of nodes are fixed and this structure is assumed as a ground structure. Presence/absence of bottom nodes, and element cross-sectional areas are selected as design variables. In topology optimization of the ground structure, tabulating of nodes is carried out based on structural symmetry: this leads to reduce complexity of design space and nodes are removed in groups of 8, 4 or 1. The presence or absence of each node group is determined by a variable (topology variable) which takes the value of 1 and 0 for the two cases, respectively. The ground structure is assumed to be supported at the perimeter nodes of the bottom grid. Therefore, these supported nodes will not be removed from the ground structure. In order to achieve a practical structure, the existence of nodes in the top grid will not be considered as a variable. This causes the load bearing areas of top layer nodes to remain constant. Also, discrete variables are used to optimize the cross-sectional area of structural members. These variables are selected from pipe sections with specified thickness and outer diameter. Therefore, in topology optimization problem, the number of design variables is the summation of the number of compressive and tensile element types and the number of topology variables. The proposed approach is successfully tested in topology optimization problem of double layer grid structure. In particular, ESO-HSA is very competitive with other metaheuristic methods recently published in literature and can always find the best design overall. Also, it is determined that HSA method can find better answer in the topology optimization of large-scale skeletal structures, in comparison to optimum structures attained by the GSA and ICA.

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In this research, an influence of topology optimization in energy absorption of lattice core sandwich beams by using ABAQUS software was an investigation. Relationships between the force and displacement at the midspan of the sandwich beams were obtained from the experiments. Two types of Steel lattice cores with three cell orientation were subjected to the low-velocity impact test under three-point bending. The core of sandwich beams was made from expanded metal sheets and a topology optimization with Solid Isotropic Microstructure with Penalization (SIMP) method was used to remove the redundant expanded metal cell. In the following, by studying the topology optimization to evaluate the impact parameters, including Specific Energy Absorption (SEA), as discussed testing purposes. The energy absorbing system can be used in the aerospace industry, shipbuilding, automotive, railway industry and elevators to absorb impact energy. Experimental and numerical results showed that topology optimization could significantly increase specific absorbed energy. Results of three-point bending crushing tests showed that the SEA of a sandwich beam with optimal core structure increased between 45% and 94% compared to the initial design structure of the core. In addition, appropriate orientation of expanded metal cell in the core of sandwich beam caused to increase the specific energy absorption by more than 90%. Finally, an appropriate optimal geometric structure with three tape of volume fraction and the best examples of criteria considered with respect to the objectives were introduced.

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Shape and topology optimization have become one of the main researches that is widely used in engineering fields. The purpose of topology optimization is to find an appropriate (optimal) distribution of materials in the design domain so that the shape and number of voids is optimized and the objective function is minimized or maximized. In recent decades, noticeable researches and various topology optimization methods were proposed. The level set method is being used successfully in structural shape and topology optimization. This method is an implicit method for moving interior and exterior boundaries, while these boundaries may join together during the process and new voids may be formed. The structural boundary is illustrated by the zero level set and nonzero in the domain. In the above context, the level set function is used as a switch to distinguish between the two domains present in the computing space. This way of illustration has an important feature by which the domain boundaries can be combined together or divided. By using the solution of Hamilton-Jacobi equation resulting from this function, the domain’s boundary starts to move. The control over movement of this boundary is done by velocity vector of Hamilton-Jacobi equation. Now, in order to use this method in topology optimization, it is sufficient to establish a relationship between velocity vector of Hamilton-Jacobi and shape derivation, which is used for optimizing objective function. It is possible to use standard level set for structural topology optimization.
In this paper, the spherical Hankel basis functions are used to optimize the structural topology using the level set method. The proposed functions are a combination of the first and second kind of Bessel functions fields as well as the polynomial ones in complex space and are derived from radial basis functions. Using the spherical Hankel functions, the dependence of the function of the level set method on the space and time is separated, which results in the transformation of the Hamilton-Jacobian partial differential equation into a conventional differential equation. In this way, the difficulties arising from solving partial differential equations are eliminated, and thus there is no need to re-set the function of the level set method in the optimization process. Further, in order to increase the speed and precision of convergence in creating an optimal design, the classic Lagrange shape functions are replaced with the spherical Hankel ones. The proposed shape functions have some properties such as infinite piecewise continuity, the Kronecker delta property, and the partition of unity. Moreover, since they satisfy all three polynomial fields and the first and second kind of Bessel ones in the complex space, they can be effective in improving the accuracy and speed of convergence, while the classic Lagrange shape functions are able to satisfy only the polynomial function fields. Finally, several numerical examples are presented to study the performance of the spherical Hankel radial basis and shape functions.
 

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A suitable design is one design can achieve to its aims with minimum cost and needing to less computing time. In civil engineering due to survey of large scale structures and large number of design variables, it is so hard achieving to such design only based on experience and therefore optimization methods came to help designer as useful tools in order to find an economic and efficient design. Structural optimization can be defined as a process of dealing with the optimal design of various structures. Ausual objective function is the weight of the structure. In general, there are three main categories in structural optimization applications, namely, size, topology and geometry (shape) optimization.Cellular automata (CA) is a computationally efficient and robust tool to simply implement complex computations. As CA is simple to be implemented and can deal with complex problems without extensive mathematical computations, it is widely used in various fields of science and engineering.In recent years, various meta-heuristic inspired optimization methods have been developed.Almost all of metaheuristic algorithms come up with an idea of employing a particular process or event in nature as a source of inspiration for the development of optimization algorithm. The Cuttlefish algorithm is inspired based on the color changing behavior of cuttlefish to find the optimal solution. The patterns and colors seen in cuttlefish are produced by reflected light from different layers of cells including (chromatophores, leucophores and iridophores) stacked together, and it is the combination of certain cells at once that allows cuttlefish to possess such a large array of patterns and colors.In this article, cuttlefish algorithm (CFA)combined with cellular automata (CA) and were used for optimization truss structures.First, cellular automata and the Moor neighboring cells are defined and to the number ofsquares of the cell number ofcellular automata lattice( )is selected from the  best population. Then, the variables vector and their objective function of selected population are placed in each cell of the cellular automata.In a Moor neighboring, nine cells are compared to each other and the best answer ( )is selected and that is used to create new population.Finally, the best person in thenew population will be selected and itreplacedwith the worst person in the cellular automata, and thus the cellular automatais updated. Some benchmark numerical examples were solved using the CFA and CA-CFA algorithms, and the results of the numerical examples showed that the enhanced algorithm performancesbetter in size and topology optimization of truss structures than cuttlefish algorithm and other methods introduced in the literature. Finally, it can be concluded that the convergence speed of the improved algorithm compared with previous approaches is higher and its ability to achieve the desired values is better too.

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In recent years, topology optimization has been used as an innovative approach to design lightweight and high-performance components. Despite the high variety of developed topology optimization approaches, only a limited number of them can be used in commercially available software programs, and in particular, for complex geometries. Three different types of these methods have been utilized and investigated in this research. In the first step, an industrial part is redesigned for topology optimization. Then, the volume of this part is reduced by 60% by three different methods of Continuous Compliance Optimization (CCO) , Discrete Compliance Optimization (DCO) , and Stress-Constrained Optimization (SCO) . Then, a number of parameters, such as the maximum stress and displacement, safety factor, error of convergence, the final weight, and the computational cost of each approach are assessed. Finally, in a nutshell, it can be concluded that despite the differences in the performance and result of each method, all of them are applicable, but the SCO method could achieve the best result due to the minimum stress concentration and final weight. It is noteworthy that topology optimization configurations have many complexities and can only be produced by additive manufacturing technologies due to their potential and flexibility.
 

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In this article, MMA which is a mathematical programing method is utilized for solving truss optimization problem. Although this method has been widely used in topology optimization problems of continuous structures, it has received less attention in the case of truss structures. The optimization problem considered here is the problem of minimizing the strain energy of the structure by considering the volume constraint. For the first time, the basics of the SIMP method and the application of the penalty exponent of the density function have been used for simultaneous optimization of topology and size of the truss members.To solve the optimization problem, analytical sensitivity analysis has been performed. Various problems have been solved at the end of the paper and the results have been discussed. The results show that if the appropriate penalty exponent is chosen, the correct optimal solution is obtained for the benchmark problems. Also, some more practical problems have been solved and the results have been discussed.

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 This research focuses on topology optimization of fluid-structure interaction (FSI) problems using the level set method. To couple the fluid and structure equations, the Arbitrary Lagrangian-Eulerian (ALE) description is employed within a monolithic formulation. The use of ALE in FSI problems, while eliminating numerical instabilities caused by the convective term, enhances the speed and accuracy of finite element solutions in fluid-structure interaction. Additionally, considering the fluid in the unsteady state allows for the interpretation of optimal topology at any given moment of the analysis. The objective function of the optimal topology design problem is to minimize the structural compliance in the dry state, subject to a fixed volume of the design domain. To determine the normal velocity in the reaction-diffusion equation (RDE), adjoint sensitivity analysis based on pointwise gradients is used. The results obtained from this approach, compared to other topology optimization methods in the literature, demonstrate higher accuracy and clearer definition of structural boundaries.