---

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.

---

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.

---

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.

---

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.

---

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.

---

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.

---

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.

---

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.

---

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.

---

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.

---

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.

---

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.

---

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.

---

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.

---

This paper aims to introduce topology optimization as a robust tool for damage detection in plane stress structures. Two objective functions based on natural frequencies and shape modes of the structure are defined to minimize discrepancy between dynamic specifications of the real damaged structure and the updating model. Damage area is assumed as a porous material where amount of porosity signifies the damage intensity. To achieve this, Solid Isotropic Material with Penalization (SIMP) model is employed. Sensitivity analysis is achieved and a mathematical based method is used for solving the optimization problems. In order to demonstrate efficiency and robustness of the method to identify various type of damages in terms of both location and intensity, several numerical examples are presented and the results are discussed.

---

In this paper, topology optimization (TO) is applied to determine the form, size and location of holes for the special form of perforated steel plate shear wall (PSPSW). The proposed model is based on the recently presented particular form of PSPSW that is called the ring-shaped steel plate shear wall. The strain energy is selected as the objective function in the optimization. Simple Isotropic Material with Penalization (SIMP) method and the solution algorithms, including sensitivity and condition-based methods are utilized in the TO. Four initial plate forms are presented in the TO with regards to the length of the connection between the plate and column. Based on the solution methods and initial forms of the plate, eight scenarios are proposed and seven different perforated plates obtained using TO. The nonlinear responses of the optimized perforated plates are compared together, and with the ring-shaped model as a benchmark. The nonlinear analysis is conducted under cyclic and monotonic loadings. Key issues include cyclic and monotonic behavior, pinching behavior, stiffness, load-carrying capacity, energy dissipation, fracture tendency and out-of-plane deformation are investigated and discussed. The results demonstrate the optimized models have better behavior than the ring-shaped model without changing the volume of the plate.

---

In this paper, topology optimization is utilized for damage detection in three dimensional elasticity problems. In addition, two mode expansion techniques are used to derive unknown modal data from measured data identified by installed sensors. Damages in the model are assumed as reduction of mass and stiffness in the discretized finite elements. The Solid Isotropic Material with Penalization (SIMP) method is used for parameterizing topology of the structure. Difference between mode shapes of the model and real structure is minimized via a mathematical based algorithm. Analytical sensitivity analysis is performed to obtain derivatives of objective function with respect to the design variables. In order to illustrate the accuracy of the proposed method, four numerical examples are presented.

---

In this paper, the bi-directional evolutionary structural optimization (BESO) method is used to find optimal layouts of 3D prestressed concrete beams. Considering the element sensitivity number as the design variable, the mathematical formulation of topology optimization is developed based on the ABAQUS finite element software package. The surface-to-surface contact with a small sliding between concrete and prestressing steels is assumed to accurately model the prestressing effects. The concrete constitutive model used is the concrete damaged plasticity (CDP) model in ABAQUS. The integration of the optimization algorithm and finite element analysis (FEA) tools is done by using the ABAQUS scripting interface. A pretensioned prestressed simply supported beam is modeled to show capabilities of the proposed method in finding optimal topologies of prestressed concrete beams. Many issues relating to topology optimization of prestressed concrete beams such as the effects of prestressing stress, geometrical discontinuities and height constraints on optimal designs and strut-and-tie models (STMs) are studied in the example. The results show that the proposed method can efficiently be used for layout optimization of prestressed concrete beams.

---

This research focuses on the effects of stiffeners and architectural opening on the steel shear wall topology optimization. To this aim, four relevant issues have been considered. The first issue is the optimality Pattern of the shear wall without stiffeners. The second is the Optimality Pattern of the shear wall with stiffeners in two directions. The third is the investigation on the topology optimization of the shear walls with fixed opening and the fourth is the multi-material topology optimization of the above issues. In the optimize process, the level set method based on the shape sensitivity and the finite element analysis for two-dimensional linear elastic problems has been used. The level set function implicitly indicated the boundaries of the design domain. Several numerical examples are used to demonstrate the optimal paths in the steel shear walls. The results show that optimal values have been changed by replacing stiffeners and creating openings in the wall, but the optimal topologies almost have a shape like a concentric bracing. Also, in the conventional shear walls (one material) the horizontal stiffeners have a significant effect on their performance.

---

The present study proposes a hybrid of the piecewise constant level set (PCLS) method and isogeometric analysis (IGA) approach for structural topology optimization. In the proposed hybrid method, the discontinuities of PCLS functions is used in order to present the geometrical boundary of structure. Additive Operator Splitting (AOS) scheme is also considered for solving the Lagrange equations in the optimization problem subjected to some constraints. For reducing the computational cost of the PCLS method, the Merriman–Bence–Osher (MBO) type of projection scheme is applied. In the optimization process, the geometry of structures is described using the Non–Uniform Rational B–Splines (NURBS)–based IGA instead of the conventional finite element method (FEM). The numerical examples illustrate the efficiency of the PCLS method with IGA in the efficiency, convergence and accuracy compared with the other level set methods (LSMs) in the framework of 2–D structural topology optimization. The results of the topology optimization reveal that the proposed method can obtain the same topology in lower number of convergence iteration.