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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 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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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 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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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.

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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.