• 대한기계학회:학술대회논문집
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• 대한기계학회 2008년도 추계학술대회A
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• pp.412-417
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• 2008

In a traditional topology optimization method, material properties are usually distributed by finite element density and visualized by a gray level image. The distribution method based on element density is adequate for a great mass of 2-D topology optimization problems. However, when it is used for 3-D topology optimization, it is always difficult to obtain a smooth model representation, and easily appears a virtualconnect phenomenon especially in a low-density domain. The 3-D structural topology optimization method has been developed using the node density instead of the element density that is based on SIMP (solid isotropic microstructure with penalization) algorithm. A computer code based on Matlab was written to validate the proposed method. When it was compared to the element density as design variable, this method could get a more uniform density distribution. To show the usefulness of this method, several typical examples of structure topology optimization are presented.

• Steel and Composite Structures
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• 제27권1호
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• pp.27-33
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• 2018

In this study, a mixed-interpolated tensorial component 4 nodes method (MITC4) is treated as a numerical analysis model for topology optimization using multiple materials assigned within Reissner-Mindlin plates. Multi-material optimal topology and shape are produced as alternative plate retrofit designs to provide reasonable material assignments based on stress distributions. Element density distribution contours of mixing multiple material densities are linked to Solid Isotropic Material with Penalization (SIMP) as a design model. Mathematical formulation of multi-material topology optimization problem solving minimum compliance is an alternating active-phase algorithm with the Gauss-Seidel version as an optimization model of optimality criteria. Numerical examples illustrate the reliability and accuracy of the present design method for multi-material topology optimization with Reissner-Mindlin plates using MITC4 elements and steel materials.

• Steel and Composite Structures
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• 제51권2호
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• pp.203-223
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• 2024

This study shows functionally graded material structural topology optimization under buckling constraints. The SIMP (Solid Isotropic Material with Penalization) material model is used and a method of moving asymptotes is also employed to update topology design variables. In this study, the quadrilateral element is applied to compute buckling load factors. Instead of artificial density properties, functionally graded materials are newly assigned to distribute optimal topology materials depending on the buckling load factors in a given design domain. Buckling load factor formulations are derived and confirmed by the resistance of functionally graded material properties. However, buckling constraints for functionally graded material topology optimization have not been dealt with in single material. Therefore, this study aims to find the minimum compliance topology optimization and the buckling load factor in designing the structures under buckling constraints and generate the functionally graded material distribution with asymmetric stiffness properties that minimize the compliance. Numerical examples verify the superiority and reliability of the present method.

• 한국전산구조공학회논문집
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• 제34권3호
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• pp.159-165
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• 2021

본 연구는 효과적인 열전도를위한 거시적 구조 구성과 단위 구조 변화의 동시 설계를 위한 위상 최적화 방법을 제시한다. 거시적 규모의 구조 내에서 위치에 따른 단위 구조의 형태 변화는 거시적 규모뿐만 아니라 미시적 단위의 설계도 가능하며 등방성 단위 구조를 사용하는 것보다 더 나은 성능을 제공할 수 있다. 이 결과로 두 구성을 결합한 기능적으로 등급의 복합 구조가 생성된다. 대표 체적 요소 (RVE) 방법은 형태 변화에 따른 다중 재료 기반 단위 구조의 다양한 열전도 특성을 얻기 위해 적용된다. RVE 분석 결과를 바탕으로 머신 러닝 기법을 이용하여 특정 형태의 단위 구조물의 물성치를 도출할 수 있다. 거시적 위상 최적화는 기존의 SIMP 방법을 사용하여 수행되며, 거시 구조를 구성하는 단위 구조는 동시 최적화 과정에 따라 열전도 성능을 향상시키기 위한 다양한 형태를 가질 수 있다. 제안된 방법의 효과를 확인하기 위해 열 컴플라이언스 최소화 문제의 수치예가 제공된다.

• Journal of Mechanical Science and Technology
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• 제17권10호
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• pp.1496-1506
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• 2003

The SIMP (solid isotropic material with penalization) approach is perhaps the most popular density variable relaxation method in topology optimization. This method has been very successful in many applications, but the optimization solution convergence can be improved when new variables, not the direct density variables, are used as the design variables. In this work, we newly propose S-shape functions mapping the original density variables nonlinearly to new design variables. The main role of S-shape function is to push intermediate densities to either lower or upper bounds. In particular, this method works well with nonlinear mathematical programming methods. A method of feasible directions is chosen as a nonlinear mathematical programming method in order to show the effects of the S-shape scaling function on the solution convergence.

• Steel and Composite Structures
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• 제46권1호
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• pp.33-51
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• 2023

This research presents a multi-material topology optimization for functionally graded material (FGM) and nonFGM with elastic buckling criteria. The elastic buckling based multi-material topology optimization of functionally graded steels (FGSs) uses a Jacobi scheme and a Method of Moving Asymptotes (MMA) as an expansion to revise the design variables shown first. Moreover, mathematical expressions for modified interpolation materials in the buckling framework are also described in detail. A Solid Isotropic Material with Penalization (SIMP) as well as a modified penalizing material model is utilized. Based on this investigation on the buckling constraint with homogenization material properties, this method for determining optimal shape is presented under buckling constraint parameters with non-homogenization material properties. For optimal problems, minimizing structural compliance like as an objective function is related to a given material volume and a buckling load factor. In this study, conflicts between structural stiffness and stability which cause an unfavorable effect on the performance of existing optimization procedures are reduced. A few structural design features illustrate the effectiveness and adjustability of an approach and provide some ideas for further expansions.

• International Journal of CAD/CAM
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• 제8권1호
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• pp.1-10
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• 2009

This paper presents a topology optimization approach using element-free Galerkin method (EFGM) for the optimal design of compliant mechanisms with geometrically non-linearity. Meshless method has an advantage over the finite element method(FEM) because it is more capable of handling large deformation resulted from geometrical nonlinearity. Therefore, in this paper, EFGM is employed to discretize the governing equations and the bulk density field. The sensitivity analysis of the optimization problem is performed by incorporating the adjoint approach with the meshless method. The Lagrange multipliers method adjusted for imposition of both the concentrated and continuous essential boundary conditions in the EFGM is proposed in details. The optimization mathematical formulation is developed to convert the multi-criteria problem to an equivalent single-objective problem. The popularly applied interpolation scheme, solid isotropic material with penalization (SIMP), is used to indicate the dependence of material property upon on pseudo densities discretized to the integration points. A well studied numerical example has been applied to demonstrate the proposed approach works very well and the non-linear EFGM can obtain the better topologies than the linear EFGM to design large-displacement compliant mechanisms.

• Structural Engineering and Mechanics
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• 제45권6호
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• pp.759-777
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• 2013

In this paper, a topology optimization method based on the element independent nodal density (EIND) is developed for continuum solids with multiple load cases and multiple constraints. The optimization problem is formulated ad minimizing the volume subject to displacement constraints. Nodal densities of the finite element mesh are used a the design variable. The nodal densities are interpolated into any point in the design domain by the Shepard interpolation scheme and the Heaviside function. Without using additional constraints (such ad the filtering technique), mesh-independent, checkerboard-free, distinct optimal topology can be obtained. Adopting the rational approximation for material properties (RAMP), the topology optimization procedure is implemented using a solid isotropic material with penalization (SIMP) method and a dual programming optimization algorithm. The computational efficiency is greatly improved by multithread parallel computing with OpenMP to run parallel programs for the shared-memory model of parallel computation. Finally, several examples are presented to demonstrate the effectiveness of the developed techniques.

• International Journal of Aeronautical and Space Sciences
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• 제14권4호
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• pp.324-333
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• 2013

The purpose of this paper is to improve the efficiency of multi-objective topology optimization using a genetic algorithm (GA) with bar-system representation. We proposed a new GA using an elite initial population obtained from a Solid Isotropic Material with Penalization (SIMP) using a weighted sum method. SIMP with a weighted sum method is one of the most established methods using sensitivity analysis. Although the implementation of the SIMP method is straightforward and computationally effective, it may be difficult to find a complete Pareto-optimal set in a multi-objective optimization problem. In this study, to build a more convergent and diverse global Pareto-optimal set and reduce the GA computational cost, some individuals, with similar topology to the local optimum solution obtained from the SIMP using the weighted sum method, were introduced for the initial population of the GA. The proposed method was applied to a structural topology optimization example and the results of the proposed method were compared with those of the traditional method using standard random initialization for the initial population of the GA.

• Smart Structures and Systems
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• 제11권3호
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• pp.241-259
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• 2013

This paper presents the optimum load-speed diagram evaluation for a linear micromotor, including multitude cantilever piezoelectric bimorphs, briefly. Each microbeam in the mechanism can be actuated in both axial and flexural modes simultaneously. For this design, we consider quasi-static and linear conditions, and a relatively new numerical method called the smoothed finite element method (S-FEM) is introduced here. For this purpose, after finding an optimum volume fraction for piezoelectric layers through a standard numerical method such as quadratic finite element method, the relevant load-speed curves of the optimized micromotor are examined and compared by deterministic topology optimization (DTO) design. In this regard, to avoid the overly stiff behavior in FEM modeling, a numerical method known as the cell-based smoothed finite element method (CS-FEM, as a branch of S-FEM) is applied for our DTO problem. The topology optimization procedure to find the optimal design is implemented using a solid isotropic material with a penalization (SIMP) approximation and a method of moving asymptotes (MMA) optimizer. Because of the higher efficiency and accuracy of S-FEMs with respect to standard FEMs, the main micromotor characteristics of our final DTO design using a softer CS-FEM are substantially improved.