• International Journal of CAD/CAM
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• 제7권1호
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• pp.41-49
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• 2007

In density-based topology optimization, 0/1 solutions are sought. Discrete topological problems are often relaxed with continuous design variables so that they can be solved using continuous mathematical programming. Although the relaxed methods are practical, grey areas appear in the optimum topologies. SIMP (Solid Isotropic Microstructures with Penalization) employs penalty schemes to suppress the intermediate densities. SRV (the Sum of the Reciprocal Variables) drives the solution to a 0/1 layout with the SRV constraint. However, both methods cannot effectively remove all the grey areas. SRV has some numerical aspects. In this work, a new scheme SIMP-SRV is proposed by combining SIMP and SRV approaches, where SIMP is employed to generate an intermediate solution to initialize the design variables and SRV is then adopted to produce the final design. The new method turned out to be very effective in conjunction with the method of moving asymptotes (MMA) when using for the stiffness topology optimization of continuum structures for minimum compliance. The numerical examples show that the hybrid technique can effectively remove all grey areas and generate stiffer optimal designs characterized with a sharper boundary in contrast to SIMP and SRV.

• 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.

• Structural Engineering and Mechanics
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• 제86권4호
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• pp.473-486
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• 2023

Nowadays, urban centers are increasingly vertical, making architects and engineers look for more efficient tools to analyze the effects of wind on tall buildings. Topology optimization can be used as an efficient tool for the design of bracing systems. Therefore, this work obtained the wind loads that act in the CAARC building, following the Brazilian standard NBR 6123/1988 and using Computational Fluid Dynamics. Four loading situations were considered, using the SIMP and BESO methods to optimize two-dimensional structures. A comparison between the SIMP and BESO methods is presented, showing the differences in the geometry of the solution found by both methods, the percentage variation in the objective function values and the dimensionless processing time. The solutions obtained through the loads obtained by the Brazilian standard are also compared with the numerical solutions obtained by CFD. The results show that the BESO method presented more rigid structures compared to the SIMP method. The bracing structures obtained with the SIMP method always present similar patterns in the distribution and quantity of bars, in contrast to the BESO method where no characteristic topology pattern was observed. It was concluded that even though the structures obtained by the BESO method presented greater stiffness, the SIMP method was less susceptible to the methodology used for the determination of wind loads. Additionally, it was evident the great potential that the combination topology optimization and computational wind engineering have in the design of bracing systems of high functional and aesthetic standards.

• Structural Engineering and Mechanics
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• 제55권6호
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• pp.1157-1176
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• 2015

This study presents optimizing structural topology patterns using regularization of Heaviside function. The present method needs not filtering process to typical SIMP method. Using the penalty formulation of the SIMP approach, a topology optimization problem is formulated in co-operation, i.e., couple-signals, with design variable values of discrete elements and a regularized Heaviside step function. The regularization of discontinuous material distributions is a key scheme in order to improve the numerical problems of material topology optimization with 0 (void)-1 (solid) solutions. The weak forms of an equilibrium equation are expressed using a coupled regularized Heaviside function to evaluate sensitivity analysis. Numerical results show that the incorporation of the regularized Heaviside function and the SIMP leads to convergent solutions. This method is tested using several examples of a linear elastostatic structure. It demonstrates that improved optimal solutions can be obtained without the additional use of sensitivity filtering to improve the discontinuous 0-1 solutions, which have generally been used in material topology optimization problems.

• 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.

• 한국전산구조공학회논문집
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• 제20권1호
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• pp.9-18
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• 2007

본 연구에서는 위상최적화 알고리즘의 수렴성을 개선하기 위해 설계영역에 초기 구멍을 도입하는 방법을 제시하는데, 이것은 경계면에 기초한 최적화 방법의 느린 수렴성을 완화하기 위해, Eschenauer et al.에 의해 고안된 버블 방법의 설계영역 안에 구멍을 도입하는 개념과 연계된다. 버블 방법과 달리, 제안된 방법에서는 최적화 과정동안 구멍의 위치를 정의하는 특성함수를 이용하지 않고, 최적화 초기화 단계에서만 초기 구멍을 도입하는데, 이러한 초기 설계영역 안의 솔리드와 보이드 영역들은 고정되는 것이 아니라 합쳐지거나 쪼개지면서 변화된다. 따라서 위상최적화 알고리즘에서 구멍의 이동에 관련된 복잡한 수치적인 계산 없이 자동적으로 설계변수의 유한변화를 더욱 강화시키기 때문에 목적함수 값의 수렴성을 개선할 수 있다. 본 논문에서는 다양한 치수와 형상의 구멍을 포함하는 초기 설계영역을 가지는 Michell형 보의 위상 최적설계를 밀도분포법으로 불리는 SIMP를 이용하여 수행하였다. 이를 통해 위상최적화의 수렴성을 개선하고 최적위상과 형상에 영향을 미치는 초기 구멍의 효과를 검증하였다.

• 대한기계학회논문집A
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• 제30권2호
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• pp.164-170
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• 2006

To prevent the numerical instabilities in topology optimization, continuous approximation of material distribution (CAMD) is proposed to the homogenization design method (HDM) and the simple isotropic material with penalization (SIMP) method. The continuous FE approximation of design variables including high order elements is applied to the formulation of SIMP method. Numerical examples are presented to compare the efficiency of CAMD both in HDM and SIMP.

• 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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• 제29권6호
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• pp.771-783
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• 2018

In this paper, topology optimization (TO) is applied to find a new configuration for the perforated steel plate shear wall (PSPSW) based on the maximization of reaction forces as the objective function. An infill steel plate is introduced based on an experimental model for TO. The TO is conducted using the sensitivity analysis, the method of moving asymptotes and SIMP method. TO is done using a nonlinear analysis (geometry and material) considering the buckling. The final area of the optimized plate is equal to 50% of the infill plate. Three plate thicknesses and three length-to-height ratios are defined and their effects are investigated in the TO. It indicates the plate thickness has no significant impact on the optimization results. The nonlinear behavior of optimized plates under cyclic loading is studied and the strength, energy and fracture tendency of them are investigated. Also, four steel plates including infill plate, a plate with a central circle and two types of the multi-circle plate are introduced with equal plate volume for comparing with the results of the optimized plate.

• Structural Engineering and Mechanics
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• 제43권6호
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• pp.711-724
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• 2012

The goal of this study is to conceptually orientate optimized layouts of outrigger belt trusses which are in widespread use today in the design of tall buildings by strut-and-tie truss models utilizing a topology optimization method. In this study unknown strut-and-tie models are realized by using a typical SIMP method of topology optimization methods. In tradition strut-and-tie model designs find the appropriate strut-and-tie trusses along force paths with respect to elastic stress distribution, and then engineers or designers determine the most proper truss models by experience and intuition. It is linked to a trial-and-error procedure based on heuristic strategies. The presented strut-and tie model design by using SIMP provides that belt truss models are automatically and robustly produced by optimal layout information of struts-and-ties conforming to force paths without any trial-and-error. Numerical applications are studied to verify that outrigger belt trusses for tall buildings are optimally chosen by the proposed method for both static and dynamic responses.