[논문]3-D Topology Optimization by a Nodal Density Method Based on a SIMP Algorithm

Kim, Cheol; Fang, Nan

[국내논문] 3-D Topology Optimization by a Nodal Density Method Based on a SIMP Algorithm
SIMP 기반 절점밀도법에 의한 3 차원 위상최적화 원문보기

Kim, Cheol (Graduate School, Dept. of Mechanical Eng, Kyungpook National University) , Fang, Nan

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.

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문제 정의

The design domain is discretized into a fixed grid of N finite elements and the design variables are constituted by the elements densities. The objective is to find an optimal material distribution subjected to some existent constraints. The minimization of a compliance function can be expressed as ⁽¹⁸⁾:
The objective of this study is to develop a simple and efficient method for 3-D topology optimization based on the SIMP method. Elements are connected by each node, and if the material densities are re-discretized from elements to nodes, a more uniform density distribution can be obtained.

제안 방법

In the references^{(2, 3)} a derivation of the SIMP method was also tried independently and the term SIMP was introduced first time in 1992. In the papers, a number of design conditions including stress constraints were applied. A similar approach was advocated by Mlejnek⁽⁹⁾and the SIMP method was coupled with the method of moving asymptotes.
Elements are connected by each node, and if the material densities are re-discretized from elements to nodes, a more uniform density distribution can be obtained. In this paper, discretized nodal densities were used to replace the element densities in order to obtain more uniform topology representation. The optimization procedure is established as following steps: First, the 3-D design domain is discretized by eight-node hexahedron elements and each element density is calculated by the SIMP algorithm.
The objective of the continuum structural topology optimization is to find out the optimum material distribution for a specified design domain, i.e., to find out an optimal load path between loading points and structural supports and provide an optimum distribution of materials after the minimization of structural compliance (i.e., maximization of stiffness). Since Bendsoe and Kikuchi ⁽¹⁾ suggested the basic theory of topology optimization based on the homogenization method in 1988, topology optimization design has been rapidly developed by in-depth study and extensive research in terms of theories, methods and applications.

대상 데이터

4(a). A mesh of 30 10 10 elements is employed and the material properties are the same as the previous model. The optimal results are shown in Figs.

이론/모형

The nodal densities are interpolated from the adjacent element densities. It must be emphasized that in this paper the design variables are element densities different from the CAMD (continuous approximation of material distribution) method ⁽²⁰⁾which used the nodal design variables. The resulting element densities are replaced by the relative nodal densities with an interpolation function.
The optimization procedure is established as following steps: First, the 3-D design domain is discretized by eight-node hexahedron elements and each element density is calculated by the SIMP algorithm. Next, nodal densities are interpolated from the adjacent element densities by an element-to-node method ⁽¹⁷⁾. A computer code has been written to implement this pre-processing.
The optimizer is based on the standard optimality criteria method like Bendsoe (19).

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[국내논문] 3-D Topology Optimization by a Nodal Density Method Based on a SIMP Algorithm
SIMP 기반 절점밀도법에 의한 3 차원 위상최적화 원문보기

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[국내논문] 3-D Topology Optimization by a Nodal Density Method Based on a SIMP Algorithm SIMP 기반 절점밀도법에 의한 3 차원 위상최적화 원문보기

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[국내논문] 3-D Topology Optimization by a Nodal Density Method Based on a SIMP Algorithm
SIMP 기반 절점밀도법에 의한 3 차원 위상최적화 원문보기

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