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Topological Optimization of Structures Using a Multilevel Nodal Density-Based Approximant

Yu Wang1, Zhen Luo1,2, Nong Zhang1

School of Electrical, Mechanical and Mechatronic Systems, The University of Technology, Sydney, NSW 2007, Australia
Corresponding Author. Tel: +61-2-9514 2994; Fax: +61-2-9514 2655. E-mail address:

Computer Modeling in Engineering & Sciences 2012, 84(3), 229-252.


This paper proposes an alternative topology optimization method for the optimal design of continuum structures, which involves a multilevel nodal density-based approximant based on the concept of conventional SIMP (solid isotropic material with penalization) model. First, in terms of the original set of nodal densities, the Shepard function method is applied to generate a non-local nodal density field with enriched smoothness over the design domain. The new nodal density field possesses non-negative and range-bounded properties to ensure a physically meaningful approximation of topology optimization design. Second, the density variables at the nodes of finite elements are used to interpolate elemental densities, as well as corresponding element material properties. In this way, the nodal density field by using the non-local Shepard function method is transformed to a practical elemental density field via a local interpolation with the elemental shape function. The low-order finite elements are utilized to evaluate the displacement and strain fields, due to their numerical efficiency and implementation easiness. So, the proposed topology optimization method is expected to be efficient in finite element implementation, and effective in the elimination of numerical instabilities, e.g. checkerboards and mesh-dependency. Three typical numerical examples in topology optimization are employed to demonstrate the effectiveness of the proposed method.


Cite This Article

Wang, Y., Luo, Z., Zhang, N. (2012). Topological Optimization of Structures Using a Multilevel Nodal Density-Based Approximant. CMES-Computer Modeling in Engineering & Sciences, 84(3), 229–252.

cc This work is licensed under a Creative Commons Attribution 4.0 International License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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