Point-wise Density Interpolation-based Topology Optimization Method And Its Applications

Over the past two decades, topology optimization has received extensive attention and has been developed as an emerged tool for a wide range of engineering applications, including aerospace, mechanical engineering and vehicle engineering. The SIMP (Solid Isotropic Material Penalization) method has become one of the most popular topology optimization approaches. The method generally takes the element-wise constant densities as design variables, typically generating optimal results with checkerboard patterns and zig-zag boundaries. With the purpose of improving the performance of the topology optimization process, the present paper developes a novel iPDI (independent Pointwise Density Dnterpolation) topology optimization methodology, and applies it to various design problems. The contents of the paper are stated as follows:(1) Nodal density-based topology optimization methods typically interpolates nodal design variables to construct density field, which can obtains optimal solutions with high description resolution. Generally, this method applies the element shape functions as the density field interpolation functions. However, if a higher-order shape function is used, the obtained densities within local regions may be negative, which violates the [0,1]-range restriction. Meanwhile, most existing topology optimization methods employed the unique mesh for both the displacement analysis field and the topological description field. This may influence the performance of the topology optimization. In order to overcome these issues, this paper proposes a novel independent pointwise density interpolation (iPDI) methodology for structural topology optimization. A rational interpolation function is used for the density field, ensuring that the obtained densities are [0,1]-range restricted. When the method is applied to the nodal density interpolation in higher-order elements, optimal results without no islanding phenomenon will be obtained. Moreover, this approach can also be used for analysis-mesh independent density interpolation, where the density value at a computational point is determined by the density design variables within a certain influence domain. The iPDI method can effectively avoid the islanding phenomenon, as well as the mesh-dependency. Moreover, the iPDI method is well-suited for optimization problems with an unregularly-shaped design domain.(2) The computational efficiency of the optimization process can be effectively improved by introducing the adaptive method into topology optimization. However, conventional adaptive topology optimization methods employ the same adaptive criteria for both the displacement analysis mesh and the density design variables. As a result, the adaptive model may generate unnecessary refinements within local regions. This paper developes an iPDI-based adaptive topology optimization (iPDI-ATOP) method. In the approach, the density field and the finite element mesh are sepratedly and independently refined, according to different refinement criteria. Newly-added density points aims to improve the boundary description quality and to reduce the ratio of the gray transitional regions in the optimal results, while the finite element mesh is refined to improve displacement accuracy. By using the iPDI-ATOP. different regions may have different refinement levels. Optimal results show that the iPDI-ATOP method enbales a high computational efficiency, a low ratio of the gray transitional regions, and a high boundary description quality.(3) Topology optimization of structures with movable embedded components has attracted increasing attention. Conventional design methods used the explicit parameter approach to represent geometries of the embedded components, which may be difficulty in describing complex-shaped ones. Moreover, the overlapping relationship between different components are hard to be detected. This paper proposes an advanced approach for the structural optimization with embedded movable components, where an iPDI-LS combined topology description model is developed to represent the embedded components and the structural topology. Here, the level set method is used to track the movement of the fixed-shaped components, and the iPDI approach is used to perform the topology optimization of the host structure. In order to prevent the overlaps between each two components, and between each component and the boundary of the design domain, we also proposes an adavanced non-overlapping constraint. The constraint has unique and explicit mathematical expression, and is applicable for detecting overlaps with arbitray number of components. We further employ the proposed combined topology description model and the non-overlapping constraint to achieve a systematic design of compliant smart strutures with movable shape-fixed PZT actuators. The layout of the actuators and the topology of the compliant structure are simultaneously optimized. Some numerical examples are presented to demonstrate the efficiency and effectiveness of the method.(4) The parametric level set method (PLSM) is introduced into the material design field. The underlying concept of the PLSM is to decouple the space-and time-related terms in the Hamilton-Jacobi equation, and then achieves topology optimization by updating the time-related design variables. Since the PLSM does not require solving the Hamilton-Jacobi partial differentical equation, and does not require the level set function to be a signed distance function, it effectively improves the numerical efficiency of the optimization process. In this paper, the PLSM is successfully incorporated into the field of metamaterial design, where the microstructures with prescribed/negative Poisson’s ratio have been obtained.