In this paper, an isogeometric density field method is proposed for the design of micro-architected materials with the specific mechanical properties, consisting of the maximum bulk and shear modulus and the Negative Poisson's ratio (NPR). Firstly, the non-uniform rational B-splines (NURBS) basis functions are employed to construct the density field function (DFF), where the Shepard function is used to improve the smoothness of the nodal densities assigned to control points. The NURBS basis functions and the Shepard function can ensure the sufficient continuity and smoothness of the DFF, owing to their significant properties. The optimization formulation for micro-architected materials is developed using the DFF, where the isogeometric analysis (IGA) is applied to evaluate the unknown structural responses. Effective macroscopic properties of materials are predicted by the asymptotic homogenization method. The same NURBS basis functions used in the IGA and the DFF can keep the consistency of the geometric model and analysis model, which provides the unique benefits. Numerical examples are used to demonstrate the effectiveness of the proposed topology optimization approach for micro-architected materials.

The discrete topology optimization method is widely used because of its high degree of freedom and high problem-solving efficiency. However, one of its main drawbacks is that the output graphic is constituted of a 0-1 matrix leading to rough boundaries and implicit structures, which are difficult to be manufactured. Aiming to transform the implicit rough boundary into the explicit smooth boundary in discrete topology optimization, an effective structural boundary processing method based on support vector machine (SVM) is proposed in this paper. In this method, SVM is used for processing the boundaries obtained by the Solid Isotropic Material with Penalization (SIMP) and discrete level-set methods, respectively. With the SVM method, the clear structural boundaries are obtained and can be used in CAD directly. In order to make structural boundaries more smooth, data filtering on boundaries is employed. The effectiveness of the proposed method is illustrated by the experimental examples. Results show that the proposed method can obtain clear and smooth structural boundaries in discrete topology optimization.

In order to address the efficient issue of the topological shape optimization problem, this paper presents a parametric level set method for the problem of continuum structures by using the Radial Basis Functions (RBFs). The level set-based method is introduced to implicitly represent the free boundary of a structure. To evolve the level set front, the Wu's compactly supported radial basis function (CSRBF) with C4 smoothness is applied. Then, the Hamilton-Jacobi partial differential equation (PDE) is transformed into a relatively easier ordinary differential equation (ODE). Furthermore, a self-adaptive moving limit scheme is incorporated into the optimality criteria to achieve a better optimal result. Finally, the numerical ex-ample is provided to show the effectiveness of the proposed method.