In order to overcome numerical instabilities such as checkerboards, meshdependence in topology optimization of continuum structures, a new implementation combined with homogenization method without introducing any additional constraint parameter is presented. To overcome the shortcoming of continuous material distribution by the introduction of finite element approximation, moving least square or modified filter functions are adopted as interpolation function. The method can be viewed as a nature extension of node-based homogenization method and named as material point homogenization method. Continuous size field and continuous density field are constructed, and structural responses' sensitivities are derived. Several representative numerical examples are presented to demonstrate the capability and the efficiency of the proposed approach against some classes of numerical instabilities. 1. Introduction Topology optimization refers to optimal design problems in which the topology of the structure is allowed to change in order to improve the performance of the structure. Relative to size optimization and shape optimization, this optimization class is regarded as one of the most challenging optimization problems in the field of structural optimization. Much research has been devoted to topology optimization over the last decades. Starting with the landmark paper of Bends？e and Kikuchi [1], numerical methods for topology optimization have been investigated extensively since the late 1980s. The area of topology optimization of continuum structures is now dominated by methods that employ the material distribution concept. The typical ones are the homogenization and variable density method. In the homogenization method, the material property of each design cell is computed by the homogenization theory, and the optimal topology is obtained by solving a material distribution problem. However, the homogenization method may not yield the intended results with infinitesimal pores in the materials that make the structure not manufacturable. An engineering alternative approach called the variable density method was introduced in [2]. Instead of using the homogenization theory to evaluate the material property for design cells, this approach assumes some explicit relationships between the density and the material property without considering their microstructures. This approach is very attractive to the engineering community because of its simplicity. The power-law approach is physically permissible as long as simple conditions on the power are satisfied (e.g., for
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