## 1 Introduction  
Pores of different shapes and sizes are inevitable constituents of practical engineering materials, such as composites, metals and synthetic materials. They significantly affect the overall elastic properties and the failure mechanism of the material [1; 2; 3; 4; 5]. Specifically, they render porous materials more suitable for medical applications [6; 7; 8; 9] because of the high controllability, lightweight, biocompatibility and easy of design. Therefore, it is important to fully understand the influence of pores on the macroscopic material properties and to model them quantitatively to design new materials that meet specific requirements and synthesize novel structures. So far, experimental testing [6; 10] is still the most common approach to investigate the pore effects, but the cost remains high, the process is time-consuming and the data obtained may present a considerable discrepancy. With the rapid development of computer hardware and computational theories, numerical simulation becomes another powerful tool to address practical problems. The finite element method (FEM) is a possible approach to quantitatively evaluate the material properties influenced by defects, like holes, cracks and many other imperfections, in the materials and the large structures in view of its low cost and relatively high efficiency [11; 12]. Many pioneering studies concerning this topic [1; 2; 3; 4; 5; 13; 14; 15; 16; 17; 18] have been conducted by now, and one critical step is to reconstruct a proper microstructure model of the porous materials. A straightforward way is adopting a commercial package, say Minics or others, to reproduce the true microstructures of porous materials at a specific scale by using the tomographic images generated by the atom-probe tomography [19], electron microscopy [20], X-ray tomography [21], or focused-ion-beam-nonatomography [22]. To be noted, the methods of this kind need a huge number of tomographic images of all directions and different scales so as to construct an accurate description of the porous materials [23] due to the random distribution of pores and their shapes and size variations. Further, the solving process may be very time-consuming and computationally challenging. The second feasible way is referred to as the equivalent computational methods, which apply certain statistics model to characterize the pore size and distribution and then generate equivalent porous microstructures according to the statistics information, including the pore size, surface-to-volume ratio and so forth, obtained experimentally. To date, these methods are classified into two main categories. The first category assumes that all micro- characteristics of porous microstructures are distributed periodically [16; 17; 24; 25; 26], hence, a minimal unit cell [1] is modeled based on the experimental observations and employed to predict the effective macroscopic elastic property of porous materials. Though the modeling process with this method is very efficient and the elastic moduli of materials are evaluated quickly ascribed to the periodic idealization, the overall material property predicted with these models may be quite different from the test data since a limited number of natural porous materials rigidly satisfy this simplification. The other category of the equivalent computational methods fully considers the randomness of the micro- characteristics of porous materials. In the early years, all pores of porous materials are deemed as regular shapes, say sphere, ellipsoid and so forth, of the same kind [13; 14; 27] and distributed randomly to facilitate the modeling and analysis. Hereafter, the relative arrangement of pores, i.e., distribution mode, among different randomly-distributed pores, is taken into account in modeling the porous microstructures in Ref. [28; 29; 30].