Otimização de materiais periódicos treliçados via método de homogeneização NIAH e metamodelo de Kriding

Abstract

The rapid technological evolution requires, more and more, the development of new engineering materials that meet multifunctional conditions. Because of this, one type of material that has attracted the attention of researchers are the so-called periodic truss materials. These are cellular materials composed by bars, forming trusses and structure with a periodic pattern. Due to such periodicity, in many cases, these materials have better physical properties when compared to materials of random structure of equal relative density. However, due to the heterogeneity of its structure (solid material and voids) in the microscale (cell scale), obtaining the macroscopic properties of the material is not straightforward. A way to obtain these properties is to use the asymptotic homogenization (AH) method. This method has a mathematical foundation based on the perturbation theory and the effective properties of periodic materials are determined through the resolution of partial differential equations defined for the unit cell. However, it may require the development of a specific computational program according to the finite element used in the discretization of the cell. As a result, some researchers have developed the new implementation of asymptotic homogenization (NIAH). NIAH uses commercial finite element software such as “black boxes” for calculations of displacements, forces, temperatures and heat flows. The use of commercial software allows any type of available element or technique to be used in the modeling of the unit cell. In addition, it allows the interaction between optimization techniques based on metamodels. Thus, this work aims to use the NIAH to obtain the elastic and thermal conductivity properties of periodic truss materials. Moreover, it is sought to optimize two initial unit cells in order to obtain better mechanical and thermal responses, keeping the relative density constant. However, there are two difficulties: the non access to the source code of the commercial program for calculating the sensitivity and the relatively large number of design variables. To overcome these difficulties, it is employed a technique based on metamodels, called Kriging, in combination with a sequential quadratic programming algorithm (SQP) used as a local optimizer, associated to an infill strategy to refine the model. The areas of the transversal sections of the bars of the unit cell are taken as design variables aiming the maximization of the shear modulus, the maximization of the thermal conductivity in the x direction, the maximization of the shear modulus combined with the thermal conductivity in the x direction and the minimization of the Poisson coefficient. With this, we obtained a set of optimal geometries and several of them are validated with results found in the literature. In problems of minimization of the Poisson coefficient were obtained with materials with typical auxetic structures. The use of NIAH allowed to obtain the properties quickly and accurate when compared to AH. The optimization technique used, based on metamodeling, was able to find good solutions. However, it faced some difficulties in the construction phase of the Kriging metamodel due to the large number of design variables and the computationally low efficiency of the infill criterion.