Extended multiscale isogeometric analysis for mechanical simulation of two-dimensional periodic heterogeneous materials

Periodic heterogeneous material (PHM) is widely used in the design of multi-functional composite structures. Embedded particles or void holes with smooth boundaries are common in PHM, and the simulation of such material needs novel multiscale simulation method. Thus, extended multiscale isogeometric analysis (EmsIGA) is developed for the mechanical analysis of planar PHM, particularly for heterogeneous materials with smooth inclusions or voids. The non-uniform rational B-spline basis function is used for geometrical description and also mechanical simulation of the fine-scale unit cells to obtain numerical heterogeneity function. By calculating the numerical heterogeneity function as a static basis function, the equivalent element stiffness matrix of the sub-grids in the coarse scale is obtained and the structural problem can be solved. Downscaling computation is applied to obtain the stress and strain of the structure. Numerical examples validate the efficiency and accuracy of the proposed EmsIGA method in the simulation of heterogeneous structures. Compared with the results of the traditional and multi-scale finite element method, the proposed method can precisely capture the geometry and concentrated stress state of the smooth inclusions and calculate the microscopic stress and strain with improved efficiency.