Analysis of heterogeneous structures of non-separated scales using curved bridge nodes

Abstract

Numerically predicting the performance of heterogeneous structures without scale separation is a challenging task owing to the critical requirements related to computational scalability and efficiency that must be satisfied. In addition, adopting a sufficiently fine mesh to consider the small-scale heterogeneities results in prohibitive computational costs, whereas neglecting them tends to drastically over-stiffen the rigidity of the structure.Thus, this study proposed an approach for constructing new material-aware shape (basis) functions per element for a coarse discretization of the structure considering each curved bridge node (CBN) that is defined along the boundaries of the elements. Rather than formulating their derivation by regarding them as a nonlinear optimization problem, the shape functions were constructed mapping the CBNs to the interior nodes and were subsequently presented in an explicit matrix form as a product of Bézier interpolation and boundary–interior transformations. The CBN shape function captures the heterogeneity of the coarse element with greater flexibility, overcomes the important and challenging issues of inter-element stiffness and displacement discontinuity across interfaces between coarse elements, and improves the analysis accuracy by several orders of magnitude. Moreover, they satisfy the basic geometric properties of shape functions, thereby avoiding non-physical analysis results. Furthermore, the performance of the proposed approach was tested and demonstrated through extensive numerical examples, including a 3D industrial example of billions of degrees of freedom, and comparisons with results obtained from classical approaches were made.