  Numerically predicting the performance of heterogenous structures without
scale separation represents a significant challenge to meet the critical
requirements on computational scalability and efficiency -- adopting a mesh
fine enough to fully account for the small-scale heterogeneities leads to
prohibitive computational costs while simply ignoring these fine
heterogeneities tends to drastically over-stiffen the structure's rigidity.
  This study proposes an approach to construct new material-aware shape (basis)
functions per element on a coarse discretization of the structure with respect
to each curved bridge nodes (CBNs) defined along the elements' boundaries.
Instead of formulating their derivation by solving a nonlinear optimization
problem, the shape functions are constructed by building a map from the CBNs to
the interior nodes and are ultimately presented in an explicit matrix form as a
product of a B\'ezier interpolation transformation and a boundary-interior
transformation. The CBN shape function accomodates more flexibility in closely
capturing the coarse element's heterogeneity, overcomes the important and
challenging issues of inter-element stiffness and displacement discontinuity
across interface between coarse elements, and improves the analysis accuracy by
orders of magnitude; they also meet the basic geometric properties of shape
functions that avoid aphysical analysis results. Extensive numerical examples,
including a 3D industrial example of billions of degrees of freedom, are also
tested to demonstrate the approach's performance in comparison with results
obtained from classical approaches.
