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.

中文翻译：

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曲桥节点非分离尺度异质结构分析

在没有尺度分离的情况下对异质结构的性能进行数值预测代表了满足计算可扩展性和效率的关键要求的重大挑战——采用足够细的网格来充分考虑小尺度异质性会导致高昂的计算成本，而仅仅忽略这些精细异质性往往会导致结构刚度过大。这项研究提出了一种方法，在结构的粗离散化上，针对沿元素边界定义的每个弯曲桥节点 (CBN)，为每个元素构建新的材料感知形状（基础）函数。而不是通过解决非线性优化问题来制定他们的推导，形状函数是通过构建从 CBN 到内部节点的映射来构建的，并最终以显式矩阵形式呈现为 B\'ezier 插值变换和边界-内部变换的乘积。CBN 形状函数在密切捕捉粗单元的异质性方面具有更大的灵活性，克服了单元间刚度和跨粗单元之间界面位移不连续性等重要且具有挑战性的问题，并将分析精度提高了几个数量级；它们还满足避免非物理分析结果的形状函数的基本几何特性。还测试了大量数值示例，包括具有数十亿自由度的 3D 工业示例，以演示该方法。