We use topological derivatives to obtain fiber-reinforced structural designs with non-periodic continuous fibers optimally arranged in specific patterns. The distribution of anisotropic fiber material within isotropic matrix material is determined for given volume fractions of void and material as well as fiber and matrix simultaneously, for maximum stiffness. In this three-phase material distribution approach, we generate a Pareto surface of stiffness and two volume fractions by adjusting the level-set plane in the topological sensitivity field. For this, we utilize topological derivatives for interchanging (i) isotropic material and void; (ii) fiber material and void; and (iii) isotropic and fiber materials, during iterative optimization. While the isotropic topological derivative is well known, the latter two required modification of the anisotropic topological derivative. Furthermore, we used the polar form of the topological derivative to determine the optimal orientation of the fiber at every point. Thus, in the discretized finite element model, we get element-wise optimal fiber orientation in the portions where fiber is present. Using these discrete sets of orientations, we extract continuous fibers as streamlines of the vector field. We show that continuous fibers are aligned with the principal stress directions as first reported by Pedersen. Three categories of examples are presented: (i) embedding fiber everywhere in the isotropic matrix without voids; (ii) selectively embedding fiber for a given volume fraction of the fiber without voids; and (iii) including voids for given volume fractions of fiber and matrix materials. We also present an example with multiple load cases. Additionally, in view of practical implementation of laying up or 3D-printing of fibers within the matrix material, we simplify the dense arrangement of fibers by evenly spacing them while retaining their specific patterns.

我们使用拓扑导数来获得纤维增强的结构设计，其中非周期性连续纤维以特定的图案最佳排列。对于最大的刚度，对于给定的空隙和材料的体积分数以及同时确定纤维和基质的各向异性，确定各向同性基质材料中各向异性纤维材料的分布。在这种三相材料分配方法中，我们通过在拓扑敏感度字段中调整水平集平面来生成刚度和两个体积分数的帕累托表面。为此，我们利用拓扑导数交换（i）各向同性的材料和空隙；（ii）纤维材料和空隙；（iii）迭代优化过程中的各向同性和纤维材料。虽然各向同性的拓扑导数是众所周知的，后两个需要对各向异性拓扑导数进行修改。此外，我们使用拓扑导数的极性形式来确定纤维在每个点的最佳取向。因此，在离散有限元模型中，我们在存在纤维的部分中获得了逐元素的最佳纤维取向。使用这些离散的方向集，我们提取连续纤维作为矢量场的流线。我们显示，连续纤维与Pedersen首次报道的主应力方向对齐。提出了三类示例：（i）将纤维嵌入到各向同性的基体中的任何位置而没有空隙；（ii）对于给定的纤维体积分数，将纤维选择性地包埋，没有空隙；（iii）包括给定体积分数的纤维和基质材料的空隙。我们还提供了一个具有多个工况的示例。此外，鉴于在基质材料内实际铺设或3D打印纤维的方法，我们通过在均匀分布的同时保留特定图案的方式简化了纤维的密集排列。