Lattice structures are of growing importance in additive manufacturing, where complex internal geometries are increasingly required for lightweight, high surface-to-volume ratios, multifunctionality, and other superior mechanical properties. Conventional lattice modeling methods typically represent struts with simple primitives, such as cylinders or cones, limiting geometric diversity and the design space. Although recent efforts have increased strut-shape complexity to address this issue, they often do so at the expense of computational efficiency and modeling robustness. As a result, achieving both rich geometric expressiveness and efficient computation remains a challenging problem. In this paper, we present an implicit modeling method that expands the design and optimization space of lattice structures while preserving the modeling robustness and efficiency of implicit representations. In our method, each strut is defined as a convolution surface over a skeletal graph, and its profile shape is controlled by a cubic Hermite curve. By exploiting the polynomial structure of both the convolution kernel and the cubic Hermite curve-controlled profile, we derive analytical expressions for efficient field evaluation, avoiding costly and unstable numerical computation. Four case studies have been conducted to validate the proposed method in terms of profile shape diversity, graded lattice modeling, as well as slicing robustness and efficiency.

翻译：格子结构在增材制造中日益重要，因其需要复杂内部几何结构以实现轻量化、高比表面积、多功能性及其他优异力学性能。传统格子建模方法通常采用圆柱或圆锥等简单基元表示支柱，限制了几何多样性与设计空间。尽管近期研究通过增加支柱形状复杂度试图解决此问题，但往往以牺牲计算效率与建模鲁棒性为代价。因此，同时实现丰富的几何表达力与高效计算仍具挑战性。本文提出一种隐式建模方法，在保持隐式表示建模鲁棒性与效率的同时，扩展格子结构的设计与优化空间。该方法中，每条支柱被定义为骨架图上的卷积曲面，其轮廓形状由三次埃尔米特曲线控制。通过利用卷积核与三次埃尔米特曲线控制轮廓的多项式结构，我们推导出高效场值评估的解析表达式，避免了昂贵且不稳定的数值计算。通过四个案例研究，从轮廓形状多样性、梯度格子建模以及切片鲁棒性与效率等方面验证了所提方法的有效性。