As additive manufacturing advances toward higher printing resolution and larger build volumes, microstructures can be designed with finer geometric features over larger physical domains. This trend poses a fundamental challenge for geometric modeling: massive geometric details must be represented compactly, while their associations across scales must be maintained consistently.Existing methods cannot scale well to this requirement. Explicit representations suffer from prohibitive memory cost, and implicit representations remain compact only when microstructures admit analytic, periodic, or otherwise concise procedural descriptions. This paper proposes a new geometric modeling method that treats microstructure modeling as an on-demand generative process, rather than requiring the full instantiation of all geometric details. We first develop ExVCC, an extended volumetric Catmull-Clark spline representation that enables local spline refinement to go beyond tensor-product topology. Built on ExVCC, we introduce new shape-coding schemes and refinement rules that compactly encode large-scale geometric details and enable their localized evaluation through on-demand hierarchical refinement. To model geometric details across scales, we further propose an isoparametric representation in which details across scales are defined over a shared parametric domain using the same family of spline bases of ExVCC. This formulation turns the ExVCC's spline refinement hierarchy into a common framework for geometry encoding, on-demand generation, and cross-scale association, allowing geometric modifications to propagate automatically across scales. The effectiveness of the proposed method is demonstrated through a series of examples and comparisons.

翻译：随着增材制造向更高打印分辨率与更大构建体积发展，微结构可在更大物理域内设计出更精细的几何特征。这一趋势对几何建模提出了根本性挑战：必须紧凑地表示海量几何细节，同时保持其跨尺度关联的一致性。现有方法难以满足这一需求。显式表示面临不可承受的内存开销，而隐式表示仅在微结构具有解析、周期或其他简洁程序化描述时才保持紧凑性。本文提出一种新型几何建模方法，将微结构建模视为按需生成过程，而非要求所有几何细节的完整实例化。我们首先开发ExVCC——一种扩展的体积Catmull-Clark样条表示，使局部样条细化能够超越张量积拓扑。基于ExVCC，我们引入新型形状编码方案与细化规则，可紧凑地编码大规模几何细节，并通过按需层次化细化实现局部化评估。为跨尺度建模几何细节，我们进一步提出等参表示：在共享参数域上，使用ExVCC的同族样条基定义跨尺度细节。该公式将ExVCC的样条细化层次结构转化为几何编码、按需生成与跨尺度关联的统一框架，使几何修改可自动跨尺度传播。通过系列实例与对比验证了所提方法的有效性。