HCC: Medium: Robust and Accurate Modeling with Multifield Geometry

用于创建、分析和修改几何图形的计算机工具是计算机辅助设计系统的核心组成部分，对于工程和产品原型、手术模拟、假体设计、牙科、建筑、计算机动画和游戏来说是不可或缺的。对高级建模工具有相当大的需求，能够直接表达用户S的意图，同时能够健壮地处理复杂的模型。曲面优化技术通过指定控制曲面行为的适当优化函数，将高质量曲面的传统从头算建模和现有几何数据的处理统一在一个自然框架中。该框架在概念上是与表示无关的：曲面可以用样条片、细分曲面、网格或隐式形式来逼近。虽然该框架对实际应用具有很高的吸引力，但由于现有的交互技术不够健壮，而更精确和可靠的方法太慢，无法交互使用，曲面优化很少进入工业应用。本研究项目开发了将曲面优化方法引入主流几何建模应用程序所必需的基本数学和算法技术。该研究正在解决以下问题：形状表示及其精度，使得建模的曲面仅依赖于用户输入，而不是特定的采样或近似基础的选择；健壮而高效的建模算法，以确保交互应用中可预测和响应的行为；多样化的控制，以允许足够的自由的任意形状操作。所研究的方法有两个关键部分：多场几何描述，建立在离散几何、高阶几何建模和有限元思想的基础上，以及新颖的多尺度数值求解器，其将粗略尺度的稳健全局算法与精细尺度的快速准确算法相结合，以获得应用所需的性能和稳健性。这些新方法是在高级建模任务的背景下探索的，包括从二维投影或图纸建模、基于特征和基于外观的编辑以及用于牙科CAD的模板拟合。

Computer tools for creating, analyzing and modifying geometry are the core component in computer-aided design systems, indispensable for engineering and product prototyping, surgical simulation, prosthesis design, dentistry, architecture, computer animation and games. There is a considerable demand for high-level modeling tools, expressing the user?s intent directly while being capable of handling complex models robustly. Surface optimization techniques unify traditional ab initio modeling of high-quality surfaces and manipulation of existing geometric data in a single natural framework by specifying suitable optimization functionals governing the surface behavior. This framework is conceptually representation-independent: the surface can be approximated by spline patches, subdivision surfaces, meshes or in implicit form. While this framework is highly appealing for practical applications, surface optimization rarely finds its way into industrial applications, as existing interactive techniques are not sufficiently robust, while the more precise and reliable methods are too slow to be used interactively.This research project develops fundamental mathematical and algorithmic techniques essential for bringing surface optimization methods to mainstream geometric modeling applications. The research is addressing the following problems: shape representation and its accuracy, such that the modeled surfaces depend solely on the user input and not the particular choice of sampling or approximation basis; robust and efficient modeling algorithms to ensure predictable and responsive behavior in interactive applications; diverse control to allow sufficient freedom for arbitrary shape manipulation. The investigated approach has two key components: multifield geometry descriptions, building on ideas from discrete geometry, high-order geometric modeling, and finite elements and novel multiscale numerical solvers that combine robust global algorithms at coarse scales with fast and accurate algorithms at fine scales to achieve the performance and robustness needed in applications. The new methods are explored in the context of high-level modeling tasks, including modeling from two-dimensional projections or drawings, feature-based and appearance-based editing and template fitting for dental CAD.