Elements:Software:Open-Source Robust Geometry Toolkit for Black-Box Finite Element Analysis

偏微分方程（PDE）的数值解在科学和工程应用中无处不在，包括弹塑性变形、流体和光散射的模拟。有限元法（FEM）是最常用的偏微分方程离散化方法，特别是在结构和热分析的背景下，由于其通用性和丰富的选择现成的商业实现。理想情况下，PDE求解器应该是一个“黑匣子”：用户提供域边界、边界条件和控制方程作为输入，代码计算输入域的一组用户指定点处的解的值。令人惊讶的是，这与所有现有的开源或商业软件的情况相去甚远，尽管在这个方向上进行了研究，并引起了学术界和工业界的广泛兴趣。在很大程度上，这是由于将网格化和FEM基础构造视为两个不相交的问题，经常使用户暴露于网格化软件与FEM基础构造的接口的技术问题，严格地说，这两者都是求解器内部的技术问题。这种情况对于需要全自动、鲁棒地处理不同大小的大量网格的应用程序来说是一个根本问题，随着大量几何数据的可用，这种情况越来越常见。该提案引入了一个集成管道，将网格和元素设计视为单一挑战，并开发了一个软件平台，以支持对表示为点云、三角形网格或CAD的复杂几何模型进行黑盒分析（计算机辅助设计）模型，本项目旨在开发一套新的形状设计软件，为新的形状设计技术在科学和工程领域的广泛应用打开大门。基于一组新方法的组件，研究人员已经开发出与合理或多精度数值表示的“过滤”使用相结合的方法，以在保持实际性能的同时处理鲁棒性问题。所提出的一组几何处理技术，虽然比现有的技术慢，但在某种意义上总是以最小的输入假设产生有效的结果，这是完全鲁棒的。几何工具包将允许自动将范围扫描、CAD模型或体素网格形式的几何数据转换为表面或体积表示，可直接用于广泛使用的开源有限元方法（FEM）包。它将包括网格生成，除了四面体网格，其他常见类型的离散化：六面体网格，和六边形为主的混合网格。关键的创新是通过仔细混合关键部分的更高精度表示来实现数值鲁棒性，同时依赖于标准的固定精度浮点表示，并设计适合这种方法的算法。由于在绝大多数情况下，需要更高的精度为一个微不足道的小部分计算，这种方法允许用户实现合理的运行时间，同时确保输出的有效性和算法的正确性不完美的，真实的世界数据。其次，将有限元基础的建立与网格质量的解耦精度结合起来。该提案中开发的软件工具包有可能在所有需要复杂几何形状物理现象计算模拟的领域产生重大影响，从而实现数据采集，重建和模拟管道的自动化。该项目的期望是，其结果不仅是减少人类的时间，而且完全自动化这一管道的机会将开辟新的研究场所。除了学术/非营利研究用途外，所有软件的发布都将促进将工作成果集成到商业软件中。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The numerical solution of partial differential equations (PDEs) is ubiquitous in science and engineering applications, including simulation of elastoplastic deformations, fluids, and light scattering. The finite element method (FEM) is the most commonly used discretization of PDEs, especially in the context of structural and thermal analysis, due to its generality and rich selection of off-the-shelf commercial implementations. Ideally, a PDE solver should be a ``black box'': the user provides as input the domain boundary, boundary conditions, and the governing equations, and the code computes the value of the solution at a set of user-specified points of the input domain. This is surprisingly far from being the case for all existing open-source or commercial software, despite the research efforts in this direction and the large academic and industrial interest. To a large extent, this is due to treating meshing and FEM basis construction as two disjoint problems, often exposing the user to the technical issues of interfacing the meshing software with FEM basis construction, both of which, strictly speaking, are technical issues internal to the solver. This state of matters presents a fundamental problem for applications that require fully automatic, robust processing of large collections of meshes of varying sizes, an increasingly common situation as large collections of geometric data become available. This proposal introduces an integrated pipeline, considering meshing and element design as a single challenge, and developing a software platform to enable black box analysis on complex geometric models represented as point clouds, triangle meshes, or CAD (Computer Aided Design) models, opening the door to new shape design technique to a wide range of new applications in sciences and engineering.This project proposes to develop a set of software components based on a set of novel approaches the investigators have developed combined with "filtered" use of rational or multi-precision numerical representations to handle robustness problems while maintaining practical performance. The proposed set of geometry processing techniques, while slower than existing ones, are fully robust in a sense of always produce a valid result with minimal assumptions on the input. The geometric toolkit will allow to automatically convert geometrical data in the form of range scans, CAD models, or voxel grids into a surface or volumetric representation, directly usable in widely used open-source finite element method (FEM) packages. It will include mesh generation, in addition to tetrahedral meshes, for other common types of discretizations: hexahedral meshes, and hex-dominant hybrid meshes. The key innovation is to achieve numerical robustness with minimal added algorithmic complexity by carefully mixing higher precision representations for the critical part, while relying on standard fixed-precision floating point representation for the rest and designing algorithms amenable to this approach. As in overwhelming majority of cases higher accuracy is needed for a vanishingly small fraction of computation, this approach allows the users to achieve sensible running time while ensuring output validity and algorithmic correctness on imperfect, real world data. Secondly, the invetigators will integrate FEM basis construction with meshing decoupling accuracy from mesh quality. The software toolkit developed in this proposal has potential for a major impact in all domains that require computational simulation of physical phenomena in complex geometries, enabling the automation of data acquisition, reconstruction, and simulation pipelines. The expectation of this project is that the outcome will not only be a reduction in human time, but the opportunity to fully automate this pipeline will open new research venues. The release of all the software with a MPL2 license will facilitate integration of the results of the work into commercial software, in addition to academic/non-profit research use.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.