The High-Order Shifted Boundary Method: A Finite Element Method for Complex Geometries without Boundary-Fitted Grids

高保真计算方法是一种非常宝贵的分析工具，在模拟和理解复杂物理现象方面有许多突破。然而，在过去的二十年里，高保真方法面临着几何形状的模拟越来越复杂的严峻挑战。增材制造和优化将设计的几何复杂性提高到了新的高度，而目前的算法已经落后。由于高保真方法的特定计算基础设施，建立设计形状的几何描述比实际计算花费更多的时间。因此，用于物理建模的高保真计算方法常常被限制于简单的设计形状。该项目旨在打破这一障碍，引入一种新的方法来计算复杂几何物体的边界表面。该项目旨在改变我们所知的计算领域，促进科学计算中高保真方法的复兴，在科学和工程的所有领域都有广泛的好处，包括模拟与人工智能和其他元算法的接口，数字孪生等。高阶有限元方法（HO-FEM）最初应用于计算物理问题，其主要目标是支持对复杂多尺度现象的科学理解。后来，HO-FEM的应用领域扩展到工程模拟，其中几何复杂的设计形状是非常常见的。在这种情况下，网格生成与曲线元素是必要的，以保持最佳的精度附近的边界。这项任务相当复杂，而且自动化程度往往很低，因此整个设计和分析周期会变慢。2018年，作为传统方法的替代方案，开发了移动边界方法（SBM）。在SBM中，它属于近似/浸入边界方法的广泛类别，应用边界条件的位置从真实边界转移到近似（替代）边界。同时，边界条件的值，应用弱，修改（移动）通过泰勒展开，以保持最佳的精度。SBM是一种简单、强大、准确且高效的算法，适用于非常复杂的几何形状，包括非水密边界表面的情况。本计画的目的是发展高阶数值模式（HO-SBM）及其数值稳定性与精确度的数学分析，适用于Poisson、Stokes、Darcy与可压缩的Euler方程。HO-SBM有几个优点：首先，它不需要弯曲的网格边缘沿着代理边界，以获得最佳的精度。复杂几何的特征在于要模拟的形状的代理边界和真实边界之间的距离。因此，HO-SBM与当前的CAD和网格生成具有灵活的集成，可以帮助降阶建模，机器学习，不确定性量化和优化方法广泛传播到复杂的工程问题。除了培养计算数学和科学的研究生外，该项目还旨在吸引对计算设计感兴趣的本科生，让他们接触简化的、易于使用的HO-SBM方法。该奖项反映了NSF的法定使命，通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

High fidelity computational methods are an invaluable tool for analysis, with many breakthroughs in the simulation and understanding of complex physics phenomena. However, over the past two decades, high-fidelity methods have faced the daunting challenge of an increasing geometric complexity of the shapes to be simulated. Additive manufacturing and optimization raised the geometric complexity of designs to new heights, and the current algorithms are lagging behind. Because of the specific computational infrastructure of a high-fidelity method, setting up the geometrical description of design shapes takes more time than the actual computation. Consequently, high-fidelity computational methods for physics modeling have often been confined to simple design shapes. This project is aimed at breaking this barrier, introducing a new way of computationally model the boundary surfaces of complex geometrical objects. This project aims to transform the field of computing as we know it, fostering a renaissance of high-fidelity methods in scientific computing, with broad benefits in all fields of science and engineering, including the interface of simulation with artificial intelligence and other meta-algorithms, digital twins, etc.High-Order Finite Element Methods (HO-FEMs) were originally applied to computational physics problems, with the primary goal of supporting the scientific understanding of complex multi-scale phenomena. Later, HO-FEMs have extended their realm of applications to engineering simulations, in which geometrically complex design shapes are very frequent. In this case, mesh generation with curvilinear elements is necessary to retain optimal accuracy near boundaries. This task is rather involved, and low levels of automation are often experienced, with a consequent slow-down of the entire design and analysis cycle. In 2018, the Shifted Boundary Method (SBM) was developed as an alternative to traditional methods. In the SBM, which belongs to the broad class of approximate/immersed boundary methods, the location where boundary conditions are applied is shifted from the true boundary to an approximate (surrogate) boundary. At the same time, the value of boundary conditions, applied weakly, is modified (shifted) by means of Taylor expansions to maintain optimal accuracy. The SBM is a simple, robust, accurate and efficient algorithm for very complex geometries, including the case of non-watertight boundary surfaces. This project aims at developing the higher-order SBM (HO-SBM) and its mathematical analysis of numerical stability and accuracy, for the Poisson, Stokes, Darcy, and compressible Euler equations. HO-SBM has several advantages: first and foremost, it does not require curved grid edges along the surrogate boundary to obtain optimal accuracy. Complex geometries are characterized by the distance between the surrogate boundary and true boundary of the shapes to be simulated. Hence, the HO-SBM has a flexible integration with current CAD and mesh generation and can help the broad diffusion of reduced-order modeling, machine learning, uncertainty quantification, and optimization methods to complex engineering problems. Together with the education of a graduate student in computational mathematics and sciences, this projects also aims at attracting undergraduate students interested using computing for design, by exposing them to simplified, easy-to-use versions of the HO-SBM method.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.