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

高阶移位边界法：一种用于无边界拟合网格的复杂几何形状的有限元方法

• 批准号: 2207164
• 负责人: Guglielmo Scovazzi
• 金额: $ 38.21万
• 依托单位: Duke University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-08-01 至 2025-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2207164&HistoricalAwards=false
• 关键词: Order; Shifted; Boundary; Method; Finite

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.

高保真度计算方法是一种无价的分析工具，在对复杂物理现象的模拟和理解中都有许多突破。然而，在过去的二十年中，高保真方法面临着迫切的挑战，即要模拟的形状的几何复杂性日益增加。增材制造和优化将设计的几何复杂性提高到了新的高度，并且当前的算法落后。由于高保真方法的特定计算基础架构，设置设计形状的几何描述比实际的计算需要更多的时间。因此，用于物理建模的高保真计算方法通常局限于简单的设计形状。该项目的目的是打破这种障碍，引入了一种新的计算方式模拟复杂几何对象的边界表面。 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复杂的多尺度现象。后来，HO-FEM将其应用领域扩展到工程模拟，其中几何复杂的设计形状非常频繁。在这种情况下，必须使用曲线元素的网格产生来保持边界接近最佳精度。这项任务相当涉及，并且经常经历低水平的自动化，因此整个设计和分析周期的下降速度减速。在2018年，开发了转移的边界方法（SBM）作为传统方法的替代方法。在属于近似/沉浸边界方法的广泛类别的SBM中，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，通过泰勒膨胀来修改（移位）边界条件的值，以保持最佳精度。 SBM是一种非常复杂的几何形状（包括非紧密边界表面的情况）的简单，健壮，准确，有效的算法。该项目旨在开发高阶SBM（HO-SBM）及其对数值稳定性和准确性的数学分析，用于Poisson，Stokes，Darcy和可压缩的Euler方程。 HO-SBM具有多个优点：首先，它不需要沿替代边界弯曲的网格边缘即可获得最佳的精度。复杂的几何形状的特征是替代边界与要模拟形状的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，可以帮助广泛扩散减少订购的建模，机器学习，不确定性量化以及对复杂工程问题的优化方法。与对计算数学和科学领域的研究生的教育，该项目还旨在吸引使用计算机进行设计的本科生感兴趣的学生，通过将其暴露于HO-SBM方法的简化，易于使用的版本中。该奖项反映了NSF的法定任务，并通过使用基础的智力效果和宽阔的范围来评估支持NSF的法定任务，并值得通过评估来进行评估。