Numerical Analysis of Smoothed Particle Hydrodynamics Type Methods via Nonlocal Models

计算流体力学是一个重要的研究领域，它在理解许多机械、流体动力学和生物物理过程中出现的流体流动方面发挥着至关重要的作用。它在计算科学的发展中占有中心地位。这项研究旨在帮助设计有效的数值算法，特别是那些与所谓的光滑粒子流体动力学(SPH)相关的算法，用于模拟复杂的流体和界面现象。总体研究目标与预测和可靠计算科学的长期愿景是一致的，在短期内，它有助于补充各学术机构和国家实验室目前正在进行的与SPH相关的方法及其应用的研究。国际和平研究所不仅致力于促进研究工作，而且还将加强对年轻学生和初级研究人员的培训和教育。他将与合作者合作，确保及时将新的理论发现转化为增强的模拟能力，用于各种应用，例如涉及与重要的国家和社会利益高度相关的地下、大气和生物物理系统、能源和高强度材料中的非均匀传输的应用。基于粒子的计算方法，如光滑粒子流体力学(SPH)及其相关方法，为数值模拟提供了极大的灵活性，并在各种科学和工程应用中得到了广泛的应用。随着这些技术被填充到大型计算科学和工程社区使用的主要模拟代码中，作为严格验证和验证过程的一部分，进行更定量的评估和数学分析势在必行。评估基于SPH的模拟是具有挑战性的，因为这些方法历史上被应用于解决复杂的问题，其中要么传统方法不能很好地工作，要么形式上的准确性是次要的。基于传统数值分析技术的研究可能并不总是产生与实践密切相关的数学结果。我们提出的研究是为了提高对SPH和相关方法的理论理解。我们方法的一个新奇之处在于利用了最近发展起来的非局部模型及其由PI小组进行的数值逼近。它通过区分和联系局部定义的空间导数的积分核表示/近似和由此产生的非局部算子的数值离散化的不同作用，为分析SPH型方法提供了新的途径。事实上，微分算子在连续谱水平上不适当的非局部松弛可能是基于粒子的模拟所固有的一些问题的根本原因。不兼容的离散化也可能导致保真度和稳定性的丧失。通过将非局部积分算子和非局部连续介质公式作为连接连续介质偏微分方程组模型和类粒子离散近似的桥梁，我们的方法与传统的直接比较离散格式和基本连续介质偏微分方程组的数值分析方法有了很大的不同。对算法稳健性的关注与SPH类方法特别相关，因为它们旨在应用于涉及多个尺度和极端操作条件的复杂系统。未来几年的具体目标包括：发展具有非局域/积分算子的类SPH方法的连续体重构；以非局域模型的离散化为桥梁研究SPH类方法。在执行拟议的工作时，我们使用集成的分析和计算方法来提供理论分析所需的数学基础设施和代码开发的实践见解。我们密切关注解决缺乏正则性或表现出强烈变化的解的技术，以及在实际实施中经常遇到的粒子分布和边界条件。

Computational fluid dynamics is an important research field that plays a crucial role in the understanding of fluid flows appearing in many mechanical, hydrodynamic and biophysical processes. It occupies a central place in the development of computational science. The proposed research intends to help designing effective numerical algorithms, particularly those related to the so-called smoothed particle hydrodynamics (SPH), for modeling complex fluids and interfacial phenomena. The overall research objective is consistent with the long term vision of predictive and reliable computational science, and in the near term, it serves to complement ongoing research on SPH related methods and their applications currently being carried out by various academic institutions and national laboratories. The PI will not only work to facilitate the research effort but also to strengthen the training and education of young students and junior researchers. He will team up with collaborators to ensure the timely translation and integration of new theoretical findings into enhanced simulation capability for a variety of applications such as those involving heterogeneous transport in underground, atmospheric and biophysical systems, energy and high-strength materials, which are highly relevant to important national and societal interests. Particle based computational methods such as the Smoothed Particle Hydrodynamics (SPH) and related methods offer great flexibility in numerical simulations and are becoming widely used in various scientific and engineering applications. As these techniques get populated into major simulation codes to be used by a large computational science and engineering community, it is imperative to carry out a more quantitative assessment and mathematical analysis as part of the rigorous validation and verification process. Assessing SPH based simulations is challenging since these methods have been historically applied to solve complex problems where either traditional methods do not work well or the formal accuracy is of secondary concern. Studies based on conventional numerical analysis techniques may not always produce mathematical findings that are strongly relevant in practice. Our proposed research is to improve the theoretical understanding of SPH and related methods. A novelty of our approach draws on the recently developed nonlocal models and their numerical approximations by the PI's group. It leads to new avenues to analyze SPH type methods by both distinguishing and relating the different roles of integral kernel representations/approximations of the locally defined spatial derivatives and numerical discretization of the resulting nonlocal operators. Indeed, inappropriate nonlocal relaxations of differential operators on the continuum level may be the root cause of some problematic issues inherent to particle based simulations. Incompatible discretization can also contribute to the loss of fidelity and stability. By taking nonlocal integral operators and nonlocal continuum formulations as bridges connecting continuum PDE models and particle like discrete approximations, our approach represents a significant departure from conventional numerical analysis that compares the discrete schemes with the underlying continuum PDEs directly. The focus on algorithm robustness is particularly relevant to SPH like methods given their intended application to complex systems involving multiple scales and extreme operating conditions. Specific objectives for the next few years include: developing continuum reformulations of SPH like methods with nonlocal/integral operators; and studying SPH type methods using discretization of nonlocal models as a bridge. In carrying out the proposed work, we use an integrated analytical and computational approach to provide both mathematical infrastructure needed for theoretical analyses and practical insight for code development. We pay close attention to techniques that work for solutions lacking regularity or exhibiting strong variations and for particle distributions and boundary conditions that are frequently encountered in practical implementations.