Numerical Analysis of Smoothed Particle Hydrodynamics Type Methods via Nonlocal Models

计算流体动力学是一个重要的研究领域，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起着至关重要的作用。它在计算科学的发展中占有中心地位。 拟议的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），复杂的流体和界面现象建模。 总体研究目标与预测性和可靠的计算科学的长期愿景一致，在短期内，它有助于补充目前正在进行的SPH相关方法及其应用的研究，目前正在由各学术机构和国家实验室进行。 PI不仅将努力促进研究工作，而且还将加强对青年学生和初级研究人员的培训和教育。他将与合作者合作，以确保新的理论研究成果及时转化和整合到各种应用的增强模拟能力中，例如涉及地下，大气和生物物理系统，能源和高强度材料中的异质传输的应用，这些应用与重要的国家和社会利益高度相关。 基于粒子的计算方法，如光滑粒子流体动力学（SPH）和相关方法提供了极大的灵活性，在数值模拟，并正在成为广泛使用的各种科学和工程应用。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，必须进行更定量的评估和数学分析，作为严格验证和验证过程的一部分。评估基于SPH的模拟是具有挑战性的，因为这些方法历来被应用于解决复杂的问题，无论是传统的方法不能很好地工作，或正式的准确性是次要的问题。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。我们的研究目的是提高对SPH及其相关方法的理论认识。我们的方法的新奇借鉴了最近开发的非局部模型和它们的数值近似PI的组。它导致了新的途径来分析SPH型方法的区别和相关的不同角色的积分核表示/近似的局部定义的空间导数和数值离散化所产生的非局部运营商。事实上，不适当的非局部松弛的微分算子在连续水平上可能是一些问题的根本原因固有的粒子模拟。不相容的离散化也会导致保真度和稳定性的损失。通过采取非局部积分算子和非局部连续介质配方作为桥梁连接连续介质PDE模型和颗粒样离散近似，我们的方法代表了一个显着偏离传统的数值分析，比较离散计划与底层连续介质PDE直接。算法鲁棒性的重点是特别相关的SPH类方法，其预期的应用程序涉及多个尺度和极端的操作条件的复杂系统。未来几年的具体目标包括：开发连续的SPH类似的方法与非本地/积分算子的重新制定;和研究SPH型方法使用离散化的非本地模型作为桥梁。 在进行拟议的工作，我们使用一个集成的分析和计算方法，提供理论分析和代码开发的实际见解所需的数学基础设施。我们密切关注的技术，工作的解决方案缺乏规律性或表现出强烈的变化，并为粒子分布和边界条件，在实际应用中经常遇到。

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.