Numerical Analysis of Smoothed Particle Hydrodynamics Type Methods via Nonlocal Models

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.