GIZMO: A New Class of Accurate, Mesh-Free Hydrodynamic Simulation Methods

We present two new Lagrangian methods for hydrodynamics, in a systematic comparison with moving-mesh, SPH, and stationary (non-moving) grid methods. The new methods are designed to simultaneously capture advantages of both smoothed-particle hydrodynamics (SPH) and grid-based/adaptive mesh refinement (AMR) schemes. They are based on a kernel discretization of the volume coupled to a high-order matrix gradient estimator and a Riemann solver acting over the volume 'overlap.' We implement and test a parallel, second-order version of the method with self-gravity & cosmological integration, in the code GIZMO: this maintains exact mass, energy and momentum conservation; exhibits superior angular momentum conservation compared to all other methods we study; does not require 'artificial diffusion' terms; and allows the fluid elements to move with the flow so resolution is automatically adaptive. We consider a large suite of test problems, and find that on all problems the new methods appear competitive with moving-mesh schemes, with some advantages (particularly in angular momentum conservation), at the cost of enhanced noise. The new methods have many advantages vs. SPH: proper convergence, good capturing of fluid-mixing instabilities, dramatically reduced 'particle noise' & numerical viscosity, more accurate sub-sonic flow evolution, & sharp shock-capturing. Advantages vs. non-moving meshes include: automatic adaptivity, dramatically reduced advection errors & numerical overmixing, velocity-independent errors, accurate coupling to gravity, good angular momentum conservation and elimination of 'grid alignment' effects. We can, for example, follow hundreds of orbits of gaseous disks, while AMR and SPH methods break down in a few orbits. However, fixed meshes minimize 'grid noise.' These differences are important for a range of astrophysical problems.Comment: 57 pages, 33 figures. MNRAS. A public version of the GIZMO code, user's guide, test problem setups, and movies are available at http://www.tapir.caltech.edu/~phopkins/Site/GIZMO.htm

Study on SPH Viscosity Term Formulations

For viscosity-dominated flows, the viscous effect plays a much more important role. Since the viscosity term in SPH-governing (Smoothed Particle Hydrodynamics) equations involves the discretization of a second-order derivative, its treatment could be much more challenging than that of a first-order derivative, such as the pressure gradient. The present paper summarizes a series of improved methods for modeling the second-order viscosity force term. By using a benchmark patch test, the numerical accuracy and efficiency of different approaches are evaluated under both uniform and non-uniform particle configurations. Then these viscosity force models are used to compute a documented lid-driven cavity flow and its interaction with a cylinder, from which the most recommended viscosity term formulation has been identified

Enhanced SPH modeling of free-surface flows with large deformations

The subject of the present thesis is the development of a numerical solver to study the violent interaction of marine flows with rigid structures. Among the many numerical models available, the Smoothed Particle Hydrodynamics (SPH) has been chosen as it proved appropriate in dealing with violent free-surface flows. Due to its Lagrangian and meshless character it can naturally handle breaking waves and fragmentation that generally are not easily treated by standard methods. On the other hand, some consolidated features of mesh-based methods, such as the solid boundary treatment, still remain unsolved issues in the SPH context. In the present work a great part of the research activity has been devoted to tackle some of the bottlenecks of the method. Firstly, an enhanced SPH model, called delta-SPH, has been proposed. In this model, a proper numerical diffusive term has been added in the continuity equation in order to remove the spurious numerical noise in the pressure field which typically affects the weakly-compressible SPH models. Then, particular attention has been paid to the development of suitable techniques for the enforcement of the boundary conditions. As for the free-surface, a specific algorithm has been designed to detect free-surface particles and to define a related level-set function with two main targets: to allow the imposition of peculiar conditions on the free-surface and to analyse and visualize more easily the simulation outcome (especially in 3D cases). Concerning the solid boundary treatment, much effort has been spent to devise new techniques for handling generic body geometries with an adequate accuracy in both 2D and 3D problems. Two different techniques have been described: in the first one the standard ghost fluid method has been extended in order to treat complex solid geometries. Both free-slip and no-slip boundary conditions have been implemented, the latter being a quite complex matter in the SPH context. The proposed boundary treatment proved to be robust and accurate in evaluating local and global loads, though it is not easy to extend to generic 3D surfaces. The second technique has been adopted for these cases. Such a technique has been developed in the context of Riemann-SPH methods and in the present work is reformulated in the context of the standard SPH scheme. The method proved to be robust in treating complex 3D solid surfaces though less accurate than the former. Finally, an algorithm to correctly initialize the SPH simulation in the case of generic geometries has been described. It forces a resettlement of the fluid particles to achieve a regular and uniform spacing even in complex configurations. This pre-processing procedure avoids the generation of spurious currents due to local defects in the particle distribution at the beginning of the simulation. The delta-SPH model has been validated against several problems concerning fluid-structure interactions. Firstly, the capability of the solver in dealing with water impacts has been tested by simulating a jet impinging on a flat plate and a dam-break flow against a vertical wall. In this cases, the accuracy in the prediction of local loads and of the pressure field have been the main focus. Then, the viscous flow around a cylinder, in both steady and unsteady conditions, has been simulated comparing the results with reference solutions. Finally, the generation and propagation of 2D gravity waves has been simulated. Several regimes of propagation have been tested and the results compared against a potential flow solver. The developed numerical solver has been applied to several cases of free-surface flows striking rigid structures and to the problem of the generation and evolution of ship generated waves. In the former case, the robustness of the solver has been challenged by simulating 2D and 3D water impacts against complex solid surfaces. The numerical outcome have been compared with analytical solutions, experimental data and other numerical results and the limits of the model have been discussed. As for the ship generated waves, the problem has been firstly studied within the 2D+t approximation, focusing on the occurrence and features of the breaking bow waves. Then, a dedicated 3D SPH parallel solver has been developed to tackle the simulation of the entire ship in constant forward motion. This simulation is quite demanding in terms of complexities of the boundary geometry and computational resources required. The wave pattern obtained has been compared against experimental data and results from other numerical methods, showing in both the cases a fair and promising agreement

Modeling free-surface solitary waves with SPH

A weakly compressible SPH solver is presented and applied to simulate free- surface solitary waves generated in a dam-break experiment. Wave propaga- tion speeds are compared with the exact solutions of the Korteweg-de Vries (KdV) equation as a first order theory and a second order approximation investigated in the literature. Test cases are constructed based on the mea- surement layouts of a dam-break experiment. Free surface shapes of different simulation cases are compared with the KdV-shapes. The simulation results show good agreement with the second order approximation of solitary wave propagation speeds

SPH-ALE Scheme for Weakly Compressible Viscous Flow with a Posteriori Stabilization

[Abstract] A highly accurate SPH method with a new stabilization paradigm has been introduced by the authors in a recent paper aimed to solve Euler equations for ideal gases. We present here the extension of the method to viscous incompressible flow. Incompressibility is tackled assuming a weakly compressible approach. The method adopts the SPH-ALE framework and improves accuracy by taking high-order variable reconstruction of the Riemann states at the midpoints between interacting particles. The moving least squares technique is used to estimate the derivatives required for the Taylor approximations for convective fluxes, and also provides the derivatives needed to discretize the viscous flux terms. Stability is preserved by implementing the a posteriori Multi-dimensional Optimal Order Detection (MOOD) method procedure thus avoiding the utilization of any slope/flux limiter or artificial viscosity. The capabilities of the method are illustrated by solving one- and two-dimensional Riemann problems and benchmark cases. The proposed methodology shows improvements in accuracy in the Riemann problems and does not require any parameter calibration. In addition, the method is extended to the solution of viscous flow and results are validated with the analytical Taylor–Green, Couette and Poiseuille flows, and lid-driven cavity test cases.This research was funded by Ministerio de Ciencia, Innovación y Universidades of the Spanish Government Grant #RTI2018-093366-B-I00, by the Consellería de Educación e Ordenación Universitaria of the Xunta de Galicia (grant#ED431C 2018/41)Xunta de Galicia; ED431C 2018/4

Modeling Free-surface Solitary Waves with Smoothed Particle Hydrodynamics

A three-dimensional weakly compressible Smoothed ParticleHydrodynamics (SPH) solver is presented and applied tosimulate free-surface solitary waves generated in a quasi twodimensionaldam-break experiment. Test cases are constructedbased on the measurement layouts of a dam-break experiment.The simulated wave propagation speeds are compared to theexact solutions of the Korteweg-de Vries (KdV) equation as afirst order theory, and to a second order iterative approximationinvestigated in the literature. Free surface shapes of differentsimulation cases are investigated as well. The results show goodagreement with the free surface shapes of the KdV equation aswell as with the second order approximation of solitary wavepropagation speeds

Lagrangian Numerical Methods for Ocean Biogeochemical Simulations

We propose two closely--related Lagrangian numerical methods for the simulation of physical processes involving advection, reaction and diffusion. The methods are intended to be used in settings where the flow is nearly incompressible and the P\'eclet numbers are so high that resolving all the scales of motion is unfeasible. This is commonplace in ocean flows. Our methods consist in augmenting the method of characteristics, which is suitable for advection--reaction problems, with couplings among nearby particles, producing fluxes that mimic diffusion, or unresolved small-scale transport. The methods conserve mass, obey the maximum principle, and allow to tune the strength of the diffusive terms down to zero, while avoiding unwanted numerical dissipation effects