Islet: Interpolation semi-Lagrangian element-based transport

Advection of trace species (tracers), also called tracer transport, in models of the atmosphere and other physical domains is an important and potentially computationally expensive part of a model's dynamical core (dycore). Semi-Lagrangian (SL) advection methods are efficient because they permit a time step much larger than the advective stability limit for explicit Eulerian methods. Thus, to reduce the computational expense of tracer transport, dycores often use SL methods to advect passive tracers. The class of interpolation semi-Lagrangian (ISL) methods contains potentially extremely efficient SL methods. We describe a set of ISL bases for element-based transport, such as for use with atmosphere models discretized using the spectral element (SE) method. An ISL method that uses the natural polynomial interpolant on Gauss-Legendre-Lobatto (GLL) SE nodes of degree at least three is unstable on the test problem of periodic translational flow on a uniform element grid. We derive new alternative bases of up to order of accuracy nine that are stable on this test problem; we call these the Islet bases. Then we describe an atmosphere tracer transport method, the Islet method, that uses three grids that share an element grid: a dynamics grid supporting, for example, the GLL basis of degree three; a physics grid with a configurable number of finite-volume subcells per element; and a tracer grid supporting use of our Islet bases, with particular basis again configurable. This method provides extremely accurate tracer transport and excellent diagnostic values in a number of validation problems. We conclude with performance results that use up to 27,600 NVIDIA V100 GPUs on the Summit supercomputer.

A Monotonic Method of Split Particles

The problem of correct calculation of the motion of a multicomponent (multimaterial) medium is the most serious problem for Lagrangian–Eulerian and Eulerian techniques, especially in multicomponent cells in the vicinity of interfaces. There are two main approaches to solving the advection equation for a multicomponent medium. The first approach is based on the identification of interfaces and determining their position at each time step by the concentration field. In this case, the interface can be explicitly distinguished or reconstructed by the concentration field. The latter algorithm is the basis of widely used methods such as VOF. The second approach involves the use of the particle or marker method. In this case, the material fluxes of substances are determined by the particles with which certain masses of substances bind. Both approaches have their own advantages and drawbacks. The advantages of the particle method consist in the Lagrangian representation of particles and the possibility of” drawbacks. The main disadvantage of the particle method is the strong non-monotonicity of the solution caused by the discrete transfer of mass and mass-related quantities from cell to cell. This paper describes a particle method that is free of this drawback. Monotonization of the particle method is performed by spliting the particles so that the volume of matter flowing out of the cell corresponds to the volume calculated according to standard schemes of Lagrangian–Eulerian and Eulerian methods. In order not to generate an infinite chain of spliting, further split particles are re-united when certain conditions are met. The method is developed for modeling 2D and 3D gas-dynamic flows with accompanying processes, in which it is necessary to preserve the history of the process at Lagrangian points.

Numerical Simulations Using Eulerian Schemes for the Vlasov–Poisson Model

The Vlasov equation describes the temporal evolution of the distribution function of particles in a collisionless plasma and, if magnetic fields are negligible, the mean electric field is prescribed by Poisson equation. Eulerian numerical methods discretize and directly solve the Vlasov equation on a mesh in phase space and can provide high accuracy with low numerical noise. In this paper, we present a comprehensive analysis and comparison between the most used Eulerian methods for the two-dimensional Vlasov–Poisson system, including finite-differences, finite-volumes and semi-Lagrangian ones. The schemes are evaluated and compared through classical problems and conclusions are drawn regarding their accuracy and performance.

Dynamic morphoskeletons in development

Morphogenetic flows in developmental biology are characterized by the coordinated motion of thousands of cells that organize into tissues, naturally raising the question of how this collective organization arises. Using only the kinematics of tissue deformation, which naturally integrates local and global mechanisms along cell paths, we identify the dynamic morphoskeletons behind morphogenesis, i.e., the evolving centerpieces of multicellular trajectory patterns. These features are model- and parameter-free, frame-invariant, and robust to measurement errors and can be computed from unfiltered cell-velocity data. We reveal the spatial attractors and repellers of the embryo by quantifying its Lagrangian deformation, information that is inaccessible to simple trajectory inspection or Eulerian methods that are local and typically frame-dependent. Computing these dynamic morphoskeletons in wild-type and mutant chick and fly embryos, we find that they capture the early footprint of known morphogenetic features, reveal new ones, and quantitatively distinguish between different phenotypes.

A Lagrangian Particle Algorithm (SPH) for an Autocatalytic Reaction Model with Multicomponent Reactants

For the numerical simulation of convection-dominated reacting flow problems governed by convection-reaction equations, grids-based Eulerian methods may cause different degrees of either numerical dissipation or unphysical oscillations. In this paper, a Lagrangian particle algorithm based on the smoothed particle hydrodynamics (SPH) method is proposed for convection-reaction equations and is applied to an autocatalytic reaction model with multicomponent reactants. Four typical Eulerian methods are also presented for comparison, including the high-resolution technique with the Superbee flux limiter, which has been considered to be the most appropriate technique for solving convection-reaction equations. Numerical results demonstrated that when comparing with traditional first- and second-order schemes and the high-resolution technique, the present Lagrangian particle algorithm has better numerical accuracy. It can correctly track the moving steep fronts without suffering from numerical diffusion and spurious oscillations.

Dynamic morphoskeletons in development

Morphogenetic flows in developmental biology are characterized by the coordinated motion of thousands of cells that organize into tissues, naturally raising the question of how this collective organization arises. Using only the Lagrangian kinematics of tissue deformation, which naturally integrates local and global mechanisms along cell paths, we can identifying the Dynamic Morphoskeletons (DM) behind morphogenesis, i.e., the evolving centerpieces of multi-cellular trajectory patterns. The DM is model and parameter-free, frame-invariant, robust to measurement errors, and can be computed from unfiltered cell velocity data. It reveals the spatial attractors and repellers of the embryo, objects that cannot be identified by simple trajectory inspection or Eulerian methods that are local and typically frame-dependent. Computing the DM underlying primitive streak formation in chicken embryo and early gastrulation in the whole fly embryo, we find that the DM captures the early footprint of known morphogenetic features, and reveals new ones, providing a geometric framework to analyze tissue organization.

Modeling Fragmentation within Pagosa Using Particle Methods

The mechanics involved in shock physics often involves materials undergoing large deformations being subjected to high strain rates and temperature variations. When considering high-velocity impacts and explosions, metals experience plastic flow, dynamic failures and fragmentation that are often too complex for a Lagrangian method, such as the finite element method, to properly resolve. Conversely, Eulerian methods are simple to setup, but often result in numerical diffusion errors [1]. These unpleasantries can be skirted by using an alternative technique that incorporates a blend of these aforementioned methods. FLIP+MPM (FLuid Implicit Particle + Material Point Method) employs Lagrangian points to track state quantities associated with materials as strength, as well as conserved quantities, such as mass. Concurrently, an Eulerian grid is used to calculate gradient fields and incorporate an algorithm that carries out the hydrodynamics [2]. By incorporating the FLIP+MPM method into Los Alamos National Laboratory’s Pagosa hydrodynamics code, massively parallel architectures may be employed to solve such problems as those including fragmentation, plastic flow and fluid-structure interaction. This paper will begin with a mathematical description of the FLIP+MPM technique and describe how it fits into Pagosa. After a description of the implementation, the capabilities of this numerical technique are highlighted by simulating fragmentation as a result of high velocity impacts and explosions. Several strength and damage models will be exercised to demonstrate the code’s flexibility. Comparison of the different models’ fragment size distributions are given and discussed.