Jeremiah Brackbill

The material-point method for granular materials

Abstract A model for granular materials is presented that describes both the internal deformation of each granule and the interactions between grains. The model, which is based on the FLIP-material point, particle-in-cell method, solves continuum constutitive models for each grain. Interactions between grains are calculated with a contact algorithm that forbids interpenetration, but allows separation and sliding and rolling with friction. The particle-in-cell method eliminates the need for a separate contact detection step. The use of a common rest frame in the contact model yields a linear scaling of the computational cost with the number of grains. The properties of the model are illustrated by numerical solutions of sliding and rolling contacts, and for granular materials by a shear calculation. The results of numerical calculations demonstrate that contacts are modeled accurately for smooth granules whose shape is resolved by the computations mesh.

A numerical method for suspension flow

Abstract Peskins ( J. Comput. Phys. 25 , 220, 1977) immersed boundary technique is modified to give a new numerical method for studying a fluid with suspended elastic particles. As before, the presence of the suspended particles is transmitted to the fluid through a force density term in the fluid equations. As a result, one set of equations holds in the entire computational domain, eliminating the need to apply boundary conditions on the surface of suspended objects. The new method computes the force density by discretizing the stress-strain constitutive equations for an elastic solid on a grid, using data provided by clusters of Lagrangian points. This approach clearly specifies the material properties of the suspended objects. A simple data structure for the Lagrangian points makes it easy to model suspended solids with arbitrary shape and size. The method is validated by comparing numerical results for elastic vibrations and particle settling in viscous fluids, with theory and analysis. The capability of the method to do a wide range of problems is illustrated by qualitative results for lubrication and cavity flow problems.

Mass matrix formulation of the FLIP particle-in-cell method

A refinement of FLIP [Brackbill and Ruppel, J. Comput. Phys. 65, 314 (1986)] is described which uses a mass matrix formulation to achieve greater accuracy and less numerical diffusion over the previous version. Without the refinement, there is a significant dissipation of energy in modeling elastic vibrations of a solid. Moreover, in modeling an initial flow discontinuity there are sub-grid-scale oscillations in the particle velocity field in the neighborhood of the discontinuity. These difficulties are eliminated using the mass matrix. In addition, the mass matrix formulation conserves kinetic energy, linear and angular momentum, and is Galilean invariant.

Two-way coupling of a global Hall magnetohydrodynamics model with a local implicit particle-in-cell model

Computational models based on a fluid description of the plasma, such as magnetohydrodynamic (MHD) and extended magnetohydrodynamic (XMHD) codes are highly efficient, but they miss the kinetic effe ...

Coordinate system control: Adaptive meshes

The discussion of the formulation and properties of the adaptive mesh algorithm is as complete as space permits. As should be clear from the discussion, adaptive gridding is most useful for singular perturbation problems where localized regions with large gradients develop spontaneously. Often, choosing the correct weight function is not trivial, and, sometimes, it is appropriate to cause the mesh to respond to structure in several dependent variables simultaneously.

Applications and generalizations of variational methods for generating adaptive meshes

Abstract : Generating computation meshes for irregular regions have been of interest to a lot of people in many areas of research for a long time. One technique that has met with sucess over the long run has been to generate methods using an elliptic equation or a system of elliptic equations. The technique in its simplest form, uses a system of Laplace equations which are solved by direct or iterative methods. As people gained more experience with this method, source terms were added to the Laplace equations to gain additional control of the mesh. In variable coefficients of the derivatives were added for further flexibility. In this paper the authors work with a method that systematically generates a set of elliptic equations without having to explicitly perturb a set of Laplace equations with source terms and variable coefficients. This technique uses the variational methods often associated with elliptic equations. Following this introduction, they briefly discuss the variational formulation in two-dimensional cartesian geometry. Then the formulation will be generalized to three dimensions. Next, several three-dimensional test problems will be shown. After displaying these three-dimensional test results, the author wil the exhibit an application of the mesh generation technique in two dimensions. This application involves generating an adaptive mesh for a supersonic flow past a step in a wind tunnel. (Author)

An implicit moment electromagnetic plasma simulation in cylindrical coordinates

An electromagnetic PIC plasma simulation code incorporating the implicit moment method and a two-dimensional cylindrical mesh, with r- and z-coordinate dependence, has been developed. The code is an extension of the VENUS code from the original two-dimensional Cartesian mesh. The physical model employed in the code will be discussed, with emphasis on aspects unique to cylindrical geometry. An application to self-generated magnetic fields and electron transport in a laser-irradiated disk is presented that highlights the usefulness of cylindrical coordinates.

Shear deformation in granular materials

An investigation into the properties of granular materials is undertaken via numerical simulation. These simulations highlight that frictional contact, a defining characteristic of dry granular materials, and interfacial debonding, an expected deformation mode in plastic bonded explosives, must be properly modeled. Frictional contact and debonding algorithms have been implemented into FLIP, a particle in cell code, and are described. Frictionless and frictional contact are simulated, with attention paid to energy and momentum conservation. Debonding is simulated, with attention paid to the interfacial debonding speed. A first step toward calculations of shear deformation in plastic bonded explosives is made. Simulations are performed on the scale of the grains where experimental data is difficult to obtain. Two characteristics of deformation are found, namely the intermittent binding of grains when rotation and translation are insufficient to accommodate deformation, and the role of the binder as a lubricant in force chains.

On energy and momentum conservation in particle-in-cell plasma simulation

Abstract Particle-in-cell (PIC) plasma simulations are a productive and valued tool for the study of nonlinear plasma phenomena, yet there are basic questions about the simulation methods themselves that remain unanswered. Here we study energy and momentum conservation by PIC. We employ both analysis and simulations of one-dimensional, electrostatic plasmas to understand why PIC simulations are either energy or momentum conserving but not both, what role a numerical stability plays in non-conservation, and how errors in conservation scale with the numerical parameters. Conserving both momentum and energy make it possible to model problems such as Jeans-type equilibria. Avoiding numerical instability is useful, but so is being able to identify when its effect on the results may be important. Designing simulations to achieve the best possible accuracy with the least expenditure of effort requires results on the scaling of error with the numerical parameters. Our results identify the central role of Gauss law in conservation of both momentum and energy, and the significant differences in numerical stability and error scaling between energy-conserving and momentum-conserving simulations.

Short Note: Boundary conditions for Maxwell solvers

which is solved in implicit plasma simulations. Compared with the standard Helmholtz equation [4, p68], (c∆t/2) replaces −(c/ω), and Eq. 1 is solved each time step (instead of for each value of ω) for E with S given. The source term, S, is neither divergence nor curl-free in general. Specifically, we ask under what conditions charge conservation is satisfied. Namely, do numerical solutions of Eq. 1 satisfy Poisson’s equation,