Jeremy Alan Templeton

A toolbox of hamilton-jacobi solvers for analysis of nondeterministic continuous and hybrid systems

Hamilton-Jacobi partial differential equations have many applications in the analysis of nondeterministic continuous and hybrid systems. Unfortunately, analytic solutions are seldom available and numerical approximation requires a great deal of programming infrastructure. In this paper we describe the first publicly available toolbox for approximating the solution of such equations, and discuss three examples of how these equations can be used in system analysis: cost to go, stochastic differential games, and stochastic hybrid systems. For each example we briefly summarize the relevant theory, describe the toolbox implementation, and provide results.

An eddy-viscosity based near-wall treatment for coarse grid large-eddy simulation

An eddy-viscosity-based near-wall treatment is proposed to enable large-eddy simulations (LES) to be performed on coarse grids. This formulation consists of imposing wall stress boundary conditions and an eddy viscosity in the near-wall region. The wall stress and eddy viscosity have a Reynolds-averaged Navier-Stokes-like character and are obtained from an averaged velocity profile of a resolved LES of channel flow at Reτ=395. Both are tabulated and are used for the instantaneous quantities. The tabulated eddy viscosity is further corrected using the resolved turbulent stress. Numerical results for flow in a channel at several Reynolds numbers ranging from Reτ=395 to Reτ=10000 are presented.

A mathematical framework for multiscale science and engineering : the variational multiscale method and interscale transfer operators.

This report is a collection of documents written as part of the Laboratory Directed Research and Development (LDRD) project A Mathematical Framework for Multiscale Science and Engineering: The Variational Multiscale Method and Interscale Transfer Operators. We present developments in two categories of multiscale mathematics and analysis. The first, continuum-to-continuum (CtC) multiscale, includes problems that allow application of the same continuum model at all scales with the primary barrier to simulation being computing resources. The second, atomistic-to-continuum (AtC) multiscale, represents applications where detailed physics at the atomistic or molecular level must be simulated to resolve the small scales, but the effect on and coupling to the continuum level is frequently unclear.

Comparison of Molecular and Primitive Solvent Models for Electrical Double Layers in Nanochannels.

In a recent article (Lee et al. J. Comput. Theor. Chem., 2012, 8, 2012-2022.), it was shown that an electrolyte solution can be modeled in molecular dynamics (MD) simulations using a uniform dielectric constant in place of a polar solvent to validate Fluid Density Functional Theory (f-DFT) simulations. This technique can be viewed as a coarse-grained approximation of the polar solvent and reduces computational cost by an order of magnitude. However, the consequences of replacing the polar solvent with an effective permittivity are not well characterized, despite its common usage in f-DFT, Monte Carlo simulation, and Poisson-Boltzmann theory. In this paper, we have examined two solvent models of different fidelities with MD simulation of nanochannels. We find that the models produce qualitatively similar ion density profiles, but physical quantities such as electric field, electric potential, and capacitance differ by over an order of magnitude. In all cases, the bulk is explicitly modeled so that surface properties can be evaluated relative to a reference state. Moreover, quantities that define the reference state, such as bulk ion density, bulk solvent density, applied electric field, and temperature, are measurable, so cases with the same thermodynamic state can be compared. Insights into the solvent arrangement, most of which can not be determined from the coarse-grained model, are drawn from the model with an explicitly described polar solvent.

On Near-Wall Dynamic Coupling of LES With RANS Turbulence Models

In this paper, the RANS/LES coupling formulation proposed in [1–3] is adapted for various RANS turbulence models. In that formulation, the LES subgrid-scale eddy-viscosity is replaced in the near-wall region with a RANS eddy-viscosity dynamically corrected with the resolved turbulent stress. The RANS eddy-viscosity is first obtained from precomputed tables. To further generalize the approach, RANS turbulence model equations (for Spalart-Allmaras and k-ω) are then solved simultaneously with the LES. Detailed results are presented for channel flow at Reτ = 395 and compared to traditional LES. The method is then applied to a serpentine passage and compared with DNS computations [4] at Reτ = 180.Copyright

Multiscale modeling for fluid transport in nanosystems.

Atomistic-scale behavior drives performance in many micro- and nano-fluidic systems, such as mircrofludic mixers and electrical energy storage devices. Bringing this information into the traditionally continuum models used for engineering analysis has proved challenging. This work describes one such approach to address this issue by developing atomistic-to-continuum multi scale and multi physics methods to enable molecular dynamics (MD) representations of atoms to incorporated into continuum simulations. Coupling is achieved by imposing constraints based on fluxes of conserved quantities between the two regions described by one of these models. The impact of electric fields and surface charges are also critical, hence, methodologies to extend finite-element (FE) MD electric field solvers have been derived to account for these effects. Finally, the continuum description can have inconsistencies with the coarse-grained MD dynamics, so FE equations based on MD statistics were derived to facilitate the multi scale coupling. Examples are shown relevant to nanofluidic systems, such as pore flow, Couette flow, and electric double layer.

Atom-to-continuum methods for gaining a fundamental understanding of fracture.

This report describes an Engineering Sciences Research Foundation (ESRF) project to characterize and understand fracture processes via molecular dynamics modeling and atom-to-continuum methods. Under this aegis we developed new theory and a number of novel techniques to describe the fracture process at the atomic scale. These developments ranged from a material-frame connection between molecular dynamics and continuum mechanics to an atomic level J integral. Each of the developments build upon each other and culminated in a cohesive zone model derived from atomic information and verified at the continuum scale. This report describes an Engineering Sciences Research Foundation (ESRF) project to characterize and understand fracture processes via molecular dynamics modeling and atom-to-continuum methods. The effort is predicated on the idea that processes and information at the atomic level are missing in engineering scale simulations of fracture, and, moreover, are necessary for these simulations to be predictive. In this project we developed considerable new theory and a number of novel techniques in order to describe the fracture process at the atomic scale. Chapter 2 gives a detailed account of the material-frame connection between molecular dynamics and continuum mechanics we constructed in order to best use atomic information from solid systems. With this framework, in Chapter 3, we were able to make a direct and elegant extension of the classical J down to simulations on the scale of nanometers with a discrete atomic lattice. The technique was applied to cracks and dislocations with equal success and displayed high fidelity with expectations from continuum theory. Then, as a prelude to extension of the atomic J to finite temperatures, we explored the quasi-harmonic models as efficient and accurate surrogates of atomic lattices undergoing thermo-elastic processes (Chapter 4). With this in hand, in Chapter 5 we provide evidence that, by using the appropriate energy potential, the atomic J integral we developed is calculable and accurate at finite/room temperatures. In Chapter 6, we return in part to the fundamental efforts to connect material behavior at the atomic scale to that of the continuum. In this chapter, we devise theory that predicts the onset of instability characteristic of fracture/failure via atomic simulation. In Chapters 7 and 8, we describe the culmination of the project in connecting atomic information to continuum modeling. In these chapters we show that cohesive zone models are: (a) derivable from molecular dynamics in a robust and systematic way, and (b) when used in the more efficient continuum-level finite element technique provide results that are comparable and well-correlated with the behavior at the atomic-scale. Moreover, we show that use of these same cohesive zone elements is feasible at scales very much larger than that of the lattice. Finally, in Chapter 9 we describe our work in developing the efficient non-reflecting boundary conditions necessary to perform transient fracture and shock simulation with molecular dynamics.

Modeling ramp compression experiments using large-scale molecular dynamics simulation.

Molecular dynamics simulation (MD) is an invaluable tool for studying problems sensitive to atomscale physics such as structural transitions, discontinuous interfaces, non-equilibrium dynamics, and elastic-plastic deformation. In order to apply this method to modeling of ramp-compression experiments, several challenges must be overcome: accuracy of interatomic potentials, length- and time-scales, and extraction of continuum quantities. We have completed a 3 year LDRD project with the goal of developing molecular dynamics simulation capabilities for modeling the response of materials to ramp compression. The techniques we have developed fall in to three categories (i) molecular dynamics methods (ii) interatomic potentials (iii) calculation of continuum variables. Highlights include the development of an accurate interatomic potential describing shock-melting of Beryllium, a scaling technique for modeling slow ramp compression experiments using fast ramp MD simulations, and a technique for extracting plastic strain from MD simulations. All of these methods have been implemented in Sandias LAMMPS MD code, ensuring their widespread availability to dynamic materials research at Sandia and elsewhere.

Enhanced molecular dynamics for simulating porous interphase layers in batteries.

Understanding charge transport processes at a molecular level using computational techniques is currently hindered by a lack of appropriate models for incorporating anistropic electric fields in molecular dynamics (MD) simulations. An important technological example is ion transport through solid-electrolyte interphase (SEI) layers that form in many common types of batteries. These layers regulate the rate at which electro-chemical reactions occur, affecting power, safety, and reliability. In this work, we develop a model for incorporating electric fields in MD using an atomistic-to-continuum framework. This framework provides the mathematical and algorithmic infrastructure to couple finite element (FE) representations of continuous data with atomic data. In this application, the electric potential is represented on a FE mesh and is calculated from a Poisson equation with source terms determined by the distribution of the atomic charges. Boundary conditions can be imposed naturally using the FE description of the potential, which then propagates to each atom through modified forces. The method is verified using simulations where analytical or theoretical solutions are known. Calculations of salt water solutions in complex domains are performed to understand how ions are attracted to charged surfaces in the presence of electric fields and interfering media.

ANear-Wall Eddy-Viscosity Formulation for LES

A near-wall eddy-viscosity formulation for LES is presented. This formulation consists of imposing a RANS eddy-viscosity dynamically corrected with the resolved turbulent stress in the near-wall region. The RANS eddy-viscosity is obtained from an averaged velocity profile of a resolved LES of channel flow at Reτ = 395 and stored in a look-up table. Results are presented for channel flow at Reτ = 395 with no-slip boundary conditions, and up to Reτ = 1, 000, 000 using a wall model.