Publications

Bond, Stephen D.; Costa, Timothy C.

Abstract not provided.

Bond, Stephen D.; Lehoucq, Richard B.

Abstract not provided.

Granzow, Brian N.; Voth, Thomas E.; Ibanez-Granados, Daniel A.; Bond, Stephen D.; Hansen, Glen H.

Abstract not provided.

Bond, Stephen D.; Lehoucq, Richard B.

Abstract not provided.

Bond, Stephen D.; Rowe, Stephen R.

Abstract not provided.

Parks, Michael L.; Lehoucq, Richard B.; Bond, Stephen D.; Summers, Randall M.; Collis, Samuel S.

Abstract not provided.

Baczewski, Andrew D.; Yarrington, Cole Y.; Bond, Stephen D.; Erikson, William W.; Lehoucq, Richard B.; Mondy, L.A.; Noble, David R.; Pierce, Flint P.; Roberts, Christine C.; Van Swol, Frank

The subject of this work is the development of models for the numerical simulation of matter, momentum, and energy balance in heterogeneous materials. These are materials that consist of multiple phases or species or that are structured on some (perhaps many) scale(s). By computational mechanics we mean to refer generally to the standard type of modeling that is done at the level of macroscopic balance laws (mass, momentum, energy). We will refer to the flow or flux of these quantities in a generalized sense as transport. At issue here are the forms of the governing equations in these complex materials which are potentially strongly inhomogeneous below some correlation length scale and are yet homogeneous on larger length scales. The question then becomes one of how to model this behavior and what are the proper multi-scale equations to capture the transport mechanisms across scales. To address this we look to the area of generalized stochastic process that underlie the transport processes in homogeneous materials. The archetypal example being the relationship between a random walk or Brownian motion stochastic processes and the associated Fokker-Planck or diffusion equation. Here we are interested in how this classical setting changes when inhomogeneities or correlations in structure are introduced into the problem. Aspects of non-classical behavior need to be addressed, such as non-Fickian behavior of the mean-squared-displacement (MSD) and non-Gaussian behavior of the underlying probability distribution of jumps. We present an experimental technique and apparatus built to investigate some of these issues. We also discuss diffusive processes in inhomogeneous systems, and the role of the chemical potential in diffusion of hard spheres is considered. Also, the relevance to liquid metal solutions is considered. Finally we present an example of how inhomogeneities in material microstructure introduce fluctuations at the meso-scale for a thermal conduction problem. These fluctuations due to random microstructures also provide a means of characterizing the aleatory uncertainty in material properties at the mesoscale.

Bolintineanu, Dan S.; Lechman, Jeremy B.; Bond, Stephen D.; Yarrington, Cole Y.; Knepper, Robert; Kramer, Sharlotte L.; Waymel, Robert W.; Quintana, Enrico C.; Long, Kevin N.

Abstract not provided.

Bond, Stephen D.; Franke, Brian C.; Lehoucq, Richard B.; Smith, John D.

Abstract not provided.

Beckwith, Kristian B.; Beckwith, Kristian B.; Beckwith, Kristian B.; Beckwith, Kristian B.; Bond, Stephen D.; Bond, Stephen D.; Bond, Stephen D.; Bond, Stephen D.; Granzow, Brian N.; Granzow, Brian N.; Granzow, Brian N.; Granzow, Brian N.; Jennings, Christopher A.; Jennings, Christopher A.; Jennings, Christopher A.; Jennings, Christopher A.; Martin, Matthew; Martin, Matthew; Martin, Matthew; Martin, Matthew; Porwitzky, Andrew J.; Porwitzky, Andrew J.; Porwitzky, Andrew J.; Porwitzky, Andrew J.; Stagg, Alan K.; Stagg, Alan K.; Stagg, Alan K.; Stagg, Alan K.; Voth, Thomas E.; Voth, Thomas E.; Voth, Thomas E.; Voth, Thomas E.

Abstract not provided.

Voth, Thomas E.; Beckwith, Kristian B.; Bond, Stephen D.; Granzow, Brian N.; Hamlin, Nathaniel D.; Hanshaw, Heath L.; Hansen, Glen H.; Jennings, Christopher A.; Love, Edward L.; Martin, Matthew; Shulenburger, Luke N.; Stagg, Alan K.

Abstract not provided.

Beckwith, Kristian B.; Bond, Stephen D.; Granzow, Brian N.; Hamlin, Nathaniel D.; Hansen, Glen H.; Hanshaw, Heath L.; Jennings, Christopher A.; Martin, Matthew; Stagg, Alan K.; Voth, Thomas E.

Abstract not provided.

Voth, Thomas E.; Beckwith, Kristian B.; Bond, Stephen D.; Granzow, Brian N.; Hamlin, Nathaniel D.; Ibanez-Granados, Daniel A.; Jennings, Christopher A.; Martin, Matthew; Stagg, Alan K.

Abstract not provided.

Beckwith, Kristian B.; Beckwith, Kristian B.; Bond, Stephen D.; Bond, Stephen D.; Granzow, Brian N.; Granzow, Brian N.; Hamlin, Nathaniel D.; Hamlin, Nathaniel D.; Martin, Matthew; Martin, Matthew; Powell, Michael P.; Powell, Michael P.; Ruggirello, Kevin P.; Ruggirello, Kevin P.; Stagg, Alan K.; Stagg, Alan K.; Voth, Thomas E.; Voth, Thomas E.

Abstract not provided.

Granzow, Brian N.; Bond, Stephen D.; Martin, Matthew; Hamlin, Nathaniel D.; Stagg, Alan K.; Voth, Thomas E.

Abstract not provided.

Voth, Thomas E.; Granzow, Brian N.; Stagg, Alan K.; Bond, Stephen D.; Hamlin, Nathaniel D.; Martin, Matthew; Shulenburger, Luke N.

Abstract not provided.

Roberts, Nathan V.; Bond, Stephen D.; Miller, Sean A.; Cyr, Eric C.

Plasma physics simulations are vital for a host of Sandia mission concerns, for fundamental science, and for clean energy in the form of fusion power. Sandia's most mature plasma physics simulation capabilities come in the form of particle-in-cell (PIC) models and magnetohydrodynamics (MHD) models. MHD models for a plasma work well in denser plasma regimes when there is enough material that the plasma approximates a fluid. PIC models, on the other hand, work well in lower-density regimes, in which there is not too much to simulate; error in PIC scales as the square root of the number of particles, making high-accuracy simulations expensive. Real-world applications, however, almost always involve a transition region between the high-density regimes where MHD is appropriate, and the low-density regimes for PIC. In such a transition region, a direct discretization of Vlasov is appropriate. Such discretizations come with their own computational costs, however; the phase-space mesh for Vlasov can involve up to six dimensions (seven if time is included), and to apply appropriate homogeneous boundary conditions in velocity space requires meshing a substantial padding region to ensure that the distribution remains sufficiently close to zero at the velocity boundaries. Moreover, for collisional plasmas, the right-hand side of the Vlasov equation is a collision operator, which is non-local in velocity space, and which may dominate the cost of the Vlasov solver. The present LDRD project endeavors to develop modern, foundational tools for the development of continuum-kinetic Vlasov solvers, using the discontinuous Petrov-Galerkin (DPG) methodology, for discretization of Vlasov, and machine-learning (ML) models to enable efficient evaluation of collision operators. DPG affords several key advantages. First, it has a built-in, robust error indicator, allowing us to adapt the mesh in a very natural way, enabling a coarse velocity-space mesh near the homogeneous boundaries, and a fine mesh where the solution has fine features. Second, it is an inherently high-order, high-intensity method, requiring extra local computations to determine so-called optimal test functions, which makes it particularly suited to modern hardware in which floating-point throughput is increasing at a faster rate than memory bandwidth. Finally, DPG is a residual-minimizing method, which enables high-accuracy computation: in typical cases, the method delivers something very close to the $L^2$ projection of the exact solution. Meanwhile, the ML-based collision model we adopt affords a cost structure that scales as the square root of a standard direct evaluation. Moreover, we design our model to conserve mass, momentum, and energy by construction, and our approach to training is highly flexible, in that it can incorporate not only synthetic data from direct-simulation Monte Carlo (DSMC) codes, but also experimental data. We have developed two DPG formulations for Vlasov-Poisson: a time-marching, backward-Euler discretization and a space-time discretization. We have conducted a number of numerical experiments to verify the approach in a 1D1V setting. In this report, we detail these formulations and experiments. We also summarize some new theoretical results developed as part of this project (published as papers previously): some new analysis of DPG for the convection-reaction problem (of which the Vlasov equation is an instance), a new exponential integrator for DPG, and some numerical exploration of various DPG-based time-marching approaches to the heat equation. As part of this work, we have contributed extensively to the Camellia open-source library; we also describe the new capabilities and their usage. We have also developed a well-documented methodology for single-species collision operators, which we applied to argon and demonstrated with numerical experiments. We summarize those results here, as well as describing at a high level a design extending the methodology to multi-species operators. We have released a new open-source library, MLC, under a BSD license; we include a summary of its capabilities as well.

Journal of Scientific Computing

Aksoylu, Burak; Bond, Stephen D.; Cyr, Eric C.; Holst, Michael

In this article, we develop goal-oriented error indicators to drive adaptive refinement algorithms for the Poisson-Boltzmann equation. Empirical results for the solvation free energy linear functional demonstrate that goal-oriented indicators are not sufficient on their own to lead to a superior refinement algorithm. To remedy this, we propose a problem-specific marking strategy using the solvation free energy computed from the solution of the linear regularized Poisson-Boltzmann equation. The convergence of the solvation free energy using this marking strategy, combined with goal-oriented refinement, compares favorably to adaptive methods using an energy-based error indicator. Due to the use of adaptive mesh refinement, it is critical to use multilevel preconditioning in order to maintain optimal computational complexity. We use variants of the classical multigrid method, which can be viewed as generalizations of the hierarchical basis multigrid and Bramble-Pasciak-Xu (BPX) preconditioners. © 2011 Springer Science+Business Media (outside the USA).

Bond, Stephen D.

Abstract not provided.

Springer Journal of Scientific Computing

Cyr, Eric C.; Bond, Stephen D.

Abstract not provided.

Tikare, Veena T.; Castillo, Andrew R.; Bond, Stephen D.; Madison, Jonathan D.; Mitchell, John A.; Rodgers, Theron R.

Abstract not provided.

Bond, Stephen D.

The availability of efficient algorithms for long-range pairwise interactions is central to the success of numerous applications, ranging in scale from atomic-level modeling of materials to astrophysics. This report focuses on the implementation and analysis of the multilevel summation method for approximating long-range pairwise interactions. The computational cost of the multilevel summation method is proportional to the number of particles, N, which is an improvement over FFTbased methods whos cost is asymptotically proportional to N logN. In addition to approximating electrostatic forces, the multilevel summation method can be use to efficiently approximate convolutions with long-range kernels. As an application, we apply the multilevel summation method to a discretized integral equation formulation of the regularized generalized Poisson equation. Numerical results are presented using an implementation of the multilevel summation method in the LAMMPS software package. Preliminary results show that the computational cost of the method scales as expected, but there is still a need for further optimization.

Costa, Timothy C.; Bond, Stephen D.; Littlewood, David J.

Abstract not provided.

Journal of Computational Physics

Roberts, Nathan V.; Bond, Stephen D.; Cyr, Eric C.; Miller, Sean T.

Kinetic gas dynamics in rarefied and moderate-density regimes have complex behavior associated with collisional processes. These processes are generally defined by convolution integrals over a high-dimensional space (as in the Boltzmann operator), or require evaluating complex auxiliary variables (as in Rosenbluth potentials in Fokker-Planck operators) that are challenging to implement and computationally expensive to evaluate. In this work, we develop a data-driven neural network model that augments a simple and inexpensive BGK collision operator with a machine-learned correction term, which improves the fidelity of the simple operator with a small overhead to overall runtime. The composite collision operator has a tunable fidelity and, in this work, is trained using and tested against a direct-simulation Monte-Carlo (DSMC) collision operator.

Multiscale Modeling and Simulation

Costa, Timothy B.; Bond, Stephen D.; Littlewood, David J.

In many applications the resolution of small-scale heterogeneities remains a significant hurdle to robust and reliable predictive simulations. In particular, while material variability at the mesoscale plays a fundamental role in processes such as material failure, the resolution required to capture mechanisms at this scale is often computationally intractable. Multiscale methods aim to overcome this difficulty through judicious choice of a subscale problem and a robust manner of passing information between scales. One promising approach is the multiscale finite element method, which increases the fidelity of macroscale simulations by solving lower-scale problems that produce enriched multiscale basis functions. In this study, we present the first work toward application of the multiscale finite element method to the nonlocal peridynamic theory of solid mechanics. This is achieved within the context of a discontinuous Galerkin framework that facilitates the description of material discontinuities and does not assume the existence of spatial derivatives. Analysis of the resulting nonlocal multiscale finite element method is achieved using the ambulant Galerkin method, developed here with sufficient generality to allow for application to multiscale finite element methods for both local and nonlocal models that satisfy minimal assumptions. We conclude with preliminary results on a mixed-locality multiscale finite element method in which a nonlocal model is applied at the fine scale and a local model at the coarse scale.