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DOE OSTI2015

Stability of the discretization of the electron avalanche phenomenon

The numerical simulation of the discharge inception is an active field of applied physics with many industrial applications. In this work we focus on the drift-reaction equation that describes the electron avalanche. This phenomenon is one of the basic building blocks of the streamer model. The main difficulty of the electron avalanche equation lies in the fact that the reaction term is positive when a high electric field is applied. It leads to exponentially growing solutions and this has a major impact on the behavior of numerical schemes. We analyze the stability of a reference finite volume scheme applied to this latter problem. The stability of the method may impose a strict mesh spacing, therefore a proper stabilized scheme, which is stable whatever spacing is used, has been developed. The convergence of the scheme is treated as well as some numerical experiments.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2015

A space–angle DGFEM approach for the Boltzmann radiation transport equation with local angular refinement

In this paper a new method for the discretization of the radiation transport equation is presented, based on a discontinuous Galerkin method in space and angle that allows for local refinement in angle where any spatial element can support its own angular discretization. To cope with the discontinuous spatial nature of the solution, a generalized Riemann procedure is required to distinguish between incoming and outgoing contributions of the numerical fluxes. A new consistent framework is introduced that is based on the solution of a generalized eigenvalue problem. The resulting numerical fluxes for the various possible cases where neighboring elements have an equal, higher or lower level of refinement in angle are derived based on tensor algebra and the resulting expressions have a very clear physical interpretation. The choice of discontinuous trial functions not only has the advantage of easing local refinement, it also facilitates the use of efficient sweep-based solvers due to decoupling of unknowns on a large scale thereby approaching the efficiency of discrete ordinates methods with local angular resolution. The approach is illustrated by a series of numerical experiments. Results show high orders of convergence for the scalar flux on angular refinement. The generalized Riemann upwinding procedure leads to stable and consistent solutions. Further the sweep-based solver performs well when used as a preconditioner for a Krylov method.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2015

An adhesive contact mechanics formulation based on atomistically induced surface traction

In this work, we have developed a novel multiscale computational contact formulation based on the generalized Derjuguin approximation for continua that are characterized by atomistically enriched constitutive relations in order to study macroscopic interaction between arbitrarily shaped deformable continua. The proposed adhesive contact formulation makes use of the microscopic interaction forces between individual particles in the interacting bodies. In particular, the double-layer volume integral describing the contact interaction (energy, force vector, matrix) is converted into a double-layer surface integral through a mathematically consistent approach that employs the divergence theorem and a special partitioning technique. The proposed contact model is formulated in the nonlinear continuum mechanics framework and implemented using the standard finite element method. With no large penalty constant, the stiffness matrix of the system will in general be well-conditioned, which is of great significance for quasi-static analysis. Three numerical examples are presented to illustrate the capability of the proposed method. Results indicate that with the same mesh configuration, the finite element computation based on the surface integral approach is faster and more accurate than the volume integral based approach. In addition, the proposed approach is energy preserving even in a very long dynamic simulation.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2016

Finite time step and spatial grid effects in δf simulation of warm plasmas

This paper introduces a technique for analyzing time integration methods used with the particle weight equations in δf method particle-in-cell (PIC) schemes. The analysis applies to the simulation of warm, uniform, periodic or infinite plasmas in the linear regime and considers the collective behavior similar to the analysis performed by Langdon for full-f PIC schemes [1,2]. We perform both a time integration analysis and spatial grid analysis for a kinetic ion, adiabatic electron model of ion acoustic waves. An implicit time integration scheme is studied in detail for δf simulations using our weight equation analysis and for full-f simulations using the method of Langdon. It is found that the δf method exhibits a CFL-like stability condition for low temperature ions, which is independent of the parameter characterizing the implicitness of the scheme. The accuracy of the real frequency and damping rate due to the discrete time and spatial schemes is also derived using a perturbative method. The theoretical analysis of numerical error presented here may be useful for the verification of simulations and for providing intuition for the design of new implicit time integration schemes for the δf method, as well as understanding differences between δf and full-f approaches to plasma simulation.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2016

Bayesian inference of nonlinear unsteady aerodynamics from aeroelastic limit cycle oscillations

A Bayesian model selection and parameter estimation algorithm is applied to investigate the influence of nonlinear and unsteady aerodynamic loads on the limit cycle oscillation (LCO) of a pitching airfoil in the transitional Reynolds number regime. At small angles of attack, laminar boundary layer trailing edge separation causes negative aerodynamic damping leading to the LCO. The fluid–structure interaction of the rigid, but elastically mounted, airfoil and nonlinear unsteady aerodynamics is represented by two coupled nonlinear stochastic ordinary differential equations containing uncertain parameters and model approximation errors. Several plausible aerodynamic models with increasing complexity are proposed to describe the aeroelastic system leading to LCO. The likelihood in the posterior parameter probability density function (pdf) is available semi-analytically using the extended Kalman filter for the state estimation of the coupled nonlinear structural and unsteady aerodynamic model. The posterior parameter pdf is sampled using a parallel and adaptive Markov Chain Monte Carlo (MCMC) algorithm. The posterior probability of each model is estimated using the Chib–Jeliazkov method that directly uses the posterior MCMC samples for evidence (marginal likelihood) computation. The Bayesian algorithm is validated through a numerical study and then applied to model the nonlinear unsteady aerodynamic loads using wind-tunnel test data at various Reynolds numbers.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2016

A multiscale coupled finite-element and phase-field framework to modeling stressed grain growth in polycrystalline thin films

A previously-developed finite-deformation- and crystal-elasticity-based constitutive theory for stressed grain growth in cubic polycrystalline bodies has been augmented to include a description of excess surface energy and grain-growth stagnation mechanisms through the use of surface effect state variables in a thermodynamically-consistent manner. The constitutive theory was also implemented into a multiscale coupled finite-element and phase-field computational framework. With the material parameters in the constitutive theory suitably calibrated, our three-dimensional numerical simulations show that the constitutive model is able to accurately predict the experimentally-determined evolution of crystallographic texture and grain size statistics in polycrystalline copper thin films deposited on polyimide substrate and annealed at high-homologous temperatures. In particular, our numerical analyses show that the broad texture transition observed in the annealing experiments of polycrystalline thin films is caused by grain growth stagnation mechanisms. - Graphical abstract: - Highlights: • Developing a theory for stressed grain growth in polycrystalline thin films. • Implementation into a multiscale coupled finite-element and phase-field framework. • Quantitative reproduction of the experimental grain growth data by simulations. • Revealing the cause of texture transition to be due to the stagnation mechanisms.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2017

A positive and entropy-satisfying finite volume scheme for the Baer–Nunziato model

We present a relaxation scheme for approximating the entropy dissipating weak solutions of the Baer–Nunziato two-phase flow model. This relaxation scheme is straightforwardly obtained as an extension of the relaxation scheme designed in for the isentropic Baer–Nunziato model and consequently inherits its main properties. To our knowledge, this is the only existing scheme for which the approximated phase fractions, phase densities and phase internal energies are proven to remain positive without any restrictive condition other than a classical fully computable CFL condition. For ideal gas and stiffened gas equations of state, real values of the phasic speeds of sound are also proven to be maintained by the numerical scheme. It is also the only scheme for which a discrete entropy inequality is proven, under a CFL condition derived from the natural sub-characteristic condition associated with the relaxation approximation. This last property, which ensures the non-linear stability of the numerical method, is satisfied for any admissible equation of state. We provide a numerical study for the convergence of the approximate solutions towards some exact Riemann solutions. The numerical simulations show that the relaxation scheme compares well with two of the most popular existing schemes available for the Baer–Nunziato model, namely Schwendeman–Wahle–Kapila's Godunov-type scheme and Tokareva–Toro's HLLC scheme . The relaxation scheme also shows a higher precision and a lower computational cost (for comparable accuracy) than a standard numerical scheme used in the nuclear industry, namely Rusanov's scheme. Finally, we assess the good behavior of the scheme when approximating vanishing phase solutions.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2017

A flexible nonlinear diffusion acceleration method for the S N transport equations discretized with discontinuous finite elements

This paper presents a flexible nonlinear diffusion acceleration (NDA) method that discretizes both the S N transport equation and the diffusion equation using the discontinuous finite element method (DFEM). The method is flexible in that the diffusion equation can be discretized on a coarser mesh with the only restriction that it is nested within the transport mesh and the FEM shape function orders of the two equations can be different. The consistency of the transport and diffusion solutions at convergence is defined by using a projection operator mapping the transport into the diffusion FEM space. The diffusion weak form is based on the modified incomplete interior penalty (MIP) diffusion DFEM discretization that is extended by volumetric drift, interior face, and boundary closure terms. In contrast to commonly used coarse mesh finite difference (CMFD) methods, the presented NDA method uses a full FEM discretized diffusion equation for acceleration. Suitable projection and prolongation operators arise naturally from the FEM framework. Via Fourier analysis and numerical experiments for a one-group, fixed source problem the following properties of the NDA method are established for structured quadrilateral meshes: (1) the presented method is unconditionally stable and effective in the presence of mild material heterogeneities if the same mesh and identical shape functions either of the bilinear or biquadratic type are used, (2) the NDA method remains unconditionally stable in the presence of strong heterogeneities, (3) the NDA method with bilinear elements extends the range of effectiveness and stability by a factor of two when compared to CMFD if a coarser diffusion mesh is selected. In addition, the method is tested for solving the C5G7 multigroup, eigenvalue problem using coarse and fine mesh acceleration. Finally, while NDA does not offer an advantage over CMFD for fine mesh acceleration, it reduces the iteration count required for convergence by almost a factor of two in the case of coarse mesh acceleration.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2017

Streamline integration as a method for two-dimensional elliptic grid generation

We propose a new numerical algorithm to construct a structured numerical elliptic grid of a doubly connected domain. Our method is applicable to domains with boundaries defined by two contour lines of a two-dimensional function. Furthermore, we can adapt any analytically given boundary aligned structured grid, which specifically includes polar and Cartesian grids. The resulting coordinate lines are orthogonal to the boundary. Grid points as well as the elements of the Jacobian matrix can be computed efficiently and up to machine precision. In the simplest case we construct conformal grids, yet with the help of weight functions and monitor metrics we can control the distribution of cells across the domain. Our algorithm is parallelizable and easy to implement with elementary numerical methods. We assess the quality of grids by considering both the distribution of cell sizes and the accuracy of the solution to elliptic problems. Among the tested grids these key properties are best fulfilled by the grid constructed with the monitor metric approach. - Graphical abstract: - Highlights: • Construct structured, elliptic numerical grids with elementary numerical methods. • Align coordinate lines with or make them orthogonal to the domain boundary. • Compute grid points and metric elements up to machine precision. • Control cell distribution by adaption functions or monitor metrics.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2017

Bayesian and variational Bayesian approaches for flows in heterogeneous random media

In this paper, we study porous media flows in heterogeneous stochastic media. We propose an efficient forward simulation technique that is tailored for variational Bayesian inversion. As a starting point, the proposed forward simulation technique decomposes the solution into the sum of separable functions (with respect to randomness and the space), where each term is calculated based on a variational approach. This is similar to Proper Generalized Decomposition (PGD). Next, we apply a multiscale technique to solve for each term (as in ) and, further, decompose the random function into 1D fields. As a result, our proposed method provides an approximation hierarchy for the solution as we increase the number of terms in the expansion and, also, increase the spatial resolution of each term. We use the hierarchical solution distributions in a variational Bayesian approximation to perform uncertainty quantification in the inverse problem. We conduct a detailed numerical study to explore the performance of the proposed uncertainty quantification technique and show the theoretical posterior concentration.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2019

Spectral methods in the presence of discontinuities

Spectral methods provide an elegant and efficient way of numerically solving differential equations of all kinds. For smooth problems, truncation error for spectral methods vanishes exponentially in the infinity norm and L 2 -norm. However, for non-smooth problems, convergence is significantly worse—the L 2 -norm of the error for a discontinuous problem will converge at a sub-linear rate and the infinity norm will not converge at all. We explore and improve upon a post-processing technique—optimally convergent mollifiers—to recover exponential convergence from a poorly-converging spectral reconstruction of non-smooth data. Furthermore this is an important first step towards using these techniques for simulations of realistic systems.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2025

Numerical schemes for 3-wave kinetic equations: A complete treatment of the collision operator

In our previous work Walton and Tran (2023), numerical schemes for a simplified version of 3-wave kinetic equations, in which only the simple forward-cascade terms of the collision operators are kept, have been successfully designed, especially to capture the long time dynamics of the equation given the multiple blow-up time phenomenon. In this second work in the series, we propose numerical treatments for the complete 3-wave kinetic equations, in which the complete, much more complicated collision operators are fully considered based on a novel conservative form of the equation. Here we then derive an implicit finite volume scheme to solve the equation. The new discretization uses an adaptive time-stepping method which allows for the simulations to be carried to very long times. Our computed solutions are compared with previously derived long-time asymptotic estimates for the decay rate of total energy of time-dependent solutions of 3-wave kinetic equations and found to be in excellent agreement.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Hardware acceleration for HPS algorithms in two and three dimensions

We provide a flexible, open-source framework for hardware acceleration, namely massively-parallel execution on general-purpose graphics processing units (GPUs), applied to the hierarchical Poincaré–Steklov (HPS) family of algorithms for building fast direct solvers for linear elliptic partial differential equations. To take full advantage of the power of hardware acceleration, we propose two variants of HPS algorithms to improve performance on two- and three-dimensional problems. In the two-dimensional setting, we introduce a novel recomputation strategy that minimizes costly data transfers to and from the GPU; in three dimensions, we modify and extend the adaptive discretization technique of Geldermans and Gillman [1] to greatly reduce peak memory usage. We provide an open-source implementation of these methods written in JAX, a high-level accelerated linear algebra package, which allows for the first integration of a high-order fast direct solver with automatic differentiation tools. We conclude with extensive numerical examples showing our methods are fast and accurate on two- and three-dimensional problems.

Fast direct solvers↗
DOE OSTI2025

Electronic density of states as the descriptor of elastic bond strength, ductility, and local lattice distortion in BCC refractory alloys

Although electronic density of states (DOS) is fundamental to materials properties, its general relationship to mechanical properties of alloys is not well established. In this paper, using density functional theory (DFT) calculations, we show that the electronic occupancy at the Fermi level, N(E f ), obtained from DOS is a key descriptor of alloy strength and ductility. Our comprehensive analysis of numerous body centered cubic (BCC) refractory high entropy alloys (RHEAs) shows an overwhelming correlation that low N(E f ) indicates strong bonds that have high stiffness resulting in high elastic constants. High bond stiffness indicates presence of covalent nature of bonds that are directional in nature resulting in resistance to deformation leading to high bulk (B) and shear (G) moduli. Consequently, N(E f ) provides a direct correlation to the tendency of alloy ductility evidenced in the Pugh ratio (G/B). As stiffer bonds result in lower local lattice distortion (LLD), N(E f ) are LLD are also found to be corelated which opens up a correlation to solid solution strengthening and yield strength. Thus, this work unveils fundamental correlations between N(E f ) and (1) elastic bond strength, (2) ductility, and (3) LLD. These correlations open opportunities for the design of high strength high ductile RHEAs.

36 MATERIALS SCIENCE↗
DOE OSTI2024

Recent advances in combustion science related to hydrogen safety

Hydrogen is a key pillar in the global Net Zero strategy. Rapid scaling up of hydrogen production, transport, distribution and utilization is expected. This entails that hydrogen, which is traditionally an industrial gas, will come into proximity of populated urban areas and in some situations handled by the untrained public. To realize all their benefits, hydrogen and its technologies must be safely developed and deployed. The specific properties of hydrogen involving wide flammability range, low ignition energy and fast flame speed implies that any accidental release of hydrogen can be easily ignited. Comparing with conventional fuels, combustion systems fueled by hydrogen are also more prone to flame instability and abnormal combustion. This paper aims to provide a comprehensive review about combustion research related to hydrogen safety. It starts with a brief introduction which includes some overview about risk analysis, codes and standards. The core content covers ignition, fire, explosions and deflagration to detonation transition (DDT). Considering that DDT leads to detonation, and that detonation may also be induced directly under special circumstances, the subject of detonation is also included for completeness. The review covers laboratory, medium and large-scale experiments, as well as theoretical analysis and numerical simulation results. While highlights are provided at the end of each section, the paper closes with some concluding remarks highlighting the achievements and key knowledge gaps.

08 HYDROGEN↗
DOE OSTI2022

The role of geographical spreaders in infectious pattern formation and front propagation speeds

The pattern formation and spatial spread of infectious populations are investigated using a kernel-based Susceptible–Infectious–Recovered (SIR) model applicable across a wide range of basic reproduction numbers R O . The goal is to examine the role of geographical spreaders on transient spatial pattern formation of infectious populations and the associated maximum invasive front speeds c max . In the simulations conducted here, geographical spreaders are defined as a portion of the infected population Φ experiencing high mobility between identical communities. The spatial organization of the infected population and c max are determined when the infections are randomly initiated in space within multiple communities. For small but finite , scaling analysis and numerical simulations in 1-dimension suggest that when the spreading kernel is Gaussian-shaped, where is the inverse of the infectious duration. This finding for agrees with a diffusion-based representation of mobility in 1-D. Numerical simulations in 2-D across wide-ranging suggest that , the variance of the spatial kernel describing mobility of long-distance geographical spreaders across communities, determines the spatial organization of infections across communities. When (long-distance mobility, where is the minimum spatial extent defining adjacent communities), the infectious population will experience a transient but spatially coherent pattern with a wavelength that can be derived from the spreading kernel properties. Moreover, the 2-D simulations for the bounded kernel suggest that attainment of is also dictated by but the magnitude is not sensitive to unlike diffusion-based models.

60 APPLIED LIFE SCIENCES↗