Publications

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This report reviews a hierarchy of formal mathematical models for describing plasma phenomena. Starting with the Boltzmann equation, a sequence of approximations and modeling assumptions can be made that progressively reduce to the equations for magnetohydrodynamics. Understanding the assumptions behind each of these models and their mathematical form is essential to appropriate use of each level of the hierarchy. A sequence of moment models of the Boltzmann equation are presented, then focused into a generalized three-fluid model for neutral species, electrons, and ions. This model is then further reduced to a two-fluid model, for which Braginskii described a useful closure. Further reduction of the two-fluid model yields a Generalized Ohm's Law model, which provides a connection to magnetohydrodynamic approaches. A verification approach based on linear plasma waves is presented alongside the model hierarchy, which is intended as an initial and necessary but not sufficient step for verification of plasma models within this hierarchy.

Performance results collected for EMPIRE plasma simulations on ATS-1, ATS-2 and Astra are shown, showing portability and areas for improvement during the FY20 L1 milestone.

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Multi-fluid plasma models, where an electron fluid is modeled in addition to multiple ion and neutral species as well as the full set of Maxwell's equations, are useful for representing physics beyond the scope of classic MHD. This advantage presents challenges in appropriately dealing with electron dynamics and electromagnetic behavior characterized by the plasma and cyclotron frequencies and the speed of light. For physical systems, such as those near the MHD asymptotic regime, this requirement drastically increases runtimes for explicit time integration even though resolving fast dynamics may not be critical for accuracy. Implicit time integration methods, with efficient solvers, can help to step over fast time-scales that constrain stability, but do not strongly influence accuracy. As an extension, Implicit-explicit (IMEX) schemes provide an additional mechanism to choose which dynamics are evolved using an expensive implicit solve or resolved using a fast explicit solve. In this study, in addition to IMEX methods we also consider a physics compatible exact sequence spatial discretization. Here, this combines nodal bases (H-Grad) for fluid dynamics with a set of vector bases (H-Curl and H-Div) for Maxwell's equations. This discretization allows for multi-fluid plasma modeling without violating Gauss' laws for the electric and magnetic fields. This initial study presents a discussion of the major elements of this formulation and focuses on demonstrating accuracy in the linear wave regime and in the MHD limit for both a visco-resistive and a dispersive ideal MHD problem.

The Unstructured Time-Domain ElectroMagnetics (UTDEM) portion of the EMPHASIS suite solves Maxwell's equations using finite-element techniques on unstructured meshes. This document provides user-specific information to facilitate the use of the code for applications of interest.

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This report documents the outcome from the ASC ATDM Level 2 Milestone 6358: Assess Status of Next Generation Components and Physics Models in EMPIRE. This Milestone is an assessment of the EMPIRE (ElectroMagnetic Plasma In Realistic Environments) application and three software components. The assessment focuses on the electromagnetic and electrostatic particle-in-cell solutions for EMPIRE and its associated solver, time integration, and checkpoint-restart components. This information provides a clear understanding of the current status of the EMPIRE application and will help to guide future work in FY19 in order to ready the application for the ASC ATDM L1 Milestone in FY20. It is clear from this assessment that performance of the linear solver will have to be a focus in FY19.

A discrete De Rham complex enables compatible, structure-preserving discretizations for a broad range of partial differential equations problems. Such discretizations can correctly reproduce the physics of interface problems, provided the grid conforms to the interface. However, large deformations, complex geometries, and evolving interfaces makes generation of such grids difficult. We develop and demonstrate two formally equivalent approaches that, for a given background mesh, dynamically construct an interface-conforming discrete De Rham complex. Both approaches start by dividing cut elements into interface-conforming subelements but differ in how they build the finite element basis on these subelements. The first approach discards the existing non-conforming basis of the parent element and replaces it by a dynamic set of degrees of freedom of the same kind. The second approach defines the interface-conforming degrees of freedom on the subelements as superpositions of the basis functions of the parent element. These approaches generalize the Conformal Decomposition Finite Element Method (CDFEM) and the extended finite element method with algebraic constraints (XFEM-AC), respectively, across the De Rham complex.

We investigate the use of implicit time integrators for finite element time domain approximations of Maxwell's equations in vacuum. We discretize Maxwell's equations in time using Crank-Nicolson and in 3D space using compatible finite elements. We solve the system by taking a single step of Newton's method and inverting the Eddy-Current Schur complement allowing for the use of standard preconditioning techniques. This approach also generalizes to more complex material models that can include the Unsplit PML. We present verification results and demonstrate performance at CFL numbers up to 1000.

The Unstructured Time-Domain ElectroMagnetics (UTDEM) portion of the EMPHASIS suite solves Maxwell’s equations using finite-element techniques on unstructured meshes. This document provides user-specific information to facilitate the use of the code for applications of interest.

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Interface-conforming elements generated by the conformal decomposition finite element method can have arbitrarily poor quality due to the arbitrary intersection of the base triangular or tetrahedral mesh with material interfaces. This can have severe consequences for both the solvability of linear systems and for the interpolation error of fields represented on these meshes. The present work demonstrates that snapping the base mesh nodes to the interface whenever the interface cuts close to a node results in conforming meshes of good quality. Theoretical limits on the snapping tolerance are derived, and even conservative tolerance choices result in limiting the stiffness matrix condition number to within a small multiple of that of the base mesh. Interpolation errors are also well controlled in the norms of interest. In 3D, use of node-to-interface snapping also permits a simpler and more robust vertex ID-based element decomposition algorithm to be used with no serious detriment to mesh quality.

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This report briefly outlines an algorithm for dividing a tetrahedron intersected by a planar interface into conforming sub-tetrahedra. The problem of conformal decomposition of tetrahedral meshes arises in enriched finite element methods; in particular, we are concerned with the Conformal Decomposition Finite Element Method (CDFEM) and variants of the eXtended Finite Element Method (XFEM). The algorithm presented is based on the paper How to Subdivide Pyramids, Prisms and Hexahedra into Tetrahedra by Dompierre, Labbe, Vallet, and Camarero (1999), and here is applied and extended to the problem of fully defining and tracking all geometric features of the sub-tetrahedra generated when a tetrahedron is cut by a planar surface.

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Surface effects are critical to the accurate simulation of electromagnetics (EM) as current tends to concentrate near material surfaces. Sandia EM applications, which include exploding bridge wires for detonator design, electromagnetic launch of flyer plates for material testing and gun design, lightning blast-through for weapon safety, electromagnetic armor, and magnetic flux compression generators, all require accurate resolution of surface effects. These applications operate in a large deformation regime, where body-fitted meshes are impractical and multimaterial elements are the only feasible option. State-of-the-art methods use various mixture models to approximate the multi-physics of these elements. The empirical nature of these models can significantly compromise the accuracy of the simulation in this very important surface region. We propose to substantially improve the predictive capability of electromagnetic simulations by removing the need for empirical mixture models at material surfaces. We do this by developing an eXtended Finite Element Method (XFEM) and an associated Conformal Decomposition Finite Element Method (CDFEM) which satisfy the physically required compatibility conditions at material interfaces. We demonstrate the effectiveness of these methods for diffusion and diffusion-like problems on node, edge and face elements in 2D and 3D. We also present preliminary work on h -hierarchical elements and remap algorithms.

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We present a new extended finite element method with algebraic constraints (XFEM-AC) for recovering weakly discontinuous solutions across internal element interfaces. If necessary, cut elements are further partitioned by a local secondary cut into body-fitting subelements. Each resulting subelement contributes an enrichment of the parent element. The enriched solutions are then tied using algebraic constraints, which enforce C0 continuity across both cuts. These constraints impose equivalence of the enriched and body-fitted finite element solutions, and are the key differentiating feature of the XFEM-AC. In so doing, a stable mixed formulation is obtained without having to explicitly construct a compatible Lagrange multiplier space and prove a formal inf-sup condition. Likewise, convergence of the XFEM-AC solution follows from its equivalence to the interface-fitted finite element solution. This relationship is further exploited to improve the numerical solution of the resulting XFEM-AC linear system. Examples are shown demonstrating the new approach for both steady-state and transient diffusion problems. © 2013 Elsevier B.V.

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