Publications

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We present a new model for closing a system of Lagrangian hydrodynamics equations for a two-material cell with a single velocity model. We describe a new approach that is motivated by earlier work of Delov and Sadchikov and of Goncharov and Yanilkin. Using a linearized Riemann problem to initialize volume fraction changes, we require that each material satisfy its own pdV equation, which breaks the overall energy balance in the mixed cell. To enforce this balance, we redistribute the energy discrepancy by assuming that the corresponding pressure change in each material is equal. This multiple-material model is packaged as part of a two-step time integration scheme. We compare results of our approach with other models and with corresponding pure-material calculations, on two-material test problems with ideal-gas or stiffened-gas equations of state.

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Algorithmic properties of the midpoint predictor-corrector time integration algorithm are examined. In the case of a finite number of iterations, the errors in angular momentum conservation and incremental objectivity are controlled by the number of iterations performed. Exact angular momentum conservation and exact incremental objectivity are achieved in the limit of an infinite number of iterations. A complete stability and dispersion analysis of the linearized algorithm is detailed. The main observation is that stability depends critically on the number of iterations performed.

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Two classical verification problems from shock hydrodynamics are adapted for verification in the context of ideal magnetohydrodynamics (MHD) by introducing strong transverse magnetic fields, and simulated using the finite element Lagrange-remap MHD code ALEGRA for purposes of rigorous code verification. The concern in these verification tests is that inconsistencies related to energy advection are inherent in Lagrange-remap formulations for MHD, such that conservation of the kinetic and magnetic components of the energy may not be maintained. Hence, total energy conservation may also not be maintained. MHD shock propagation may therefore not be treated consistently in Lagrange-remap schemes, as errors in energy conservation are known to result in unphysical shock wave speeds and post-shock states. That kinetic energy is not conserved in Lagrange-remap schemes is well known, and the correction of DeBar has been shown to eliminate the resulting errors. Here, the consequences of the failure to conserve magnetic energy are revealed using order verification in the two magnetized shock-hydrodynamics problems. Further, a magnetic analog to the DeBar correction is proposed and its accuracy evaluated using this verification testbed. Results indicate that only when the total energy is conserved, by implementing both the kinetic and magnetic components of the DeBar correction, can simulations in Lagrange-remap formulation capture MHD shock propagation accurately. Additional insight is provided by the verification results, regarding the implementation of the DeBar correction and the advection scheme.

A common attribute of capturing schemes used to find approximate solutions to the Euler equations is a sub-linear rate of convergence with respect to mesh resolution. Purely nonlinear jumps, such as shock waves produce a first-order convergence rate, but linearly degenerate discontinuous waves, where present, produce sub-linear convergence rates which eventually dominate the global rate of convergence. The classical explanation for this phenomenon investigates the behavior of the exact solution to the numerical method in combination with the finite error terms, often referred to as the modified equation. For a first-order method, the modified equation produces the hyperbolic evolution equation with second-order diffusive terms. In the frame of reference of the traveling wave, the solution of a discontinuous wave consists of a diffusive layer that grows with a rate of t1/2, yielding a convergence rate of 1/2. Self-similar heuristics for higher-order discretizations produce a growth rate for the layer thickness of Δt1/(p+1) which yields an estimate for the convergence rate as p/(p + 1) where p is the order of the discretization. In this paper we show that this estimated convergence rate can be derived with greater rigor for both dissipative and dispersive forms of the discrete error. In particular, the form of the analytical solution for linear modified equations can be solved exactly. These estimates and forms for the error are confirmed in a variety of demonstrations ranging from simple linear waves to multidimensional solutions of the Euler equations. © 2008 Elsevier Inc.

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ALEGRA is an arbitrary Lagrangian-Eulerian (multiphysics) computer code developed at Sandia National Laboratories since 1990. The code contains a variety of physics options including magnetics, radiation, and multimaterial flow. The code has been developed for nearly two decades, but recent work has dramatically improved the code's accuracy and robustness. These improvements include techniques applied to the basic Lagrangian differencing, artificial viscosity and the remap step of the method including an important improvement in the basic conservation of energy in the scheme. We will discuss the various algorithmic improvements and their impact on the results for important applications. Included in these applications are magnetic implosions, ceramic fracture modeling, and electromagnetic launch. Copyright © 2008 by the American Institute of Aeronautics and Astronautics, Inc.

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