Arbitrary Lagrangian Eulerian Hydrodynamics Research Articles

Interface-unaware sub-scale dynamics closure model for multimaterial cells in cell-centered arbitrary Lagrangian-Eulerian hydrodynamics

We develop new interface-unaware sub-scale dynamics (IU-SSD) multimaterial cell closure models for the Lagrangian stage of cell-centered arbitrary Lagrangian-Eulerian method. The IU-SSD closure models do not require information about the interface in the multimaterial cell and consist of two stages. The first stage makes use of a constant volume fraction model, wherein the main ingredient is the construction of a nodal Riemann solver that is capable of handling multimaterial cells. This model provides the numerical flux at cell face and introduces the mechanism of velocity relaxation. The sub-scale interactions of the materials in a multimaterial cell are taken into account by incorporating the pressure relaxation model in the second stage. Two types of pressure relaxation models are designed utilizing the linear relation between viscous pressure and volume strain rate. The pressure relaxation models are implemented via a multi-step time marching procedure such that they satisfy the physically justified constraints: positivity of volume, positivity of internal energy and smooth relaxation of pressure. The performances of new IU-SSD models are demonstrated via a series of numerical tests involving both Lagrangian and arbitrary Lagrangian-Eulerian hydrodynamics calculations.

A High Resolution Simulation of a Single Shock-Accelerated Particle

Abstract Particle drag models, which capture macroviscous and pressure effects, have been developed over the years for various flow regimes to enable cost effective simulations of particle-laden flows. The relatively recent derivation by Maxey and Riley has provided an exact equation of motion for spherical particles in a flow field based on the continuum assumption. Many models that have been simplified from these equations have provided reasonable approximations; however, the sensitivity of particle-laden flows to particle drag requires a very accurate model to simulate. To develop such a model, a two-dimensional axisymmetric Navier–Stokes direct numerical simulation of a single particle in a transient, shock-driven flow field was conducted using the hydrocode FLAG. FLAGs capability to run arbitrary Lagrangian-Eulerian hydrodynamics coupled with solid mechanic models makes it an ideal code to capture the physics of the flow field around and in the particle as it is shock-accelerated—a challenging regime to study. The goal of this work is twofold: to provide a validation for FLAGs Navier–Stokes and heat diffusion solutions and to provide a rationale for recent experimental particle drag measurements.

A positivity-preserving and conservative intersection-distribution-based remapping algorithm for staggered ALE hydrodynamics on arbitrary meshes

We introduce new intersection-distribution-based remapping tools for indirect staggered arbitrary Lagrangian-Eulerian (ALE) simulations of multi-material shock hydrodynamics on arbitrary meshes. In addition to conserving momentum and total energy, the three-stage remapper proposed in this work preserves non-negativity of the internal energy. At the first stage, we construct slope-limited piecewise-linear reconstructions of all conserved quantities on zones of the source mesh and perform intersection-based remap to obtain bound-preserving zonal quantities on the target mesh. At the second stage, we define bound-preserving nodal quantities of the staggered ALE discretization as convex combinations of corner quantities. The nodal internal energy is corrected in a way which keeps it non-negative, while providing exact conservation of total energy. At the final stage, we distribute the non-negative nodal internal energy to corners, zones and materials using non-negative weights. Proofs of positivity preservation are provided for each stage.This work is a natural extension of our paper [14] in which a similar intersection-distribution-based remapping procedure was employed. The original version used a nodal kinetic energy fix which did not provably ensure positivity preservation for the zonal internal energy after the final distribution stage. The new algorithm cures this potential drawback by using ‘coordinated’ limiters for piecewise-linear reconstructions, remapping the internal energy to nodes and correcting it before redistribution. The effectiveness of the new nodal fix is illustrated by numerical examples.

Intersection-distribution-based remapping between arbitrary meshes for staggered multi-material arbitrary Lagrangian-Eulerian hydrodynamics

We present a new intersection-distribution-based remapping method between arbitrary polygonal meshes for indirect staggered multi-material arbitrary Lagrangian-Eulerian hydrodynamics. All cell-centered material quantities are conservatively remapped using intersections between the Lagrangian (old, source) mesh and the rezoned (new, target) mesh. The new nodal masses are obtained by conservative distribution of all material masses in each new cell to the cell's corners and then collecting those corner masses at new nodes. This distribution is done using a local constrained optimization approach for each cell in the new mesh. In order to remap nodal momentum we first define cell-centered momentum for each cell in the old mesh, conservatively remap this to the new mesh and then conservatively distribute the new zonal momentum to each cell's bounding nodes, again using local constrained optimization. Our method also conserves total energy by applying a new nodal kinetic energy correction that relies on a process similar to that used for remapping nodal mass and momentum. Cell-centered kinetic energy is computed, conservatively remapped and then distributed to nodes. The discrepancy between this conservatively remapped and actual nodal kinetic energy is then conservatively distributed to the internal energies of the materials in the cells surrounding each node. Unlike conventional cell-based corrections of this type, this new nodal kinetic energy correction has not been observed to drive material internal energy negative in any of our testing. Unlike flux based remapping, our new intersection-distribution method can be applied to remapping between source and target meshes that are arbitrarily different, which provides superior flexibility in the rezoning strategy. Our method is accurate, essentially conservative and essentially bounds preserving.

Simulation-driven optimization of high-order meshes in ALE hydrodynamics

In this paper we propose tools for high-order mesh optimization and demonstrate their benefits in the context of multi-material Arbitrary Lagrangian-Eulerian (ALE) compressible shock hydrodynamic applications. The mesh optimization process is driven by information provided by the simulation which uses the optimized mesh, such as shock positions, material regions, known error estimates, etc. These simulation features are usually represented discretely, for instance, as finite element functions on the Lagrangian mesh. The discrete nature of the input is critical for the practical applicability of the algorithms we propose and distinguishes this work from approaches that strictly require analytical information. Our methods are based on node movement through a high-order extension of the Target-Matrix Optimization Paradigm (TMOP) of [1]. The proposed formulation is fully algebraic and relies only on local Jacobian matrices, so it is applicable to all types of mesh elements, in 2D and 3D, and any order of the mesh. We discuss the notions of constructing adaptive target matrices and obtaining their derivatives, reconstructing discrete data in intermediate meshes, node limiting that enables improvement of global mesh quality while preserving space-dependent local mesh features, and appropriate normalization of the objective function. The adaptivity methods are combined with automatic ALE triggers that can provide robustness of the mesh evolution and avoid excessive remap procedures. The benefits of the new high-order TMOP technology are illustrated on several simulations performed in the high-order ALE application BLAST [2].

The distribution-based remapping of the nodal mass and momentum between arbitrary meshes for staggered arbitrary Lagrangian-Eulerian hydrodynamics

We present a new distribution-based remapping of the nodal mass and momentum between arbitrary source (Lagrangian) and arbitrary target (rezoned) meshes for indirect staggered arbitrary Lagrangian-Eulerian hydrodynamics. The method is based on the following ideas: define cell-centered momentum and mass on the source mesh; conservatively remap those cell-centered quantities from source to target meshes; and use local constrained optimization for each cell of the target mesh to conservatively distribute cell mass and momentum to the nodes of the cell. This new method is efficient, conservative, accurate and bounds preserving.

Numerical Modeling of Complex Targets for High-Energy- Density Experiments with Ion Beams and other Drivers

We explore the simulation challenges and requirements for experiments planned on facilities such as the NDCX-II ion accelerator at LBNL, currently undergoing commissioning. Hydrodynamic modeling of NDCX-II experiments include certain lower temperature effects, e.g., surface tension and target fragmentation, that are not generally present in extreme high-energy laser facility experiments, where targets are completely vaporized in an extremely short period of time. Target designs proposed for NDCX-II range from metal foils of order one micron thick (thin targets) to metallic foam targets several tens of microns thick (thick targets). These high-energy-density experiments allow for the study of fracture as well as the process of bubble and droplet formation. We incorporate these physics effects into a code called ALE-AMR that uses a combination of Arbitrary Lagrangian Eulerian hydrodynamics and Adaptive Mesh Refinement. Inclusion of certain effects becomes tricky as we must deal with non-orthogonal meshes of various levels of refinement in three dimensions. A surface tension model used for droplet dynamics is implemented in ALE-AMR using curvature calculated from volume fractions. Thick foam target experiments provide information on how ion beam induced shock waves couple into kinetic energy of fluid flow. Although NDCX-II is not fully commissioned, experiments are being conducted that explore material defect production and dynamics.

Large-eddy and unsteady RANS simulations of a shock-accelerated heavy gas cylinder

Two-dimensional numerical simulations of the Richtmyer–Meshkov unstable “shock-jet” problem are conducted using both large-eddy simulation (LES) and unsteady Reynolds-averaged Navier–Stokes (URANS) approaches in an arbitrary Lagrangian–Eulerian hydrodynamics code. Turbulence statistics are extracted from LES by running an ensemble of simulations with multimode perturbations to the initial conditions. Detailed grid convergence studies are conducted, and LES results are found to agree well with both experiment and high-order simulations conducted by Shankar et al. (Phys Fluids 23, 024102, 2011). URANS results using a k–L approach are found to be highly sensitive to initialization of the turbulence lengthscale L and to the time at which L becomes resolved on the computational mesh. It is observed that a gradient diffusion closure for turbulent species flux is a poor approximation at early times, and a new closure based on the mass-flux velocity is proposed for low-Reynolds-number mixing.

Monotonicity in high‐order curvilinear finite element arbitrary Lagrangian–Eulerian remap

SummaryThe remap phase in arbitrary Lagrangian–Eulerian (ALE) hydrodynamics involves the transfer of field quantities defined on a post‐Lagrangian mesh to some new mesh, usually generated by a mesh optimization algorithm. This problem is often posed in terms of transporting (or advecting) some state variable from the old mesh to the new mesh over a fictitious time interval. It is imperative that this remap process be monotonic, that is, not generate any new extrema in the field variables. It is well known that the only linear methods that are guaranteed to be monotonic for such problems are first‐order accurate; however, much work has been performed in developing non‐linear methods, which blend both high and low (first) order solutions to achieve monotonicity and preserve high‐order accuracy when the field is sufficiently smooth. In this paper, we present a set of methods for enforcing monotonicity targeting high‐order discontinuous Galerkin methods for advection equations in the context of high‐order curvilinear ALE hydrodynamics. Published 2014. This article is a U.S. Government work and is in the public domain in the USA.

Constrained optimization framework for interface-aware sub-scale dynamics closure model for multimaterial cells in Lagrangian and arbitrary Lagrangian–Eulerian hydrodynamics

A systematic description of the new interface-aware sub-scale-dynamics (IA-SSD) closure model for the Lagrangian stage of multimaterial arbitrary Lagrangian–Eulerian methods is presented. The IA-SSD closure model consists of two stages. During the first, bulk, stage, the well known equal compressibility model is used. During the second stage, sub-scale interactions of the materials inside the multimaterial cell are taken into account. At this stage, information about the topology of the materials inside the multimaterial cell is utilized, allowing the orientations of internal interfaces to be included in the model. Each material interacts in a pair-wise fashion with the materials with which it has a common boundary. The interactions are based on the solution of the acoustic Riemann problem between each pair of materials and is limited using physically justified constraints: positivity of volume, positivity of internal energy and controlled rate of pressure relaxation. To determine the values of the limiter coefficients, a constrained-optimization framework is employed using a quadratic objective function with linear constraints. The algorithm guarantees the positivity of the material volume and internal energy as well as the smooth relaxation of the pressure – this allows a significant increase in the robustness of the overall algorithm.The results of comprehensive testing of the new model have been presented for one- and two-dimensional multimaterial Lagrangian hydrodynamics along with representative results for 2D multimaterial arbitrary Lagrangian–Eulerian (ALE) calculations. The numerical tests have shown that in most cases the new IA-SSD closure model produces better results compared to the well known Tipton's closure model.

Compatible, total energy conserving and symmetry preserving arbitrary Lagrangian–Eulerian hydrodynamics in 2D rz – Cylindrical coordinates

We present a new discretization for 2D arbitrary Lagrangian–Eulerian hydrodynamics in rz geometry (cylindrical coordinates) that is compatible, total energy conserving and symmetry preserving.In the first part of the paper, we describe the discretization of the basic Lagrangian hydrodynamics equations in axisymmetric 2D rz geometry on general polygonal meshes. It exactly preserves planar, cylindrical and spherical symmetry of the flow on meshes aligned with the flow. In particular, spherical symmetry is preserved on polar equiangular meshes. The discretization conserves total energy exactly up to machine round-off on any mesh. It has a consistent definition of kinetic energy in the zone that is exact for a velocity field with constant magnitude.The method for discretization of the Lagrangian equations is based on ideas presented in [2,3,7], where the authors use a special procedure to distribute zonal mass to corners of the zone (subzonal masses). The momentum equation is discretized in its “Cartesian” form with a special definition of “planar” masses (area-weighted).The principal contributions of this part of the paper are as follows: a definition of “planar” subzonal mass for nodes on the z axis (r=0) that does not require a special procedure for movement of these nodes; proof of conservation of the total energy; formulated for general polygonal meshes.We present numerical examples that demonstrate the robustness of the new method for Lagrangian equations on a variety of grids and test problems including polygonal meshes. In particular, we demonstrate the importance of conservation of total energy for correctly modeling shock waves.In the second part of the paper we describe the remapping stage of the arbitrary Lagrangian–Eulerian algorithm. The general idea is based on the following papers [25–28], where it was described for Cartesian coordinates. We describe a distribution-based algorithm for the definition of remapped subzonal densities and a local constrained-optimization-based approach for each zone to find the subzonal mass fluxes.In this paper we give a systematic and complete description of the algorithm for the axisymmetric case and provide justification for our approach.1 The ALE algorithm conserves total energy on arbitrary meshes and preserves symmetry when remapping from one equiangular polar mesh to another.The principal contributions of this part of the paper are the extension of this algorithm to general polygonal meshes and 2D rz geometry with requirement of symmetry preservation on special meshes.We present numerical examples that demonstrate the robustness of the new ALE method on a variety of grids and test problems including polygonal meshes and some realistic experiments. We confirm the importance of conservation of total energy for correctly modeling shock waves.

Symmetry-preserving momentum remap for ALE hydrodynamics

In this paper, a symmetry-preserving remapping algorithm for vectors respecting their local bounds by components and in magnitude is presented. First, description of the Vector Image Polygon (VIP) limiter for a piece-wise linear velocity reconstruction is presented. Numerical fluxes obtained from this reconstruction lead to symmetry- and bounds- preserving remap of momentum for staggered Arbitrary Lagrangian-Eulerian (ALE) hydrodynamical methods. A novel bounds definition for vectors and corresponding modification of the VIP limiter is introduced, which fixes undershoots in a radial velocity component for polar meshes. Comparison with standard scalar-based limiters is given. Cyclic remapping is performed to numerically verify properties of the methods.

Velocity gradient induced line splitting in x-ray emission accompanying plasma–wall interaction

High-resolution x-ray spectroscopy was applied to study the ion back-scattering in plasma–wall interaction. In experiments carried out on the Prague Asterix Laser System (PALS), the energetic ions produced at thin laser burnt-through Al foil impacted the secondary C target where they were decelerated, trapped and scattered in radial and backward directions. The environmental conditions in near-wall plasmas were analyzed using the high-dispersion x-ray spectroscopy and temporally-resolved x-ray imaging. The radial expansion of ions back-scattered in the near-wall region was directly observed via the Doppler effect induced splitting of the Al Heα intercombination line. The experimentally observed line splitting was modeled using a combination of the 2D arbitrary Lagrangian Eulerian hydrodynamics code PALE with the multifluid code MULTIF and spectrally postprocessed using HULLAC atomic model builder and CRETIN collisional-radiative code.

3D unstructured mesh ALE hydrodynamics with the upwind discontinuous finite element method

We describe a numerical scheme to solve 3D Arbitrary Lagrangian-Eulerian (ALE) hydrodynamics on an unstructured mesh using discontinuous finite element space and an explicit Runge-Kutta time discretization. This scheme combines the accuracy of a higher-order Godunov scheme with the unstructured mesh capabilities of finite elements that can be explicitly evolved in time. The spatial discretization uses trilinear isoparametric elements (tetrahedrons, pyramids, prisms and hexahedrons) in which the primitive variables (mass density, velocity and pressure) are piecewise trilinear. Upwinding is achieved by using Roe's characteristic decomposition of the inter-element boundary flux depending on the sign of characteristic wave speeds. The characteristics are evaluated at the Roe average, of variables on both sides of the inter-element boundary, for a general equation of state. An explicit second order Runge-Kutta time stepping is used for the time discretization. To capture shocks, we have generalized van Leer's 1D nonlinear minmod slope limiter to 3D using a quadratic programming scheme. For very strong shocks we find it necessary to supplement this with a Godunov stabilization where the trilinear representation of the variables is reduced to its constant average value. The resulting numerical scheme has been tested on a variety of problems relevant to ICF (inertial confinement fusion) target design and appears to be robust. It accurately captures shocks and contact discontinuities without unstable oscillations and has second-order accuracy in smooth regions. Object-oriented programming with the C++ programming language was used to implement our numerical scheme. The object-oriented design allows us to remove the complexities of an unstructured mesh from the basic physics modules and thereby enables efficient code development.