The standard Lax-Wendroff scheme with the conservative Lax-Friedrichs nodal predictor on highly non-uniform meshes produces serious oscillations, making it useless on such meshes. Wendroff and White (1989) [13] proposed two versions (WW and WWJp) with different predictors which work robustly on such meshes. Both WW and WWJp are second order accurate. We investigate how these methods behave on highly non-uniform meshes of three types (Pike, cluster and van der Corput) for 1D smooth solutions of the Burgers and the Euler equations. The WW and WWJp methods are extended to 2D and tested on smooth solutions of the Euler equations on 2D meshes created by the Cartesian product of 1D highly non-uniform meshes. We have not been able to find any significant difference between the WW and WWJp results, thus the simpler WW should be preferred. (C) 2021 IMACS. Published by Elsevier B.V. All rights reserved.

We propose a general methodology and practical implementation of arbitrary Equations of State (EOS) evaluation for Lagrangian and ALE hydrodynamic simulations. This approach is based on higher-order interpolations of the Helmholtz free energy (HFE) and derived quantities. We also discuss several pitfalls related to thermodynamic consistency, physical relevance and robustness of the EOS calculations and demonstrate them for realistic values of temperature and density. The developed library HerEOS has been tested and used in various hydrodynamic codes for practical laser plasma simulations, some of which are presented here.

The edge viscosity of Caramana, Shashkov and Whalen is known to fail on the Noh problem in an initially rectangular grid. We present a simple change that significantly improves the behavior in that case. We also show that added energy exchange between cells improves the symmetry of both edge viscosity and the tensor viscosity of Campbell and Shashkov. As suggested by Noh, this addition also reduces the wall heating effect.

We present an artificial viscous force for two-dimensional axi-symmetric r–z geometry and logically rectangular grids that is dissipative, conserves the z-component of momentum and preserves spherical symmetry on an equi-angular polar grid. The method turns out to be robust and performs well for spherically symmetric problems on various grid types, without any need for problem- or grid-dependent parameters.

This work is focused on the issue of symmetry preservation, energy and volume conservation and other important properties of staggered Lagrangian hydrodynamic schemes in cylindrical geometry. Typical advantages and drawbacks of existing area-weighted (AW) and genuinely r-z schemes will be pointed out. With quadrilateral cells it is known that, in r-z, spherical symmetry preservation, perfect satisfaction of GCL, and total energy conservation are incompatible [9].Being aware of this, we propose a staggered approach that conserves energy by construction and tries to do its best by diminishing the GCL error to the order of entropy error. In particular, we correct the volume consistent forces from [5] so that spherical symmetry is preserved. This idea is similar to the approach from [8], where we suggested a new r-z artificial viscosity that preserves symmetry and is (unlike typical AW viscosities) strictly dissipative.A practical implementation our symmetrization term will be presented. Its comparison to the existing methods from [1] and [5] will be demonstrated on a convergence study of the adiabatic Coggeshall test, and the effect of the symmetrization term on accuracy will be assessed using the Sedov blast wave test.

SUMMARYWe present a novel artificial viscosity for staggered Lagrangian schemes in 2D axi‐symmetric r‐z geometry on logically rectangular grids. The suggested viscous force is dissipative by construction, conserves both components of momentum, and preserves spherical symmetry on an equiangular polar grid. This method turns out to be robust and performs well for spherically symmetric problems on various grid types (symmetric, perturbed polar, rectangular), without any need for tinkering with problem‐dependent or grid‐dependent parameters. The results are compared with the outcome of the area‐weighted approach using the popular tensor viscosity by Campbell and Shashkov. Copyright © 2014 John Wiley & Sons, Ltd.

Jets of energetic ions launched at laser-burnt-through foils represent an efficient tool for investigation of plasma interaction with solid surfaces (plasma-wall interaction, PWI) and for description of transient phenomena occurring close to the walls. Highly charged ions approaching the secondary target interpenetrate the near surface layer, collide with the counter-propagating matter and capture a large number of electrons. This results in a creation of atoms in highly excited Rydberg states or hollow ions with multiple inner vacancies; plasma jet and target ions may also undergo charge exchange (CE) processes. We report PWI experiments with Al/Si(PMMA) and Al/C targets irradiated at normal or oblique laser incidence. The distinct dip structures observed in red wings of Al Lyγ self-emission is interpreted in terms of CE between C6+ and Al12+ in the near-wall zone. The spectroscopic identification of CE phenomena is supported by results of analytical and numerical calculations.

SUMMARYThe aim of the present work is the 3D extension of a general formalism to derive a staggered discretization for Lagrangian hydrodynamics on unstructured grids. The classical compatible discretization is used; namely, momentum equation is discretized using the fundamental concept of subcell forces. Specific internal energy equation is obtained using total energy conservation. The subcell force is derived by invoking the Galilean invariance and thermodynamic consistency. A general form of the subcell force is provided so that a cell entropy inequality is satisfied. The subcell force consists of a classical pressure term plus a tensorial viscous contribution proportional to the difference between the node velocity and the cell‐centered velocity. This cell‐centered velocity is an extra degree of freedom solved with a cell‐centered approximate Riemann solver. The second law of thermodynamics is satisfied by construction of the local positive definite subcell tensor involved in the viscous term. A particular expression of this tensor is proposed. A more accurate extension of this discretization both in time and space is also provided using a piecewise linear reconstruction of the velocity field and a predictor‐corrector time discretization. Numerical tests are presented in order to assess the efficiency of this approach in 3D. Sanity checks show that the 3D extension of the 2D approach reproduces 1D and 2D results. Finally, 3D problems such as Sedov, Noh, and Saltzman are simulated. Copyright © 2012 John Wiley & Sons, Ltd.

This talk is focused on the issue of symmetry preservation, energy and volume conservation and other important properties of Lagrangian hydrodynamic schemes in cylindrical geometry. Existing approaches to construct schemes in r-z will be reviewed, and the way they deal with basic physical requirements studied. Ideas will be given on how to overcome their drawbacks while leveraging their strengths, and examples of practical implementations will be shown.

We present a new flux-corrected approach for remapping of velocity in the framework of staggered arbitrary Lagrangian–Eulerian methods. The main focus of the paper is the definition and preservation of coordinate invariant local bounds for velocity vector and development of momentum remapping method such that the radial symmetry of the radially symmetric flows is preserved when remapping from one equiangular polar mesh to another. The properties of this new method are demonstrated on a set of selected numerical cyclic remapping tests and a full hydrodynamic example.

Production of sharply collimated high velocity outflows - plasma jets from massive planar targets by a single laser beam at PALS facility is clarified via numerical simulations. Since only a few experimental data on the intensity distribution in the interaction beam near the focus are available for the PALS facility, the laser beam profile was calculated by a numerical model of the laser system and the interaction optics. The obtained intensity profiles are used as the input for plasma dynamic simulations by our cylindrical two-dimensional fluid code PALE. Jet formation due to laser intensity profile with a minimum on the axis is demonstrated. The outflow collimation improves significantly for heavier elements, even when radiative cooling is omitted. Using an optimized interaction beam profile, a homogeneous jet with a length exceeding its diameter by several times may be reliably generated for applications in laboratory astrophysics and impact ignition studies.

Interest in laser-driven plasma jets is justified by their relevance for high-energy-density laboratory astrophysics and for the fusion directed research. In experiments carried out on the iodine laser system PALS, interactions of plasma jets with solid surfaces are studied in context with phenomena accompanying the material erosion and migration at plasma facing components, i.e., an issue of paramount importance for development of future fusion devices. The energetic ions were produced at burnt-through foils with low-to-high atomic numbers (Al, Ag, Ta). The formation of the outflow plasma was investigated using the three-frame interferometry, the observed density distribution was complemented by numerical modeling of the plasma parameters based on the Arbitrary Lagrangian Eulerian code PALE. The found optimum conditions for the jet production were used in a design of alternate experimental configurations. The interaction of the directional plasma flows with secondary targets was studied via x–ray imaging, optical and high–resolution x–ray spectroscopy. The examples of jet applications for investigating the transition phenomena at surfaces of plasma-exposed solids are presented.

A new optimization-based synchronized flux corrected conservative interpolation (remapping) of mass, momentum and energy for arbitrary Lagrangian Eulerian method is developed. Fluxes of conserved variables (mass, momentum and total energy) are limited in a synchronous FCT-like way to preserve local bounds in density, velocity and specific internal energy.

New method for weighted condition number smoothing of general unstructured computational meshes is presented. Its core, proper discretization of weighted smoothness functional, is detailed, options of particular implementation are discussed and demonstrated on general convex polygonal cells in two dimensions. Possible applications of this algorithm are suggested, namely solution-sensitive mesh adaptation (respecting the variation of some variable) and prevention of unwanted smoothing effects on polar meshes.

Simulations of laser-produced plasmas are essential for laser-plasma interaction studies and for inertial confinement fusion (ICF) technology. Dynamics of such plasmas typically involves regions of large scale expansion or compression, which requires to use the moving Lagrangian coordinates. For some kind of flows such as shear or vortex the moving Lagrangian mesh however tangles and such flows require the use of arbitrary Lagrangian Eulerian (ALE) method. We have developed code PALE (Prague ALE) for simulations of laser-produced plasmas which includes Lagrangian and ALE hydrodynamics complemented by heat conductivity and laser absorption. Here we briefly review the numerical methods used in PALE code and present its selected applications to modeling of laser interaction with targets.

The aim of the present work is to develop a general formalism to derive staggered discretizations for Lagrangian hydrodynamics on two-dimensional unstructured grids. To this end, we make use of the compatible discretization that has been initially introduced by E. J. Caramana et al., in J. Comput. Phys., 146 (1998). Namely, momentum equation is discretized by means of subcell forces and specific internal energy equation is obtained using total energy conservation. The main contribution of this work lies in the fact that the subcell force is derived invoking Galilean invariance and thermodynamic consistency. That is, we deduce a general form of the sub-cell force so that a cell entropy inequality is satisfied. The subcell force writes as a pressure contribution plus a tensorial viscous contribution which is proportional to the difference between the nodal velocity and the cell-centered velocity. This cell-centered velocity is a supplementary degree of freedom that is solved by means of a cell-centered approximate Riemann solver. To satisfy the second law of thermodynamics, the local subcell tensor involved in the viscous part of the subcell force must be symmetric positive definite. This subcell tensor is the cornerstone of the scheme. One particular expression of this tensor is given. A high-order extension of this discretization is provided. Numerical tests are presented in order to assess the efficiency of this approach. The results obtained for various representative configurations of one and two-dimensional compressible fluid flows show the robustness and the accuracy of this scheme.

In this work we develop a general framework to derive and analyze staggered numerical scheme devoted to solve hydrodynamics equations.

A new optimization-based synchronized flux-corrected conservative interpolation (remapping) of mass and momentum for arbitrary Lagrangian–Eulerian hydro methods is described. Fluxes of conserved variables – mass and momentum – are limited in a synchronous way to preserve local bounds of primitive variables – density and velocity.

We develop a general framework to derive and analyze staggered numerical schemes devoted to solve hydrodynamics equations in 2D. In this framework a cell-centered multi-dimensional approximate Riemann solver is used to build a form of artificial viscosity that leads to a conservative, compatible and thermodynamically consistent scheme. A second order extension in space and time for this scheme is proposed in this work and we prove on numerical examples the validity of this approach.

Three Arbitrary Lagrangian-Eulerian codes are compared on a set of test problems. Code CHIC is based on cell-centered Lagrangian scheme while codes ALE INC and PALE are based on staggered Lagrangian scheme. Pure Lagrangian methods are first tested on Sod and Sedov problems. The full ALE methods are then compared on two more advanced problems, namely triple point and shock bubble interaction problems, which cannot be treated by pure Lagrangian method due to severe distortion of moving Lagrangian mesh.