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Basic principles of numerical radiation hydrodynamics and the RALEF code M. M. Basko ELI, Prague 10 July 2014 Keldysh Institute of Applied Mathematics, Moscow, Russia

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Main constituents of the RALEF-2D package 1.Hydrodynamics 2.Thermal conduction 3.Radiation transport 4.EOS and opacities 5.Laser absorption The RALEF-2D code has been developed with the primary goal to simulate high-temperature laser plasmas. Its principal constituent blocks are

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Equations of hydrodynamics – energy deposition by thermal conduction (local), – energy deposition by radiation (non-local), – eventual external heat sources. The RALEF-2D code is based on a one-fluid, one-temperature hydrodynamics model in two spatial dimensions (either x,y, or r,z): ideal hydrodynamics

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Selection of principal dependent variables Illustration with the 1D planar example: Divergent form: ρ, ρu, ρE Non-divergent form: ρ, u, e Naturally leads to a conservative numerical scheme Easier to avoid unphysical values of the internal energy e and temperature T RALEF

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Selection of principal independent variables Divergent Eulerian form: t, xNon-divergent Lagrangian form: t, m Eulerian simulation: spatial mesh in x,y,z is fixed in time. Lagrangian simulation: Lagrangian mesh in m=∫ρdx is “attached” to the fluid; typically, the simulation time is severely limited by mesh “collapse”.

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The ALE technique (adaptive mesh) The Arbitrary Lagrangian-Eulerian approach allows free motion of the x,y,z mesh – independent of the motion of the fluid! Implementation in RALEF: every time step consists of 2 substeps: 1.a purely Lagrangian step – the mesh is “attached” to the fluid, followed by 2.a rezoning step, where a completely new x,y,z mesh is constructed, and the principal dynamic variables are remapped to the new mesh. At substep 2, a new mesh is constructed by solving the Dirichlet problem for a separate Poisson-like differential equation with a user-defined weight function. In RALEF, there are quite a number of user-accessible parameters for control over the mesh dynamics – which is often the most tricky part of a successful simulation!

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Mesh topology in RALEF Computational mesh consists of blocks with common borders. Each block is topologically equivalent to a rectangle. Common block faces have equal number of cells. Cartesian (x,y) cylindrical (r,z) Mesh geometry: Mesh library: Different mesh options, distinguished by the value of variable igeom, are combined into a mesh library, which is being continuously expanded.

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Computational mesh: block structure Every block can be subdivided into n p1 n p2 topologically rectangular parts; different parts can be composed of different materials.

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N cyc = 1N cyc = 100 RALEF: the ALE technique in action

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Limitations of the ALE technique Example: laser irradiation of a Cu foil laser Cu foil t = 1.6 ns fixed-pressure boundary Simulation stops because a singularity develops at the plasma-vacuum boundary !

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Boundary conditions To obtain a particular solution of the hydrodynamics equations, we need to define the initial state, and to impose certain boundary conditions. The initial state is conceptually simple and fully free to be set by the user: either equilibrium or non-equilibrium. Setting up a physically consistent and computationally efficient combination of boundary conditions is a considerably more complex task (requires basic understanding of hydrodynamics and some physical intuition). Basic types of boundary conditions in RALEF: reflective (symmetry) boundary (fixed in space → Eulerian) prescribed pressure (moves in space → Lagrangian) prescribed velocity (moves in space → Lagrangian) free outflow (fixed in space → Eulerian) prescribed inflow (fixed in space → Eulerian) Additional degree of freedom: the boundary conditions and the ALE mode can be changed at will at any time during the simulation.

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Laser irradiation of a thin Sn disk reflective (symmetry) boundary free-outflow boundary

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Different materials By finding an appropriate combination of boundary conditions and ALE options, one can adequately simulate practically any 2D problem with a single material. Multiple materials pose an additional challenge: AlSn In RALEF, mixing of different materials within a single mesh cell is not allowed hence, any material interface must be treated as a Lagrangian curve, which usually tends to get tangled: as a result, the simulation stops.

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Equation of state For ideal hydrodynamics (without thermal conduction, viscosity, etc) we need an equation of state in the form RALEF can easily accommodate any thermodynamically stable EOS. However, because the principal variable is E=e+u 2 /2 rather than e (or T ), the numerical scheme is not positive with respect to T sporadic appearance of negative temperatures! An additional conceptual difficulty arises for a van- der-Waals type of EOS in the region of liquid-vapor phase coexistence – i.e. below the binodal, where we have two different branches of the same EOS: a metastable branch, and an absolutely stable equilibrium branch, obtained by means of Maxwell construction. This latter problem has not been resolved yet !

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Implicit and explicit numerical hydrodynamics Numerical stability of explicit schemes requires the Courant–Friedrichs–Lewy condition (CFL condition) to be fulfilled Explicit scheme Implicit scheme Implicit schemes are numerically stable and allow large time steps, but are difficult to implement. which for certain problems may incur prohibitively long computing times. RALEF has explicit hydrodynamics, i.e. is oriented on simulating fast processes.

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Numerical scheme for hydrodynamics The numerical scheme for the 2D hydrodynamics is built upon the CAVEAT-2D (LANL, 1990) hydrodynamics package and has the following properties: it uses cell-centered principal variables on a multi-block structured quadrilateral mesh (either in the x-y or r-z geometry); is fully conservative and belongs to the class of second-order Godunov schemes (no artificial viscosity is needed); the mesh is adapted to the hydrodynamic flow by applying the ALE (arbitrary Lagrangian-Eulerian) technique; the numerical method is based on a fast non-iterative Riemann solver (J.K.Dukowicz, 1985), well adapted for handling arbitrary equations of state.

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The Godunov numerical method i, u i, E i No artificial viscosity is needed ! Principal variables: Equation of state: ― all assigned to the cell centers ! Lagrangian phase: the Riemann problem is solved at each cell face fluxes F of momentum u and total energy E new u i (t+dt) and E i (t+dt) ; mass is conserved. ← 1 st order A fast non-iterative Riemann solver by J.K.Dukowicz (1985) is used. A “snag”: node velocities u vi !

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Importance of the 2-nd order + ALE RALEF: 1-st orderRALEF: 2-nd order Non-linear stage of the Rayleigh-Taylor instability of a laser-irradiated thin carbon foil

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Thermal conduction (a test bed for radiation transport)

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Equations of hydrodynamics – energy deposition by thermal conduction (local), – energy deposition by radiation (non-local), – eventual external heat sources. The newly developed RALEF-2D code is based on a one-fluid, one-temperature hydrodynamics model in two spatial dimensions (either x,y, or r,z):

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Implicit versus explicit algorithms for conduction Explicit scheme Fully implicit scheme An explicit scheme is stable under the condition, which practically always incurs prohibitively long computing times. A fully implicit scheme is always stable but requires iterative solution – which can be implemented for thermal conduction, but becomes absolutely unrealistic for radiation transport. Symmetric semi-implicit (SSI) scheme RALEF is based on the SSI algorithm for both the thermal conduction and radiation transport.

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The SSI method for thermal conduction The numerical scheme for thermal conduction (M.Basko, J.Maruhn & A.Tauschwitz, J.Com.Phys., 228, 2175, 2009) has the following features: it uses cell-centered temperatures from the FVD (finite volume discretization) hydrodynamics on distorted quadrilateral grids, is fully conservative (based on intercellular fluxes with an SSI energy correction for the next time step), (almost) unconditionally stable, space second-order accurate on all grids for smooth , symmetric on a local 9-point stencil, computationally efficient. The key ingredient to the RALEF-2D code is the SSI (symmetric semi-implicit) method of E.Livne & A.Glasner (1985), used to incorporate thermal conduction and radiation transport into the 2D Godunov method (in order to avoid costly matrix inversion required by fully implicit methods).

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Time step control in the SSI method Constraints on t are based on usual approximation-accuracy considerations, but here we need two separate criteria with two control parameters: T s is a problem-specific sensitivity threshold for temperature variations.

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Test problems: linear steady-state solutions Time convergence to the steady state The linear solution is reproduced exactly on all grids, unless special constraints to ensure positiveness are imposed on strongly distorted grids. For sufficiently large time steps t, the SSI method does not converge to the steady-state solution an indication of its conditional stability (neutral for large t). Piecewise linear solutions with -jumps are reproduced exactly only with harmonic mean f,ij, and only on rectangular grids.

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Test problems: non-linear wave into a cold wall Parameters: Initial condition: Boundary condition: Solution: Test results: This solution cannot be simulated with the harmonic-mean f, whereas excellent results are obtained with the arithmetic- mean f : the front position is reproduced with an error of 0.1% for . Square grid 100x100 Temperature in all cells

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Standard grids for test problems Only the Kershaw grid is strongly distorted in the sense that some of the coefficients (1 )(1 ) become negative.

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Test problems: non-linear steady-state solution Here we test against a steady-state solution of the form with the source term. Our scheme clearly demonstrates the 2 nd order convergence rate on all grids, and is no less accurate than the best published schemes of Morel et al. (1992), Shashkov et al. (1996), Breil & Maire (2007).

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Radiation transport

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Equations of hydrodynamics – energy deposition by thermal conduction (local), – energy deposition by radiation (non-local), – eventual external heat sources. The newly developed RALEF-2D code is based on a one-fluid, one-temperature hydrodynamics model in two spatial dimensions (either x,y, or r,z):

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Radiation transport Transfer equation for radiation intensity I in the quasi-static approximation (the limit of c → ): Quasi-static approximation: radiation transports energy infinitely fast (compared to the fluid motion) the energy residing in radiation field at any given time is infinitely small ! Radiation transport adds 3 extra dimensions (two angles and the photon frequency) the 2D hydrodynamics becomes a 5D radiation hydrodynamics ! Coupling with the fluid energy equation: In the present version, the absorption coefficient k and the source function B = B (T) are calculated in the LTE approximation.

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Not to be mixed up with spectral multi-group diffusion In many cases the term “radiation hydrodynamics” (RH) is applied to hydrodynamic equations augmented with the multi-frequency diffusion equation for the spectral radiation energy density ; the coupling term to the fluid is Here the information about the angular dependence of the radiation field is lost; one simply has to solve some 30 – 100 additional mutually independent diffusion equations. Not much of a challenge for computational physics: there already exist numerically stable, positive and conservative numerical schemes on distorted (non-rectangular) grids. In the RALEF code we have such a scheme implemented for the thermal conduction.

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Challenges for computational RH Ideally, the numerical scheme for solving the radiation transfer equation must possess the following important properties: it must be positive in the sense that non-negative boundary values of I ν guarantee I ν ≥ 0 at all collocation points ― provided that B ν ≥ 0 ; it must be conservative in the sense that both locally and globally the finite- difference equivalent of the Gauss theorem is fulfilled, it must reproduce the diffusion limit on arbitrary quadrilateral (or triangular) grids in optically thick regions ― especially when the optical thickness of individual grid cells becomes >> 1 a high-order accuracy scheme is needed. To the best of our knowledge, no such scheme is known in the open literature. The difficulty of constructing such a scheme has been recognized early in the theory of neutron transport: K.D.Lathrop, J.Comp.Phys., 4 (1969) 475.

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Calculation of I ν : the method of short characteristics with the method of short characteristics (A.Dedner, P.Vollmöller, JCP, 178, 263, 2002). Mesh nodes are chosen as collocation points for the radiation intensity I. angular directions are discretized by using the S n method with n(n+2) fixed photon propagation directions over the 4π solid angle; for each angular direction and frequency, the radiation field I is found by solving the transfer equation Advantages: even on strongly distorted meshes, it is guaranteed that light rays pass through each mesh cell; the algorithm is generally computationally more efficient than that with long characteristics. Disadvantages: a significant amount of numerical diffusion in space.

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Conservativeness: flux conservation in vacuum Consider a scheme where the radiation intensity is assigned to mesh vertices. Consider one cell with a negligibly small absorption ( k 0 ): In our example on the left we have Conclusion: a numerical scheme based on nodal collocation points is generally non-conservative ! Conservativeness can be restored by assigning the intensity I to cell faces (R.Ramis, 1992) ― equivalent to using the fluxes H as independent variables But then the high-order of finite-difference approximation, needed for the diffusion limit, is lost !

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The diffusion limit The diffusion limit is the limit of In this limit we can apply the asymptotic expansion: In the quasi-static LTE case the diffusion limit is equivalent to the approximation of radiative thermal conduction: k >> | ln T|, | ln |, | ln k |

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Heating by two opposite beamlets in the diffusion limit Consider two opposite beamlets propagating along an infinitely thin column h: In the diffusion limit we have Having combined the two opposite beams, we get Conclusion: our finite-difference scheme must correctly reproduce the 2 nd spatial derivatives of the unknown function I ν – i.e. to be of high order accuracy ! The latter is a major challenge for non-rectangular grids in 2 and 3 dimensions – and especially so when the cell optical thickness Δτ >> 1 !

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Achievements We have developed an original numerical scheme for radiation transport which is strictly positive, from the practical point of view ― reproduces the diffusion limit with a fair degree of accuracy on not too strongly distorted meshes, works robustly in both (x,y) and (r,z) geometries. Our numerical scheme is not conservative (violates the finite-difference version of the Gauss theorem), and has no strict convergence in the diffusion limit. Failures

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Radiation transport: test problems

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Numerical diffusion: a searchlight beam in vacuum The short-characteristic method produces a significant amount of numerical diffusion for light beams with sharp edges. For thermal radiation a certain amount of numerical diffusion may be more an advantage than a drawback.

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Test problems: radiative cooling of a slab Exact solution: Computational mesh: Typical accuracy in problems with optically thin mesh cells is 1 2%. S2S2 S4S4 S6S6 S8S8 S 12 error L 2 23%7.6%2.3%1.9%1.15% Convergence of the S n method

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Diffusion limit in a slab No convergence, the error level depends on the level of mesh distortion.

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Test “fireball” (the limit of radiative heat conduction) Consider an initially hot unit sphere with T 0 = 1 and k = 100/T. Propagation of a radial heat wave is governed by the conduction equation which admits an analytical self-similar solution with the front position

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Fireball expansion in xy-geometry By t = the exact position of the front must be at r = 1.5.

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Final state: the conduction solution By t = the exact position of the front must be at r = 1.5.

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Final state: the transport solution with k 0 = 100 From practical point of view, in this particular case we observe a slow convergence to the exact diffusion solution.

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Final state: the transport solution with cell = 10 In this example we observe no convergence to the exact diffusion solution. Numerical energy “leak” is 40% !

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S N method: the “ray effect” Consider radiation transport from a central “hot” rod across a vacuum cavity. S6S6 R 2 /R 1 =4 S6S6 S 12 S 24 S 48 S 96 error max/min41%18%6.8%1.9%0.47% error L 2 15%8.5%2.4%0.52%0.23% Convergence of the S n method

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Opacities

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Opacity options in RALEF-2D Here we profit from many years of a highly qualified work at KIAM (Moscow) in the group of Nikiforov-Uvarov-Novikov (the THERMOS code based on the Hartree-Fock- Slater atomic modeling). Opacity options: 1.power law, 2.ad hoc analytical, 3.inverse bremsstrahlung (analytical), 7.GLT tables (source opacities from Novikov)

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Laser absorption in RALEF-2D An arbitrary number of monochromatic cylindrical laser beams is allowed: for every beam the transfer equation is solved by the same method of short characteristics as for thermal radiation. No refraction, no reflection, the inverse bremsstrahlung absorption coefficient, artificially enhanced absorption beyond the critical surface. Present model:

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Illustrative problems

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Importance of radiation transport for laser plasmas Peak absorbed laser flux: F 00 = 1.4 W/cm 2 Sn foil: Δz foil = 0.42 µm, R foil = 75 μm, R mesh = 120 μm. Temporal profile (normalized) Ring-Gaussian spatial profile (normalized) z CO 2 laser r Sn foil

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No radiative transfer: thermal conduction only

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Radiation transport: S 8 with 8 spectral groups

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Irradiation by a short (20ps) YAG ( = μm) pulse Nominal laser energy: E las-absorbed = 1 mJ (reflection ignored) Sn microsphere: R Sn = 17 μm, R mesh = 150 μm. Gaussian temporal profile (normalized)Gaussian spatial profile (normalized) z laser Absorbed laser energy: E las-absorbed = 0.4 mJ 0.6 mJ missed the target

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Sn droplet irradiated by CO 2 -laser ( = 10.6 μm) Sn microsphere: R Sn = 17 μm, R mesh = 150 μm. Gaussian temporal profile (normalized)Gaussian spatial profile (normalized) z laser Absorbed laser energy: E las-absorbed = 10 mJ Peak absorbed laser flux: F 00 = 6.2 10 9 W/cm 2

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Spectral Sn opacities In band: 2 -groups

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Click to see the movie: Time is in nanoseconds

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Evolution after the laser is turned off: equilibrium EOS

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The effect of metastable EOS Fully equilibrium EOS (Maxwell construction in the two-phase region) Metastable EOS (van der Waals loops in the two-phase region)

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Summary Our agenda for future work: laser deposition with refraction and reflection, hydrodynamics with metastable → stable phase transitions, material mixing, non-LTE radiation transport.

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