Presentation on theme: "06/04/2017 Basic principles of numerical radiation hydrodynamics and the RALEF code M. M. Basko Slide 1: GSI 2014 Keldysh Institute of Applied Mathematics,"— Presentation transcript:
106/04/2017Basic principles of numerical radiation hydrodynamics and the RALEF codeM. M. BaskoSlide 1: GSI 2014Keldysh Institute of Applied Mathematics, Moscow, RussiaELI, Prague 10 July 2014
2Main constituents of the RALEF-2D package 06/04/2017Main constituents of the RALEF-2D packageThe RALEF-2D code has been developed with the primary goal to simulate high-temperature laser plasmas. Its principal constituent blocks areHydrodynamicsThermal conductionRadiation transportEOS and opacitiesLaser absorptionSlide 2: GSI 2014
3Equations of hydrodynamics 06/04/2017Equations of hydrodynamicsThe RALEF-2D code is based on a one-fluid, one-temperature hydrodynamics model in two spatial dimensions (either x,y, or r,z):ideal hydrodynamicsSlide 2: ILE 2010– energy deposition by thermal conduction (local), – energy deposition by radiation (non-local), – eventual external heat sources.
4Selection of principal dependent variables Illustration with the 1D planar example:Divergent form: ρ, ρu, ρENon-divergent form: ρ, u, eNaturally leads to a conservative numerical schemeEasier to avoid unphysical values of the internal energy e and temperature TRALEF
5Selection of principal independent variables Divergent Eulerian form: t, xNon-divergent Lagrangian form: t, mEulerian 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”.
6The 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:a purely Lagrangian step – the mesh is “attached” to the fluid, followed bya 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!
7Mesh topology in RALEF Mesh geometry: Cartesian (x,y) 06/04/2017Mesh topology in RALEFComputational mesh consists of blocks with common borders. Each block is topologically equivalent to a rectangle. Common block faces have equal number of cells.Mesh geometry:Cartesian (x,y)cylindrical (r,z)Slide 8: ILE 2010Mesh library:Different mesh options, distinguished by the value of variable igeom, are combined into a mesh library, which is being continuously expanded.
8Computational mesh: block structure 06/04/2017Computational mesh: block structureEvery block can be subdivided into np1np2 topologically rectangular parts;different parts can be composed of different materials.Slide 9: ILE 2010
9RALEF: the ALE technique in action 06/04/2017RALEF: the ALE technique in actionNcyc = 1Ncyc = 100Slide 11: ILE 2010
10Limitations of the ALE technique Example: laser irradiation of a Cu foilt = 1.6 nsfixed-pressure boundaryCu foillaserSimulation stops because a singularity develops at the plasma-vacuum boundary !
11Boundary conditionsTo obtain a particular solution of the hydrodynamics equations, we needto define the initial state, andto 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.
12Laser irradiation of a thin Sn disk free-outflow boundaryreflective (symmetry) boundary
13Different materialsBy 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: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.AlSn
14Equation of stateFor ideal hydrodynamics (without thermal conduction, viscosity, etc) we need an equation of state in the formRALEF can easily accommodate any thermodynamically stable EOS.However, because the principal variable is E=e+u2/2 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, andan absolutely stable equilibrium branch, obtained by means of Maxwell construction.This latter problem has not been resolved yet !
15Implicit and explicit numerical hydrodynamics Explicit schemeImplicit schemeImplicit schemes are numerically stable and allow large time steps, but are difficult to implement.Numerical stability of explicit schemes requires the Courant–Friedrichs–Lewy condition (CFL condition) to be fulfilledwhich for certain problems may incur prohibitively long computing times.RALEF has explicit hydrodynamics, i.e. is oriented on simulating fast processes.
16Numerical scheme for hydrodynamics 06/04/2017Numerical scheme for hydrodynamicsThe 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.Slide 7: ILE 2010
17The Godunov numerical method 06/04/2017The Godunov numerical methodPrincipal variables:i, ui, Ei― all assigned to the cell centers !Equation of state:Lagrangian phase: the Riemann problem is solved at each cell face fluxes F of momentum u and total energy E new ui(t+dt) and Ei(t+dt) ; mass is conserved.Slide 10: ILE 2010← 1st orderA fast non-iterative Riemann solver by J.K.Dukowicz (1985) is used.No artificial viscosity is needed !A “snag”: node velocities uvi !
18Importance of the 2-nd order + ALE 06/04/2017Importance of the 2-nd order + ALENon-linear stage of the Rayleigh-Taylor instability of a laser-irradiated thin carbon foilRALEF: 1-st orderRALEF: 2-nd orderSlide 12: ILE 2010
19(a test bed for radiation transport) 06/04/2017Thermal conduction(a test bed for radiation transport)Slide 17: ILE 2010
20Equations of hydrodynamics 06/04/2017Equations of hydrodynamicsThe 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):Slide 2: ILE 2010– energy deposition by thermal conduction (local), – energy deposition by radiation (non-local), – eventual external heat sources.
21Implicit versus explicit algorithms for conduction Explicit schemeFully implicit schemeAn 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) schemeRALEF is based on the SSI algorithm for both the thermal conduction and radiation transport.
22The SSI method for thermal conduction 06/04/2017The SSI method for thermal conductionThe 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).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.Slide 18: ILE 2010
23Time step control in the SSI method 06/04/2017Time step control in the SSI methodConstraints on t are based on usual approximation-accuracy considerations, but here we need two separate criteria with two control parameters:Slide 24: ILE 2010Ts is a problem-specific sensitivity threshold for temperature variations.
24Test problems: linear steady-state solutions 06/04/2017Test problems: linear steady-state solutionsTime convergence to the steady stateThe linear solutionis reproduced exactly on all grids, unless special constraints to ensure positiveness are imposed on strongly distorted grids.Slide 25: ILE 2010For 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.
25Test problems: non-linear wave into a cold wall 06/04/2017Test problems: non-linear wave into a cold wallParameters:Square grid 100x100Initial condition:Boundary condition:Solution:Slide 26: ILE 2010Test 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 .Temperature in all cells
26Standard grids for test problems 06/04/2017Standard grids for test problemsOnly the Kershaw grid is strongly distorted in the sense that some of the coefficients (1)(1) become negative.Slide 27: ILE 2010
27Test problems: non-linear steady-state solution 06/04/2017Test problems: non-linear steady-state solutionHere we test against a steady-state solution of the formwith the source termSlide 28: ILE 2010Our scheme clearly demonstrates the 2nd 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).
29Equations of hydrodynamics 06/04/2017Equations of hydrodynamicsThe 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):Slide 2: ILE 2010– energy deposition by thermal conduction (local), – energy deposition by radiation (non-local), – eventual external heat sources.
3006/04/2017Radiation transportTransfer 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 !Slide 3: ILE 2010In the present version, the absorption coefficient k and the source function B = B(T) are calculated in the LTE approximation.Coupling with the fluid energy equation:Radiation transport adds 3 extra dimensions (two angles and the photon frequency) the 2D hydrodynamics becomes a 5D radiation hydrodynamics !
31Not to be mixed up with spectral multi-group diffusion 06/04/2017Not to be mixed up with spectral multi-group diffusionIn many cases the term “radiation hydrodynamics” (RH) is applied to hydrodynamic equations augmented with the multi-frequency diffusion equationfor the spectral radiation energy density ; the coupling term to the fluid isSlide 4 : GSI 2013Here 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.
32Challenges for computational RH 06/04/2017Challenges for computational RHIdeally, 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 Iν guarantee Iν ≥ 0 at all collocation points ― provided that Bν ≥ 0 ;it must be conservative in the sense that both locally and globally the finite- difference equivalent of the Gauss theorem is fulfilled,Slide 6: GSI 2013it 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.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.To the best of our knowledge, no such scheme is known in the open literature.
33Calculation of Iν: the method of short characteristics 06/04/2017Calculation of Iν: the method of short characteristicsangular directions are discretized by using the Sn 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 equationwith 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 .Slide 36: ILE 2010Advantages: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.
34Conservativeness: flux conservation in vacuum 06/04/2017Conservativeness: flux conservation in vacuumConsider 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 haveSlide 7: GSI 2013Conclusion: 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 variablesBut then the high-order of finite-difference approximation, needed for the diffusion limit, is lost !
35The diffusion limit k >> | ln T|, | ln |, | ln k | 06/04/2017The diffusion limitThe diffusion limit is the limit ofk >> | ln T|, | ln |, | ln k |In this limit we can apply the asymptotic expansion:Slide 8: GSI 2013In the quasi-static LTE case the diffusion limit is equivalent to the approximation of radiative thermal conduction:
36Heating by two opposite beamlets in the diffusion limit 06/04/2017Heating by two opposite beamlets in the diffusion limitConsider two opposite beamlets propagating along an infinitely thin column h:In the diffusion limit we haveSlide 9: GSI 2013Having combined the two opposite beams, we getConclusion: our finite-difference scheme must correctly reproduce the 2nd spatial derivatives of the unknown function Iν – 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 !
37Achievements Failures 06/04/2017AchievementsWe have developed an original numerical scheme for radiation transport whichis 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.Slide 11: GSI 2013FailuresOur numerical schemeis not conservative (violates the finite-difference version of the Gauss theorem), andhas no strict convergence in the diffusion limit.
39Numerical diffusion: a searchlight beam in vacuum 06/04/2017Numerical diffusion: a searchlight beam in vacuumThe short-characteristic method produces a significant amount of numerical diffusion for light beams with sharp edges.Slide 40: ILE 2010For thermal radiation a certain amount of numerical diffusion may be more an advantage than a drawback.
40Test problems: radiative cooling of a slab 06/04/2017Test problems: radiative cooling of a slabComputational mesh:Exact solution:Slide 41: ILE 2010Convergence of the Sn methodS2S4S6S8S12error L223%7.6%2.3%1.9%1.15%Typical accuracy in problems with optically thin mesh cells is 12%.
41Diffusion limit in a slab 06/04/2017Diffusion limit in a slabSlide 15: GSI 2013No convergence, the error level depends on the level of mesh distortion.
42Test “fireball” (the limit of radiative heat conduction) 06/04/2017Test “fireball” (the limit of radiative heat conduction)Consider an initially hot unit sphere with T0 = 1 and k = 100/T. Propagation of a radial heat wave is governed by the conduction equationwhich admits an analytical self-similar solution with the front positionSlide 16: GSI 2013
43Fireball expansion in xy-geometry 06/04/2017Fireball expansion in xy-geometryBy t = the exact position of the front must be at r = 1.5.Slide 17: GSI 2013
44Final state: the conduction solution 06/04/2017Final state: the conduction solutionBy t = the exact position of the front must be at r = 1.5.Slide 18: GSI 2013
45Final state: the transport solution with k0 = 100 06/04/2017Final state: the transport solution with k0 = 100Slide 19: GSI 2013From practical point of view, in this particular case we observe a slow convergence to the exact diffusion solution.
46Final state: the transport solution with cell = 10 06/04/2017Final state: the transport solution with cell = 10Slide 20: GSI 2013In this example we observe no convergence to the exact diffusion solution. Numerical energy “leak” is 40% !
47SN method: the “ray effect” 06/04/2017SN method: the “ray effect”Consider radiation transport from a central “hot” rod across a vacuum cavity.S6Slide 43: ILE 2010Convergence of the Sn methodR2/R1=4S6S12S24S48S96error max/min41%18%6.8%1.9%0.47%error L215%8.5%2.4%0.52%0.23%
49Opacity options in RALEF-2D 06/04/2017Opacity options in RALEF-2DHere 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:power law,ad hoc analytical,inverse bremsstrahlung (analytical),Slide 47: ILE 2010GLT tables (source opacities from Novikov)
50Laser absorption in RALEF-2D Present model: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.
51Illustrative problems 06/04/2017Illustrative problemsSlide 44: ILE 2010
86The 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)
87Summary Our agenda for future work: 06/04/2017SummaryOur agenda for future work:laser deposition with refraction and reflection,hydrodynamics with metastable → stable phase transitions,material mixing,non-LTE radiation transport.Slide 51: ILE 2010