Basko, M.

Faik, S.

Maruhn, J.

Tauschwitz, Anna

GSI Repository

New laser energy deposition algorithm for the RALEF-2D code∗S. Faik1, An. Tauschwitz1, J. A. Maruhn1, and M. M. Basko21Goethe University, Frankfurt am Main, Germany; 2KIAM, Moscow, RussiaCorrect modeling of the laser beam evolution and powerdeposition on unstructured grids is a computationally andnumerically challenging task. The first goal for a new al-gorithm for the radiation-hydrodynamics code RALEF-2D[1] is the calculation of the refracted laser light distribu-tion with minimal numerical diffusion. Therefore a longcharacteristics approach is applied. The incoming laserbeam intensity is spatially discretized into single rays be-ing traced through the computational grid. On the basisof the eikonal equation in geometrical optics [2] the equa-tion of motion of a ray [3] in the undercritical regime,ne ≪ nc, becomes d2~x/dt2 = −(c2/2)(~∇ne/nc). Con-tinuous transitions of the ray trajectories up to the first spa-tial derivative between the numerical cells are guaranteedby the division of the original two-dimensional quadrilat-eral grid into triangles. Within each triangle the free elec-tron density ne(~x) then is uniquely defined piecewise linearby the vertex-centered valuesne and the gradients ~∇ne. In-side each triangle a ray with incoming power P0 depositsthe power Pdep = P0 (1− e−κ) given the optical depthκ = ∆s Im (σ). Here ∆s is the length of the ray segment,∆s =∫ ∆t0 c√1− ne [~x (t)] /nc dt, ∆t the correspond-ing transition time, and σ the complex refractive index (perunit length) of the associated quadrilateral cell calculatedby Kramers’ inverse bremsstrahlung formula.Figure 1 shows the mesh setup and two simulated ray tra-jectories for a quadratic density trough test case [3]. Withthe density profile ne(y) = (2nc/l2y)(y2− lyy+ l2y/2), themesh heigth ly = 100 mm, the temperature dependenceT (y) ∝ (ne(y)/nc)2/3, and a fixed Coloumb logarithmand ionization, analytical solutions for the cosine-shapedtrajectories and their optical depth exist. The dimensionlesstrajectory error ǫT at the mesh exit point and the relative er-ror ǫP in the summed-up deposited powers as functions ofthe number of quadrilateral cells in the y-direction Ny fora quarter cosine-wavelength are shown in Figure 2.The second goal for the new algorithm is the calculationof the deposited and reflected powers and the angular dis-tribution of the reflected laser light close to and above thecritical free electron density (ne & nc), e.g. at the sur-face of a solid metal. Therefore the raytracing algorithm isaugmented by an one-dimensional wave optics solver forthe wave propagation and energy deposition in a stratifiedmedium [2]. A geometrical optics ray can split up into an”overcritical” wave optics ray propagating perpendicular tothe critical surface and a reflected geometrical optics ray.Further test simulations will be conducted soon.∗Work supported by BMBF (Project 05P12RFFTR), HIC for FAIRFigure 1: Mesh setup and two simulated ray trajectories fora quadratic density trough test case [3] with mesh heigthly = 100 mm. Trajectory entry points: ya = 20, 80 mm;analytical trajectory exit point: yane = 50 mm.Figure 2: Dimensionless trajectory error ǫT at the meshexit point and relative error ǫP in the summed-up depositedpowers as functions of the number of quadrilateral cells inthe y-direction Ny for a quarter cosine-wavelength.References[1] M. M. Basko et al., GSI Report 2010-1 410.[2] M. Born, E. Wolf, ”Principles of Optics”, 7th Ed., 2005.[3] T. B. Kaiser, Phys. Rev. E 61 (2000) 895.GSI SCIENTIFIC REPORT 2014 APPA-MML-PP-13DOI:10.15120/GR-2015-1-APPA-MML-PP-13 287

New laser energy deposition algorithm for the RALEF-2D code

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New laser energy deposition algorithm for the RALEF-2D code

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