Section: Application Domains

High performance simulation dedicated to ITER project

Participants : Rémi Abgrall, Pierre Ramet.

In the context of a previous ANR project called ASTER (Adaptive MHD Simulation of Tokamak Elms for iteR), we have established a collaboration with the physicists of the CEA/DRFC group. The magneto-hydrodynamic instability called ELM for Edge Localized Mode is commonly observed in the standard tokamak operating scenario. The energy losses the ELM will induce in ITER plasmas are a real concern. However, the current understanding of what sets the size of these ELM induced energy losses is extremely limited. Recently, encouraging results on the simulation of an ELM cycle have been obtained with the JOREK code developed at CEA but at reduced toroidal resolution. The JOREK code uses a fully implicit time evolution scheme in conjunction with the PaStiX sparse matrix library.

To improve the order of the spatial representation of the variables and their gradients, the so-called Bezier finite elements have been developed and implemented in the JOREK code. This allows an accurate alignment of the finite elements with the magnetic geometry of tokamak plasmas. This alignment is necessary due to the large anisotropy of the physics behavior along and perpendicular to the magnetic fieldlines. The Bezier elements, an extension of the standard cubic Hermite elements, allow the local refinement of the elements. During a postdoctoral position, H. Sellama has implemented an adaptive refinement and successfully applied to a tearing instability test case and to the injection of pellets in the plasma.

The fully implicit time evolution scheme in the JOREK code leads to large sparse matrices which have to be solved at every time step. The MHD model leads to very badly conditioned matrices. In principle the PaStiX library can solve these large sparse using the direct method. However, for large 3D problems the CPU time for the direct solver becomes too large. Iterative solution methods require a preconditioner adapted to the problem. Many of the commonly used preconditioners have been tested but no satisfactory solution has been found. Instead, a physics based preconditioner has been constructed by using the diagonal block for each of the Fourier modes in the toroidale direction. This means the preconditoner represents the linear part of each harmonic but neglects the interaction between harmonics. This scheme leads independent matrices that are factorized and solved in parallel using the PaStiX solver. A GMRES iterative solver with the preconditioner has proved to be an efficient solver for the non-linear MHD code. The developments of the JOREK code in the ASTER project have allowed simulating ELMs with a much improved accuracy in a real 3D geometry. The typical problem size has increased from $5\times {10}^5$ unknowns to $9\times {10}^6$ unknowns. The largest cases have been run on 1500 processors.