ANNEX XII Plasma multiphysics fluid analysis utilising an adaptive mesh generator

For the plasma multiphysics fluid analysis, an adaptive mesh generator is necessary that will include all carriers and that will produce accurate results at reduced computational cost, and it is explained below. ADAPTIVE MESH GENERATION A new adaptive mesh generator has been developed, reducing computational time and computer memory needs significantly (1). The new adaptive mesh generator, uses normalised error indicators, varying from 0 to 1, to guarantee optimal mesh resolution for all carriers involved in the analysis. Furthermore, it uses h- and r- refinement techniques such as mesh jiggling, edge swapping and node addition/removal operations to develop an element quality improvement algorithm that maximises the mesh quality, and a fast and accurate interpolation between meshes algorithm. Implementation of the adaptive mesh generator: The main steps used to implement the new adaptive mesh generator are shown in Fig. 1 in the form of a flow chart. The first step is to create a coarse initial mesh that will be used as a reference mesh for future refinement and coarsening of the meshes. The second step is to apply the initial conditions on the coarse initial mesh and calculate the amount of refinement for the coarse initial mesh by multiplying the error by a constant factor in each element. Then a newly created mesh is developed using a freely available two-dimensional software package which uses the "divide and conquer" method to refine elements. The elements that are created from this package though are of bad element equality and are treated first by using h-refinement techniques, such as edge swapping and node additions/removals to improve the interconnectivity between adjacent nodes. Secondly, the r-refinement technique, which involves jiggling of the mesh, follows, i.e. the movement of nodes around the geometry in such a controlled way that the overall element quality is improved. Once the adapted new mesh is created and can be used for the solution of the differential equations, an interpolation of the results from the coarse initial mesh to the adapted new mesh is performed. Having defined the new mesh and the corresponding values that all the variables have at the nodes of this mesh, the simulation proceeds forward in time, until an indicator signifies that the solution has reached the outer boundary of the refined region, and a re-meshing operation needs to be performed. The indicator calculates the maximum distance that a particle travels and ensures that it does not exceed the geometric tolerance of the initial coarse mesh, thereby it is guaranteed that the correct resolution is provided at all times. The electrons are generally 100 times faster than the positive and negative ions, thereby only the maximum distance travelled by the electrons since the last mesh refinement is used as an indicator. Then, in order to decide on the new mesh to run the simulations, the results are interpolated from the adapted new mesh back to the initial coarse mesh and then the error is calculated to create a new mesh1. Then this mesh is treated as above, using edge swapping and node additions/removals operations to improve the interconnectivity between adjacent nodes, and finally mesh jiggling operations to improve the overall element quality, creating the adapted new mesh1. Then one needs not to interpolate from the initial coarse mesh to the created adapted new mesh1, but instead from the created adapted new mesh to the created adapted new mesh1. This operation is performed due to the fact that interpolation is a source of diffusion on the results, and by interpolating just once, instead of twice, makes the results during the interpolation processes more accurate. Then having decided on the adapted new mesh1 and having interpolated the results, one runs the simulation forward in time and the procedure is repeated many times. This completes the implementation of the new adaptive mesh generator. For the implementation of the above adaptive mesh generator, three tools are necessary, which are the error calculation, the element quality improvement algorithm, and the interpolation between meshes tools, and are explained below. Error calculation tool: The error indicator decides upon the amount of refinement needed for the coarse mesh by calculating the gradients of the parameters being analysed, such as the electron, positive and negative ion fluxes, using the following equation: 2