Magnetic potential based formulation for linear and non-linear 3D RF sheath simulation

Abstract

Abstract This paper reports a new numerical scheme to simulate the radio-frequency (RF) induced RF sheath, which is suitable for a large 3D simulation. In the RF sheath boundary model, the tangential component of the electric field ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mi>E</mml:mi> <mml:mrow> <mml:mi mathvariant="normal">t</mml:mi> </mml:mrow> </mml:msub> </mml:math> ) is given by the gradient of a scalar electric field potential. We introduce two additional scalar potentials for the tangential components of the magnetic field, which effectively impose the normal electric displacement ( D n ) on the plasma sheath boundary condition via in-homogeneous Neumann boundary condition and constrain the tangential electric field on the surface as curl-free ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi mathvariant="normal">∇</mml:mi> <mml:mo>×</mml:mo> <mml:msub> <mml:mi>E</mml:mi> <mml:mrow> <mml:mi mathvariant="normal">t</mml:mi> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:math> ). In our approach, the non-linear sheath impedance is formulated as a natural extension of the large thickness (or asymptotic) sheath limit ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mi>D</mml:mi> <mml:mrow> <mml:mi mathvariant="normal">n</mml:mi> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:math> ), allowing for handling both asymptotic and non-linear regimes seamlessly. The new scheme is implemented using the Petra-M finite element method analysis framework and is verified with simulations in the literature. The significance of non-linearity is discussed in various plasma conditions. An application of this scheme to asymptotic RF sheath simulation on the WEST ICRF antenna side limiters is also discussed.

Topics & Concepts