Quasi-static free-boundary equilibrium of toroidal plasma with CEDRES++: Computational methods and applications

Abstract

We present a comprehensive survey of the various computational methods in CEDRES++ for finding equilibria of toroidal plasma. Our focus is on free-boundary plasma equilibria, where either poloidal field coil currents or the temporal evolution of voltages in poloidal field circuit systems are given data. Centered around a piecewise linear finite element representation of the poloidal flux map, our approach allows in large parts the use of established numerical schemes. The coupling of a finite element method and a boundary element method gives consistent numerical solutions for equilibrium problems in unbounded domains. We formulate a new Newton method for the discretized nonlinear problem to tackle the various non-linearities, including the free plasma boundary. The Newton method guarantees fast convergence and is the main building block for the inverse equilibrium problems that we can handle in CEDRES++ as well. The inverse problems aim at finding either poloidal field coil currents that ensure a desired shape and position of the plasma or at finding the evolution of the voltages in the poloidal field circuit systems that ensure a prescribed evolution of the plasma shape and position. We provide equilibrium simulations for the tokamaks ITER and WEST to illustrate the performance of CEDRES++ and its application areas.Here again, this is a straightforward and simple calculation for all mappings except one: the mapping J P,h that is related to the non-linear current profile in the plasma domain. The mapping J P,h is given by

Plasma Magnetic Control in Tokamak Devices

“…The Grad-Shafranov partial differential equation is solved by using numerical codes [3,50,54,57], and returns the poloidal flux w, and in particular its value in the conductive structures that surrounds the plasma, both passive and active (i.e., the PF coils). This information can be used to compute the parameters of the following nonlinear lumped parameters circuital model, that can be obtained under axisymmetric assumption and describes the behaviour of the plasma and of the currents that flow in the surrounding conductive structures (more details can be found in [4 • y(t) are the outputs to be controlled; other than the plasma current and the current in the active and passive structures, this vector may contain plasma shape and position descriptors, e.g.…”

Towards Automated Magnetic Divertor Design for Optimal Heat Exhaust

“…Hence, if we want to use a fully adjoint method to compute the gradient of our reduced objective functional we need to have explicit expressions of the derivatives of c pe , c ct , c eq , and I with respect to the control ϕ and state variables q eq , q ct , and q pe . The derivatives B ϕ c eq pϕ, q eq q and B qeq c eq pϕ, q eq q of the equilibrium problem are available both for the continuous as well as for the discretized problem [8]. Also the derivative B qpe c pe pq ct , q pe q was given explicitly in [20].…”

“…We are solving this FBE problem by the numerical methods outlined in [8,17]. The poloidal flux is approximated by a finite dimensional function that is piecewise linear with respect to an unstructured triangular mesh.…”

“…Such problems are frequently solved with optimal design methods. We refer to [7] and [8] for examples of adjoint based optimization for tokamak control applications in the framework of sequential quadratic programming (SQP) [9]. Similar algorithms are used for real time plasma reconstruction codes [10][11][12][13], that identify the magnetic configuration that matches best with some observed measurement data.…”

“…the magnetic field lines, for a given current configuration. We are using an implementation of the numerical methods described in [8,17] to find approximate solutions of the FBE problem.…”