Advanced FI²TD algorithms for transient eddy current problems

Abstract

Transient eddy current formulations based on the Finite Integration Technique (FIT) for the magneto‐quasistatic regime are extended to include motional induction effects of moving conductors with simple geometries by different approaches. A new regularization of the formulation using discrete grad‐div augmentation of the curlcurl formulation is presented and tested. To improve the implicit time integration process, several schemes for an error controlled variable time step selection are presented and for the repetitive solution of the arising large sparse systems of equations a sparse direct solver is compared to iterative methods such as a preconditioned conjugate gradient method and a new algebraic multigrid solver, which is aware of the curlcurl nullspace.

The development and modelling of devices and paradigms for transcranial magnetic stimulation

“…In the operating frequencies of TMS (< 10 kHz), the quasi-static approximation allows that the spatial distribution of the magnetic field can be decoupled from its temporal dynamics (Bossetti, Birdno, & Grill, 2008). Numerically, the calculation of induced currents tends to instability as is known from other disciplines (Beck, Hiptmair, & Wohlmuth, 1999; Biro & Preis, 1990; Biro, Preis, Buchgraber, & Ticar, 2004; Clemens, Wilke, & Weiland, 2001; Hollaus & Biro, 2000; Soleimani, Lionheart, Peyton, Xiandong, & Higson, 2006; Wong & Cendes, 1989). In magnetic stimulation, stability is further complicated by the nonideal geometry in case of realistic models and because the coil currents and the induced currents in the brain differ by at least six orders of magnitude but are numerically tightly coupled in Maxwell’s equations as well as the double-curl equation that is derived from them using the Helmholtz decomposition.…”

3-D transient eddy-current simulations using FI/sup 2/TD schemes with variable time-step selection

“…The other employed time integrator is a singly diagonally implicit four stage Runge-Kutta method of order three with an embedded solution of order two [SDiRK3(2)] [2]. In contrast to the ROW3(2) method used in [8], this method of order three as well as the embedded one of order two are stiffly accurate and L-stable.…”

“…The influence of the absolute tolerance parameter on the adaptive scheme is large [10, p. 131], [8], because our problems are not scaled to receive algebraic solutions with a 2-norm in the interval -a procedure usually undertaken in the mathematical literature. To determine appropriate values for the sensitive absolute tolerance , an a priori linear magnetostatic calculation can be performed for the highest excitation currents in the whole time interval of the transient simulation.…”

Numerical analysis of a magnetic recording write head benchmark problem using the finite integration technique

“…Considering the strongly varying device dimensions, this results in discrete problem sizes, which exceed the number of unknowns of the unstructured grids used for the FEM codes presented in [3]. Considering the asymptotical complexity of iterative SSOR-CG or ICCG solution methods of number of DoF for the linear systems of equations [12], this amounts to a severe drawback concerning the required times for simulations of the read write head. On the other hand, the regular grid structure allows for a very simple and fast grid generation, and it supports the implementation of fast data structures for matrix vector multiplications, which were shown in [7] to outperform those of general matrix storage formats.…”

“…The same code implementation on a SUN 3500 Enterprise (336 MHz) without using the optimized routines did not show advantages when compared to the SSOR-CG-solver for the smaller discretization (Problem B in Table I), which still required 1783 Mb memory. These high storage requirements yet prohibit to use these solution methods, for which an asymptotical optimality (work ) was shown in [12], within the nonlinear transient simulations of the SRC read write head. This left only a memory efficient, but less advanced SSOR-CG implementation available for the presented simulations.…”