2,364 research outputs found
A Posteriori Error Estimation for the p-curl Problem
We derive a posteriori error estimates for a semi-discrete finite element
approximation of a nonlinear eddy current problem arising from applied
superconductivity, known as the -curl problem. In particular, we show the
reliability for non-conforming N\'{e}d\'{e}lec elements based on a residual
type argument and a Helmholtz-Weyl decomposition of
. As a consequence, we are also able to derive an a
posteriori error estimate for a quantity of interest called the AC loss. The
nonlinearity for this form of Maxwell's equation is an analogue of the one
found in the -Laplacian. It is handled without linearizing around the
approximate solution. The non-conformity is dealt by adapting error
decomposition techniques of Carstensen, Hu and Orlando. Geometric
non-conformities also appear because the continuous problem is defined over a
bounded domain while the discrete problem is formulated over a weaker
polyhedral domain. The semi-discrete formulation studied in this paper is often
encountered in commercial codes and is shown to be well-posed. The paper
concludes with numerical results confirming the reliability of the a posteriori
error estimate.Comment: 32 page
Minimax Current Density Coil Design
'Coil design' is an inverse problem in which arrangements of wire are
designed to generate a prescribed magnetic field when energized with electric
current. The design of gradient and shim coils for magnetic resonance imaging
(MRI) are important examples of coil design. The magnetic fields that these
coils generate are usually required to be both strong and accurate. Other
electromagnetic properties of the coils, such as inductance, may be considered
in the design process, which becomes an optimization problem. The maximum
current density is additionally optimized in this work and the resultant coils
are investigated for performance and practicality. Coils with minimax current
density were found to exhibit maximally spread wires and may help disperse
localized regions of Joule heating. They also produce the highest possible
magnetic field strength per unit current for any given surface and wire size.
Three different flavours of boundary element method that employ different basis
functions (triangular elements with uniform current, cylindrical elements with
sinusoidal current and conic section elements with sinusoidal-uniform current)
were used with this approach to illustrate its generality.Comment: 24 pages, 6 figures, 2 tables. To appear in Journal of Physics D:
Applied Physic
A Posteriori Error Analysis for the Optimal Control of Magneto-Static Fields
This paper is concerned with the analysis and numerical analysis for the
optimal control of first-order magneto-static equations. Necessary and
sufficient optimality conditions are established through a rigorous Hilbert
space approach. Then, on the basis of the optimality system, we prove
functional a posteriori error estimators for the optimal control, the optimal
state, and the adjoint state. 3D numerical results illustrating the theoretical
findings are presented.Comment: Keywords: Maxwell's equations, magneto statics, optimal control, a
posteriori error analysi
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