Adjoint variable method for time-harmonic Maxwell equations. (English) Zbl 1177.78064
Summary: Purpose - The purpose of this paper is to study the optimization problem of low-frequency magnetic shielding using the adjoint variable method (AVM). This method is compared with conventional methods to calculate the gradient.
Design/methodology/approach - The equation for the vector potential (eddy currents model) in appropriate Sobolev spaces is studied to obtain well-posedness. The optimization problem is formulated in terms of a cost functional which depends on the vector potential and its rotation. Convergence of a steepest descent algorithm to a stationary point of this functional is proved. Finally, some numerical results for an axisymmetric induction heater are presented.
Findings - Using Friedrichs’ inequality, the existence and uniqueness of the vector potential, its gradient and the corresponding adjoint variable can be proved. From the numerical results, it is concluded that the AVM is advantageous if the number of parameters to optimize is larger than two.
Research limitations/implications - The AVM is only faster than conventional methods if the gradients can be calculated with sufficient accuracy.
Originality/value - Theoretical results for eddy currents model are often based on a non-vanishing conductivity. The theoretical value of this paper is the presence of non-conducting materials in the domain. From a practical viewpoint, it has been demonstrated that the AVM can yield a significant reduction of computational time for advanced optimization problems.
Design/methodology/approach - The equation for the vector potential (eddy currents model) in appropriate Sobolev spaces is studied to obtain well-posedness. The optimization problem is formulated in terms of a cost functional which depends on the vector potential and its rotation. Convergence of a steepest descent algorithm to a stationary point of this functional is proved. Finally, some numerical results for an axisymmetric induction heater are presented.
Findings - Using Friedrichs’ inequality, the existence and uniqueness of the vector potential, its gradient and the corresponding adjoint variable can be proved. From the numerical results, it is concluded that the AVM is advantageous if the number of parameters to optimize is larger than two.
Research limitations/implications - The AVM is only faster than conventional methods if the gradients can be calculated with sufficient accuracy.
Originality/value - Theoretical results for eddy currents model are often based on a non-vanishing conductivity. The theoretical value of this paper is the presence of non-conducting materials in the domain. From a practical viewpoint, it has been demonstrated that the AVM can yield a significant reduction of computational time for advanced optimization problems.
MSC:
| 78M50 | Optimization problems in optics and electromagnetic theory |
| 78A30 | Electro- and magnetostatics |
| 49M15 | Newton-type methods |
| 90C31 | Sensitivity, stability, parametric optimization |
References:
| [1] | DOI: 10.1137/S0036139998348979 · Zbl 0978.35070 · doi:10.1137/S0036139998348979 |
| [2] | DOI: 10.1088/0266-5611/23/2/007 · Zbl 1115.35146 · doi:10.1088/0266-5611/23/2/007 |
| [3] | DOI: 10.1016/j.physb.2007.08.036 · doi:10.1016/j.physb.2007.08.036 |
| [4] | DOI: 10.1017/S0956792503005151 · Zbl 1051.78004 · doi:10.1017/S0956792503005151 |
| [5] | DOI: 10.1109/TMAG.2003.819460 · doi:10.1109/TMAG.2003.819460 |
| [6] | Sergeant, P., Cimrák, I., Melicher, V., Dupré, L. and van Keer, R. (2008), ”Adjoint variable method for the study of combined active and passive magnetic shielding”, Mathematical Problems in Engineering, Article ID 369125. · Zbl 1152.78305 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
