Electromagnetic Devices with Moving Parts—Simulation with FEM/BEM Coupling
2. Continuous Formulation
2.1. Eddy-Current Model
2.2. Weak Form
- Perfect electric conductor (PEC) boundary conditions imply (and therefore ). In this case, one usually makes use of the function space , that is the subspace of (12) with vanishing on the boundary. Application of the weighted-residual method in this subspace does not yield any boundary term.
- Perfect magnetic conductor (PMC) boundary conditions state , which means that the boundary term can be dropped from Equation (19).
- Dirichlet-to-Neumann maps allow for a (global) functional relation between and . In this case, the boundary term is either approximated by absorbing or asymptotic boundary conditions or represented accurately by means of boundary integral equations as in the following. For an overview of these techniques, see .
2.3. Boundary Integral Equations
3.2. Spatial Discretisation
3.3. Time Integration
3.4. Newton Method
4. Solution of Linear System of Equations
- The matrix , that is, the linearisation of at , has a large kernel due to the fact that . In case there are non-conducting regions, or in other words , this deficiency carries over to the entire top-left block of the system matrix. In case of a direct solver, one would need to classify the degrees of freedom on the finite element mesh and eliminate the redundant ones, see . For iterative solvers, on the other hand, careful preconditioning is needed as discussed below.
- The matrices , , and are fully populated. This implies a quadratic complexity for storage and cost of matrix-vector products. Since a direct solution is complicated from the onset—see the previous point—it can be ruled out completely when considering its cubic complexity due to the boundary element matrices. In case of an iterative solution, only the actions of these matrices on a given vector need to be calculated.
4.1. Fast Multipole Method
5. Force Computation
6.1. Force between Permanent Magnets
6.2. TEAM 24
6.3. Eddy-Current Brake
6.4. Electromagnetic Launcher
6.5. Magnetic Metal Forming
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
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Rüberg, T.; Kielhorn, L.; Zechner, J. Electromagnetic Devices with Moving Parts—Simulation with FEM/BEM Coupling. Mathematics 2021, 9, 1804. https://doi.org/10.3390/math9151804
Rüberg T, Kielhorn L, Zechner J. Electromagnetic Devices with Moving Parts—Simulation with FEM/BEM Coupling. Mathematics. 2021; 9(15):1804. https://doi.org/10.3390/math9151804Chicago/Turabian Style
Rüberg, Thomas, Lars Kielhorn, and Jürgen Zechner. 2021. "Electromagnetic Devices with Moving Parts—Simulation with FEM/BEM Coupling" Mathematics 9, no. 15: 1804. https://doi.org/10.3390/math9151804