The use of trigonometric polynomials as Lagrange multipliers in the harmonic mortar method enables an efficient and elegant treatment of relative motion in the stator-rotor coupling of electric machine simulation. Explicit formulas for the torque computation are derived by energetic considerations, and their realization by harmonic mortar finite element and isogeometric analysis discretizations is discussed. Numerical tests are presented to illustrate the theoretical results and demonstrate the potential of harmonic mortar methods for the evaluation of torque ripples.
In this work, we introduce a new space-time variational formulation of the secondorder wave equation, where integration by parts is also applied with respect to the time variable, and a modified Hilbert transformation is used. For this resulting variational setting, ansatz and test spaces are equal. Thus, conforming finite element discretizations lead to Galerkin-Bubnov schemes. We consider a conforming tensor-product approach with piecewise polynomial, continuous basis functions, which results in an unconditionally stable method, i.e., no CFL condition is required. We give numerical examples for a one-and a two-dimensional spatial domain, where the unconditional stability and optimal convergence rates in space-time norms are illustrated.
We analyze the finite element discretization of distributed elliptic optimal control problems with variable energy regularization, where the usual L 2 (Ω) norm regularization term with a constant regularization parameter is replaced by a suitable representation of the energy norm in H −1 (Ω) involving a variable, mesh-dependent regularization parameter (x). It turns out that the error between the computed finite element state u h and the desired state u (target) is optimal in the L 2 (Ω) norm provided that (x) behaves like the local mesh size squared. This is especially important when adaptive meshes are used in order to approximate discontinuous target functions. The adaptive scheme can be driven by the computable and localizable error norm u h − u L 2 (Ω) between the finite element state u h and the target u. The numerical results not only illustrate our theoretical findings, but also show that the iterative solvers for the discretized reduced optimality system are very efficient and robust.
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