Band 28, Heft 6

- Inductive heating is a technological process where a steel workpiece is surrounded by an electromagnetic coil to which currents at various frequencies and time-varying amplitudes are applied. The amplitudes are considered as the controls and the objective is to heat the workpiece up to a desired temperature profile at the final time of the heating process. The workpiece is then quenched, which due to a phase transition in the crystallographic structure of the steel leads to a hardening of the surface of the workpiece. For the inductive heating process, the state equations represent a coupled system of nonlinear partial differential equations consisting of the eddy current equations in the coil, the workpiece, and the ambient air, and a heat equation in the workpiece. The nonlinearity stems from the temperature-dependent nonlinear material laws for steel, both with regard to its electromagnetic and thermal behaviour. Following the principle ’Discretize first, then optimize’, we consider a semi-discretization in time by the implicit Euler scheme, which leads to a discretetime optimal control problem. We prove the existence of a minimizer for the discrete-time optimal control problem and derive the first-order necessary optimality conditions.

- Applying the formalism of the Tsallis nonadditive statistical mechanics, a derivation of hydrodynamic and quasi-hydrodynamic equations is considered based on the generalized BGK kinetic equation. In particular, those equations are intended for construction of more appropriate macroscopic and mesoscopic models of transport systems related to the so-called anomalous systems where the corresponding phase space possesses a complicated (fractal) structure. The classic Boltzmann-Gibbs statistics (or Maxwell’s distribution of velocities in the case of kinetic theory) is violated in such systems and hence the additivity of extensive variables related to it does not hold either. The application of the approach developed in this paper does not change the structure of hydrodynamic and quasihydrodynamic equations, but the modified thermal and calorific state equations and also the transfer coefficients contain two additional free parameters, which are the parameter q of the non-additivity of the system and the fractional dimension D of the phase space. Along with the smoothing parameter t, these parameters can be determined empirically in each particular case from statistical or experimental data, which allows us to simulate the actual variable traffic situation within a continual approach both in regular and crisis cases.

- This paper is concerned with the derivation of conforming and non-conforming functional a posteriori error estimates for elliptic boundary value problems in exterior domains. These estimates provide computable and guaranteed upper and lower bounds for the difference between the exact and the approximate solution of the respective problem. We extend the results from [5] to non-conforming approximations, which might not belong to the energy space and are just considered to be square integrable. Moreover, we present some numerical tests.

- We present a method of deriving fully guaranteed bounds of the difference between the exact and approximate solutions of variational inequalities generated by problems in the theory of elasticity with nonlinear boundary conditions (e.g., unilateral and friction type boundary conditions). These estimates are obtained with the help of the duality technique in the calculus of variations. They do not contain mesh-dependent constants and are valid for any function from the corresponding energy space compared with the exact solution. We prove that the majorants of deviations are continuous and vanish if and only if approximate solutions coincide with the exact one. Several numerical tests demonstrate the quality of the estimates.