In this article, we use numerical simulation to investigate transient temperature
phenomena during sublimation growth of SiC single crytals via physical
vapor transport (also called the modified Lely method). We consider the evolution
of temperatures at the SiC source and at the SiC seed crystal, which
are highly relevant to the quality of the grown crystals, but inaccessible to
direct measurements. The simulations are based on a transient mathematical
model for the heat transport, including heat conduction, radiation, and radio
frequency (RF) induction heating. Varying the position of the induction coil
as well as the heating power, it is shown that the measurable temperature difference
between the bottom and the top of the growth apparatus can usually
not be used as a simple indicator for the respective temperature difference
between SiC source and seed. Moreover, it is shown that there can be a time
lack of 1.5 hours between the heating of the temperature measuring points
and the heating of the interior of the SiC source.
We use a numerical optimization method to determine the control parameters
frequency, power, and coil position for the radio frequency (RF) induction
heating of the growth apparatus during sublimation growth of SiC single crystals
via physical vapor transport (PVT) (also called the modified Lely method). The
control parameters are determined to minimize a functional, tuning the radial
temperature gradient on the single crystal surface as well as the vertical temperature
gradient between SiC source and seed, both being crucial for high-quality
growth. The optimization is subject to constraints with respect to a required
temperature difference between source and seed, a required temperature range at
the seed, and an upper bound for the temperature in the entire apparatus. The
numerical computations use a stationary mathematical model for the heat transport,
including heat conduction, radiation, and RF heating to solve the forward
problem, and a Nelder-Mead method for optimization. A minimal radial temperature
gradient is found to coincide with a minimal temperature at the single
crystal surface, and a maximal temperature gradient between source and seed is
found to coincide with a low coil position.
Error estimates for the numerical approximation of boundary semilinear elliptic control problems
(2004)
We study the numerical approximation of boundary optimal control problems governed
by semilinear elliptic partial differential equations with pointwise constraints on the control.
The analysis of the approximate control problems is carried out. The uniform convergence of discretized
controls to optimal controls is proven under natural assumptions by taking piecewise constant
controls. Finally, error estimates are established.
Regular Lagrange multipliers for control problems with mixed pointwise control-state constraints
(2004)
A class of quadratic optimization problems in Hilbert spaces is considered, where
pointwise box constraints and constraints of bottleneck type are given. The main focus is to prove the
existence of regular Lagrange multipliers in L2-spaces. This question is solved by investigating the
solvability of a Lagrange dual quadratic problem. The theory is applied to different optimal control
problems for elliptic and parabolic partial differential equations with mixed pointwise control-state
constraints.
This paper addresses the regularization of pointwise state constraints in optimal
control problems. By analyzing the associated dual problem, it is shown that the regularized problems
admit Lagrange multipliers in L2-spaces. Under a certain boundedness assumption, the solution of
the regularized problem converges to the one of the original state constrained problem. The results
of our analysis are confirmed by numerical tests.
The aim of this paper is to study the behaviour of a weak solution to Navier-Stokes equations for isothermal fluids with a nonlinear stress tensor for time going to infinity. In an analogous way as in [18], we construct a suitable function which approximates the density for time going to infinity. Using properties of this function, we can prove the strong convergence of the density to its limit state. The behaviour of the velocity field and kinetic energy is mentioned as well.
In this paper, we investigate the decay rate of stabilization of the solution of the system of partial differential equations governing the dynamics of martensitic phase transitions in shape memory alloys under the presence of a viscous stress. The corresponding free energy is assumed in Landau-Ginzburg form and nonconvex as a function of the order parametr. We prove that for appropriate constants, which appear in the above-mentioned model, we can decide upon the exponencial decrease of the solution to its attractor for time tending to infinity.
We prove an optimal regularity result in the two dimensional theory of soft ferromagnetic films. The associate Euler-Lagrange equation
is given by a microlocally degenerate variational inequality involving
fractional derivatives. A difference quotient type argument based on a dual formulation in terms of magnetostatic potentials yields a H\"older estimate for the uniquely determined gradient projection of the magnetization field.
A primal-dual interior point method for state-constrained parabolic optimal control problems is considered. By a Lavrentiev type regularization, the state constraints are transformed to mixed control-state constraints which, after a simple transformation, can be handled as control constraints. Existence and convergence of the central path are shown. Moreover, the convergence of a short step interior point algorithm is proven in a function space setting. The theoretical properties of the algorithm are confirmed by numerical examples.
This paper concerns hyperbolic systems of two linear first-order PDEs in one space dimension with periodicity conditions in time and reflection boundary conditions in space.
The coefficients of the PDEs are supposed to be time independent, but allowed to be discontinuous with respect to the space variable. We construct two scales of Banach spaces (for the solutions and for the right hand sides of the equations, respectively) such that the problem can be modeled by means of Fredholm operators of index zero between corresponding spaces of the two scales.
The main tools of the proofs are separation of variables, integral representation of the solutions of the corresponding boundary value problems of the ODE systems and an abstract criterion for Fredholmness which seems to be new.
A Lavrentiev type regularization technique for
solving elliptic boundary control problems with pointwise state
constraints is considered. The main concept behind this
regularization is to look for controls in the range of the adjoint
control-to-state mapping. After investigating the analysis of the
method, a semismooth Newton method based on the optimality
conditions is presented. The theoretical results are confirmed by
numerical tests. Moreover, they are validated by comparing the
regularization technique with standard numerical codes based on the
discretize-then-optimize concept.
We consider an adaptive finite element method (AFEM) for obstacle problems associated with linear second order elliptic boundary value problems and prove a reduction in the energy norm of the discretization error which leads to $R$-linear convergence. This result is shown to hold up to a consistery error due to the extension of the discrete multipliers (point functionals) to $H^{-1}$ and a possible mismatch between the continuous and discrete coincidence and noncoincidence sets. The AFEM is based on a residual-type error estimator consisting of element and edge residuals. The a posteriori error analysis reveals that the significant difference to the unconstrained case lies in the fact that these residuals only have to be taken into account within the discrete noncoincidence set. The proof of the error reduction property uses the reliability and the discrete local efficiency of the estimator as well as a perturbed Galerkin orthogonality. Numerical results are given illustrating the performance of the AFEM.
Some aspects of reachability for parabolic boundary control problems with control constraints
(2009)
A class of one-dimensional parabolic optimal boundary control problems
is considered. The discussion includes Neumann, Robin, and Dirichlet
boundary conditions. The reachability of a given target state in final
time is discussed under box constraints on the control. As a mathematical
tool, related exponential moment problems are investigated. Moreover,
based on a detailed study of the adjoint state, a technique is presented
to find the location and the number of the switching points of optimal
bang-bang controls. Numerical examples illustrate this procedure.
Optimality conditions for a class of optimal control problems with quasilinear elliptic equations
(2008)
A class of optimal control problems for quasilinear elliptic
equations is considered, where the coefficients of the elliptic
differential operator depend on the state function. First- and
second-order optimality conditions are discussed for an associated
control-constrained optimal control problem. In particular, the
Pontryagin maximum principle and second-order sufficient
optimality conditions are derived. One of the main difficulties is
the non-monotone character of the state equation.
Second-order sufficient optimality conditions are established for the optimal control
of semilinear elliptic and parabolic equations with pointwise constraints on the control and the state. In
contrast to former publications on this subject, the cone of critical directions is the smallest possible in the
sense that the second-order sufficient conditions are the closest to the associated necessary ones. The theory
is developed for elliptic distributed controls in domains up to dimension three. Moreover, problems of elliptic
boundary control and parabolic distributed control are discussed in spatial domains of dimension two and one,
respectively.
We prove the existence, uniqueness, regularity and smooth dependence
of the weak solution on the initial data for a certain class of semilinear first order
dissipative hyperbolic systems with spacially discontinuous coefficients. Such
kind of hyperbolic problems have succesfully been used to describe the dynamics
of distributed feedback multisection semiconductor lasers in recent years. We
show that in a suitable function space of continuous functions the weak solutions
generate a smooth semiflow.
We revisit here the situation of a thin liquid film driven up an
inclined substrate by a thermally induced Marangoni shear stress against
the counter-acting parallel component of gravity. In contrast to previous
studies, we focus here on the meniscus region, in the case where the substrate
is nearly horizontal, so there is a significant contribution from the
normal component of gravity. Our numerical simulations show that the
time-dependent lubrication model for the film profile can reach a steady
state in the meniscus region that is unlike the monotonic solutions found
in [MÄunch, SIAM J. Appl. Math., 62(6):2045-2063, 2002]. A systematic
investigation of the steady states of the lubrication model is carried out by
studying the phase space of the corresponding third order ODE system. We
find a rich structure of the phase space including multiple non-monotonic
solutions with the same far-field film thickness.
An optimal control problem for a 2-d elliptic equation is investigated with pointwise control constraints.
This paper is concerned with discretization of the control by piecewise linear functions. The state and the
adjoint state are discretized by linear finite elements. Approximation of order h in the L1-norm is proved in the
main result.
We consider a sequence of curved rods which consist of isotropic material and which are clamped on the lower base or on both bases. We study the asymptotic behaviour of the stress tensor and displacement under the assumptions of linearized elasticity when the cross-sectional diameter of the rods tends to zero and the body force is given in the particular form. The analysis covers the case of a non-smooth limit line of centroids. We show how the body force and the choice of the approximating curved rods can affect the strong convergence and the limit form of the stress tensor for the curved rods clamped on both bases.
We investigate the dewetting rates of thin liquid films using a lubrication
model that describes the dewetting process of polymer melts on hydrophobized
substrates. We study the effect of different boundary conditions at the liquid/solid
interface, in particular, of the no-slip and the Navier slip boundary condition, and
compare our numerical solutions for the no-slip and the slip dominated cases to
available results that originate from scaling arguments, simplified
ow assumptions
and energy balances. We furthermore consider these issues for an extended lubrication
model that includes nonlinear curvature.