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Application Area
- C (20)
In this paper a nonlocal phase-field model for non-isothermal phase transitions
with a non-conserved order parameter is studied. The paper complements
recent investigations by S. Zheng and the second author and treats
the case when the part of the free energy density forcing the order parameter
to attain values within the physically meaningful range [0; 1] is not given
by a logarithmic expression but by the indicator function of [0; 1] . The resulting
field equations form a system of integro-partial differential inclusions
that are highly nonlinearly coupled. For this system, results concerning global
existence, uniqueness and large-time asymptotic behaviour are derived. The
main results are proved by first transforming the system of inclusions into an
equivalent system of equations in which hysteresis operators occur, and then
employing techniques similar to those recently developed by the authors for
phase-field systems involving hysteresis operators.
We study the optimization of three dimensional curved rods and of shells
under minimal regularity assumptions for the geometry. The results that we
establish concern the existence of optimal shapes and the sensitivity analysis.
We also compute several numerical examples for the curved rods. The models
that we use have been investigated in our previous work [11], [16] and a
complete study of the Kirchhoff-Love arches and their optimization has been
performed in [10].
We prove new properties for the linear isotropic elasticity system and for
thickness minimization problems. We also present very recent results concerning
shape optimization problems for three-dimensional curved rods and
for shells. The questions discussed in this paper are related to the control
variational method and to control into coefficients problems.
In this paper, the one-dimensional equation for the transversal vibrations of an elastoplastic beam is derived from a general
three-dimensional system. The plastic behavior is modeled using the classical
three-dimensional von Mises plasticity model. It turns out that this single-yield model leads after a dimensional reduction to a multi-yield one-dimensional hysteresis model,
given by a hysteresis operator of Prandtl-Ishlinskii type whose density
function can be determined explicitly. This result indicates that the use
of Prandtl-Ishlinskii hysteresis operators in the modeling of elastoplasticity
is not just a questionable phenomenological approach, but in fact quite natural. In addition to the derivation of the model, it is shown that the resulting partial differential equation with hysteresis can be transformed into an equivalent system for which the existence and uniqueness
of a strong solution is proved. The proof employs techniques from the mathematical theory of hysteresis operators.
In this paper, we study an optimal control problem for a singular system of partial differential equations that models a nonisothermal phase transition with a nonconserved order parameter. The control acts through a third boundary condition for the absolute temperature and plays the role of the outside temperature. It is shown that the corresponding control-to-state mapping is well defined, and the existence of an optimal control and the first-order optimality conditions for a quadratic cost functional of Bolza type are established.
We prove the existence, uniqueness, thermodynamic consistency,
global boundedness from both above and below, and continuous data
dependence for a strong solution to an
integrodifferential model for nonisothermal phase transitions
under nonhomogeneous mixed boundary conditions.
The specific heat is allowed to depend on the order parameter,
and the convex component of the free energy may or may not
be singular.
We propose a model for non-isothermal phase
transitions with non-conserved order parameter driven by
a spatially nonlocal free energy with respect to both the
temperature and the order parameter. The resulting system of equations
is shown to be thermodynamically consistent and to admit a strong
solution.
We study a diffusion model of phase field type, consisting of
a system of two partial differential equations encoding the balances
of microforces and microenergy; the two unknowns are the order parameter
and the chemical potential. By a careful development of uniform estimates and
the deduction of certain useful boundedness properties, we
prove existence and uniqueness of a global-in-time
smooth solution to the associated initial/boundary-value problem;
moreover, we give a description of the relative $\omega$-limit set.
Mathematical modeling of Czochralski type growth processes for semiconductor bulk single crystals
(2012)
This paper deals with the mathematical modeling and simulation of
crystal growth processes by the so-called Czochralski method and related methods,
which are important industrial processes
to grow large
bulk single crystals of semiconductor materials such as, e.g., gallium arsenide
(GaAs) or silicon (Si) from the melt.
In particular, we investigate a recently developed
technology in which traveling magnetic fields are applied in order to
control
the behavior of the turbulent melt flow. Since numerous different physical effects
like electromagnetic fields, turbulent melt flows, high temperatures, heat transfer via
radiation, etc., play an important role in the process, the corresponding mathematical
model leads to an extremely difficult system of initial-boundary value problems for
nonlinearly coupled partial differential equations. In this paper, we describe a mathematical
model that is under use for the simulation of real-life growth scenarios, and we give an overview
of mathematical results and numerical simulations that have been obtained for it in recent years.