| part of | DFG Priority program 1253: | "Optimierung mit partiellen Differenzialgleichungen" | run time: | June 2006 - |
| project leader: | PD Dr. Luise Blank | Prof. Dr. Harald Garcke |
| address: | Lehrstuhl für Mathematik VIII
Universität Regensburg 93040 Regensburg Germany |
Summary
(as pdf)
The aim of this project is to develop efficient numerical methods
to control interface evolution governed by
Cahn-Hilliard variational inequalities.
The applications range from quantum dot formation
in crystal growth of heteroepitaxial thin films
and grain growth to void
evolution in microelectronic devices. In all these applications
a certain location of phases or special properties
of the interface distribution are of importance.
The Cahn-Hilliard model is a conserved phase
field model based on a diffuse (not sharp) interface.
The underlying free energy is related to the interfacial thickness and
includes an obstacle potential.
The model is usually formulated
as a non-standard variational inequality of
fourth order.
The current state of the art provides numerical simulation
mainly to study the evolution and various
appearing phenomena.
They are not fast enough
to be used as a simulation solver
for control purposes.
Hence an efficient
method has to be developed for the Cahn-Hilliard model.
The semi-implicit time discretization of this Cahn-Hilliard model
can be viewed itself as a control
problem with possibly nonlinear constraints, control box
constraints
and a highly complex cost function, which includes
semi-norms and non local behaviour
of the control function.
Hence the evolution of interfaces
can be simulated by a sequence of
optimization problems, where the inputs change with time.
The control of interface evolution can be seen as a process of
nested optimization.
This will be the starting point for our numerical approach.
In order to solve the control problem corresponding to the
Cahn-Hilliard model
we wish to study the application of the
primal-dual active set strategy
and/or a semi-smooth Newton method.
While on the first glance the optimization formulation falls into the class,
where convergence is guaranteed, it is
not obvious whether all convergence conditions are fulfilled.
This is one of the first issues which has to be clarified.
Here, due to
the time stepping local convergence is sufficient.
The next question to clarify is how
to apply the primal-active set strategy efficiently due
to the semi-norms in the cost function.
Issues as preconditioning, adaptivity and efficient time stepping
must be approached.
The goal is to derive a mesh independent,
superlinear convergent method where the dependence on the
interfacial thickness is moderate.
In practical applications the Cahn-Hilliard variational inequality
has to be coupled either to an elliptic system
or to a nonlinear heat equation.
Hence, when efficient methods
are developed for Cahn-Hilliard variational inequalities, these have to
be generalised to the extended versions.
The final goal is to solve optimal control problems in
which the extended versions of the
Cahn-Hilliard variational inequality act as
constraints.
In addition to the highly nonlinear constraints the cost functional
is often non-convex and gradient based.
First we plan to study
this optimal control problem, which falls into the class of MPEC's,
analytically.
We wish to derive
first and second order optimality conditions as well as
the existence of
Lagrange multipliers.
In a second step
it is planned to design and implement a fast solver
for the optimal control problem
with superlinear convergence.
Since it does not seem promising to obtain smoothness of
the solution operator concerning the variational inequality
and of the remaining constraints,
we target the application of a semi-smooth Newton-type method also
for this overall control problem.
Optimal design problems involving Cahn-Hilliard
variational inequalities shall also be an issue.
Last update: 05.09.2006