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The aim of this project is to introduce new concepts
into the area of
automatic mesh adaptivity for inequality constrained problems by
incorporating an appropriate optimization framework based on, for
example, nonlinear complementarity approaches and merit functions, and
by carefully handling the error estimates.
The following presentation is organized like this:
Outreach
Research
Plan
For a detailed
description of the project, click here
(PDF file).
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Outreach
The scientific merits of the project will be
- a systematic approach to goal oriented mesh
adaptivity for the numerical solution of constrained optimal control
and structural optimization problems associated with partial
differential equations,
- the implementation of the developed a posteriori
error estimators within adaptive multilevel finite element codes.
The technological impact will be the
- availability of efficient and reliable algorithmic
tools for the improvement of the design and the functionality of
technologically relevant devices and systems.
The broader educational impact of the project will be
- to introduce graduate students to state-of-the-art
numerical methods and optimization methods,
- enabling them to perform essential work on goal
oriented mesh adaptivity including implementional issues,
- using the material developed in this project in
courses on numerical methods for partial differential equations,
optimal control for partial differential equations, and structural
optimization.
- offering special topic courses and seminars on
adaptive finite element methods, control and state constrained optimal
control problems, and equality and inequality constrained optimization
problems related to scientific issues that are relevant for this
project.
Research Plan
The detailed research plan for this project is as follows:
First Year:
- Development and analysis of error estimators for
control constrained elliptic optimal control problems with distributed
and boundary controls,
- Implementation of the algorithmic tools and
application to model elliptic control problems.
Second Year:
- Development and analysis of error estimators for
equality and inequality constrained structural optimization problems,
- Development and analysis of error estimators for
state constrained elliptic optimal control problems with distributed
and boundary controls,
- Implementation of the algorithmic tools and
application to model structural optimization and elliptic control
problems..
Third Year:
- Development and analysis of a posteriori error
estimators for control and state constrained parabolic optimal control
problems,
- Implementation of the algorithmic tools and
application to model parabolic control problems.
The research teams of the principal investigators have extensive
experiences in the development, analysis, and implementation of
adaptive finite element methods on the basis of efficient and reliable
a posteriori error estimators and of state-of-the-art numerical methods
for optimal control and structural optimization problems. In
particular, the PI and his collaborators have developed
- <>residual type and hierarchical a posteriori
error
estimators and error estimators based on local averaging for various
finite element discretizations of partial differential equations
including nonconforming and mixed methods,
- efficient numerical solution techniques for equality
and inequality constrained shape and topology optimization problems
using state-of-the-art primal-dual Newton interior point methods.
They have applied the developed algorithmic tools in the
- numerical simulation of technologically relevant
problems in computational fluid dynamics [42],
Micro-Electro-Mechanical-Systems (MEMS), materials science, and
computational electromagnetics,
- optimization of devices and systems in high power
electronics and materials science.
The co-PI and his collaborators have developed
- semismooth Newton techniques for inequality
constrained optimal control problems, SQP-semismooth Newton methods,
and augmented Lagrangian SQP methods for the efficient numerical
treatment of inequality constraints,
- Newton-type approaches for shape and topology
optimization,
- the applications include control constrained optimal
control of the Navier-Stokes system, control of a simplified
Ginzburg-Landau model, optimal control of elasto-hydrodynamic
lubrication problems, shape optimization techniques in image
segmentation.
The principal investigators maintain co-operations with various
- scientific partners (R. Glowinski, Y. Kuznetsov
(Univ. of Houston), M.Heinkenschloss (Rice University), M. Ulbrich
(Univ. of Hamburg), V. Schulz (Univ. of Trier), K. Kunisch (Univ. of
Graz), M. Hinze (Univ. of Dresden), M. Masmoudi (Univ. of Toulouse), M.
Bergounioux (Univ. of Orleans), D. Ralph and S. Scholtes (Univ. of
Cambridge) ),
- industrial partners (Infineon Technologies, Siemens
AG, Schlumberger Oil-field Services)
on issues related to this project. Moreover, they are involved in the
organization of international workshops, conferences and minisymposia
on topics in optimal control and optimization for partial differential
equations (e.g., Math. Research Center
Oberwolfach (February 2003), GAMM Annual Meeting (March 2004), ECCOMAS
2004 (Jyv¨askyl¨a, Finland), ICIAM 2003 (Sydney, Australia),
SIAM Annual meeting 2002 (Philadelphia, USA), INFORMS meeting (Miami,
2001)).
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