The professorship for mathematics in civil engineering, lead by Prof. Dr. Kathrin Welker, is performing research to the topics of mathematical optimization, modeling, theory and numerics of partial differential equations. Specific focus is put on shape optimization in shape spaces, whereby stochastic aspects are also included in the modeling process.
Shape optimization problems are not only considered from an analytical point of view, but also numerical methods to solve them are developed. Additionally, shape spaces and their structure are one of the main research topics, especially with respect to Riemannian manifolds and diffeological spaces.
Goals of research are providing shape spaces with different structures, modeling of shape optimization problems as optimization problems on corresponding shape spaces, formulation of efficient numerical methods, and numerical algorithms which use the different structures of shape spaces.
Current Research Projects
Structural Health Monitoring (funded by dtec.bw)
The overarching goal of the SHM project is to develop new and innovative methods to monitor infrastructure buildings and to continuously evaluate their structural conditions. In this project, an interdisciplinary team consisting of engineers and mathematicians works together with industrial companies. This projects aims to develop methods which are eligible for the detection of any kind of damage in various building structures. The developed methods should allow the reliability-based evaluation of existing infrastructure buildings using sensor data.
Permeable Breakwaters for the Protection of Built Structures and Goods
As the threat to densely populated coastal regions from storm surges increases with climate change, the overarching goal of the interdisciplinary project “Permeable Breakwaters for the Protection of Built Structures and Goods” is to optimize breakwaters and thus ensure better coastal protection. The shape of a breakwater and also the shape of the slots (slanted, curved, different sizes) in the breakwater are crucial for efficiency: For example, the shape of the slots determines how water flows through, so that specifically shaped slots can dissipate more energy from the wave. The subproject of the Professorship for the Mathematics of Civil Engineering aims to develop mathematical methods to optimize breakwaters.
Former Research Projects
Semi-Smooth Newton Methods on Shape Spaces (funded by the German Research Foundation within DFG Priority Programme SPP 1962/2)
The main aim of this project is to set up an approach for investigating analytically and solving computationally shape optimization problems constrained by variational inequalities (VI) in shape spaces. Shape optimization problem constraints in the form of VIs are challenging, since classical constraint qualifications for deriving Lagrange multipliers generically fail.
In this project, we consider Newton-shape derivatives instead of classical shape derivatives in order to formulate first-order necessary optimality conditions. Setting up a Newton-shape derivative scheme is the guiding principle for the analytical and numerical investigations within this project. More precisely, the resulting scheme enables the analytical and computational treatment of shape optimization problems constrained by VIs which are non-shape differentiable in the classical sense such that these can handled and solved without any regularization techniques leading often only to approximated shape solutions. Further goals of this project are investigations in the area of shape optimization for VIs regarding appropriate shape space formulations, existence and well-posedness of solutions including stationary concepts in shape spaces, semi-smooth Newton methods in shape spaces, mesh independent algorithmic approaches, robust treatment of uncertainties and solution approaches to application problems like, e.g., from the field of (thermo-)mechanics.
Simulation-Based Design Optimization of Dynamic Systems under Uncertainties (funded by Landesforschungsförderung Hamburg)
The main aim of the project is to develop new innovative simulation methods for the robust optimization of complex components. By combining methods from applied mathematics and theoretical mechanical engineering, mathematical models, which involve dynamic operating conditions and uncertain manufacturing processes, will be developed. In particular, a robust design is important for maintenance-intensive and maintenance-free products from the Hamburg aviation and medical technology environment.
Current Job Opportunities
Student research assistants (on request; if interested, then please contact Prof’in Welker via E-mail to [email protected])