Projects & thesis topics
This page collects concrete work within my four research areas. Ongoing projects are already active. Open topics are bounded questions that can be shaped into a Bachelor’s or Master’s thesis. Exploratory questions describe longer-term directions and are not immediate thesis advertisements.
ONGOING active research · OPEN available student topic · EXPLORATORY larger question
Currently open thesis topics
Curvature of discrete and higher-order curves
Compare local and reconstructed curvature and measure their convergence on known curves.
Stabilized approximation of curvature
Implement and evaluate stabilized reconstructions of curvature and the shape operator.
Penalty methods for tangential fields
Study accuracy, constraint errors, parameter scaling, and conditioning on surfaces.
Shape characterization with Minkowski tensors
Implement tensor-valued shape measures and test their stability under refinement.
A. Geometry-aware finite-element methods
ONGOING
Stabilized curvature and shape-operator reconstruction
Direct differentiation of an approximated surface loses accuracy with each derivative. This project studies finite-element reconstructions of mean curvature and the shape operator that add edge-jump stabilization. Initial experiments show improved convergence. Current work concerns analysis, higher-order variants, conditioning, and comparison with HHJ-based reconstruction.
Related work: odd behaviour of even geometries · dune-curvedgrid
OPEN
Curvature of discrete and higher-order curves
Compare element-wise curvature with a global reconstruction based on changes of tangent angle. Extend the comparison from polygonal curves to higher-order approximations, compute errors on known curves, and measure convergence rates. The project combines differential geometry, numerical experiments, and a focused literature study.
Prerequisites: basic numerical analysis and programming. Prior finite-element experience is helpful but not required.
OPEN
Stabilized finite-element approximation of curvature
Implement stabilized reconstructions of mean curvature and the shape operator in one and two dimensions. Compare them with direct differentiation of the discrete geometry, study approximation errors and convergence rates, and estimate the conditioning of the resulting systems. A higher-order extension can be included if the initial study progresses well.
Prerequisites: a finite-element course, numerical linear algebra, and confident programming in C++ or a comparable language.
Related project: stabilized curvature and shape-operator reconstruction
EXPLORATORY Compatible discrete geometry
HHJ and Regge spaces provide complementary representations of the shape operator and its cofactor. A larger question is whether these quantities can be approximated so that a meaningful discrete analogue of Gauss-Codazzi compatibility is retained. This direction needs further analytical preparation before it becomes a defined project.
B. Finite elements for geometrically constrained fields
ONGOING
Symmetry, length, and tangential constraints
This joint project with Hanne Hardering and doctoral researcher Valentin Neumann studies vector and tensor fields with several interacting constraints. Current work compares exact tangentiality with penalized unit length against exact unit length with penalized tangentiality. The aim is to understand approximation, stability, penalty scaling, and the effect of the discrete surface geometry.
Related work: tangential errors of tensor surface finite elements · Area B overview
OPEN
Penalty methods for tangential vector fields
Implement a vector Laplace problem on the sphere and investigate how the tangentiality error depends on mesh size, surface approximation, and penalty parameter. The work should separate tangential and normal errors, check predicted convergence rates, and report the effect of the parameter choice on conditioning.
Prerequisites: finite elements, numerical linear algebra, and scientific programming.
Related work: tangential errors of tensor surface finite elements
EXPLORATORY Finite elements for bundle-valued fields
A tangential field takes values in spaces that vary with position. Extending geometric finite elements to more general fibre bundles would require new definitions of interpolation, lifting, and conformity when both the domain and target fibres are approximated. This is a longer-term mathematical and computational direction, not a current student topic.
C. Structure-preserving geometric evolution
ONGOING
Squared Gaussian-curvature flow
The gradient flow of the squared Gaussian-curvature energy involves the shape operator, its cofactor, Gaussian curvature, and their derivatives. The current project develops mixed formulations that avoid a direct discretization of very high derivatives and compares alternative curvature approximations. It is still in the research phase.
Related work: geometry-aware finite-element methods · structure-preserving evolution
ONGOING
Evolution controlled by surface Minkowski measures
Surface Minkowski tensors quantify rotational symmetry through global, tensor-valued shape measures. Current analysis and experiments ask whether a curve on a surface can evolve toward a prescribed symmetry or optimize such a measure. This branch is separate from the curvature-energy project and remains exploratory.
Related work: surface Minkowski tensors · SurfaceMinkowski.jl
OPEN
Characterizing planar shapes with Minkowski tensors
Implement translation-invariant Minkowski tensors for polygonal curves from their edge lengths and normal vectors. Test the resulting eigenvalues and eigenvectors on families of known shapes and study their stability under mesh refinement and perturbation. The project provides a self-contained introduction to tensor-valued shape measures.
Prerequisites: linear algebra, basic geometry, and programming in Julia, Python, or C++.
Related software: SurfaceMinkowski.jl
EXPLORATORY Coupled field and geometry evolution
If a surface evolves, its tangent spaces and the constraints on attached fields evolve as well. A future direction is to couple structure-preserving surface motion with constrained vector or tensor fields and determine which continuous identities should survive discretization.
D. Computational abstractions for advanced finite elements
ONGOING
Higher-order geometry and transformed finite elements in DUNE
Curvature calculations and Piola-transformed elements on curved cells require derivatives that are not yet part of the general DUNE geometry interface. Current work develops suitable interfaces for higher derivatives, tensor-valued ranges, and transformed elements. The goal is reusable support for HHJ, Regge, Arnold-Winther, and related finite elements.
Related software: DUNE and AMDiS · dune-curvedgrid · dune-tensor
EXPLORATORY Interoperable descriptions of advanced finite elements
Libraries such as DUNE, FEniCSx, NGSolve, deal.II, Basix, DefElement, and Symfem represent advanced elements in different ways. A longer-term question is whether element definitions or generated basis data can be exchanged without losing the transformations and tensor structure required by an implementation.
Working on a topic
A student project normally starts with focused reading and a small reproducible experiment. We then refine the question, agree on milestones, and meet regularly to discuss results and obstacles. Code, notes, parameters, and figures should remain reproducible from the beginning.
To inquire about an open topic, email contact@simon-praetorius.me with your degree program, current stage, relevant coursework or programming experience, and preferred timeframe. If your interests fit one of the research areas but not a listed topic, you are also welcome to get in touch. A suitable project still needs a clear and realistic scope.
For results and implementations connected to these projects, continue to selected publications and scientific software.