Talks

In [Mugnolo, Rhandi, DCDS-S (2022)] the semigroup generated by the Ornstein-Uhlenbeck operator on metric
star graphs was studied. In this talk, we consider a so-called regular rooted tree $T$
and analyse the Ornstein-Uhlenbeck operator on $L_2(T)$. In particular, leveraging
the group of symmetries of $T$, we obtain a decomposition of $L_2(T)$ into orthogonal
sum of subspaces that reduce the Ornstein-Uhlenbeck operator.
This talk is based on ongoing work with Marjeta Kramar Fijavž, Delio Mugnolo,
and Abdelaziz Rhandi.

Projekt mit Institut V-10 (Prof. A. Penn, Dr. S. Benders)

We model the behavior of a solid body under external pressure or traction. The deformation of the body depends on its material. We focus on elastic material models, which are used in deformation simulations of rubber or biological tissue. In general, the deformation of an elastic body is nonlinear with respect to the external pressure or traction. Analytical solutions are rarely available for nonlinear elastic deformation problems. Therefore, iterative methods such as Newton’s method are used to linearize the problem.

High-order finite element discretizations (p-FEM) provide accurate numerical solutions to the deformation problem. Based on a benchmark problem in computational mechanics, we demonstrate that the computation time for p-FEM discretizations is dominated by the numerical integration and assembly of the global stiffness matrices. The convergence of Newton’s method depends on an initial guess. Thus, to converge with large displacements, boundary conditions must sometimes be applied incrementally to ensure that the initial guess for Newton’s method is good enough.

Building on the traditional incremental load step approach, we introduce a new approach that exploits the hierarchical high-order finite element discretization. Instead of increasing the load on the full model, we iterate on reduced-order models in early stages of the computation. Furthermore, the early load steps are solved with a relaxed tolerance for the termination criterion in Newton’s method. We demonstrate that the new approach has the potential to reduce computation time to 40 − 60% of the original CPU time, depending on the geometry of the problem.

This talk contains the results obtained in my dissertation.

Zoomlink:
https://tuhh.zoom.us/j/81920578609?pwd=TjBmYldRdXVDT1VkamZmc1BOajREZz09

Geometrically unfitted finite element methods such as CutFEM, Finite Cell, XFEM or unfitted DG methods have been developed and applied successfully in the last decades to a variety of problems, ranging from scalar PDEs on stationary domains to systems of PDEs on moving domains and PDEs on level set surfaces. These approaches combined with established tools of finite element methods allowed to apply and analyse unfitted methods in many fields. In this talk, we deal with an elliptic interface problem for the time-harmonic quasi-magnetostatic Maxwells equations.

Here the material function, the magnetic permeability, can jump at an interface. Such problems are considered in low-frequency applications. Standard unfitted Nitsche methods are not robust with respect to the parameter k, proportional to the wavenumber. For example, a standard Nitsche discretization for the curl-curl-operator introduces terms that do no longer vanish for gradient fields.
In this talk, we will use a vectorial finite element discretization based on H(Curl) conforming functions. We will tackle the problem of robustness by using a mixed formulation and a Nitsche formulation. Additionally, we apply a careful tailored ghost penalization term. We will also give a short discussion of possible preconditioners.