Vorträge

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.

Tumor perfusion and vascular properties are important determinants of a cancer’s
response to therapy. Being able to determine those parameters from patient-specific data
collected at the bedside would allow for better, more individual tumor treatment.
Models describing the transport of contrast agent based on advection-diffusion equations
are commonly used, but lack the ability to derive physically accurate solutions
for the transportation of tracer through an organ. Therefore, Sourbron proposed a
two-compartment model, where the flow of contrast agent is modeled by separating the
arterial and venous flows into a system of transport equations, coupled by a transfer
coefficient function which describes the exchange of the contrast agent from arteries to
veins through capillaries.
In this talk we discuss the parameter identification problem, i.e., how to estimate
flow velocities and the conversion coefficient function, given the concentration of contrast
agent over time, which will be obtained via 3D dynamic contrast-enhanced ultrasound
measurements. We derive adjoint equations for efficient gradient computation, discuss
the discretization of state and adjoint equation and the use of Leray projection within the
optimization algorithm to ensure a divergence free velocity field, and present numerical
results for artificial and ultrasound data.

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