Abstract: 

In this talk,  we consider bi-directionally and one-directionally coupled mixed dimensional problems. The resulting systems result from a dimension reduction. Of special interest are bi-directionally coupled problems of co-dimension two.   Due to the lack of suitable trace results the coupling is performed via a suitable lifting. As an example, we consider the well-posedness analysis of a highly non-linear time dependent 1D-3D PDE system and track the influence of the coupling parameter. The mathematical model is then enriched by stochastic components mimicking the dynamic growth of capillary systems in case of angiogenesis. The  1D flow and transport process is given by nonlinear and linear hyperbolic PDEs suitable for larger arteries with highly elastic vessel walls and smaller ones with stiffer vessel walls, respectively. Numerical results illustrate the effect of the random growth algorithm but also how the system can be closed by 0D models if the flow in the arterioles and the peripheral circulation is not fully resolved.  Additionally, we consider two examples of one-directionally coupled 1D-3D problems. The conceptual difference between the two settings is which dimension affects the solution of the other one. The first example shows how the ground-motion triggers the vibration of 1D structures.  The ground-motion itself is realized by  a 3D elasto-acoustic coupled wave-propagation model, discretized in space by a spectral multi-patch DG approach. The motion is initiated by a seismic source  given by a fault plane calibrated from the magnitude Izmir 2020 earthquake. A challenging task is to resolve the multiple length scales that are present due to the huge differences in size between the close by dam building structure and the area of interest. The second example illustrates the influence of a 1D folded coil within an aneurysm on the 3D non-Newtonian blood flow discretized by a Lattice Boltzmann scheme. Alternatively to the fully resolved 2D flow one can use a one domain approach in which homogenized porosity and permeability enters into a porous media approach.

This talk is partly joint work with J.T. Oden and P. Jha (University of Texas at Austin), I. Mazzieri  (Politecnico di Milano), M. Stupazzini (Munich RE), V. Nicoli\'c (Radboud University) and M. Fritz,  T.K\"oppl, M. Muhr, M. Horvat,  A. Wagner (TU Munich).