0301 Mathematical and Mechanical Aspects of Mixed-Dimensional Coupling Problems

  • Alexander Popp, University Of The Bundeswehr Munich
  • Barbara Wohlmuth, Technical University of Munich
  • Jan Martin Nordbotten, University of Bergen

Many applications concerning continuum mechanics in both natural and artificial materials involve the presence of high aspect ratio features. As examples, such features may take the form of both cavities (e.g. fractures in solids or thin conduits such as the vasculature in tissue) as well as composite inclusions (e.g. reinforcing fibers or reinforcing sheets). When considering an ambient 3-dimensional domain, it is natural to consider these features as either "rod-like" or "shell-like", and correspondingly model the features on manifolds of topological dimension 1 and 2, respectively.

Mixed-dimensional couplings therefore arise when the high aspect ratio features are embedded into an ambient domain, and when the physical processes on domains of different dimensionality have non-negligible interactions. Continuing on the examples above, we understand that despite their small cross section, cavities may be major conduits of flow, and similarly, fibers may carry significant tensile load.

Traditionally, dimensionally reduced models have typically been considered in and of themselves, with the relation to full 3D modeling being primarily for the purpose of validating approximation properties. Truly coupled mixed-dimensional problems are currently gaining much attention, driven in part by their importance in many practical engineering applications, e.g. geological (fractured porous materials) and biomedical (vasculature in tissue) applications to name only two examples. Last but not least, from a more abstract mathematical viewpoint, mixed-dimensional coupling problems also emerge when imposing complex interface conditions on embedded lower-dimensional subspaces, e.g. with the definition of a Lagrange multiplier.

In this minisymposium, we aim to address the forefront of current research, both in terms of the mathematical properties of mixed-dimensional equations, the approximation properties of mixed-dimensional models as compared to their equi-dimensional counterparts, and dedicated numerical methods in terms of both discretization and solvers. We wish to attract contributions not only from the applications exemplified above, but also scientists working on related problems from other applications.

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