### Immersed methods for multiscale and multiphysics problems (IMMEDIATE)

Project Type: Prin 2022

Funded by: MUR

Period: Sep 28, 2023 – Sep 27, 2025

Budget:

Principal Investigator: Paolo Zunino (Politecnico di Milano)

Local coordinator: Luca Heltai (Università di Pisa)

##### Description

Real-life multiscale and multiphysics problems present outstanding challenges, whose practical resolution often requires unconventional numerical methods. Classical numerical approaches require - at the very least - large computational power, excellent meshing tools, and the fulfillment of hypotheses which are often too stringent for real-life applications.
A particularly relevant example of problems that are severely limited by available computational technologies are biphasic systems with embedded interfaces such as vascularized biological tissues or fiber-reinforced materials. These systems are characterized by the intrinsic dimension of the immersed phase (for example, the blood vasculature) that is orders of magnitude smaller than the system scale, so that resolving the full geometrical complexity becomes unfeasible. Computational and mathematical tools can be used to analyze the functioning of these systems and to establish quantitative relationships between phenomena occurring on larger and smaller scales, leading to insights that could not be obtained otherwise. This modeling standpoint, bridging larger and smaller scale phenomena, is called the mesoscale. In this context, our reference applications are blood flow, transport phenomena and mechanics in complex vascular networks, such as the ones perfusing the brain. As a result, this project may significantly impact society by promoting the in-silico investigation of cancer and age-related neurological disorders.
From the mathematical standpoint, this research project focuses on the development and analysis of non-matching and immersed computational methods combined with complexity reduction methods, to enable the mesoscale resolution of problems with embedded structures. This objective entails the formulation, analysis, and approximation of Mixed-Dimensional PDEs (MD-PDEs), more precisely, coupled PDEs defined on embedded domains of different dimensionality, including the case of high-dimensionality gap. The immersed finite element method and the coupling through Lagrange multipliers are numerical techniques at the forefront of research in the context of numerical approximation of PDEs, which look ideally suited to tackle these problems. This project will hinge around these techniques to build a comprehensive approximation platform for MD-PDEs. The application of this platform to realistic and biologically relevant problems involving complex vascular networks embedded into permeable and deformable biological tissues will also be provided in the form of reproducible benchmarks.
The research units in Milano, Trieste and Trento share a consolidated expertise on advanced discretization schemes based on variational approaches, such as finite elements, immersed methods, and model order reduction combined with a vivid activity on the modeling of blood flow problems and its applications to various areas of medicine, with primary interests to cardiology, oncology and neurology.