Real life multi-phase and multi-physics problems coupled across different scales present outstanding challenges, whose practical resolution often require unconventional numerical methods. When some of the intrinsic dimensions of one of the two phases (herein referred to as “the immersed phase”) is orders of magnitude smaller than the system scale, the numerical simulation of a fully resolved geometrical model becomes unfeasible. Yet, complex phenomena that happen at the smaller scale have an effect on the global behavior of the material, and one is often interested in the coupling between the two phases without homogenising away the immersed phase.
Non-matching and immersed methods can be used to enable the mesoscale resolution of problems with embedded structures. In this talk, I will discuss how to leverage immersed finite element methods and (distributed) Lagrange multipliers to formulate coupled multi-physics and multi-dimensional problems in a mathematically consistent way, analyze the model error deriving from the dimensionality reduction, and provide some relevant numerical examples. Multiscale modeling of vascularized tissues via nonmatching immersed methods. International Journal for Numerical Methods in Biomedical Engineering.
- G. Alzetta and L. Heltai. Multiscale modeling of fiber reinforced materials via non-matching immersed methods. Computers & Structures, 239:106334, Oct. 2020.
- L. Heltai and A. Caiazzo. Multiscale modeling of vascularized tissues via non-matching immersed methods. International Journal for Numerical Methods in Biomedical Engineering, 35(12):e3264, 2019.
Luca Heltai is an Associate Professor in Numerical Analysis at the International School for Advanced studies, in the Applied Mathematics Laboratory "mathLab" since 2012. He has been a research associate at Penn State College of Earth and Mineral Sciences, State College (USA) in the group of Prof. Francesco Costanzo in 2006 and 2007, and he obtained his PhD in Mathematics and Scientific Computing at the University of Pavia, under the supervision of Prof. Daniele Boffi, in 2006.
He was a Fulbright scholar at the Courant Institute of Mathematical Sciences in New York (USA), and a visiting faculty at the Ecole Polytechnique de Paris, at the Universitat Polytechnica de Catalunia, at the Weierstrass Institute for Applied Analysis and Stochastics, and at the King Abdullah University of Science and Technology.
He has been the director and co-founder of the joint SISSA-ICTP Master's program in High Performance Computing (https://www.mhpc.it/) from 2014 to 2022, and He contributed to the design and establishment of the Master Degree in Data Science and Scientific Computing offered jointly by SISSA, ICTP, University of Trieste, and University of Udine (https://dssc.units.it).
He is an expert in the development of numerical methods and algorithms for the solution of biological, physiological, and industrial problems modeled by partial differential equations, and he is one of the main developer of the open source library deal.II (https://www.dealii.org) for the high performance computing solution of partial differential equations.