Novel Techniques for Interface Treatment in Mechanics

Modern computational mechanics increasingly requires the accurate and robust representation of interfaces between materials, phases, and physical domains. Classical finite element methods (FEMs), which are based on conforming meshes aligned with the geometry, often reach their limit of performance and robustness when interfaces evolve, exhibit discontinuities, or require high geometric fidelity without excessive remeshing effort. Geometric representations of interfaces may sometimes be imperfect, if obtained by means of scanning/imaging techniques, which are emerging as a prime tool in the development of Digital Twins (DTs) of mechanical systems. Applications such as fracture mechanics, multi-phase flows, additive manufacturing, composite materials, and contact problems rely fundamentally on numerical strategies that can resolve complex interfaces with optimal computational efficiency.

This course addresses these challenges by discussing advanced interface treatment. The topics covered represent state-of-the-art approaches in computational science and engineering, including third-media contact formulations; the Finite Cell Method; phase-field models for diffuse interfaces; unfitted discretization using CutFEM, the Shifted Boundary/Interface Method, and Optimization-Based Methods for embedded geometries and interfaces, as representative of the broad class of unfitted-mesh techniques. Together, these methods enable accurate simulations in situations where classical mesh-conforming strategies become impractical or break down entirely. Students will gain both conceptual insight and hands-on experience in these modern techniques, preparing them for current and emerging research challenges.

The primary objective of this course is to provide a comprehensive understanding of numerical interface treatment in the context of numerical methods. The course consists of an introductory discussion on: The classification of interface phenomena (geometric, physical, kinematic) and selection of appropriate numerical strategies for their treatment and the mathematical/numerical challenges of representing discontinuities, evolving boundaries, and complex geometries in advanced technologies based on FEM. Emphasis will be given on variational problem formulation, weak imposition of boundary conditions, numerical stability, matrix conditioning, and algebraic solution techniques. The following topics will be discussed in detail: