Stable discretization methods and scalable solvers for embedded fiber/solid coupling

• Applicant: Dr.-Ing. Matthias Mayr
• Subject Area: Applied Mechanics, Statics and Dynamics
• Term: since 2023
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 528397555

Structures and systems reinforced by thin fibers play an increasingly important role in many applications. Embedded fibers can positively influence the mechanical, but also the functional properties of a component or system. For example, short steel fibers embedded in concrete components increase the tensile strength (even in the post-cracking regime) as well as the impact resistance. Further applications are found in light-weight structures in aerospace engineering or the design of medical devices. In order to quantitatively characterize the influence of such fibers on the system behavior and, above all, to predict and optimize the system’s behavior, a detailed understanding of the interaction of the fibers with the surrounding solid material is imperative. So far, the behavior of a single fiber can be determined experimentally as well as by fully resolved simulation models. However, if systems with several or even many fibers are to be analyzed, existing simulation methods reach their limits. As remedy, mixed-dimensional modeling approaches drastically reduce the size of a simulation model while maintaining comparable modeling quality, which also allows the analysis of systems with thousands of embedded fibers. The embedding solid material is still considered a volume problem, whereas the embedded fibers are modeled by slender one-dimensional structural models (beams). In this way, the meshing of the volume model and the fiber model is decoupled from each other during mesh generation. Both are coupled with each other in the numerical model by embedded mesh approaches. So far, the coupling constraints are enforced using so-called penalty methods, which penalize deviations from the exact constraint fulfillment. Penalty methods are conceptually easy to implement but come with mathematical disadvantages such as inexact constraint enforcement and pose immense challenges to the solution algorithms. This is exactly where the proposed research project comes in: By using Lagrange multipliers for constraint enforcement, the coupling conditions can be satisfied exactly and mathematical deficiencies and ill-conditioning from penalty methods are avoided. Yet, the discretization of the Lagrange multiplier field requires special consideration and demands suitable solution techniques. This project will deliver stable and robust discretization schemes for Lagrange multipliers for mixed-dimensional fiber/solid coupling. By enrichment with mechanical and discretization insight, novel and custom-built multilevel solvers and block preconditioning methods will be developed for the problem class of mixed-dimensional fiber/solid coupling. This clever combination of physical knowledge with numerics and simulation technology will allow to analyze realistic systems efficiently and scalably on modern parallel high-performance computers. This will open the door to further analyses (e.g. for the quantification of uncertainties or optimization problems).

• DFG Programme: Research Grants
• Co-Investigator: Professor Dr.-Ing. Alexander Popp