Speaker
Description
Abstract
What grants a theory its predictive power?
The numerical simulation of fracture in brittle materials poses fundamental challenges due to the non-convex and irreversible nature of the underlying variational models. Evolution problems governed by energetic principles may admit multiple equilibria, making the computed fracture path strongly dependent on algorithmic choices and stability criteria.
In this talk I share a series of remarks on the role of stability in the construction and computation of fracture evolution in gradient-damage models.
Considering a prototype brittle variational model with an irreversibility constraint, we analyse equilibrium states using the full second variation of the energy. Equilibrium maps obtained via branch-following reveal the coexistence of stable and unstable branches and show that quasi-Newton methods, relying on approximate Hessians, may miss bifurcation points and select non-physical paths.
To address these issues, I discuss a computational framework based on three nonlinear variational inequality solvers: a hybrid solver for constrained equilibrium, a bifurcation solver based on a projected Hessian spectrum, and a cone-constrained stability solver using a projection–scaling algorithm. Implemented in a modular Python framework built on DOLFINx, PETSc and SLEPc, these tools enable scalable PDE simulations and provide a general approach for detecting bifurcations, localisation and irreversible transitions in nonlinear systems.
This strategy provides a robust approach for tracking fracture paths in nonlinear damage models and clarifies the interplay between numerical algorithms, stability criteria, and the emergence of localisation and fracture patterns in brittle systems.