2015/08/12 by Marco Artina, Artina, Marco, Filippo Cagnetti +5 · 1 citation
Chemical Engineering · Computer Science · Physics and Astronomy · #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Numerical Analysis (math.NA) #Rheology and Fluid Dynamics Studies #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.1508.02965
openalex publication_date 2015/08/12 · openalex created_date 2022/10/04 · openalex updated_date 2026/07/28
We introduce a novel constructive approach to define time evolution of\ncritical points of an energy functional. Our procedure, which is different from\nother more established approaches based on viscosity approximations in infinite\ndimension, is prone to efficient and consistent numerical implementations, and\nallows for an existence proof under very general assumptions. We consider in\nparticular rather nonsmooth and nonconvex energy functionals, provided the\ndomain of the energy is finite dimensional. Nevertheless, in the infinite\ndimensional case study of a cohesive fracture model, we prove a consistency\ntheorem of a discrete-to-continuum limit. We show that a quasistatic evolution\ncan be indeed recovered as a limit of evolutions of critical points of finite\ndimensional discretizations of the energy, constructed according to our scheme.\nTo illustrate the results, we provide several numerical experiments both in one\nand two dimensions. These agree with the crack initiation criterion, which\nstates that a fracture appears only when the stress overcomes a certain\nthreshold, depending on the material.\n