A spatio-temporal adaptive phase-field fracture method

We present an energy-preserving mechanic formulation for dynamic quasi-brittle fracture in Eulerian–Lagrangian formulation, where a second-order phase-field equation controls the damage evolution. The numerical adapts space and time to bound errors, solving mesh-bias issues these models typically suffer. time-step adaptivity estimates temporal truncation error of partial differential that governs solid equilibrium. generalized- ? time-marching scheme evolves system. estimate by extrapolating first-order approximation solution using previous ones with backward difference formulas; compares extrapolation solution. use adaptive built on residual minimization space. spatial enriching discretization elemental bubbles; then, we localize indicator norm guide mesh refinement as propagates. combined allows us low-order linear elements problems involving complex stress paths. efficiently robustly discretizations while avoiding bias structured unstructured meshes . demonstrate method’s efficiency experiments feature crack branching, capacity space–time is apparent. method delivers accurate reproducible paths fewer elements. • An spatio-temporal processes presented. physical problem stated thermodynamical consistent framework. efficient time-error based simple formulas proposed. A space-error ideas formulated. Mesh-independent examples branching are