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

翻译：我们在Eulerian-Lagrangian的配方中为动态准裂痕提供了一种节能机械化配方,这种配方为动态的准裂痕提供了一种节能机械化配方,在Eulerian-Lagrangian-Lagrangian的配方中,第二阶阶段阶段阶段阶段阶段阶段阶段阶段阶段阶段阶段阶段性方程式控制着损害的演变。数字配方在空间和时间上进行调整以约束错误,解决这些模型通常会遇到的网状-线性问题。时间阶段性适应性估计估计对调节固态平衡的局部差分方方方程的时间疏漏错误进行估计。第二阶级通用-美元-alpha$(alpha$)时间性计划使动态系统演变演变演变。我们通过使用前一阶级精确的偏差公式对当前时间级解决方案进行外推,估算出的时间误差与时间上的外推法比较。我们使用一个基于空间剩余最小最小最小度的公式的适应性方案,我们用一个高效和稳健健的平的机型空间偏差性调整方法来展示了空间偏差的平时空心性调整。