This paper presents computationally feasible rank-one relaxation algorithms for the efficient simulation of a time-incremental damage model with nonconvex incremental stress potentials in multiple spatial dimensions. While the standard model suffers from numerical issues due to the lack of convexity, the relaxation by rank-one convexification prevents non-existence of minimizers and mesh dependence of the solutions of finite element discretizations. By the combination, modification and parallelization of the underlying convexification algorithms, the novel approach becomes computationally feasible. A descent method and a Newton scheme enhanced by step-size control prevent stability issues related to local minima in the energy landscape and the computation of derivatives. Numerical techniques for the construction of continuous derivatives of the approximated rank-one convex envelope are discussed. A series of numerical experiments demonstrates the ability of the computationally relaxed model to capture softening effects and the mesh independence of the computed approximations. An interpretation in terms of microstructural damage evolution is given, based on the rank-one lamination process.

翻译：本文提出了计算可行的秩一松弛算法，用于高效模拟具有非凸增量应力势的多维空间时间增量损伤模型。标准模型因缺乏凸性而存在数值问题，而通过秩一凸化松弛可避免极小值不存在及有限元离散解对网格的依赖性。通过基础凸化算法的组合、改进与并行化，该新方法在计算上变得可行。结合步长控制的下降法与牛顿法克服了能量景观中局部极小值与导数计算相关的稳定性问题。本文讨论了连续导数近似的秩一凸包构造数值技术。系列数值实验表明，计算松弛模型能捕捉软化效应，且计算近似解具有网格无关性。基于秩一分层过程，给出了微观损伤演化的物理解释。