In this work, goal-oriented adjoint-based a posteriori error estimates are derived for a nonlinear phase-field discontinuity problem in which a scalar-valued displacement field interacts with a scalar-valued smoothed indicator function. The latter is subject to an irreversibility constraint, which is regularized using a simple penalization strategy. The main advancements in the current work are error identities, resulting estimators, and two-sided estimates employing the dual-weighted residual method, which address the influence of the phase-field regularization, penalization, and spatial discretization parameters. Some numerical tests accompany our derived estimates.

中文翻译：

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惩罚相场狭缝不连续问题的双加权残差后验误差估计

在这项工作中，针对非线性相场不连续性问题导出了基于目标伴随的后验误差估计，其中标量值位移场与标量值平滑指示函数相互作用。后者受不可逆约束的约束，该约束使用简单的惩罚策略进行正则化。当前工作的主要进步是误差识别、结果估计量和采用双加权残差方法的两侧估计，它们解决了相场正则化、惩罚和空间离散化参数的影响。一些数值测试伴随着我们得出的估计。