Reliability of efficient approximation schemes for material discontinuities described by functions ofbounded variation

有界变差函数空间提供了一个描述材料不连续性(如损伤和断裂)的吸引人的框架。近年来，相应的演化模型问题建立了适当的解的概念和一般的存在性理论，并发展了离散化和迭代求解含全变差范数的变分问题的数值方法并进行了分析。在项目的第一个资助期内，分析了耦合速率相关/速率无关系统的抽象存在性结果、粘弹性动力学中的分层过程以及损伤演化的相场描述。在迭代求解有界变差函数的模型问题、基于完全可计算的后验误差估计的不连续函数的自适应逼近以及BV正则化损伤模型的收敛有限元模拟等方面做出了算法贡献。在第二个资助期内，这些结果将被合并、改进并扩展到描述损伤和裂缝演变的复杂模型。设想的模型可以直接捕获BV中的材料不连续性，也可以将其作为比例限制。计划中的研究包括推导先验和后验误差估计，构造与基于存在理论和Gamma收敛的结果交织在一起的自适应逼近格式，以及它们在具体的力学基准问题中的实现和应用。将特别注意有效方法的可靠性，例如基于变分技术的适当的时间步长和自适应近似方案的收敛。

Spaces of functions of bounded variations provide an attractive framework to describe material discontinuities such as damage and fracture. Recently, suitable notions of solution and general existence theories for corresponding evolutionary model problems have been established and numerical methods for discretizing and iteratively solving variational problems involving the total variation norm have been developed and analyzed.In the first funding period of the project, abstract existence results for coupled rate-dependent/rate-independent systems, delamination processes in visco-elastodynamics, and phasefield descriptions of damage evolutions have been investigated analytically. Algorithmic contributions have been made to the iterative solution of model problems on functions of bounded variation, the adaptive approximation of discontinuous functions based on fully computable a~posteriori error estimates, and the convergent finite element simulation of a BV-regularized damage model. Within a second funding period these results will be combined, refined, and extended to complex models describing damage and fracture evolution. The envisaged models capture material discontinuities in BV either directly or as a scaling limit. The planned research includes the derivation of a priori and a posteriori error estimates, the construction of adaptive approximation schemes intertwined with results based on existence theory and Gamma-convergence, and their implementation and application to specific benchmark problems in mechanics. Particular attention will be paid to the reliability of efficient methods, e.g., convergence of suitable time-stepping and adaptive approximation schemes based on variational techniques.