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

有界变分函数的空间提供了一个有吸引力的框架来描述材料的不连续，如损伤和断裂。近年来，针对相应的演化模型问题建立了合适的解的概念和一般存在性理论，并发展和分析了涉及全变分范数的变分问题的离散和迭代求解的数值方法。在项目的第一个资助阶段，对耦合速率依赖/速率无关系统的抽象存在结果、粘弹性动力学中的分层过程以及损伤演化的相场描述进行了分析研究。在有界变分函数模型问题的迭代求解、基于完全可计算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、细化并扩展到描述损伤和裂缝演变的复杂模型中。设想的模型直接或作为缩放限制捕获BV中的材料不连续。计划的研究包括先验和后验误差估计的推导，基于存在论和伽玛收敛的结果交织的自适应逼近方案的构建，以及它们在具体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，适当的时间步进和基于变分技术的自适应逼近方案的收敛性。

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.