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

由有界变差函数描述的材料不连续性的有效近似方案的可靠性

• 批准号: 255461777
• 负责人: Professor Dr. Sören Bartels
• 金额: --
• 依托单位: Institut für Mathematik
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2014
• 资助国家: 德国
• 起止时间: 2013-12-31 至 2023-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/255461777?language=en
• 关键词: Reliability; efficient; approximation; schemes; material

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.

有界变差函数空间为描述材料的不连续性（如损伤和断裂）提供了一个有吸引力的框架。近年来，相应的演化模型问题的解的概念和一般存在性理论已经建立，涉及全变分范数的变分问题的离散化和迭代求解的数值方法也得到了发展和分析。在该项目的第一个资助期，耦合率相关/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，和损伤演化的相场描述进行了分析研究。在有界变差函数模型问题的迭代求解、基于完全可计算的后验误差估计的间断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都做出了重要贡献。在第二个资助期内，这些结果将被合并、完善并扩展到描述损伤和断裂演化的复杂模型。设想的模型捕捉材料的不连续性BV直接或作为一个缩放限制。计划的研究包括推导先验和后验误差估计，自适应逼近方案的建设与基于存在理论和伽玛收敛的结果交织在一起，以及它们在力学中的具体基准问题的实施和应用。 将特别注意有效方法的可靠性，例如，基于变分技术的合适的时间步进和自适应近似方案的收敛性。