An adaptive hyperreduced domain decomposition approach for nonlinear heterogeneous structures

非线性异质结构的自适应超简化域分解方法

• 批准号: 394350870
• 负责人: Dr.-Ing. Jörg F. Unger
• 金额: --
• 依托单位: Abteilung 7: Bauwerkssicherheit
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2017
• 资助国家: 德国
• 起止时间: 2016-12-31 至 2022-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/394350870?language=en
• 关键词: adaptive; hyperreduced; domain; decomposition; approach

Almost all engineering materials are intrinsically multiscale. The often heterogeneous fine scales are important to describe the macroscopic behavior with physics based models. An increase of the accuracy by dissolving finer scales is accompanied with a significant increase of the computational cost. The resolution of the fine scale for a whole realistic building structure is not possible due to limited computational resources. The purpose of the proposal is to develop new computationally efficient discretization and solution techniques for complex, heterogeneous structures.Many multiscale methods are based on FE² type of approaches with a nested solution strategy. The finer scale is represented by a representative volume element (RVE) evaluated in each integration point of the coarse scale structure to compute the material response. As a result, the computational effort increases with the complexity of the RVE. Especially in case of concrete, where the RVE is relatively large and a scale separation can often not be achieved, these computational homogenization techniques are very expensive. Alternative approaches are domain decomposition techniques. In this case, the structure is divided into sub-domains resulting in a high number of small problems which can be solved in parallel. The FETI method couples each disconnected sub-domain using Lagrange multipliers. Using the advantage of parallel computing, these kind of methods are computationally very efficient. Furthermore, model reduction techniques have been developed in the last decades and are recently used in a variety of applications. Model reduction is a popular and powerful tool to decrease the computational effort of complex numerical simulations. The key idea is the projection of the system to a lower dimensional space which represent the overall behavior in a best possible way.The idea of the proposal is a combination of model reduction techniques with domain decomposition to solve complex and realistic heterogeneous mesoscale problems. The macroscopic domain is decomposed into representative subdomains. Based on offline simulation with periodic boundary conditions, a global set of basis functions for each subdomain is determined. In an adaptive scheme based on clustering techniques, a subset of these basis functions is determined to accurately approximate the displacement field. In this way, the order of reduction is directly linked to the nonlinearity of the solution - from linear elastic solutions with only six degrees of freedom per sub-domain up to a discretization with the complete set of basis functions. In addition, a hyperreduction approach is used to ensure an efficient speedup of the reduced order model for parallel implementations.The coupling of the reduced sub-domains is based on the FETI framework by enforcing a weak constraint of the displacement field at the interfaces. The applicability of the method will be shown for specific examples.

几乎所有的工程材料都是本质上的多尺度。通常异质的细尺度对于用基于物理的模型来描述宏观行为很重要。通过溶解较细的尺度来增加准确性，计算成本显着提高。由于计算资源有限，无法解决整个现实建筑结构的精细规模。该提案的目的是为复杂的异质结构开发新的计算有效离散化和解决方案技术。许多嵌套解决方案策略基于FE²的方法类型。更细的量表由在粗尺度结构的每个集成点中评估的代表体积元素（RVE）表示，以计算材料响应。结果，计算工作随RVE的复杂性而增加。尤其是在混凝土的情况下，RVE相对较大并且通常无法实现尺度分离，这些计算均质化技术非常昂贵。替代方法是域分解技术。在这种情况下，结构分为子域，导致大量的小问题可以并行解决。 Feti方法使用Lagrange乘法器耦合了每个分离的子域。利用并行计算的优势，这些方法在计算上非常有效。此外，在过去的几十年中已经开发了模型的技术，最近在各种应用中使用。减少模型是一种流行而有力的工具，可减少复杂数值模拟的计算工作。关键思想是系统对较低维空间的投影，该空间以最佳方式代表整体行为。该建议的想法是模型还原技术与域分解的结合，以解决复杂而现实的异质性中尺度问题。宏观域分解为表示子域。基于具有周期性边界条件的离线模拟，确定每个子域的全局基础函数集。在基于聚类技术的自适应方案中，确定了这些基础函数的子集以准确近似位移场。这样，还原顺序与解决方案的非线性直接相关 - 从每个子域的线性弹性解决方案只有六个自由度，到具有完整基集函数的离散化。此外，使用超级还原方法来确保对并行实现的减少订单模型的有效加速。还原子域的耦合是基于FETI框架，通过在接口处执行位移场的弱约束。该方法的适用性将用于特定示例。