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本质上都是多尺度的。通常不均匀的精细尺度对于用基于物理的模型描述宏观行为是重要的。通过溶解更精细的尺度来提高精度伴随着计算成本的显著增加。由于有限的计算资源，整个现实建筑结构的精细尺度的分辨率是不可能的。该提案的目的是开发新的计算效率高的离散化和解决技术的复杂，异质结构。许多多尺度方法是基于FE²类型的方法与嵌套的解决方案策略。较细尺度由在粗尺度结构的每个积分点中评估的代表性体积元素（RVE）表示，以计算材料响应。因此，计算工作量随着RVE的复杂性而增加。特别是在混凝土的情况下，其中RVE相对较大并且通常不能实现尺度分离，这些计算均匀化技术非常昂贵。另一种方法是区域分解技术。在这种情况下，结构被划分为子域，导致大量的小问题可以并行解决。FETI方法使用拉格朗日乘子来耦合每个断开的子域。利用并行计算的优势，这类方法在计算上非常有效。此外，模型降阶技术在过去几十年中得到了发展，最近被用于各种应用中。模型降阶是减少复杂数值模拟计算量的一种常用而有力的工具。其核心思想是将系统投影到一个低维空间，以最好的方式表示系统的整体行为，该方案的思想是将模型降阶技术与区域分解相结合，以解决复杂和现实的非均匀中尺度问题。宏观域被分解成具有代表性的子域。基于离线模拟与周期性边界条件，为每个子域的基函数的整体设置确定。在基于聚类技术的自适应方案中，确定这些基函数的子集以精确地近似位移场。通过这种方式，降阶阶与解的非线性直接相关-从每个子域只有六个自由度的线性弹性解到具有完整基函数集的离散化。此外，为了保证降阶模型在并行实现中的有效加速，本文采用了超约简方法，通过在界面处施加位移场的弱约束，实现了基于FETI框架的降阶子域耦合。该方法的适用性将显示为具体的例子。