An adaptive hyperreduced domain decomposition approach for nonlinear heterogeneous structures

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。通过溶解更细的尺度来提高精度的同时，计算成本也显著增加。由于计算资源有限，无法对整个真实建筑结构进行精细尺度的解析。该建议的目的是开发新的计算高效的离散化和解决技术，复杂的，非均质结构。许多多尺度方法是基于FE²类型的方法和嵌套的解决策略。细尺度由粗尺度结构的每个积分点的代表性体积元（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，RVE相对较大，通常无法实现尺度分离，这些计算均质化技术非常昂贵。可选的方法是领域分解技术。在这种情况下，该结构被划分为子域，从而产生大量可以并行解决的小问题。FETI方法利用拉格朗日乘法器对每个断开的子域进行耦合。这些方法利用并行计算的优势，计算效率很高。此外，模型简化技术在过去几十年中得到了发展，最近在各种应用中得到了应用。模型约简是减少复杂数值模拟计算量的一种流行而有力的工具。关键思想是将系统投影到一个较低维度的空间，以最好的方式代表整体行为。该建议的思想是将模型约简技术与领域分解相结合，以解决复杂和现实的异构中尺度问题。将宏观域分解为具有代表性的子域。在具有周期边界条件的离线仿真基础上，确定了各子域的全局基函数集。在基于聚类技术的自适应方案中，确定这些基函数的子集来精确地逼近位移场。通过这种方式，简化的顺序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，还使用了一种超约方法来确保并行实现的降阶模型的有效加速。通过在界面处施加位移场的弱约束，简化子域的耦合是基于fei框架的。具体实例将说明该方法的适用性。

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.