Goal-oriented adaptive finite elements for parameter identification of conventional and additive micromorphic continuum models

由于局部连续体的不足，特别是有限元解在数值上的病理性网格依赖，以及在理论上不能模拟长度尺度相关的问题，广义连续体理论有着广泛的应用领域。在这个项目中，我们讨论了一类微态连续体，包括微极(Cosserat)连续体和微应变连续体作为重要的特例。在上一个项目中，我们处理了面向目标的微形态弹性和塑性的自适应有限元，并对正问题进行了误差控制模拟。此外，还提出了一种新的连续体类型，称为加性微形态连续体，用于有限应变弹性，允许不同连续体之间的灵活转换，并根据需要对它们进行不同的加权。在此基础上，本课题主要解决了以下问题：·在传统的埃林根意义下的微态理论中，我们考虑了线弹性中的尺寸效应。将设计新的实验来识别相关微观模型的参数，其中使用不同大小的缺口来激活冷箱铸造砂样的尺寸效应。·新的加法微观模型将扩展到包括损伤在内的弹塑性。由于它包含微极和微应变连续体作为特例，因此可以通过基于实验数据的反问题自动选择合适的连续体。通过模拟冷箱砂破坏的全过程进行了说明。新的添加剂模型的优点是在适当的权重下考虑了砂粒的旋转和粘结剂的变形。·将根据实验数据处理传统和添加剂微观模型的参数识别逆问题。由于要求处于非均匀变形状态，因此将在有限元离散变分公式的基础上进行灵敏度分析。为了提高参数辨识的数值效率，将采用自适应有限元方法进行有效的网格划分(时间相关问题的时空离散化)。具有挑战性的部分是开发适当的面向目标的误差估计器来驱动相应的自适应算法。

Due to deficiencies of local continua, in particular pathological mesh dependency of FE solutions on the numerical side and failure to simulate length scale dependent problems on the theoretical side, generalized continuum theories have wide areas of applications. In this project, we address the class of micromorphic continua, containing micropolar (Cosserat) continua and microstrain continua as important special cases. In the previous project, we dealt with goal-oriented adaptive FEM for micromorphic elasticity and plasticity towards an error-controlled simulation of the direct problem. In addition, a novel continuum type labeled as additive micromorphic continuum has been proposed for finite strain elasticity, enabling a flexible transition of different continua and weightingthem differently if required. On this basis, this project primarily addresses the following issues:• With conventional micromorphic theories in the sense of Eringen, we consider size effects in linear elasticity. New experiments will be designed for identifying parameters of related micromorphic models, where notches of varying sizes are used to activate size effects in sand specimens for cold-box casting.• The novel additive micromorphic model will be extended to elastoplasticity including damage. Since it contains micropolar and microstrain continua as special cases, a selection of a proper continuum is expected to be done automatically via an inverse problem based on experimental data. It will be illustrated by simulating the whole damaging process of a cold-box sand. The advantage of the new additive model is to account for the rotation of sand particles and the deformation of the binder with proper weights.• Inverse problems for parameter identification will be handled for both conventional and additive micromorphic models based on experimental data. As a heterogeneous deformation state is required, a sensitivity analysis will be carried out on the basis of discretized variational formulations for the FEM. Special care will be paid to the additive model with a time-dependent character.• To enhance the numerical efficiency of the parameter identification, adaptive FEM will be developed for an effective meshing (spatial and temporal discretization for time-dependent problems). The challenging part is to develop appropriate goal-oriented error estimators to drive the corresponding adaptive algorithms.