Variational model order reduction for the prediction of plastic events in disordered materials

无序材料，如玻璃和无定形固体，在工程和日常生活中普遍存在。无序固体的复杂力学由发生在纳米尺度上的基本非弹性事件控制，并在材料的局部微观图像中解码。因此，无序材料的基本行为只能通过分子描述来捕捉，这在很大程度上限制了在更大范围内描述此类材料的能力。因此，本项目旨在开发一种描述无序固体力学响应的降阶模型。我们的目标建立在这样一个想法上，即在基本的非弹性事件中，只有几种响应模式，即所谓的软振动模式被激活。由于高度的非线性和这种缓慢驱动的动力学的不可预测性，我们将使用一种新的机械探测技术来预测对局部事件敏感的区域，我们将使用机器学习算法来增强该技术。这样，人们就可以在机械加载之前预测这些基本的非弹性事件，识别所谓的局部变化模式。用经验插值法在数学上实现降阶模型，而用摄动理论和Grassmann外推法考虑由于系统的高度非线性而引起的海森本征模得到的降阶基数的变化。通过局部变分模式对降阶基进行扩展，建立了功能强大的高保真降维模型。该方法将首先在二维基准分子模型上进行测试，然后在更大的三维无序材料上进行测试。

Disordered materials, such as glasses and amorphous solids, are ubiquitous in engineering and everyday life. The complex mechanics of disordered solids is governed by elementary inelastic events that happen on the nanoscale and are decoded in the local microscopic picture of the material. Consequently, the essential behavior of disordered materials can only be captured using molecular descriptions, which considerably limits the capability to describe such materials on larger scales. Thus, this project aims to develop a reduced-order model for the mechanical response description of disordered solids. Our objective builds on the idea that only a few response patterns, so-called soft vibrational modes, are activated during elementary inelastic events. Due to the high level of nonlinearity and the unpredictability of such slowly driven dynamics, we will predict the zones susceptible to local events using a new mechanical probing technique, which we will enhance using machine learning algorithms. This way, one can predict these elementary inelastic events before mechanical loading, identifying so-called local variation modes. A reduced order model will mathematically be realized using the Empirical Interpolation method, while the change in the reduced basis obtained by the eigenmodes of the Hessian occurring due to the high nonlinearity of the system will be considered using the perturbation theory and Grassmann extrapolation. The powerful high-fidelity reduced order model will be formulated by extending the reduced order basis by the local variation modes. The method will, firstly, be tested on two-dimensional benchmark molecular models and then on larger three-dimensional disordered materials.