Shearlet approximation of brittle fracture evolutions (C03* [A10])

一种易碎的材料，在外力作用下，首先发生弹性变形，然后在没有任何中间阶段的情况下断裂。Francfort和Marigo提出了脆性断裂模型，其中材料位移通常是一个平滑函数，除了一个相对平滑的跳跃集决定断裂。这种方法的优点是不需要预先定义裂纹路径，但缺点是任何网格离散化都是几何偏差。与有限元不同，shearlets是一类脆性断裂模型典型解具有旋转不变性和最优非线性逼近性质的框架。该项目的主要目标是实现脆性断裂的鲁棒数值模拟，实现基于shearlet离散化的可证明收敛且几何无偏的自适应框架方案。

A brittle material, subjected to a force, first deforms itself elastically, then it breaks without any intermediate phase. A model of brittle fractures was proposed by Francfort and Marigo, where the material displacement is typically a smooth function except on a relatively smooth jump set determining the fracture. This approach has the advantage not to require a pre-defined crack path, but has the drawback that any mesh discretization is a geometrical bias. Differently from finite elements, shearlets are frames with rotation invariance and optimal nonlinear approximation properties for the class of typical solutions of the brittle fracture model. The main goal of the project is the robust numerical simulation of brittle fractures, implementing a provably convergent and geometrically unbiased adaptive frame scheme based on shearlet discretizations.