Efficient Energy Release Rate Computations for Cracks with Arbitrary Location and Geometry

这个断裂力学项目的目标是利用拓扑导数来近似与任意参数化几何形状的（小）裂纹相关的能量释放率（ERR）场。该方法与传统的有限元法、广义有限元法和扩展有限元法处理断裂问题完全不同，而且效率更高。事实上，它消除了对开裂结构部件进行离散化的需要。相反，它通过简单地评估加载的未破裂部件中的应力场，为位于固体中任何位置的任意几何形状的裂纹提供了近似的ERR。这种分析只需要进行一次，并且使用的有限元网格比模拟裂纹试件所需的网格粗得多（因为不需要捕捉裂纹前沿的应力集中）。此外，它使用传统的有限元方法，从而消除了对特殊元件的需要。最终，它将用于生成有限元等高线图，以说明整个车身的关键裂纹几何形状。因此，潜在的故障位置和关键的检查地点可以很容易地确定。在本提案中，仅限于二维均匀各向同性线弹性结构的初步结果将扩展到非均质、各向异性、非线性三维结构。此外，将一阶近似扩展到二阶精度。该项目的成功完成将导致一种新型的计算设计工具，其效率大大高于传统的计算设计工具。通过极大地促进基于断裂的结构部件分析，该方法有望在所有行业中产生重大影响，因为这些行业的结构部件设计是由关键缺陷的存在决定的。该方法也将被纳入断裂力学研究生课程，吸引了来自工程学院的学生。通过伊利诺伊大学工程在线教育办公室，本课程将在校外提供，特别是少数民族服务机构（MSI）。

The goal of this fracture mechanics project is to utilize the topological derivative to approximate the energy release rate (ERR) field associated with a (small) crack of arbitrary parameterized geometry. The proposed method is radically different from, and substantially more efficient than, conventional finite element method (FEM), generalized FEM and extended FEM treatment of fracture problems. Indeed, it eliminates the need to discretize the cracked structural component. Rather it provides an approximation of the ERR for a crack of arbitrary geometry located anywhere in the solid by simply evaluate the stress field present in the loaded un-cracked component. This analysis needs to be conducted only once and with a finite element mesh substantially coarser than that needed to model the cracked specimen (since there is no need to capture the stress concentration at a crack front). Moreover it uses conventional FEA methods and thereby it eliminates the need for specialty elements. Ultimately it will be used to generate finite element contour plots that illustrate critical crack geometries throughout the body. Whence, potential failure locations and critical inspection sites can be readily identified. In this proposal, the preliminary results that are limited to 2-D homogenous isotropic linear elastic structures will be extended to heterogeneous, anisotropic, nonlinear three-dimensional structures. Moreover, the first-order approximation will be extended to achieve second-order accuracy. The successful completion of this project will lead to a novel computational design tool that is substantially more efficient than conventional. By substantially facilitating the fracture-based analysis of structural components, the method is expected to have a major impact in all industries for which the design of structural components is dictated by the presence of critical flaws. The method will also be incorporated in the graduate course on fracture mechanics that attracts students from across the College of Engineering. Through the office of Engineering On-Line Education at the University of Illinois, this course will be made available outside the University, and especially to Minority Serving Institutions (MSI).