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。 所提出的方法是从根本上不同的，并大大更有效地比，传统的有限元法（FEM），广义有限元和扩展有限元断裂问题的处理。 事实上，它消除了离散化裂纹结构部件的需要。 相反，它提供了一个近似的ERR的任意几何形状的裂纹位于固体中的任何地方，通过简单地评估应力场存在于加载的无裂纹组件。该分析仅需进行一次，并且有限元网格比对裂纹试样建模所需的网格粗糙得多（因为不需要捕获裂纹前沿的应力集中）。此外，它使用传统的有限元分析方法，从而消除了对专业元素的需要。最终，它将被用来生成有限元等高线图，说明整个身体的关键裂纹几何形状。因此，可以很容易地确定潜在的故障位置和关键的检查部位。在这个建议中，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匀，各向异性，非线性的三维结构。 此外，一阶近似将被扩展到实现二阶精度。 这个项目的成功完成将导致一个新的计算设计工具，是比传统的更有效。 通过极大地促进结构部件的基于缺陷的分析，该方法预计将在所有行业中产生重大影响，对于这些行业，结构部件的设计取决于关键缺陷的存在。该方法也将被纳入断裂力学的研究生课程，吸引来自工程学院的学生。通过工程在线教育在伊利诺伊大学的办公室，这门课程将在大学外提供，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).