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

该断裂力学项目的目标是利用拓扑导数来逼近与任意参数几何形状的(小)裂纹相关的能量释放率(ERR)场。该方法与传统的有限元方法、广义有限元方法和扩展有限元方法处理断裂问题有着根本的不同，并且大大提高了计算效率。事实上，它消除了将破裂的结构部件离散化的需要。相反，它通过简单地评估加载的未开裂组件中存在的应力场，提供了位于固体中任何位置的任意几何裂纹的误差的近似值。这种分析只需要进行一次，并且有限元网格比模拟裂纹试件所需的网格要粗糙得多(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，从而消除了对专业单元的需要。最终，它将被用来生成有限元等值线图，以说明整个身体的关键裂纹几何形状。从那里可以很容易地识别潜在的故障位置和关键检查位置。在这一建议中，将局限于二维均匀各向同性线弹性结构的初步结果推广到非均匀、各向异性、非线性的三维结构。此外，一阶近似将被扩展到二阶精度。这个项目的成功完成将导致一种新的计算设计工具，它比传统的设计工具效率高得多。通过极大地促进结构部件的断裂分析，该方法预计将在结构部件设计取决于关键缺陷存在的所有行业产生重大影响。这种方法还将被纳入断裂力学研究生课程，该课程吸引了整个工程学院的学生。通过伊利诺伊大学工程在线教育办公室，这门课程将在大学以外的地方提供，特别是少数族裔服务机构(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).