A Partition of Unity Boundary Element Method for Fracture and Fatigue Analysis

这项建议的目的是产生一种新的计算建模算法，用于分析含有裂纹的工程部件。最近一些备受瞩目的事故清楚地表明，当由于裂缝而发生断裂故障时，它们往往是突然的、灾难性的和没有警告的。裂纹在循环载荷作用下扩展到临界尺寸的过程称为疲劳。这项工作提出了一种模拟断裂力学和疲劳裂纹扩展的新方法。断裂和疲劳模拟的基础是确定裂纹附近材料中的应力和位移。求解这类问题已有成熟的计算方法，但在存在裂纹的情况下，基于传统的有限元和边界元方法的传统方法会变得繁琐和低效。近年来，为了克服这些困难，人们提出了许多不同的方法，其中包括两种方法的结合：“对偶边界元方法”(DBEM)和“单位划分法”(PUM)。DBEM克服了对非常精细的计算模型的需求，尽管存在一些缺陷，使得该方法更难实现。PUM预计将允许进一步减少所需的模型精化。PUM是基于在我们的近似模型中包含一组函数，该函数源自裂纹尖端材料的局部理论行为。这已成功地与断裂力学的有限元方法相结合。然而，我们在达勒姆的团队在用于波传播模拟的边界元算法中的PUM取得了巨大的成功。在这里，我们发现误差比传统的边界单元减少了8个数量级，并允许问题的大小显著减小。由于PUM似乎非常适合边界单元的实现，并且边界单元以其对裂纹问题的适用性而闻名，我们预计该项目将有很好的成功几率，并可能对机械和航空航天工程中的断裂和疲劳模拟产生重大影响。

The aim of this proposal is to produce a novel computational modelling algorithm for the analysis of engineering components containing cracks. A number of recent, high profile accidents clearly demonstrate that when fracture failures occur due to cracking, they often do so suddenly, catastrophically and without warning. The process of cracks growing to a critical size under cyclical loading is termed fatigue. The work proposes a new method of simulating both fracture mechanics and fatigue crack growth.Simulations in fracture and fatigue are based around the determination of the stresses and displacements in the material in the vicinity of the crack. There are well established computational methods of finding this type of solution, but in the presence of cracks the traditional approaches based on conventional 'finite element' and 'boundary element' methods can become cumbersome and inefficient. In recent years, a number of variants of these techniques have been proposed to overcome some of these difficulties.The proposed work involves the combination of two such strategies: the 'Dual Boundary Element Method' (DBEM) and the 'Partition of Unity Method' (PUM). The DBEM overcomes the need for a very refined computational model, though there are some drawbacks making the method more difficult to implement. The PUM is expected to allow for further reductions in the required model refinement.The PUM is based on the inclusion in our approximate model of a set of functions derived from the theoretical behaviour of the materials locally at the crack tip. This has been successfully used with the finite element method for fracture mechanics. However, our own group in Durham has achieved great success with the PUM in a boundary element algorithm for wave propagation simulation. Here we have found up to 8 orders of magnitude reduction in errors over conventional boundary elements and allows for a considerable reduction in the size of the problem.Since the PUM seems well suited to a boundary element implementation, and boundary elements are well known for their suitability for crack problems, we anticipate this project to have a good probability of success, and could have a significant impact on fracture and fatigue simulations in mechanical and aerospace engineering.