1. Parallel Simulations of Fracture and Fragmentation I. Arias, J. Knap and M. Ortiz California Institute of Technology Figure 1. The tractions at each side of the cohesive surface are related to the opening displacement through a cohesive law. A typical choice of cohesive law linking the effective displacement and the effective traction is depicted in Fig. 1 Cohesive Finite Elements Cohesive theories of fracture fit naturally within the finite element framework. Changes to existing finite element software necessary to accommodate cohesive finite elements are minimal (Fig. 1). Figure 2. Seamless integration of cohesive elements with the quadratic tetrahedral element. Adaptive Cohesive Element Insertion The cohesive law dictates a criterion for the insertion of cohesive surfaces. Each cohesive element insertion introduces major topological changes in the vicinity of the two adjacent elements: insertion of nodes, edges or faces, which, in turn, leads to the formation of free surfaces. Coalescence of these newly formed cracks may lead eventually to the formation of new bodies (fragmentation). Figure 3. An example of these topological changes for the tetrahedral mesh is shown in Figure 4. Four cases can be distinguished there: The new cohesive element is embedded entirely within the body (red). The new cohesive element has one edge on the boundary (yellow). The new cohesive element has two edges on the boundary (green). The new cohesive element has all edges (three) on the boundary (blue), which leads to complete separation of two bodies. Nodal Connectivity Graphs The essence of this approach is the development of efficient data structures capable of describing the connectivity in the two-dimensional (2D) triangular finite element mesh: The connectivity of the elements attached to a node is represented by a graph (Fig. 6). The links in the graph encode the state of the element interfaces. When the cohesive element is inserted, the corresponding link in the graph is removed (Fig. 7). Due to subsequent insertions of cohesive elements, the parent graph splits into two disjoint children graphs. This corresponds to the subdivision of the node (Fig. 8). The extension of nodal connectivity graphs to domain-decomposition parallel finite elements is straightforward: Shared connectivity graphs are introduced for nodes on the sub-domain boundaries. These graphs contain local information as well as data from neighboring sub-domains. The nodal graphs are built in the initial configuration of the mesh. The fracture criterion is evaluated for each local link in all nodal graphs. The removed local link data in the shared graphs is collected and exchanged with neighboring sub-domains sharing a particular boundary graph. Figure 6. Figure 7.Figure 8. Fracture and Parallel Processing The modifications introduced by the development of new surfaces are local and confined to a set of elements sharing topological entities with the newly inserted cohesive elements. In the context of domain decomposition parallel finite element calculations, these modifications may no longer be local, as illustrated in Fig. 5. Figure 5. The red line represents a partition boundary separating two sub- domains, with an open edge indicating a new cohesive element. The discretizations of both sub- domains are no longer compatible. Moreover, any operation designed to remedy this situation will be non- local in nature. Cases as in Fig. 5 can be handled by the approach based on nodal connectivity graphs. Branching instability Crack deflection and subsequent penetration for an interface angle of 45 o Crack deflection and subsequent penetration for an interface angle of 60 o time = 330 µstime = 235.8.5 µs time = 73 µs time = 129.5 µs Validation Material: Homalite -100, 450 x 254 x 9.5 (mm) Field of view interface Wedge  Material: PMMA, 3 x 3 (mm) Cohesive Theory of Fracture Fineberg et al. `92 Xu et al. `03 Performance and Scaling on LANL’s HP/Compaq, QSC Procedures with highest CPU cycle consumption –element_driver 14.9% –assemble 13.9% –NewNeohookean 8.12% –Only ~ 10% spend in fragmentation routines! Much room for improvement on our I/O performance dumping to the parallel file system (/scratch[1,2]) Levels of subdivision 450K to 2.4M elements 61K→100K 450K85K→1.8M 2.4M 0.01 0.1 1 567 Processors (Log 2) seconds/iteration level 6 level 7 level 8 level 9 level 10 level 11 level 12