An adaptive space-time phase field formulation for dynamic fracture of brittle shells based on LR NURBS

We present an adaptive space-time phase field formulation for dynamic fracture of brittle shells. Their deformation is characterized by the Kirchhoff–Love thin shell theory using a curvilinear surface description. All kinematical objects are defined on the shell’s mid-plane. The evolution equation for the phase field is determined by the minimization of an energy functional based on Griffith’s theory of brittle fracture. Membrane and bending contributions to the fracture process are modeled separately and a thickness integration is established for the latter. The coupled system consists of two nonlinear fourth-order PDEs and all quantities are defined on an evolving two-dimensional manifold. Since the weak form requires C1-continuity, isogeometric shape functions are used. The mesh is adaptively refined based on the phase field using Locally Refinable (LR) NURBS. Time is discretized based on a generalized-α method using adaptive time-stepping, and the discretized coupled system is solved with a monolithic Newton–Raphson scheme. The interaction between surface deformation and crack evolution is demonstrated by several numerical examples showing dynamic crack propagation and branching

Managing supply disruption in a three-tier supply chain with multiple suppliers and retailers

© 2014 IEEE. In this paper, a supply disruption management model is introduced in a three-tier supply chain with multiple suppliers and retailers, where the system may face sudden disruption in its raw material supply. At first, we formulated a mathematical model for ideal conditions and then reformulated it to revise the supply, production and delivery plan after the occurrence of a disruption, for a future period, to recover from the disruption. Here, the objective is to minimize the total cost during the recovery time window while being subject to supply, capacity, demand, and delivery constraints. We have also proposed an efficient heuristic to solve the model and the results have been compared, with another established solution approach, for a good number of randomly generated test problems. The comparison showed the consistent performance of our developed heuristic. This paper also presents some numerical examples to explain the usefulness of the proposed approach