INTRODUCTION

The research presented in this thesis is sponsored by ASCI/ASAP Center for the Simulation of Advanced Rockets (CSAR) whose main objective is the detailed, integrated, whole-system simulation of solid propellant rockets under both normal and abnormal operating conditions Heath et al., (2000).The development of numerical tools needed to simulate in an adaptive fashion accident scenarios involving the propagation of one or more cracks in the solid propellant (or grain) or along the grain/case interface constitutes the primary objective of the research work summarized hereafter.

Fracture events taking place in the grain during the flight of a solid propellant rocket often have detrimental effects on the performance of the rocket. As the crack propagates in the grain or along the grain/case interface, it creates additional burning surfaces, generating an excess of hot gas, which, in turn, may strongly affect the pressure history in the rocket chamber and sometimes lead to a catastrophic failure. A classical example of a catastrophic solid rocket failure that involved the propagation of a crack along the grain/case interface is that of the Titan IV grain collapse accident that took place on April 1, 1991 (Wilson et al., 1990; Chang et al. 1994) The Titan IV accident scenario is schematically illustrated in Figure 1.1. Due to the aerodynamic effects associated with the grain shape near a slot and the interaction between core and cross flows, a region of lower pressure developed along the downstream portion of the grain, leading to its progressive deformation into the rocket chamber and resulting in a dramatic increase in the head end pressure. Associated with the grain deformation, a crack is believed to have initiated from the aft segment stress relief groove, extended to the adjacent casebond and propagated along the interface at speeds exceeding $m/s$ (Wilson et al., 1992).The propagation of the interface crack accentuated the grain slumping process, which eventually led to the choking of the core flow and the explosion of the rocket.

Figure 1.1: Illustration of the fracture process associated with the Titan IV SRMU grain collapse accident (taken from Chang et al., (1994)).

Although progress has been made over the past four decades in understanding the complex physical phenomena associated with this class of fracture events (Kuo and Kooker, 1990; Lu and Kuo, 1994; Smirnov, 1985), no truly predictive numerical tools are currently available to capture adequately the failure process and its effect on the rocket performance. The simulation of dynamic fracture events taking place while the grain is burning constitutes a major computational challenge for various reasons. Firstly, the constitutive and failure responses of the solid propellant are quite complex and often involve large deformations and rate dependence, which must be accounted for in the constitutive, failure and kinematic descriptions of the continuum. Secondly, the geometry of the problem changes substantially during the fracture event due to the rapid propagation of the crack and the deformation and progressive burning of the grain. Thirdly, this problem is characterized by a complex fluid/structure interaction due to the aeroelastic deformations of the grain and to the pressurization of the newly created fracture surfaces by the reacting gas. This interaction is particularly hard to model in the vicinity of the advancing crack front where the geometry of the corresponding fluid domain is especially complex and new fluid regions are continuously added due to the crack motion. Finally, the problem is highly transient, as the speed of the crack has been shown to be sometimes of the order of tens of meters per second, possibly resulting in failure events lasting a fraction of a second.

As described in Geubelle et al., (2001), the key component of the multi-physics fluid/structure code to be used in the simulation of dynamic failure in "live" solid propellant is an explicit Arbitrary/Lagrangian Eulerian (ALE) form of the Cohesive/Volumetric Finite Element (CVFE) scheme, specially developed for the simulation of dynamic fracture events in structural domains with regressing boundaries. The CVFE scheme relies on a combination of conventional (volumetric) elements and of interfacial (cohesive) elements to capture the constitutive and failure responses of the material, respectively. The numerical method, which is described in detail in Chapter 2, has been shown to be quite successful in the simulation of various dynamic fracture problems involving spontaneous crack initiation, propagation and arrest. It was used, for example, by Camacho and Ortiz (1996) to simulate impact damage in ceramic materials, and by Needleman (1997) to model dynamic failure events in brittle materials. Siegmund and Needleman (1997) have used the CVFE scheme to study rate-dependence in the dynamic failure of elasto-plastic materials. Geubelle and Baylor (1998) have simulated impact-induced delamination of composites, and, more recently, Bi et al., (2001) have used the CVFE scheme to capture dynamic fiber push-out in model composites.

However, in its current implementation derived from the work of Geubelle and Baylor (1998), the CVFE scheme relies on the initial "static" introduction of the cohesive elements in the finite element mesh. In other words, the analyst provides at the onset of the simulation a set of possible paths for the dynamic crack(s) that would result from the dynamic loading of the structure. This approach is particularly attractive for its simplicity: once the cohesive/volumetric mesh is created, the dynamic fracture simulation proceeds without the need to modify the structural model. However, it suffers from two important limitations: firstly, the presence of the interfacial elements in the finite element mesh greatly increases the number of nodes, and, therefore, the number of degrees of freedom. This increase in the problem size often has substantial impact on the computational cost of the simulation. Secondly, and perhaps more importantly, the presence of a large number of cohesive elements may adversely affect the precision of the numerical solution. As shown by Baylor (1997), the additional compliance associated with the cohesive elements may lead to under predicting the stress fields in the discretized structure. And since the failure process is stress-based, this error on the stress value may affect the precision of the fracture prediction.

To address these two issues, we propose to develop and implement in this project an adaptive CVFE scheme, for which the cohesive elements are not introduced initially in the finite element mesh but are inserted dynamically during the simulation itself. This approach will not only substantially reduce the number of nodal degrees of freedom, especially during the loading phase of the dynamic problem during which little failure takes place, but also will guarantee a more precise capture of the dynamic stress field before and during the fracture event.

The development and implementation of the adaptive CVFE scheme presents various challenges that need to be addressed. These challenges are concerned with 1) the management of the database containing the evolving finite element discretization, 2) the criterion to be used to insert cohesive elements adaptively, 3) the mechanical perturbation created by the dynamically inserted cohesive elements, and 4) the load imbalance inherently present in the parallel implementation of the adaptive CVFE scheme. The approach adopted in the present study to address these challenges is summarized in Chapter 2.

The present project also addresses the issue of adaptivity of the CVFE scheme at another important level. As shown by Baylor (1997), one important difficulty associated with the CVFE scheme is the fact that it often requires the use of very small time steps typically representing a small fraction (3 to 5%) of the Courant limiting value characterizing explicit dynamic schemes. This limitation greatly impacts the computational effort of such simulations, as tens or hundreds of thousands of time steps are often needed. The need to use very small time steps for the entire mesh seems especially wasteful when only a small number of cohesive elements are used in the analysis, as it is the case with the proposed adaptive CVFE scheme. A natural way to address this issue is the nodal explicit subcycling scheme proposed by Smolinski (1989), which allows for the use of distinctly different time step values in various parts of the finite element domain.

The application of the nodal subcycling scheme to the adaptive CVFE scheme is also described in Chapter 2, followed, in Chapter 3, by a one-dimensional (1-D) study of the adaptive CVFE scheme, performed because of its simplicity and its ability to provide useful insight on various stability issues. Finally, we summarize in Chapter 4 various implementation issues associated with the more complex 2-D case and present the results of various 2-D simulations performed with the adaptive CVFE code.