Adaptive Multi-Time-step Methods for Dynamic Crack Propagation

Mriganabh Boruah, Purdue University

Abstract

Problems in structural dynamics that involve rapid evolution of the material at multiple scales of length and time are challenging to solve numerically. One such problem is that of a structure undergoing fracture, where the material in the vicinity of a crack front may experience high stresses and strains while the remainder of the structure may be unaffected by it. Usually, such problems are solved using numerical methods based on a finite element discretization in space and a finite difference time-stepping scheme to capture dynamic response. Regions of interest within the structure, where high transients are expected, are usually modeled with a fine discretization in space and time for better accuracy. In other regions of the model where the response does not change rapidly, a coarser discretization suffices and helps keep the computational cost down. This variation in spatial and temporal discretization is achieved through domain decomposition and multi-time-step coupling methods which allow the use of different levels of mesh discretization and time-steps in different regions of the mesh. For problems where the region of interest evolves in time, such as those with an advancing crack front, the discretization of the problem domain must evolve as well so that the region of fine spatial and temporal discretization is able to track the region of interest. This need for adaptively refining and coarsening different regions of a numerical model presents several challenges. First, one must establish criteria based on the physical characteristics of the dynamic response of the material to identify the regions of interest that require a fine discretization in space and time. Then, one must devise a strategy to ensure that the region of fine discretization encompasses the region of interest at all times during the simulation without necessarily having to constantly modify the discretization at every time step. As the mesh discretization changes dynamically, an accurate and efficient mapping algorithm must also be implemented to transfer the state variables from the coarse mesh to the fine mesh and vice versa. Lastly, quantities associated with the domain decomposition and multi-time-step coupling method must also be updated every time the mesh discretization changes. These include the subdomain mass, stiffness and damping matrices and the interface coupling matrices needed to ensure spatial continuity of the solution between the subdomains. In this study, all the challenges associated with dynamically evolving fine and coarse mesh discretizations are addressed by developing an adaptive multi-time-step (AMTS) domain decomposition and coupling method. Some of the challenges associated with adaptivity are also addressed in existing literature on adaptive mesh refinement, for example, but the approach described in this study is distinct from existing methods. First, two different criteria are investigated for identifying the regions of interest in a mesh. One is based on identifying regions of high spatial gradients in the solution, which would be regions of high stress and strain in the model, while the other is based on identifying the location of a dynamically propagating crack front in a brittle material. These regions are encapsulated with a fine discretization, which is periodically updated as the physical characteristics of the solution evolve.

Degree

M.Sc.

Advisors

Prakash, Purdue University.

Subject Area

Mathematics

COinS