Fine numerical analysis of the crack-tip position for a Mumford–Shah minimizer

Abstract

A new algorithm to determine the position of the crack (discontinuity set) of certain minimizers of Mumford-Shah functional in situations when a crack-tip occurs is introduced. The conformal mappingz = √ z in the complex plane is used to transform the free discontinuity problem to a new type of free boundary problem, where the symmetry of the free boundary is an additional constraint of a non-local nature. Instead of traditional Jacobi or Newton iterative methods, we propose a simple iteration method which does not need the Jacobian but is way fast than the Jacobi iteration. In each iteration, a Laplace equation needs to be solved on an irregular domain with a Dirichlet boundary condition on the fixed part of the boundary; and a Neumann type boundary condition along the free boundary. The augmented immersed interface method is employed to solve the potential problem. The numerical results agree with the analytic analysis and provide insight into some open questions in free discontinuity problems.

Effective matrix-free preconditioning for the augmented immersed interface method

“…Figure 5 gives an illustration. Additional plots of the final shape of the boundary can be found in [25]. (17)- (19) in [25], where Γ 1 is a free boundary and Γ 2 is fixed.…”

“…From the analysis in [25], g(t) is the solution to the following non-linear equation which depends on u(x, y): (t 2 + g 2 (t))g (t) (1 + g 2 (t)) 3 2…”

“…We then show a comprehensive test in terms of a free boundary problem [25], including all the stages of an entire simulation.…”

From Iim to Augmented Iim: A Powerful Tool for Complex Problems Using Cartesian Meshes

“…For a variable coefficient β(x), the conclusions are still true if h is small enough, that is, in the asymptotic sense. New applications of the augmented IIM include multi-physics simulations with different governing equations on different regions such as the Stokes or Navier-Stokes equations coupled with the Darcy's law in (Li, 2015;Li, Lai, Peng, & Zhang, 2018), determine the crank tips for a Mumford-Shah minimizer of a free boundary value problem in (Li & Mikayelyan, 2016); the 3D compressible bubbles in a incompressible fluid (Li, Xiao, Cai, Zhao, & Luo, 2015); and an efficient preconditioner for the Schur complement matrix resulted from AIIM in (Angot & Li, 2017;Xia, Li, & Ye, 2015).…”