Abstract

The phase-field modeling and its adaptive moving mesh solution for studying initiation and propagation of brittle fracture will be presented. Challenges such as non-smoothness of the energy functional, violation of fracture boundary conditions, and the need for mesh adaptation, and possible remedies for these challenges will be discussed. In particular, a moving mesh finite element method will be presented for the numerical solution of the phase-field model for brittle fracture.

Abstract

Abstract

We consider a 2D parabolic Monge-Ampère equation, and aim at tracking stationary solutions. A time-stepping algorithm is advocated, together with a finite element space approximation. A mesh adaptation technique is considered to track the singularities of the solutions. The approximate error indicator is based on a linearization of the Monge-Ampère operator. Numerical experiments show the numerical efficiency of the approach on several examples.

Abstract

Cardiac muscle tissue has a unique, network-like structure. Three-dimensional models of this structure are needed for simulations of cardiac electrophysiology and mechanics. We developed an algorithm to produce such models artificially, using an implicit surface expressed on a tailored unstructured multi-domain mesh to define the cell membranes. The algorithm first creates a random network of cell centers, observing angle and distance criteria inferred from real tissue. The space around the network edges is assigned to the cellular domains based on the nearest half-edge. The network is then immersed in a regular tetrahedral mesh which is refined to fit the domain boundaries and to offer sufficient density around the cell membrane. The refinements are alternated with mesh improvement operations to maintain an acceptable mesh quality. On the refined mesh a level-set function is expressed that defines the cell membrane. The remeshing code Mmg3d is then used to discretize the level set while retaining the domains, and to improve the quality of the final mesh. A serial implementation of the algorithm was able to produce meshes of a few hundreds of cardiac cells in 15 minutes, but we are still facing difficulties in the remesher, likely resulting from the unusual complexity of these meshes. It was still possible, however, to correctly mesh a small network of cells that was designed to be replicated by successive mirroring. This allowed us to build models of upto 1 cm3of tissue (10 million cells and 370 billion tetrahedra) that now serve in performance tests of a large-scale simulation code.

Abstract