Thermoelectrical effects in fractured media are encountered in many situations of practical interest; and the numerical treatment of this physical phenomenon has run into difficulties due to the temperature and voltage discontinuity.
In this presentation, a general computational procedure, which is based on the eXtended finite element method (XFEM), is proposed to efficiently deal with transient and nonlinear problems, and to estimate field distributions in cracked medium.
The finite element approximation is enriched in order to take into account the crack discontinuities due to the jump and the asymptotic near-tip function, using the partition of unity method. And the discretization results in a nonlinear system that is solved using the Newton-Raphson algorithm.
Different numerical examples show the high accuracy and the robustness of the proposed computational procedure in efficiently capturing the temperature and voltage jumps across the crack.
In this talk we consider a hybrid asymptotic-numerical analysis of reaction diffusion systems with sharp transitions across interfaces formed by closed curves. Without certain criterion on the kinetics, these solutions may undergo strong amplitude perturbations on an O(1) time scale, leading to a dissociation of the curve into localized spots. However, if the kinetics are such that these interface solutions are stable, we may cast the problem for the slower dynamics by formulating an effective Stefan problem for the interface. We present preliminary results for conditions of O(1) time stability as well as the Stefan conditions that move the interface.
Projection methods can be used to enforce a divergence constraint on a velocity field. Our aim is to find a velocity field satisfying a divergence constraint with a source term represented by a Dirac delta function along a front. The projection involves the resolution of a Poisson equation with a singularity on the right hand side term over an irregular domain. A description of the Cartesian grid embedded boundary method and of the regularization of the singularity source for solving this equation will be given. An application of this method is a forest fire model that includes the feedback of the heat release at the front on the local wind.
The immersed boundary (or IB) method is a proven approach for simulating fluid-structure interaction problems that involve a highly-deformable elastic structure immersed in an incompressible fluid. The method has primarily been used in applications from biofluid dynamics, although increasingly it is finding use in other problems from suspension flows, aerodynamics, etc.
In this talk, I will illustrate the versatility and accuracy of the IB method via two idealized problems that are inspired by the study of biofilm dynamics: namely, sedimentation in particle suspensions, and deformation of flexible cantilevers in response to a shearing flow. Both problems have been very well-studied in the engineering literature and so there are many numerical and experimental results available in the literature. Our primary goals here are to carefully validate the IB approach for these two applications involving passive structures in a simple 2D geometry, and to identify the advantages and disadvantages of the method compared with other approaches. This work then sets the stage for future computational studies of more complex problems with dynamically evolving self-propelled biofilm structures in 3D.