### Numerical Stabilization of the Melt Front for Laser Beam Cutting [chapter]

Torsten Adolph, Willi Schönauer, Markus Niessen, Wolfgang Schulz
2010 Numerical Mathematics and Advanced Applications 2009
It is well known that high order numerical schemes exhibit oscillations around shocks but are very efficient for smooth solutions. In flow fields where both shocks and weak features exist, both the shock capturing capability and high order accuracy are required. We develop a hybrid scheme consisting of a combination of a second order MUSCL scheme and a high order finite difference scheme. The hybrid scheme is constructed in such a way that we can prove that it is stable. We can also show that
more » ... is high order accurate in smooth domains and oscillatory free close to the shock. The order of accuracy is measured locally to see its appropriate size. Numerical experiments will be done for simple linear problems, the Burger's equation, and for the Euler equations with high Mach numbers. 24 This contribution deals with the modeling of soft biological tissues in the physiological as well as in the supra-physiological (overstretched) domain. In order to capture the softening hysteresis as observed in cyclic uniaxial tension tests, we introduce a scalar-valued damage variable into the strain-energy function for the embedded fibers such that remnant strains are obtained in the fibers for the natural state after unloading the material. This can be reasonably included by assuming an additively decoupled energy function into an undamaged isotropic part, for the ground substance, and superimposed transversely isotropic parts, for the embedded fiber families. A saturation function is considered accounting for converging stress-strain curves in cyclic tension tests at fixed load levels. As a numerical example we consider a circumferentially overstretched atherosclerotic arterial wall. We develop a mathematical model of air flow and pollution transport in 2D street canyon. The model is based on Navier-Stokes equations for viscous incompressible flow and convection-diffusion equation describing pollution transport. The solution is obtained by means of Finite Element Method (FEM). We use non-conforming Cruzeix Raviart elements for velocity, piecewise constant elements for pressure, and linear Lagrange elements for concentration. The resulting linear systems are solved alternatively by the Multigrid method and Krylov subspace methods. We treat two types of boundary conditions: Dirichlet and natural boundary condition. We present computational studies of the given problem.