Discrete multi physic model for the Rayleigh collapse of a single cavity

In this thesis, a Discrete Multi-Physics model based on Smoothed Particle Hydrodynamics is developed to simulate a Rayleigh collapse of a single bubble. All the simulations were run on a modified version of the open source software LAMMPS and visualised on OVITO. Initially a 2D model is validated by simulating a phenomenon that shares many similarities with a collapse mechanism, the interaction of a shock wave with a discrete gas inhomogeneity, showing similar performance to classic mesh based CFD. The model is then used to simulate a 2D Rayleigh collapse and validated against the 2D Rayleigh-Plesset equation for both empty and gas filled cavity. The validated model is used to investigate the role of heat diffusion at the gas-liquid interface of the cavity, and to study non-symmetrical collapse induced by the presence of a nearby surface. Enabling heat diffusion at the gas-liquid interface allowed to identify five different possible behaviours that range from isothermal to adiabatic, while the results of non symmetric collapse show that the surface is hit by a stronger shock when distance between the center of the cavity and the surface is zero while showing more complex double peaks behaviour for other distances. In the final chapter a 3D model is used to model an attached non-symmetrical collapse and its hydrodynamic is compared with the equivalent 2D case

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described