Seminar

Abstract:

The Navier-Stokes equations (NSE) model the motion of incompressible viscous fluids and are widely used in engineering and physics, etc. In many applications, the time-averaged information is very often the quantity of interest, so this talk focuses on finding the numerical solutions  of the steady Navier-Stokes equations (SNSE). We introduce a few improvements to the commonly used Picard iteration for solving the SNSE  from different aspects, like shortening the required number of iterations and reducing the computational cost for each iteration. 

Abstract:

Irreversible, dissipative processes can be often naturally modeled as gradient flows; under certain assumptions, flow in deformable porous media is such a process. In this talk, we formulate the problem of linear poro-(visco-)elasticity as generalized gradient flow. By exploiting this structure, the analysis of well-posedness and construction of numerical solvers can be performed in a rather straight-forward and abstract manner. For this, we apply results from gradient flow theory and convex optimization.

Hopefully we know that $\displaystyle \sum_{i \geq 0} x^i $ $= \frac{1}{1-x}$. Similarly one computes that $ \displaystyle \sum_{i \leq 1} x^i $ $= \frac{x^2}{x-1}$. Interestingly, $ \frac{1}{1-x} + \frac{x^2}{x-1} = 1 + x$ which is the sum corresponding to the integers in the interval $[0, 1] = [0, \infty] \cap [-\infty, 1]$. We will explain generalization of this (called Brion's theorem) to integer points in convex polytopes of arbitrary dimension.

When a young mathematician told his advisor that he had found a way to turn a sphere inside out (without making any creases), his advisor told him that this was impossible to do so, and gave him a proof. His advisor was wrong. That young mathematician (Smale) went on to win the highest mathematical prize (the Fields Medal). His solution is very non-intuitive. Today, there are several excellent videos on YouTube that show how to turn a sphere inside out, and this talk will explain some of them. 

We will discuss telescoping, Tannery's Theorem and other techniques for calculating infinite sums.

Collisions between two masses $m_{1}$ and $m_{2}$ have two important quantities: momentum and energy, respectively given by

$$ p = m_{1}v_{1} + m_{2}v_{2}$$

and

$$E = \frac{1}{2}m_{1}v_{1}^2 + \frac{1}{2}m_{2}v_{2}^2.$$