José Abell's research blog


Finite Elements for Shallow Water Equations

The linear shallow water equations (SWE) are used extensively to model propagation of waves in situations where lateral domain dimensions are much greater than fluid height and, at the same time, wave perturbation height are much smaller than the fluid height. This finds useful applications in the case of tsunami-wave propagation modeling in deep-sea. The equations, after heavy linearization, are given by:

$$\begin{aligned} \pardiff{u}{t} - f v = -g \pardiff{h}{x} \\ \pardiff{v}{t} + f v = -g \pardiff{h}{y} \\ \pardiff{h}{t} = -H \pare{ \pardiff{u}{x} + \pardiff{v}{y} }\end{aligned}$$

Where

  • \(u\) is the speed …

NTS-02. On Rayleigh damping coefficients for FE analysis

Note to self. How to compute Rayleigh damping coefficients for given damping ratios $\xi_1$ and $\xi_2$ at frequencies $f_1$ and $f_2$.

This is textbook content, I just need to remind myself too often how this is done and end up re-deriving the equations.

Given the second-order system of differential equations representing the FE model

$$ M \ddot{u} + C \dot{u} + K u = F(t) $$

The damping matrix $C$ can be written as a Rayleigh damping matrix:

$$ C = a_0 M + a_1 K $$

$a_0$ and $a_1$ are Rayleigh damping coefficients found by solving

$$ \left[ \begin{array}{cc} \dfrac{1}{2\pi f_1} & 2 \pi f_1 \ \dfrac{1}{2\pi f_2} & 2 \pi f_2 \end{array} \right] \left[ \begin{array}{c} a_0 \ a_1 \end{array} \right] = \left[ \begin{array}{c} \xi_1 \ \xi_2 \end{array} \right] $$

Which I do in the following code: