Developers Card

Collection of showcase applications that I have developed over the years.

🚧 This repository is under constant development 🚧

Ordinary differential equations (ODE) solvers in Julia

The damped harmonic oscillator

A classical physical example of an ODE is the damped harmonic oscillator. Consider the following equation that describes the motion of a damped spring

$$m \frac{d^2x}{dt^2} = -kx -c\frac{dx}{dt}.$$

This equation can be rewritten in the following way

$$\frac{d^2x}{dt^2} + \omega^2 x + 2\zeta \omega \frac{dx}{dt} = 0 ,$$

where $$\omega = \sqrt{k/m}$$ and $$\zeta = c / 2 / \sqrt{mk}$$, and solved to using the Julia package DifferentialEquations.jl. The movie below shows the results obtained by the script dampedHarmonicOscillator.jl

Partial differential equation (PDE) solvers

Hydrothermal fluid flow

Porous convection is an important physical process that describes the dynamics of fluid extraction from an essentially undeforming solid in a multiphase aggregate. Applications include industrial filtration systems, subsurface hydrogeological reservoirs, or melt extraction below a volcano. The movie below shows porous convection across a 2D domain. To calculate the fluid motion through the pore space, Darcy's equation is coupled to the Heat equation. The solution is obtained iteratively applying a pseudo-transient time stepping method.

Attribute	Value
Physics	Darcy's law + Heat equation
Method	central finite differences
Time integration	fully implicit
Advection scheme	upwind (first order)
Programming language	Julia
Parallelization	no
Rayleigh-No	1000
Porosity	10 % (const.)

PC_impl_BConFlux_mov.mov

Analytical solutions

For benchmarking purposes it is useful to compare the numerical solution to analytical solutions. For some equations it is possible to derive a solution analytically for others they do not exist. In this section, I showcase a 1D Diffusion-Convection equation for which I have derived an analytical solution assuming an initial normal (or Gaussian) distribution of a scalar quantity and homogeneous Dirichlet boundary conditions. As shown in the movie below the numerical fits the analytical solution up to an acceptable tolerance.

DiffusionConvection.mov

Data Analysis

Fitting data using neural networks

The following figure shows an example of fitting the temperature-dependent rock density using neural networks. The linear regression algorithm was developed using the Flux.jl package.

Numerical Methods

Newton-Raphson iterative solution procedure

This technique allows to find the root of any function $f(x)$. Solution procedure starts by defining an initial guess ($x_n$), ideal already close to the solution. Determining the derivate $f'(x_n)$ and its root gives a new guess $x_{n+1}$ which is closer to the solution than the guess before. Repeat this procedure either until $|x_{n+1} - x_n| < \varepsilon$ or $f(x_{n+1}) \approx 0$. An advantage of the Newton-Raphson iteration is that convergence is quadratically in the vicinity of the root, reducing the computational cost. The example below shows the Newton-Raphson iterative solution procedure to find the root of $f(x) = x^2 - 2$. The derivative $f'(x_n)$ is calculated (1) analytically, (2) numerically as difference between $x_n$ and a perturbation of $x_n$ approximation, (3) via the Julia package Zygote.jl which enables automatic differentiation of any function. Note the scatter plot in the right panel indeed shows a quadratic convergence behaviour.

NewtonRaphson.mov

This and similar methods are widely used in optimization algorithms and PDE solvers. I have also used this technique to find the liquidus temperature of geomaterials as a function of pressure and chemical composition.