The book Modellbildung und Simulation: Eine anwendungsorientierte Einführung contains interesting topics such as road traffic, scheduling, chaos, and moleculardynamics. It describes the modeling and simulation ascpects. But it gives only some implementation ideas, and contains no program code. To further dive into the simulations, this repo holds implementations of the presented models in Python with visualization in pygame or with matplotlib.
The folder model_and_simulate contains individual projects. All simulations start from a single menu. Simply run the model_and_simulate/main script to start the GUI.
Moleculardynamics is the implementation of Chapter 13. As model the Lennard-Jones Potential is used to get forces (attraction and repulsion) between all molecules. The accelerations get calculated by using the Velocity-Störmer-Verlet algorithm. To avoid quadratic runtime scaling, a Linked-Cells-Datastructure is implemented.
The simulation has a menu to set parameters such as number of molecules, molecule radius (Sigma), and the Cell grid structure:
Step size dt is the resolution of the algorithm in the numerical solution of the equations of motion.
6 different distributions can be chosen to draw the initial molecule positions from:
Chapter 12 describes chaotic systems and shows simulations as bifurcation diagrams.
One common attractor is the Lorenz-Attractor (Butterfly effect). The corresponding three-dimensional
system of nonlinear differential equations is not in the book. An implementation with
Runge-Kutta procedure from scipy.integrate and matplotlib visualization produces a rotatable figure:
Another interesting ode system corresponds to the Aizawa attractor. Formulas, parameters, and initial values come from
algosome.
However, the plot does look different:
Maybe there is an error in the equations...
Microscopic modeling of road traffic is the implementation of Chapter 8. The rules of NaSch-Model describe the movement of cellular automatas on a road. Roads are only one-dimensional, collision-free, and all vehicles have equal characteristics such as length, dawdling factors, or maximum velocities.
The density of the traffic on the road as well as the probability for dawdling can be set in a Pygame GUI. Furthermore, the total length of the road is settable. The implementation can simulate up to 461 cars.
A Pygame visualization shows a 2-D plot in real-time, which is the section on the x-Axis over time:
It shows the same traits as Figures 8.4 - 8.7 in Chapter 8 of the book.
I would like to acknowledge the work of Hans-Joachim Bungartz, Stefan Zimmer and Dirk Pflüger. The book Modellbildung und Simulation: Eine anwendungsorientierte Einführung (eXamen.press) is very instructive, inspiring and understandable. It's really enjoyable to read!
The course 01610 at FernUniversität in Hagen uses the above mentioned book to explain the basics of simulations at a university level. The course is very exciting: Simulation This repo does not contain any solutions to exercises of the course.
The Lorenz-Attractor equations are from wikipedia: Lorenz, Edward Norton (1963). "Deterministic nonperiodic flow". Journal of the Atmospheric Sciences. 20 (2): 130–141.
The Aizawa-Attractor equations are from algosome: Langford, William Finlay (1984). "NumericalStudies of Torus Bifurcations". International Series of Numerical Mathematics. 70: 285–295.
I searched looperman to get some fancy music loops for pygame. I would like to thank Rasputin(Brisk Bass Sequence, Goa Trance Gated Vox), SUPERSAMPLE(KORG 2), and soaren(Temptation Playground Trippy Bells). Some sound effects were taken from freesound. Thanks to michael_kur95(Hit02), Leszek_Szary(jumping), Tissman(hit3). The sounds are under CC0 1.0 Universal license.
