It is infinite in the Z axis, so it can be reduce to a 2 dimensional problem. In this case, the prism is just a box within a box. Personally, I found the 'Prism' name a bit confusing since it is usually associated with a triangular shape. Middle side is how the potential is distributed after a few iterations. Left side is the input and right side is the output. Note that all three methods will have the same output. The top and bottom were described as non-conducting, so a voltage drop equation was implemented to calculate the potential.īelow is the metal box geometry being solved by Jacobi method. The metal box has the leftmost and rightmost columns set as the boundaries, so -1 and +1 V in this case.

The 3 methods were tested using three different geographies that were described in the book: SOR simply multiplies the value received by GS by alpha where alpha is between 1 and 2. I drew an example of 2 GS iterations with Inksapce below. GS uses the same matrix to store and update the values. This is done until delta v is below a predefined treshold. Then the first matrix is set to equal the second matrix.

These values are then updated in the second matrix, not the first. In matrix one, loop through all the elements and get the average of the neighbouring points for each matrix element. The smaller the delta v value, the more accurate your solution is. This is the sum of differences between the old potential points and the new. The three methods differ in how each point is updated. In all 3 methods, the matrix is updated one-by-one from top left to bottom right. Poisson’s equation is important because it is at the core of this project and, in the words of Richard Feynman, “The entire subject of electrostatics, from a mathematical point of view, is merely a study of the solutions of the single equation. It can be used to model gravity, fluid dynamics, steady-state heat transfer and electrostatics. Poisson’s equation has a wide range of applications in physics and engineering. Laplace equation is a special case of Poisson’s equation. It can be used to model a wide variety of objects such as metal prisms, wires, capacitors, inductors and lightning rods. Laplace equation models the electric potential of regions with no electric charge. Most of the work here was heavily inspired by 'Computational Physics' book by Giordano and Nakanishi. This repo will focus on three iterative relaxation methods: There is one restriction: there must be a boundary in the geometry that you're testing and all the maxima and minima points (max and min potential) must be located at the boundary points. The solution of the Laplace equation for a given electrostatic geometry can be used to model the distribution of potential in a general way.

Gauss seidel python code