1. 𝝏𝑳 𝝏 𝒙 𝒊 =𝟎⟹ 𝒋=𝟏,𝒋≠𝒊 𝑵 𝒙 𝒊 − 𝒙 𝒋 𝒙 𝒊 − 𝒙 𝒋 𝒔+𝟐 = 𝒇 𝒊 𝒙 𝒊
Computational Mathematics & Minimizing Energy A Visualization Approach Department of Mathematics. Student: Lingxi Wu. Advisor: Peter Dragnev
Abstract
Goal and Methodology
The Math …
Minimal energy configurations have wide ranging applications in various fields of science, such as crystallography, nanotechnology, material science, information theory, wireless communications, etc. Whether one studies electrons in equilibrium, large carbon fullerenes, orifices of pollen grain, or complexity of computation algorithms, distributing points on the sphere in some uniform way plays significant role.
In this presentation we shall introduce our mathematical model of optimizing the spherical configurations, show the programming procedures in Maple, and demonstrate the visualizations of the minimization process.
Study the movements of a set of n (2<n<60) randomly distributed points on a unit sphere.
They repel each other and they are free to move on the surface of the unit sphere.
Demonstrate how they reach a stable state using calculated animations.
N points on a unit sphere represented by 3D vectors:
𝑿= 𝒙 𝟏 , 𝒙 𝟐 , 𝒙 𝟑 … 𝒙 𝑵 and | 𝒙 𝒊 | 𝟐 -1=0.
We minimize their total Riesz Energy:
𝟏≤𝒊≠𝒋≤𝑵 𝟏 𝒙 𝒊 − 𝒙 𝒋 𝒔
Apply Lagrange Multiplier Method (Multivariate Calculus):
𝑳( 𝒙 𝟏 , … 𝒙 𝒏 , 𝝀 𝟏 , … 𝝀 𝒏 ) = 𝟏≤𝒊≠𝒋≤𝑵 𝟏 𝒙 𝒊 − 𝒙 𝒋 𝒔 + 𝒊=𝟏 𝑵 𝝀 𝒊 (| 𝒙 𝒊 | 𝟐 −𝟏)
Force Equation:
𝝏𝑳 𝝏 𝒙 𝒊 =𝟎⟹ 𝒋=𝟏,𝒋≠𝒊 𝑵 𝒙 𝒊 − 𝒙 𝒋 𝒙 𝒊 − 𝒙 𝒋 𝒔+𝟐 = 𝒇 𝒊 𝒙 𝒊
Math
Program
Computational Mathematics
Background
Pseudo-code snippet
Energy minimization is the process of arranging a set of points(atoms, electrons, pollens grains, and etc) in some spaces such that the potential energy of that arrangement is minimal.
Ubiquitous in nature.
Studying the math behind these beautiful patterns will give us insights into other science and engineering fields
Math Model
Theoretical Foundations.
Build proofs in mathematical languages.
Algorithm
Define functions, procedures, and methods.
Program workflow.
Pseudo-code.
Application
Programming scripts.
Animation modules.
// 1. Generate N random points on a sphere. Initial configuration.
// 2. Core calculations: points start moving . Record each new configuration.
while(current_iteration < target_iteration){
for(i=1;i<=Number_of_points;i++){
for(j=1;j<=Number_of_points;j++){
if(i!=j){
// a. Get force vector from point j to point i; }
// b. Calculate net force vector from points other than point i.
// c. Add point i vector and net force vector together.
// d. Project that total vector onto the surface of the unit sphere.
// e. Update the current configuration with the new point i.
// f. After all points been calculated, save the new configuration into a config list .
//3. Graph all the configurations in the list continuously to simulate the motions.
32 Electrons
Montreal Biosphère
Carbon C60
Basic ideas
Generate Points
Calculate forces
Net force
More examples…
Tammes problem
Hives
Lion paw Bucky ball
Nano Technology
Puffer Fish
Conclusion
When s=1, the Maple script based on our model and algorithm can successfully animate the motions of up to 60 points.
It has limitations but runs successfully in most cases.
Peter D. Dragnev (2010). Electrons, Buckyballs, and Orifices: Nature's Way of Minimizing Energy.