The Mathematics of Sand Piles: Eikonal Solution over an Elliptic Constraint

This Demonstration plots rectangular and elliptic solutions to the eikonal equation and a novelty known as a cross-figure solution together with their surface areas and volumes.

The eikonal equation is a nonlinear PDE that simulates the physical laws governing the piling of sand on a plate of a given shape held over a boundless chasm. With no loss of generality, the eikonal equation may be written as

.

The boundary conditions, being represented physically by the shaped plate, are dominated by a stretch factor that describes the elongation in a particular direction of the plane. Although the choice of direction along which applies is arbitrary, here are the conventions used in this Demonstration:

The three snapshots show different levels of elongation along the axis. With , the axis component is elongated and vice versa for . At , the stretches in each directions are equal and we recover a cone from the elliptic boundary condition solution and a pyramid from the rectangular boundary condition solution.

Some graphical errors may exist due to the large amount of information that must be processed at each slider setting. Users are welcome to reproduce true images by working off the supplied code. Additionally, due to the latency attributed to the number of computations being performed at each step, only image resizing is recommended.

Reference

[1] I. P. Stavroulakis and S. A. Tersian, Partial Differential Equations: An Introduction with Mathematica and Maple, Singapore: World Scientific, 2004.