Material elasticity 3
Spring systems
Last updated
Spring systems
Last updated
In this section we will reuse the Cartesian product force application approach we developed earlier for n-body simulation but with springs instead of gravity forces. Visually this setup captures beautifully the idea of Cartesian product being the equivalent of complete graphs [] in the graph theoretic sense.
The setup for this type of simulation is quite straight forward: We produce a random number of points in the plane, set their velocities initially to all zero, and their masses all set to one unit. In addition, all spring rest lengths are set to be equal.
The spring force computation is significantly simplified as we just need to take one Cartesian product between all particle’s positions and compute the bi-lateral spring forces exerted between one another. To collapse the grafted tabular data structure we use the summation component and flatten its output.
One aspect worth considering is that both a square, as a regular polygon in the plane result of 4-node complete graph, and the tetrahedron in space, come to a stop after a few iterations. The internal forces of the particle spring system have been balanced in both cases.
However, while the springs of the tetrahedron can all acquire their desired lengths, this is not the case for its planar equivalent, because we cannot possibly construct a square where all sides and diagonals are equal. It is geometrically impossible.
The square therefore is under balance of non-zero compression and tension forces. While the tetrahedron has the degrees of freedom to achieve its desired state, the planar square can only find a good enough compromise.
Practice
Experiment with evolving shapes with higher number of nodes and edges. Notice that not always the result configuration conforms with the regular polygon idea. Explain under which condition this takes place.
You may apply a constraint such as projecting the particles onto surfaces such as spheres or splines to produce interesting configurations confined by their local environment.
In this session we developed concepts and techniques related to deformable bodies using particle and spring physics. The point of interest to take away from this session is in the ability to express desired outcomes under potentially conflicting conditions. Using the aggregate propagation of forces through the particle as a system overtime we are able to arrive to equilibrium conditions may those be arriving at an exact type of solution or a well balanced compromise.
Using particle spring relaxation is in fact a typical way to visualize graphs including social networks, where nodes represent people and edges their relationships, with various degrees of connectivity and strength [].
What is beautiful about this construction is that it illustrates the relationship between regular planar polygons as well as spatial polyhedra, simplexes in general [] with complete graphs. In the animation below we evolve the tetrahedron [] which is the simplest three dimensional prism and one the five Platonic solids [].
