A tutorial for the course Digital World 2015 demonstrating a bare simple particle spring simulation for Processing/Python. To run the code below, download Processing and install the Python editing mode. The tutorial contains a series of suggestions/challenges for improving this basic template.
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#-- #-- Particle Spring Dynamics Simulation #-- #-- Window Dimensions #-- dx = 800 dy = 600 #-- Definition of Complex Types --------------------------------------------- #-- #-- As not sure if you are familiar with such concepts as a class or a tuple, #-- I use below vanilla lists to express semantically complex data forms such #-- as a vector, a particle and a spring. #-- #-- The convention used is: Any complex type of information will be stored in #-- a list each element thereof representing a property in orderly manner. #-- #-- The name of the property will be globally defined as a constant containing #-- the index of the property value in the list. #-- #-- Example: A point/vector is a complex entity which contains a number of #-- coordinates. For 2D vectors we need two entries the first one being X #-- and second Y coordinate. Below is the definition of X = 0 and Y = 1 #-- Therefore I can get/set the X coordinate of point P using P[X] #-- #-- Vector Definition ------------------------------------------------------- #-- #-- A vector represents a position, displacement or force in two dimensions. #-- Property definitions (X,Y coordinates) and basic algebra for vectors are #-- expressed below as global functions. #-- #-- CHALLENGE: Modify the definion to support 3D points/vectors (easy) #-- X = 0 Y = 1 #-- Create a New Vector #-- def vnew( x, y ): return [x, y] #-- Add Two Vectors #-- def vadd( u, v ): return vnew( u[X] + v[X], u[Y] + v[Y] ) #-- Subtract Two Vectors #-- def vsub( u, v ): return vnew( u[X] - v[X], u[Y] - v[Y] ) #-- Vector Scale #-- def vmul( u, s ): return vnew( u[X] * s, u[Y] * s ) #-- Vector Dot Product #-- def vdot( u, v ): return u[X] * v[X] + u[Y] * v[Y] #-- Vector Length #-- def vlen( u ): return sqrt( vdot( u, u ) ) #-- Particle Definition ----------------------------------------------------- #-- #-- A particle represent a dimensionless mass with internal dynamics state of #-- position and velocity. The force acting upon the particle can be also #-- though of as its acceleration. In other words the particle captures a set #-- of information about a vector valued function and its first and second #-- derivatives. #-- MASS = 0 POSITION = 1 VELOCITY = 2 FORCE = 3 #-- Create Particle #-- def pnew( m, p, v, f ): return [m, p, v, f] #-- Spring Definition ------------------------------------------------------- #-- A spring represents an idealized elastic link between two particles that #-- obays Hooke's Law of springyness: F = k * d, where F is the force magnitude #-- k is the spring constant, and d is the displacement. #-- #-- Alternatively you can think of this as: F = k * ( RestLength - CurrentLength ) #-- where RestLength is the length at which the spring excerts zero force to both #-- particles (aka ends of the spring locations) and CurrentLength is the length #-- which is either bigger (spring is stretched) or smaller (spring is compressed) #-- than the rest length (if they are equal then F = 0). #-- #-- In terms of properties we store the source and target locations (the end-points) #-- of the spring such that we can connect them with other springs or fix them etc. #-- We obviously need the spring constant and rest length. #-- #-- NOTE: The source and target properties are expressed as indices (integers) #-- instead of particle objects (list of properties). The conventions is #-- made such that we can store multiple particles in one list (each having #-- a known index) and define springs that associate/point-into the particle #-- collection by index (instead of duplicating particles per spring). #-- #-- CHALLENGE #1: Can we change the representation from the high-school notion #-- of the spring, to the proper engineering idea of elasticity? (easy) #-- HINT: google stress/strain and replace k with the elastic modulus e #-- #-- CHALLENGE #2: The spring, apart from its direct reaction to loading expressed #-- by Hooke's Law, it also has a force component, not used above, which #-- captures its internal damping. Otherwise a spring would oscillate #-- indefinately if think about it. Can express this also (medium) #-- HINT: google linear spring damping #-- SOURCE = 0 #-- Source Particle INDEX TARGET = 1 #-- Target Particle INDEX STIFFNESS = 2 #-- Spring Stiffness Constant LENGTH = 3 #-- Rest Length def snew( s, t, k, l ): return [s, t, k, l] #-- Particle and Spring Lists ----------------------------------------------- #-- particles = [] springs = [] #-- Create Initial Model ---------------------------------------------------- #-- #-- This function is required by processing in order to prepare the program. #-- It will be the first thing to be executed before any drawing takes place. #-- In here we will construct a chain of particles and connect them with #-- springs. They will be stored in the global variables "particles" and #-- "springs" for later use in the simulation/animation cycle aka draw( ) #-- #-- NOTE: There is a conceptual leap here... it is only a convention or some #-- sort of approximation involved in the assumption that a sequentially #-- connected set of particles represents a "physical" chain. In the #-- fashion we may "define" a rectangular grid of connected particles #-- as a "fabric" sheet. More on that later, meanwhile think of other #-- "things" that can be represented by such configurations of particles #-- as springs. #-- def setup( ): #-- Window Size #-- size( dx, dy ) frameRate( 24 ) #-- Import Globals #-- global particles global springs #-- Number of Particles #-- count = 12 #-- Construct Particles --------------------------------- #-- Create a sequence of particles along the x-direction #-- NOTE: We need to provide mass, possition and velocity #-- The last two items are important as they define #-- the dynamics state of the particle. You can set #-- velocity to non-zero if you like. #-- for index in range( 0, count ): mass = 20.0 x = index * 50 + 100 y = 100 position = vnew( x, y ) velocity = vnew( 0, 0 ) force = vnew( 0, 0 ) particles.append( pnew( mass, position, velocity, force ) ) #-- Construct Springs ----------------------------------- #-- Connect previously constructed particles in order #-- for index in range( 1, count ): source_index = index - 1 target_index = index - 0 stiffness = 15.0 rest_length = 50.0 springs.append( snew( source_index, target_index, stiffness, rest_length ) ) #-- By own convention, particles with zero mass are assumed fixed in place #-- There are other ways to define such a constraints but for simplicity #-- we define here as such. The first and last particles of the chain are fixed. #-- particles[ 0][MASS] = 0.0 particles[count - 1][MASS] = 0.0 #-- Draw Window Contents ----------------------------------------------------- #-- This is also a standard function required by processing. It is executed #-- every some odd milliseconds and it is supposed to draw the screen (aka FPS) #-- #-- In here we perform two tasks: 1. We run the simulation of particles and #-- springs. 2. We draw them in the window as we are supposed. #-- #-- Logic of Simulation: the steps taken for simulation are... #-- 1. Force Calculation/Accumulation F = m * a -> #-- a = F / m #-- #-- 2. Velocity update from acceleration a = dv / dt -> #-- dv = a * dt -> #-- v_new - v_old = a * 1.0 -> (!!! dv = v_new - v_old ) #-- v_new = v_old + a #-- #-- 3. Position update from velocity v = dp / dt -> #-- dp = v * dt -> #-- p_new - p_old = v * 1.0 -> (!!! dp = p_new - p_old ) #-- p_new = p_old + v #-- #-- 4. Enforce Constraints (such that particle dont fall off the screen) #-- #-- NOTE: The above logic is called forward explicit Newton integration. #-- The places with (!!!) are important because there are assumptions #-- of approximating differential equations with their first linear #-- term only (which will cause noticable inaccuracy, unless dt is small) #-- #-- Newton integration is simple and fast but tends to spiral out of #-- control due to numerical errors (we are just following the tangent #-- line forward instead of the curved true trajectory, picture it). #-- #-- The second assumption is that the time step dt = 1.0 which is a #-- simplification that couples the animation FPS with simulation time. #-- You may slow down the simulation by changing the FPS or actually #-- multiplying velocity and acceleration by a small scalar eg. 0.0001 #-- Scaling the time step (dt) is advisable as it will improve accuracy. #-- #-- CHALLENGE: Add a proper time step component. HINT: basic vector math (easy) #-- #-- CHALLENGE: There other more sophisticated integration methods which are #-- also explicit but more accurate eg. Mid-Point (easy), Verlet (easy) #-- RK4 (tricky), as well as implicit methods using linear algebra (hard) #-- #-- CHALLENGE: So we know from theory that the shape of handing chain is #-- expressed-by / has-a closed form equation known as the catenoid. #-- Draw it over the simulated shape (same parameters) and compare the #-- results. What is the significance of the error? #-- #-- CHALLENGE: So we have been using abstract units for space (pixels), mass(units) #-- time (fps). What if you apply real units eg. meters, kilograms, seconds #-- such that you can compare the real shape of a physical chain with the #-- simulation. How does the real shape (take photo and overlay), theory #-- and simulated form compare? #-- def draw( ): #-- Import Globals #-- global particles global springs #-- Reset Force Accumulator --------------------------------- #-- for particle in particles: particle[FORCE] = vnew( 0.0, 0.0 ) #-- Accumulate Gravity Forces ------------------------------- #-- Assuming earth gravity constant but note y is positive #-- because y-screen direction is pointing downwards already. #-- GRAVITY = 9.81 for particle in particles: gravity = vnew( 0, GRAVITY * particle[MASS] ) particle[FORCE] = vadd( particle[FORCE], gravity ) #-- Accumulate Damping Forces ------------------------------- #-- Fictional linear drag opposite to particle's velocity #-- for stability. Air drag is quadratic to velocity and #-- proportional to fluid density as well as surface area. #-- #-- CHALLENGE: Change this unrealistic drag force to a proper #-- air drag force (easy). HINT: google drag force #-- when is doubt about areas & constants use 1.0 #-- DAMPING = 2.00 for particle in particles: damping = vmul( particle[VELOCITY], -DAMPING ) particle[FORCE] = vadd( particle[FORCE], damping ) #-- Accumulate Spring Forces ------------------------------------------- #-- for spring in springs: #-- Extract Particle Indices #-- source_index = spring[SOURCE] target_index = spring[TARGET] #-- Extract Particles (themselves) #-- source_particle = particles[source_index] target_particle = particles[target_index] #-- Extract Positions #-- source_position = source_particle[POSITION] target_position = target_particle[POSITION] #-- Hook's Law (over-simplified) #-- The proper formulation includes a damping #-- component proportional to velocity along the #-- spring direction. NOTE: I am dividing by the #-- distance between the spring's end-points aka #-- the current length, because the force magnitude #-- is assuming a normalized directional vector #-- stiffness = spring[STIFFNESS] rest_length = spring[LENGTH] direction = vsub( target_position, source_position ) distance = vlen( direction ) magnitude = stiffness * ( distance - rest_length ) / distance source_particle[FORCE] = vadd( source_particle[FORCE], vmul( direction, magnitude ) ) target_particle[FORCE] = vadd( target_particle[FORCE], vmul( direction, -magnitude ) ) #-- Newton Integration ----------------------------------------- #-- Discrete integration by assuming time step of one unit. #-- For better accuracy and stability either scale acceleration #-- and velocity vectors by a time step eg. dt = 0.001 or use #-- better integration approximation. #-- for particle in particles: #-- Zero mass particles are ignored #-- mass = particle[MASS] if( mass > 0.0 ): #-- F = ma <=> a = F / m #-- acceleration = vmul( particle[FORCE], 1.0 / mass ) #-- acceleration = dv/dt ~> v_new = v_old + acceleration * dt #-- particle[VELOCITY] = vadd( particle[VELOCITY], acceleration ) #-- velocity = dp/dt ~> p_new = p_old + velocity * dt #-- particle[POSITION] = vadd( particle[POSITION], particle[VELOCITY] ); #-- Constraint Enforcement ----------------------------------- #-- #-- Force particles to remain within window boundaries by #-- reversing their position to previous and flipping velocity #-- #-- This is kind of a perfect elestic collision responce since #-- velocity is preserved 100% #-- #-- The reason for reversing position to previous is because #-- the particle may have escaped significantly outside. So #-- better safe than sorry. #-- #-- Constraints are better expressed as "force constraints" #-- rather than "penalties". If you think about it, the code #-- below is not 100% realistic because every time we move #-- particles back and flip their velocity direction we are #-- fiddling with the energy/momentum of the system in ways #-- that may violate constitutional laws of coservation. #-- #-- CHALLENGE: Read about the notion of "virtual work" and #-- implement a constraint such as particle-on-circle (hard) #-- for particle in particles: #-- Particle Properties #-- position = particle[POSITION] velocity = particle[VELOCITY] radius = particle[MASS] * 0.5 #-- Drawing artifact, see below #-- Bound Checking #-- if( ( position[X] < radius ) or ( position[X] > ( dx - radius ) ) ): position[X] = position[X] - velocity[X] velocity[X] =-velocity[X] if( ( position[Y] < radius ) or ( position[Y] > ( dy - radius ) ) ): position[Y] = position[Y] - velocity[Y] velocity[Y] =-velocity[Y] #-- Prepare Frame ------------------------------------------- #-- Clears the background, otherwise it will look psychedelic #-- try commenting out the background line below and enjoy #-- fill( 0 ) background( 255 ) #-- Draw Springs -------------------------------------------- #-- for spring in springs: source_index = spring[SOURCE] target_index = spring[TARGET] source_particle = particles[source_index] target_particle = particles[target_index] source_position = source_particle[POSITION] target_position = target_particle[POSITION] stroke( 0, 0, 255 ) line( source_position[X], source_position[Y], target_position[X], target_position[Y] ); #-- Draw Particles --------------------------------------------- #-- noStroke( ) index = 0 for particle in particles: position = particle[POSITION] mass = particle[MASS] #-- Fixed particles (with zero mass) have green color #-- RGB=(0,255,0), unconstrained particles have red #-- color (0,255,0) and radius = half their mass #-- if( mass > 0.0 ): radius = 0.5 * mass fill( 255, 0, 0 ) else: fill( 0, 255, 0 ) radius = 10 ellipse( position[X], position[Y], radius, radius ) #-- Display Particle Index #-- fill( 0 ) text( str( index ), position[X] + 10, position[Y] - 10 ) index = index + 1 |