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mass_spring_system.html
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<!doctype html>
<html class="no-js" lang="en">
<head>
<meta charset="utf-8">
<style>
body {font-family: Helvetica, sans-serif;}
table {background-color:#CCDDEE;text-align:left}
</style>
<script type="text/x-mathjax-config">
MathJax.Hub.Config({
extensions: ["tex2jax.js"],
jax: ["input/TeX", "output/HTML-CSS"],
tex2jax: {
inlineMath: [ ['$','$'], ["\\(","\\)"] ],
displayMath: [ ['$$','$$'], ["\\[","\\]"] ],
processEscapes: true
},
"HTML-CSS": { fonts: ["TeX"] }
});
</script>
<script type="text/javascript" aync src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.4/MathJax.js"></script>
<title>Mass Spring System</title>
</head>
<body>
<main>
<h1 style="text-align:center">Mass Spring System</h1>
<table style="align_center;border-radius: 20px;padding: 20px;margin:auto">
<col width="1100">
<col width="400">
<tr>
<td>
<canvas id="simCanvas" width="1024" height="768" style="border:2px solid #000000;border-radius: 20px;background-color:#EEEEEE">Your browser does not support the HTML5 canvas tag.</canvas>
</td>
<td>
<table>
<col width="180" style="padding-right:10px">
<col width="100">
<tr>
<td><label>Current time</label></td>
<td><span id="time">0.00</span> s</td>
</tr>
<tr>
<td><label>Time per sim. step</label></td>
<td><span id="timePerStep">0.00</span> ms</td>
</tr>
<tr>
<td><label># particles</label></td>
<td><span id="numParticles">0</span></td>
</tr>
<tr>
<td><label># springs</label></td>
<td><span id="numSprings">0</span></td>
</tr>
<tr>
<td><label for="widthInput">Width</label></td>
<td><input onchange="gui.restart()" id="widthInput" type="number" value="40" step="1"></td>
</tr>
<tr>
<td><label for="heightInput">Height</label></td>
<td><input onchange="gui.restart()" id="heightInput" type="number" value="30" step="1"></td>
</tr>
<tr>
<td><label for="fixedParticlesInput"># fixed particles</label></td>
<td><select onchange="gui.restart()" id="fixedParticlesInput">
<option value="1">1</option>
<option value="2" selected="selected">2</option>
<option value="4">4</option>
</select></td>
</tr>
<tr>
<td><label for="timeStepSizeInput">Time step size</label></td>
<td><input onchange="gui.sim.timeStepSize=parseFloat(value)" id="timeStepSizeInput" type="number" value="0.005" step="0.001"></td>
</tr>
<tr>
<td><label for="stiffnessInput">Stiffness</label></td>
<td><input onchange="gui.sim.stiffness=parseFloat(value)" id="stiffnessInput" type="number" value="5000.0" step="1.0"></td>
</tr>
<tr>
<td><label for="dampingInput">Damping</label></td>
<td><input onchange="gui.sim.damping=parseFloat(value)" id="dampingInput" type="number" value="10.0" step="0.1"></td>
</tr>
<tr>
<td><label for="gravityInput">Gravity</label></td>
<td><input onchange="gui.sim.gravity=parseFloat(value)" id="gravityInput" type="number" value="-9.81" step="0.01"></td>
</tr>
<tr>
<td><label for="massInput">Mass</label></td>
<td><input onchange="gui.sim.mass=parseFloat(value)" id="massInput" type="number" value="0.5" step="0.01"></td>
</tr>
<tr>
<td></td>
<td><button onclick="gui.restart()" type="button" id="restart">Restart</button></td>
</tr>
<tr>
<td></td>
<td><button onclick="gui.doPause()" type="button" id="Pause">Pause</button></td>
</tr>
</table>
</td>
</tr>
<tr><td>
<h2>Mass spring algorithm:</h2>
This example shows a mass spring system that consists of particles linked by damped springs:
<ol>
<li>compute spring forces</li>
<li>time integration to get new particle positions and velocities</li>
</ol>
Note that the simulation is only conditionally stable since a conditionally stable explicit time integration method is used.
<h3>1. Compute spring forces</h3>
<p>In general a spring force can be obtained for any holonomic constraint.
In this example we use distance constraints
$$C_i(\mathbf{x}_{i_1}, \mathbf{x}_{i_2}) = \| \mathbf x_{i_1} -\mathbf x_{i_2} \|-d,$$
where $d$ is the rest length between particles $\mathbf{x}_{i_1}$ and $\mathbf{x}_{i_2}$.</p>
<h4>Potential energy</h4>
<p>For a scalar constraint we can define a potential energy as:
$$E(\mathbf x) = \frac k 2 C(\mathbf x)^2,$$
where $k$ is the stiffness of the spring.
<h4>General spring force</h4>
The spring force for a particle $j$ is then determined by the negative gradient of the potential energy function:
$$\mathbf F_j = - \frac{\partial E(\mathbf x)}{\partial \mathbf x_j} = -k \frac{\partial C(\mathbf x)}{\partial \mathbf x_j} C(\mathbf x).$$
<h4>General damping force</h4>
The corresponding damping force is obtained by using the time derivative of the constraint function:
$$\mathbf F^D_j = -\mu \frac{\partial C(\mathbf x)}{\partial \mathbf x_j} \dot{C}(\mathbf x).$$
<h4>Constraint gradients:</h4>
<p>To compute the spring and damping forces, the constraint gradients are required which are computed as: </p>
$$\begin{align*}
\frac{\partial C_i}{\partial \mathbf x_{i_1}} &= \frac{\mathbf x_{i_1} -\mathbf x_{i_2}}{\| \mathbf x_{i_1} -\mathbf x_{i_2} \|} \\
\frac{\partial C_i}{\partial \mathbf x_{i_2}} &= - \frac{\mathbf x_{i_1} -\mathbf x_{i_2}}{\| \mathbf x_{i_1} -\mathbf x_{i_2} \|}
\end{align*}$$
<h4>Spring force</h4>
So finally we get the following spring forces for a distance constraint $C_i(\mathbf{x}_{i_1}, \mathbf{x}_{i_2})$:
$$\begin{align*}
\mathbf F_{i_1} = -k (\| \mathbf x_{i_1} -\mathbf x_{i_2} \|-d) \frac{\mathbf x_{i_1} -\mathbf x_{i_2}}{\| \mathbf x_{i_1} -\mathbf x_{i_2} \|} \\
\mathbf F_{i_2} = +k (\| \mathbf x_{i_1} -\mathbf x_{i_2} \|-d) \frac{\mathbf x_{i_1} -\mathbf x_{i_2}}{\| \mathbf x_{i_1} -\mathbf x_{i_2} \|}.
\end{align*}$$
<h4>Damping force</h4>
The damping forces for a distance constraint $C_i(\mathbf{x}_{i_1}, \mathbf{x}_{i_2})$ are determined as:
$$\begin{align*}
\mathbf F^D_{i_1} = -k \left ((\mathbf v_{i_1}- \mathbf v_{i_2}) \cdot \frac{\mathbf x_{i_1} -\mathbf x_{i_2}}{\| \mathbf x_{i_1} -\mathbf x_{i_2} \|} \right ) \frac{\mathbf x_{i_1} -\mathbf x_{i_2}}{\| \mathbf x_{i_1} -\mathbf x_{i_2} \|} \\
\mathbf F^D_{i_2} = +k \left ((\mathbf v_{i_1}- \mathbf v_{i_2}) \cdot \frac{\mathbf x_{i_1} -\mathbf x_{i_2}}{\| \mathbf x_{i_1} -\mathbf x_{i_2} \|} \right ) \frac{\mathbf x_{i_1} -\mathbf x_{i_2}}{\| \mathbf x_{i_1} -\mathbf x_{i_2} \|}.
\end{align*}$$
<h3>2. Time integration</h3>
Finally, the particles are advected by numerical time integration. In our case we use a symplectic Euler method:
$$\begin{align*}
\mathbf v(t + \Delta t) &= \mathbf v(t) + \frac{\Delta t}{m} \left (\mathbf F(t) + \mathbf F^D + \mathbf F^{\text{ext}} \right ) \\
\mathbf x(t + \Delta t) &= \mathbf x(t) + \Delta t \mathbf v(t + \Delta t),
\end{align*}$$
where $\mathbf F^{\text{ext}}$ are the external forces.
</td></tr>
</table>
</main>
<script id="simulation_code" type="text/javascript">
class Particle
{
constructor (x, y)
{
this.x = x;
this.y = y;
this.fx = 0.0;
this.fy = 0.0;
this.vx = 0.0;
this.vy = 0.0;
}
}
class Spring
{
constructor (index1, index2, restLength)
{
this.index1 = index1;
this.index2 = index2;
this.length = restLength;
}
}
class Simulation
{
constructor(width, height, fixed)
{
this.springs = [];
this.particles = [];
this.width = width;
this.height = height;
this.stiffness = 200.0;
this.damping = 5.0;
this.mass = 0.5;
this.gravity = -9.81;
this.timeStepSize = 0.005;
this.time = 0;
this.numFixedParticles = fixed;
this.init();
}
init()
{
// create particles
let i;
let j;
let w = this.width;
let h = this.height;
let s = 0.1;
for (i = 0; i < h; i++)
for (j = 0; j < w; j++)
this.particles.push(new Particle(s*2*j, -s*2*i))
// create horizontal springs
for (i = 0; i < h; i++)
for (j = 0; j < w - 1; j++)
this.springs.push(new Spring(i*w + j, i*w + j + 1, s*2.0));
// create vertical springs
for (i = 0; i < h-1; i++)
for (j = 0; j < w; j++)
this.springs.push(new Spring(i*w + j, i*w + j + w, s*2.0));
// create diagonal springs
for (i = 0; i < h-1; i++)
{
for (j = 0; j < w-1; j++)
{
this.springs.push(new Spring(i*w + j, i*w + j + w + 1, s*Math.sqrt(8)));
this.springs.push(new Spring(i*w + j +1, i*w + j + w, s*Math.sqrt(8)));
}
}
}
isActive(i)
{
// particle 0 and the selected particle are fixed
if (this.numFixedParticles == 1)
return (i != 0) && (i != gui.selectedParticle);
else if (this.numFixedParticles == 2)
return (i != 0) && (i != this.width-1) && (i != gui.selectedParticle);
else
return (i != 0) && (i != this.width-1) && (i != (this.height-1)*this.width) && (i != this.height*this.width-1) && (i != gui.selectedParticle);
}
computeSpringForces()
{
// reset forces
for (let i = 0; i < this.particles.length; i++)
{
let p = this.particles[i];
p.fx = 0.0;
p.fy = 0.0;
}
for (let i = 0; i < this.springs.length; i++)
{
let index1 = this.springs[i].index1;
let index2 = this.springs[i].index2;
let p1 = this.particles[index1];
let p2 = this.particles[index2];
// compute current length of spring
let dx = p1.x - p2.x;
let dy = p1.y - p2.y;
let dl = Math.sqrt(dx*dx + dy*dy)
// compute displacement: ||x1-x2|| - restLength
let C = dl - this.springs[i].length;
// determine gradient
let gradC_x = 0.0;
let gradC_y = 0.0;
if (dl > 0.001)
{
gradC_x = dx/dl;
gradC_y = dy/dl;
// compute spring force
let Fs_x = -this.stiffness * C * gradC_x;
let Fs_y = -this.stiffness * C * gradC_y;
// compute damping force
let dot = (p1.vx - p2.vx) * gradC_x + (p1.vy - p2.vy) * gradC_y; // (v1-v2)*gradC
let Fd_x = -this.damping * dot * gradC_x;
let Fd_y = -this.damping * dot * gradC_y;
// add forces to particles
if (this.isActive(index1))
{
p1.fx += Fs_x + Fd_x;
p1.fy += Fs_y + Fd_y;
}
if (this.isActive(index2))
{
p2.fx -= Fs_x + Fd_x;
p2.fy -= Fs_y + Fd_y;
}
}
}
}
// simulation step
simulationStep()
{
let dt = this.timeStepSize;
this.computeSpringForces();
// compute preview using symplectic Euler
for (let i = 0; i < this.particles.length; i++)
{
if (this.isActive(i))
{
let p = this.particles[i];
// integrate velocity considering gravitational acceleration and spring forces
p.vx = p.vx + dt * p.fx / this.mass;
p.vy = p.vy + dt * (p.fy / this.mass + this.gravity);
// integrate position
p.x = p.x + dt * p.vx;
p.y = p.y + dt * p.vy;
}
}
// update simulation time
this.time = this.time + dt;
}
}
class GUI
{
constructor()
{
this.canvas = document.getElementById("simCanvas");
this.c = this.canvas.getContext("2d");
this.requestID = -1;
this.timeSum = 0.0;
this.counter = 0;
this.pause = false;
this.origin = { x : this.canvas.width / 2, y : this.canvas.height/2-200};
this.zoom = 50;
this.particleRadius = 0.025;
this.selectedParticle = -1;
// register mouse event listeners (zoom/selection)
this.canvas.addEventListener("mousedown", this.mouseDown.bind(this), false);
this.canvas.addEventListener("mousemove", this.mouseMove.bind(this), false);
this.canvas.addEventListener("mouseup", this.mouseUp.bind(this), false);
this.canvas.addEventListener("wheel", this.wheel.bind(this), false);
}
// set simulation parameters from GUI and start mainLoop
restart()
{
window.cancelAnimationFrame(this.requestID);
let w = parseInt(document.getElementById('widthInput').value);
let h = parseInt(document.getElementById('heightInput').value);
let f = parseInt(document.getElementById('fixedParticlesInput').value);
delete this.sim;
this.sim = new Simulation(w, h, f);
this.timeSum = 0.0;
this.counter = 0;
this.sim.stiffness = parseFloat(document.getElementById('stiffnessInput').value);
this.sim.damping = parseFloat(document.getElementById('dampingInput').value);
this.sim.gravity = parseFloat(document.getElementById('gravityInput').value);
this.sim.timeStepSize = parseFloat(document.getElementById('timeStepSizeInput').value);
this.sim.mass = parseFloat(document.getElementById('massInput').value);
document.getElementById("numParticles").innerHTML = this.sim.particles.length;
document.getElementById("numSprings").innerHTML = this.sim.springs.length;
this.mainLoop();
}
drawCoordinateSystem()
{
// draw x-axis
this.c.strokeStyle = "#FF0000";
this.c.beginPath();
this.c.moveTo(this.origin.x, this.origin.y);
this.c.lineTo(this.origin.x+1*this.zoom, this.origin.y);
this.c.stroke();
// draw y-axis
this.c.strokeStyle = "#00FF00";
this.c.beginPath();
this.c.moveTo(this.origin.x, this.origin.y);
this.c.lineTo(this.origin.x, this.origin.y-1*this.zoom);
this.c.stroke();
}
draw()
{
this.c.clearRect(0, 0, this.canvas.width, this.canvas.height);
this.drawCoordinateSystem();
// draw springs as lines
this.c.strokeStyle = "#666666";
for (let i = 0; i < this.sim.springs.length; i++)
{
let index1 = this.sim.springs[i].index1;
let index2 = this.sim.springs[i].index2;
let p1 = this.sim.particles[index1];
let p2 = this.sim.particles[index2];
this.c.beginPath();
this.c.moveTo(this.origin.x + p1.x*this.zoom, this.origin.y - p1.y*this.zoom);
this.c.lineTo(this.origin.x + p2.x*this.zoom, this.origin.y - p2.y*this.zoom);
this.c.stroke();
}
// draw particles as circles
for (let i = 0; i < this.sim.particles.length; i++)
{
let p = this.sim.particles[i];
let r = this.particleRadius;
if (i == this.selectedParticle)
{
// draw selected particle in red with larger radius
this.c.fillStyle = "#FF0000";
r = 3*r;
}
else
this.c.fillStyle = "#0000FF";
let px = this.origin.x + p.x * this.zoom;
let py = this.origin.y - p.y * this.zoom;
this.c.beginPath();
this.c.arc(px, py, r * this.zoom, 0, Math.PI*2, true);
this.c.closePath();
this.c.fill();
}
}
mainLoop()
{
// perform multiple sim steps per render step
for (let i=0; i < 8; i++)
{
let t0 = performance.now();
this.sim.simulationStep();
let t1 = performance.now();
this.timeSum += t1 - t0;
this.counter += 1;
if (this.counter % 50 == 0)
{
this.timeSum /= this.counter;
document.getElementById("timePerStep").innerHTML = this.timeSum.toFixed(2);
this.timeSum = 0.0;
this.counter = 0;
}
document.getElementById("time").innerHTML = this.sim.time.toFixed(2);
}
this.draw();
if (!this.pause)
this.requestID = window.requestAnimationFrame(this.mainLoop.bind(this));
}
doPause()
{
this.pause = !this.pause;
if (!this.pause)
this.mainLoop();
}
mouseDown(event)
{
// left mouse button down
if (event.which == 1)
{
let mousePos = this.getMousePos(this.canvas, event);
for (let i = 0; i < this.sim.particles.length; i++)
{
let p = this.sim.particles[i];
let px = this.origin.x + p.x * this.zoom;
let py = this.origin.y - p.y * this.zoom;
let dx = px - mousePos.x
let dy = py - mousePos.y
let dist2 = Math.sqrt(dx * dx + dy * dy)
if (dist2 < 10)
{
this.selectedParticle = i;
break;
}
}
}
}
getMousePos(canvas, event)
{
let rect = canvas.getBoundingClientRect();
return {
x: event.clientX - rect.left,
y: event.clientY - rect.top
};
}
mouseMove(event)
{
if (this.selectedParticle != -1)
{
let mousePos = this.getMousePos(this.canvas, event);
this.sim.particles[this.selectedParticle].x = (mousePos.x - this.origin.x) / this.zoom;
this.sim.particles[this.selectedParticle].y = -(mousePos.y - this.origin.y) / this.zoom;
}
}
mouseUp(event)
{
this.selectedParticle = -1;
}
wheel(event)
{
event.preventDefault();
this.zoom += event.deltaY * -0.05;
if (this.zoom < 1)
this.zoom = 1;
}
}
gui = new GUI();
gui.restart();
</script>
</body>
</html>