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sph_approximation_user_fct.html
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sph_approximation_user_fct.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>
<script src="https://cdn.plot.ly/plotly-2.5.1.min.js"></script>
<title>SPH Approximation</title>
</head>
<body>
<main>
<h1 style="text-align:center">SPH Approximation</h1>
<table style="align_center;border-radius: 20px;padding: 20px;margin:auto">
<col width="400">
<col width="400" height="400">
<tr>
<td>
<div id="plotOutput" style="width: 600px; height: 600px;border:2px solid #000000;border-radius: 0px;background-color:#EEEEEE"></div>
</td>
<td>
<div id="plotOutput2" style="width: 600px; height: 600px;border:2px solid #000000;border-radius: 0px;background-color:#EEEEEE"></div>
</td>
</tr>
<tr>
<td><table style="margin:20px">
<col width="200" style="padding-right:10px">
<col width="300">
<tr>
<td><label for="function">Function f(x,y)</label></td>
<td><input onchange="plot.reset()" id="function" value="Math.sqrt(x*x +y*y)" type="input"></td>
</tr>
<tr>
<td><label>Particle radius</label></td>
<td><span>0.025</span></td>
</tr>
<tr>
<td><label for="supportRadius">Support radius</label></td>
<td><input onchange="plot.reset()" id="supportRadius" value="0.1" type="input"></td>
</tr>
<tr>
<td><label for="randomOffset">Add random offset</label></td>
<td><input onchange="plot.reset()" id="randomOffset" type="checkbox"></td>
</tr>
<tr>
<td><label for="shepardFilter">Use Shepard filter</label></td>
<td><input onchange="plot.plotFunctions()" id="shepardFilter" type="checkbox"></td>
</tr>
</table></td>
</tr>
<tr><td style="padding-left:20px">
(Note you can use expressions like Math.sin or Math.sqrt in the function field.)
</td></tr>
<tr><td colspan="2">
<h2>SPH Approximation of different functions:</h2>
In this example a user-defined function is discretized using SPH with the cubic spline kernel [KBST19]. The left plot shows the particle sampling pattern while the right plot shows the exact functions and their SPH approximations. To compute the SPH approximations the function values are sampled along the red line in the left plot. That means that for each point on the line the neighbors are determined and the SPH approximation formula is applied:
$$\langle f(x,y) \rangle = \sum_j \frac{m_j}{\rho_j} f(x_j,y_j) W_{ij}$$
<h3>Shepard filter</h3>
The SPH approximation of a fluid quantity is error-prone if the particle neighborhood is only partially filled. To consider this in the computation the smoothing kernel can be corrected by a factor $s_i$:
$$\tilde {W}_{ij} = s_i W_{ij} = \frac{1}{\sum_k \frac{m_k}{\rho_k} W_{ik}} W_{ij}.$$
<h3>References</h3>
<ul>
<li>[KBST19] Dan Koschier, Jan Bender, Barbara Solenthaler, Matthias Teschner. Smoothed Particle Hydrodynamics for Physically-Based Simulation of Fluids and Solids. Eurographics Tutorial, 2019</li>
</ul>
</td></tr>
</table>
</main>
<script id="simulation_code" type="text/javascript">
class Particle
{
constructor (x, y)
{
this.x = x; // position
this.y = y;
this.density = 0; // density
}
}
class Simulation
{
constructor(width, height, sRadius, rand)
{
this.particles = [];
this.particleRadius = 0.025;
this.supportRadius = sRadius; // support radius
this.density0 = 1000.0; // rest density of water
this.diam = 2.0 * this.particleRadius;
this.mass = this.diam*this.diam*this.density0; // mass = area * rest density
// constants for kernel computation
this.kernel_k = 40.0 / (7.0 * (Math.PI*this.supportRadius*this.supportRadius));
this.kernel_l = 240.0 / (7.0 * (Math.PI*this.supportRadius*this.supportRadius));
this.kernel_0 = this.cubicKernel2D(0);
this.width = width;
this.height = height;
this.numParticles = 0;
this.init(rand);
}
// initialize scene: generate a block of particles
init(rand)
{
// create particles
let i;
let j;
let w = this.width;
let h = this.height;
// generate a block of fluid particles
for (i = 0; i < h; i++)
{
for (j = 0; j < w; j++)
{
this.particles.push(new Particle(
-w*0.5*this.diam + j*this.diam + this.particleRadius,
i*this.diam + this.particleRadius));
if (rand)
{
this.particles[this.particles.length-1].x -= 0.01 + Math.random()*0.02;
this.particles[this.particles.length-1].y -= 0.01 + Math.random()*0.02;
}
}
}
this.numParticles = this.particles.length;
// Compute densities
this.computeDensity();
}
// compute the norm of a vector (x,y)
norm(x, y)
{
return Math.sqrt(x*x + y*y);
}
// Cubic spline kernel 2D
cubicKernel2D(r)
{
let res = 0.0;
let q = r / this.supportRadius;
if (q <= 1.0)
{
let q2 = q*q;
let q3 = q2*q;
if (q <= 0.5)
res = this.kernel_k * (6.0*q3 - 6.0*q2 + 1.0);
else
res = this.kernel_k * (2.0*Math.pow(1.0 - q, 3));
}
return res;
}
// Gradient of cubic spline kernel 2D
cubicKernel2D_Gradient(rx, ry)
{
let res = [0,0];
let rl = this.norm(rx, ry);
let q = rl / this.supportRadius;
if (q <= 1.0)
{
if (rl > 1.0e-6)
{
let gradq_x = rx * (1.0 / (rl*this.supportRadius));
let gradq_y = ry * (1.0 / (rl*this.supportRadius));
if (q <= 0.5)
{
res[0] = this.kernel_l*q*(3.0*q - 2.0) * gradq_x;
res[1] = this.kernel_l*q*(3.0*q - 2.0) * gradq_y;
}
else
{
let factor = (1.0 - q) * (1.0 - q);
res[0] = this.kernel_l*(-factor) * gradq_x;
res[1] = this.kernel_l*(-factor) * gradq_y;
}
}
}
return res;
}
// Find all neighbors including the particle itself by a brute force search.
find_neighbors(x, y)
{
let neighbors = [];
for (let i = 0; i < this.numParticles; i++)
{
let p = this.particles[i];
let dist2 = (x-p.x)*(x-p.x) + (y-p.y)*(y-p.y);
let r2 = this.supportRadius * this.supportRadius;
if (dist2 - r2 < 1.0e-6)
neighbors.push(i);
}
return neighbors;
}
// compute the density of all particles using the SPH formulation
computeDensity()
{
for (let i = 0; i < this.numParticles; i++)
{
let p_i = this.particles[i];
let neighbors = this.find_neighbors(p_i.x, p_i.y);
p_i.density = 0.0;
let nl = neighbors.length;
for(let j=0; j < nl; j++)
{
let nj = neighbors[j];
let p_j = this.particles[nj];
// compute W (xi-xj)
let Wij = this.cubicKernel2D(this.norm(p_i.x - p_j.x, p_i.y - p_j.y));
p_i.density += this.mass * Wij;
}
}
}
}
class Plot
{
constructor()
{
this.fct = "0.5*(x+y)";
}
reset()
{
let sr = parseFloat(document.getElementById('supportRadius').value);
this.fct = document.getElementById('function').value;
let r = document.getElementById('randomOffset').checked;
delete this.sim;
this.sim = new Simulation(30, 30, sr, r);
this.plotParticles();
this.plotFunctions();
}
user_function(x, y)
{
let s = this.fct;
s.replaceAll("x", x);
s.replaceAll("y", y);
return eval(s);
}
plotFunctions()
{
let useShepard = document.getElementById('shepardFilter').checked;
let x = -0.75;
let y = 0.5;
let num_steps = 200;
let xValues = [];
let yValues_orig = [];
let yValues = [];
for (let i = 0; i <= num_steps; i++)
{
xValues.push(x);
yValues_orig.push(this.user_function(x,y));
let neighbors = this.sim.find_neighbors(x, y);
let val = 0.0;
// Compute Shepard filter factor
let s = 1.0;
if (useShepard)
{
s = 0.0;
for (let j = 0; j < neighbors.length; j++)
{
let p_j = this.sim.particles[neighbors[j]];
// compute W (xi-xj)
let Wij = this.sim.cubicKernel2D(this.sim.norm(x - p_j.x, y - p_j.y));
s += this.sim.mass / p_j.density * Wij;
}
if (Math.abs(s) > 1.0e-6)
s = 1.0/s;
else
s = 1.0;
}
for (let j = 0; j < neighbors.length; j++)
{
let p_j = this.sim.particles[neighbors[j]];
// compute W (xi-xj)
let Wij = this.sim.cubicKernel2D(this.sim.norm(x - p_j.x, y - p_j.y));
val += s * this.sim.mass / p_j.density * this.user_function(p_j.x, p_j.y) * Wij;
}
yValues.push(val);
x += 1.5 / num_steps;
y += 1.0 / num_steps;
}
var trace_orig = {
x: xValues,
y: yValues_orig,
name: "original"
};
var trace = {
x: xValues,
y: yValues,
name: "SPH"
};
var data = [trace_orig, trace];
var layout = {
title: 'Functions',
width: 600,
height: 600
};
Plotly.newPlot('plotOutput2', data, layout);
}
plotParticles()
{
this.x = [];
this.y = [];
this.shapes = [];
for (let i = 0; i < this.sim.numParticles; i++)
{
let p = this.sim.particles[i];
this.x.push(p.x);
this.y.push(p.y);
this.shapes.push({
type: 'circle',
xref: 'x',
yref: 'y',
x0: p.x-0.025,
y0: p.y-0.025,
x1: p.x+0.025,
y1: p.y+0.025,
fillcolor: 'rgba(0, 70, 200, 0.7)',
line: {
width: 1,
color: 'rgba(0, 30, 150, 0.7)'
}
});
}
var trace1 = {
x: this.x,
y: this.y,
mode: 'text',
type: 'scatter',
};
var trace2 = {
x: [-0.75, 0.75],
y: [0.5,1.5],
line: {
dash: 'solid',
color: 'rgba(200,0,0,1)',
width: 4
}
};
var layout = {
title: 'Particles',
width: 600,
height: 600,
xaxis: {
autorange: [0,2],
},
yaxis: {
autorange: [0,2]
},
showlegend: false,
shapes: this.shapes
};
var data = [trace2, trace1];
Plotly.newPlot('plotOutput', data, layout);
Plotly.react('plotOutput', [trace2, trace1], layout);
}
}
plot = new Plot();
plot.reset();
</script>
</body>
</html>