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sph_kernel.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 Kernel</title>
</head>
<body>
<main>
<h1 style="text-align:center">SPH Kernel</h1>
<table style="align_center;border-radius: 20px;padding: 20px;margin:auto">
<col width="1000">
<tr>
<td>
<div id="plotOutput" style="width: 1000px; 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="100">
<tr>
<td><label for="supportRadius">Support radius</label></td>
<td><input type="text" id="textInput" value="0.2" readonly></td>
</tr>
<tr>
<td></td>
<td><input onchange="document.getElementById('textInput').value=this.value;plot.reset()" id="supportRadius" value="0.2" type="range" min="0.01" max="1" step="0.01"></td>
</tr>
<tr>
<td><label for="kernelFct">Kernel function</label></td>
<td><select onchange="plot.reset()" id="kernelFct" size="1">
<option selected="selected">Cubic spline</option>
<option>Wendland C2</option>
<option>Quintic Spline</option>
</select>
</td>
</tr>
</table></td>
</tr>
<tr><td>
<h2>SPH kernel functions:</h2>
<p>In the following we investigate different SPH kernel functions and their derivatives. A kernel function $W(\mathbf{r}, h)$ with the smoothing length $h$ must have the following properties:</p>
<ul>
<li>Normalization condition: $\int_{\mathbb{R}^d} W(\mathbf x - \mathbf x^*, h) d \mathbf x^* = 1 $</li>
<li>Symmetry condition: $W(\mathbf x - \mathbf x^*, h) = W(\mathbf x^* - \mathbf x, h)$</li>
<li>Dirac-$\delta$ condition: $\lim_{h \rightarrow 0} W(\mathbf x - \mathbf x^*, h) = \delta(\mathbf x - \mathbf x^*)$</li>
<li>Non-negative condition: $W(\mathbf x - \mathbf x^*, h) \geq 0$</li>
<li>Compact support condition with support radius $r$: $W(\mathbf x - \mathbf x^*, h) = 0 \quad \text{if} \quad \|\mathbf x - \mathbf x^*\| > r$</li>
</ul>
<p>Furthermore, a kernel function should be at least twice continuously differentiable to enable a consistent discretization of 2nd-order partial
differential equations (PDEs).
The smoothing length controls defines how strongly a value at position $\mathbf x$ is influenced by the values in its neighborhood. Hence, a larger value $h$ yields a larger smoothing effect.</p>
<h3>Popular kernel functions</h3>
<h4>Cubic spline kernel [Mon92]</h4>
$$W(q) = \alpha_d \begin{cases}
\frac23 - q^2 + \frac12 q^3 & 0 \leq q < 1 \\
\frac16 (2-q)^3 & 1 \leq q < 2 \\
0 & q \geq 2,
\end{cases}$$
where $q = \frac{\|r\|}{h}$ the kernel normalization factors $\alpha_d$ for the respective
dimensions $d$ are $\alpha_1 = \frac1h$, $\alpha_2 = \frac{15}{7\pi h^2}$, and $\alpha_3 = \frac{3}{2\pi h^3}$.
<h4>Wendland C2 kernel for 1D [Wen95]</h4>
$$W(q) = \alpha_d \begin{cases}
\left (1-\frac{q}{2} \right )^3 \left (\frac32 q+1 \right ) & 0 \leq q \leq 2 \\
0 & q > 2,
\end{cases}$$
where the kernel normalization factor is $\alpha_1 = \frac{5}{8h}$.
<h4>Quintic spline kernel [LL10]</h4>
$$W(q) = \alpha_d \begin{cases}
(3-q)^5 - 6(2-q)^5 + 15 (1-q)^5 & 0 \leq q \leq 1 \\
(3-q)^5 - 6(2-q)^5 & 1 < q \leq 2 \\
(3-q)^5 & 2 < q \leq 3 \\
0 & q > 3,
\end{cases}$$
where the kernel normalization factors are $\alpha_1 = \frac{1}{120h}$, $\alpha_2 = \frac{7}{478 \pi h^2}$, and $\alpha_3 = \frac{1}{120 \pi h^3}$.
<h3>Derivatives</h3>
<p>The gradient of the kernel function is
$$\nabla W = \frac{\partial W}{\partial \mathbf r} = \frac{\partial W(q)}{\partial q} \cdot \frac{\partial q}{\partial \mathbf r} = \frac{\partial W(q)}{\partial q} \frac{\mathbf r} {h \| \mathbf r \|}$$
and the Laplacian
$$\nabla^2 W = \frac{\partial^2 W(q)}{\partial q^2} \frac{1}{h^2} + \frac{\partial W(q)}{\partial q} \nabla^2 q$$
</p>
<h3>References</h3>
<ul>
<li>[Mon92] J. Monaghan. Smoothed Particle Hydrodynamics. Annual Review of Astronomy and Astrophysics, 1992, 30, 543-574</li>
<li>[Wen95] H. Wendland. Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree. Advances in computational Mathematics 4.1, 389-396, 1995</li>
<li>[LL10] M. Liu, G. Liu. Smoothed Particle Hydrodynamics (SPH): an Overview and Recent Developments. Archives of Computational Methods in Engineering, 2010, 17, 25-76</li>
</ul>
</td></tr>
</table>
</main>
<script id="simulation_code" type="text/javascript">
class Plot
{
constructor()
{
this.supportRadius = 0.1; // support radius
this.reset();
}
reset()
{
this.supportRadius = parseFloat(document.getElementById('supportRadius').value);
this.kernelFct = document.getElementById('kernelFct').value;
this.plot();
}
// Cubic spline kernel 1D
// W(r) = alpha * (2/3 - q^2 + 1/2 q^3) if 0 <= q < 1
// = alpha * 1/6 * (2-q)^3 if 1 <= q < 2
// = 0 otherwise
// q = |r|/h
cubicKernel_1D(r)
{
let h = 0.5 * this.supportRadius;
let q = Math.abs(r) / h;
let alpha = 1.0/h; // normalization factor for 1D
if ((q >= 0.0) && (q < 1.0))
return alpha * (2.0/3.0 - q*q + 0.5 * Math.pow(q,3));
else if ((q >= 1.0) && (q < 2.0))
return alpha/6.0 * Math.pow(2.0 - q, 3);
else
return 0.0;
}
// Gradient of cubic spline kernel 1D
// ∂W/∂q = alpha * (-2 q + 1.5 q^2) if 0 <= q < 1
// = -alpha * 0.5 * (2-q)^2 if 1 <= q < 2
// = 0 otherwise
// q = |r|/h
// grad W(r) = ∂W/∂q * grad q
// grad q = r / (|r| h)
gradCubicKernel_1D(r)
{
let h = 0.5 * this.supportRadius;
let q = Math.abs(r) / h;
let gradq = r / (Math.abs(r)* h);
let dW_dq = 0.0;
let alpha = 1.0/h;
if ((q >= 0.0) && (q < 1.0))
dW_dq = alpha * (-2.0*q + 1.5 * q*q);
else if ((q >= 1.0) && (q < 2.0))
dW_dq = -alpha/2.0 * Math.pow(2.0 - q, 2);
return dW_dq * gradq;
}
// Laplacian of cubic spline kernel 1D
// ∂^2 W/∂^2 q = alpha * (-2 + 3 q) if 0 <= q < 1
// = alpha * (2-q) if 1 <= q < 2
// = 0 otherwise
// q = |r|/h
// in 1D: ∂^2 W/∂^2 r = ∂^2 W/∂^2 q * 1/h^2
laplaceCubicKernel_1D(r)
{
let h = 0.5 * this.supportRadius;
let q = Math.abs(r) / h;
let d2W_d2q = 0.0;
let alpha = 1.0/h;
if ((q >= 0.0) && (q < 1.0))
d2W_d2q = alpha * (-2.0 + 3.0*q);
else if ((q >= 1.0) && (q < 2.0))
d2W_d2q = alpha * (2.0-q);
return d2W_d2q * 1.0/(h*h);
}
// Wendland C2 kernel 1D
// W(r) = alpha * (1 - 0.5 q)^3 (1.5 q + 1) if 0 <= q <= 2
// = 0 otherwise
// q = |r|/h
WendlandC2_1D(r)
{
let h = 0.5 * this.supportRadius;
let q = Math.abs(r) / h;
let alpha = 5.0/(8.0*h);
if ((q >= 0.0) && (q <= 2.0))
//return alpha * Math.pow(1.0-0.5*q, 3)*(1.5*q + 1.0);
return alpha * (1.0 - 1.5*q*q + q*q*q - 3.0/16.0*q*q*q*q);
else
return 0.0;
}
// Gradient of Wendland C2 kernel 1D
// ∂W/∂q = alpha * (-3q + 3q^2 - 0.75q^3) if 0 <= q <= 2
// = 0 otherwise
// q = |r|/h
// grad W(r) = ∂W/∂q * grad q
// grad q = r / (|r| h)
gradWendlandC2_1D(r)
{
let h = 0.5 * this.supportRadius;
let q = Math.abs(r) / h;
let alpha = 5.0/(8.0*h);
let gradq = r / (Math.abs(r)* h);
let dW_dq = 0.0;
if ((q >= 0.0) && (q <= 2.0))
//dW_dq = alpha * (-1.5 * Math.pow(1.0-0.5*q, 2)*(1.5*q + 1.0) + Math.pow(1.0-0.5*q, 3) * 1.5);
dW_dq = alpha * (-3.0*q + 3.0 * q*q - 0.75 * q*q*q);
return dW_dq * gradq;
}
// Laplacian of Wendland C2 kernel 1D
// ∂^2 W/∂^2 q = alpha * (-3 + 6q - 2.25 q^2) if 0 <= q <= 2
// = 0 otherwise
// q = |r|/h
// in 1D: ∂^2 W/∂^2 r = ∂^2 W/∂^2 q * 1/h^2
laplaceWendlandC2_1D(r)
{
let h = 0.5 * this.supportRadius;
let q = Math.abs(r) / h;
let d2W_d2q = 0.0;
let alpha = 1.0/h;
if ((q >= 0.0) && (q <= 2.0))
d2W_d2q = alpha * (-3.0 + 6.0*q - 9.0/4.0*q*q);
return d2W_d2q * 1.0/(h*h);
}
// Quintic Spline kernel 1D
// W(r) = alpha * ((3-q)^5 - (2-q)^5 + 15(1-q)^5) if 0 <= q <= 1
// = alpha * ((3-q)^5 - (2-q)^5) if 1 < q <= 2
// = alpha * (3-q)^5 if 2 < q <= 3
// = 0 otherwise
// q = |r|/h
QuinticSpline_1D(r)
{
let h = 0.5 * this.supportRadius;
let q = Math.abs(r) / h;
let alpha = 1.0/(120.0*h);
if ((q >= 0.0) && (q <= 1.0))
return alpha * (Math.pow(3.0-q, 5) - 6.0*Math.pow(2.0-q, 5) + 15.0*Math.pow(1.0-q, 5));
else if ((q > 1.0) && (q <= 2.0))
return alpha * (Math.pow(3.0-q, 5) - 6.0*Math.pow(2.0-q, 5));
else if ((q > 2.0) && (q <= 3.0))
return alpha * (Math.pow(3.0-q, 5));
else
return 0.0;
}
// Gradient of Quintic Spline kernel 1D
// ∂W/∂q = alpha * (-5(3-q)^4 + 30(2-q)^4 - 75(1-q)^4) if 0 <= q <= 1
// = alpha * (-5(3-q)^4 + 30(2-q)^4) if 1 < q <= 2
// = alpha * (-5(3-q)^4) if 2 < q <= 3
// = 0 otherwise
// q = |r|/h
// grad W(r) = ∂W/∂q * grad q
// grad q = r / (|r| h)
gradQuinticSpline_1D(r)
{
let h = 0.5 * this.supportRadius;
let q = Math.abs(r) / h;
let alpha = 1.0/(120.0*h);
let gradq = r / (Math.abs(r)* h);
let dW_dq = 0.0;
if ((q >= 0.0) && (q <= 1.0))
dW_dq = alpha * (-5.0*Math.pow(3.0-q, 4) + 30.0*Math.pow(2.0-q, 4) - 75.0*Math.pow(1.0-q, 4));
else if ((q > 1.0) && (q <= 2.0))
dW_dq = alpha * (-5.0*Math.pow(3.0-q, 4) + 30.0*Math.pow(2.0-q, 4));
else if ((q > 2.0) && (q <= 3.0))
dW_dq = alpha * (-5.0*Math.pow(3.0-q, 4));
return dW_dq * gradq;
}
// Laplacian of Quintic Spline kernel 1D
// ∂^2 W/∂^2 q = alpha * (20(3-q)^3 - 120(2-q)^3 + 300(1-q)^3) if 0 <= q <= 1
// = alpha * (20(3-q)^3 - 120(2-q)^3) if 1 < q <= 2
// = alpha * (20(3-q)^3) if 2 < q <= 3
// = 0 otherwise
// q = |r|/h
// in 1D: ∂^2 W/∂^2 r = ∂^2 W/∂^2 q * 1/h^2
laplaceQuinticSpline_1D(r)
{
let h = 0.5 * this.supportRadius;
let q = Math.abs(r) / h;
let d2W_d2q = 0.0;
let alpha = 1.0/(120.0*h);
if ((q >= 0.0) && (q <= 1.0))
d2W_d2q = alpha * (20.0*Math.pow(3.0-q, 3) - 120.0*Math.pow(2.0-q, 3) + 300.0*Math.pow(1.0-q, 3));
else if ((q > 1.0) && (q <= 2.0))
d2W_d2q = alpha * (20.0*Math.pow(3.0-q, 3) - 120.0*Math.pow(2.0-q, 3));
else if ((q > 2.0) && (q <= 3.0))
d2W_d2q = alpha * (20.0*Math.pow(3.0-q, 3));
return d2W_d2q * 1.0/(h*h);
}
plot()
{
let x = -1.1 * this.supportRadius;
let num_steps = 1000;
let stepSize = 2.2 * this.supportRadius / (num_steps-1);
let xValues = [];
let yValues = [];
let yValues_grad = [];
let yValues_lap = [];
// compute function values for num_steps
for (let i = 0; i <= num_steps; i++)
{
xValues.push(x);
if (this.kernelFct == "Wendland C2")
{
yValues.push(this.WendlandC2_1D(x));
yValues_grad.push(0.1*this.gradWendlandC2_1D(x));
yValues_lap.push(0.001 * this.laplaceWendlandC2_1D(x));
}
else if (this.kernelFct == "Quintic Spline")
{
yValues.push(this.QuinticSpline_1D(x));
yValues_grad.push(0.1*this.gradQuinticSpline_1D(x));
yValues_lap.push(0.001 * this.laplaceQuinticSpline_1D(x));
}
else
{
yValues.push(this.cubicKernel_1D(x));
yValues_grad.push(0.1*this.gradCubicKernel_1D(x));
yValues_lap.push(0.001 * this.laplaceCubicKernel_1D(x));
}
x += stepSize;
}
let trace_cubic = {
x: xValues,
y: yValues,
name: "kernel"
};
let trace_cubic_grad = {
x: xValues,
y: yValues_grad,
name: "gradient (* 0.1)"
};
let trace_cubic_lap = {
x: xValues,
y: yValues_lap,
name: "Laplacian (* 0.001)"
};
let data = [trace_cubic, trace_cubic_grad, trace_cubic_lap];
let title = this.kernelFct;
let layout = {
title: title,
width: 1000,
height: 600,
xaxis: {
range: [-0.5,0.5],
},
yaxis: {
range: [-8.6,8.6],
}
};
// plot values
Plotly.newPlot('plotOutput', data, layout);
}
}
plot = new Plot();
plot.reset();
</script>
</body>
</html>