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<article id="content">
<header>
<h1 class="title"><code>acceleration</code> module</h1>
</header>
<section id="section-intro">
<p>Defines the possible routines for computing the gravitational forces in the
simulation.</p>
<p>All the methods in these file require a position (n, 3) vector, a mass (n, )
vector and an optional softening scale float.</p>
<details class="source">
<summary>Source code</summary>
<pre><code class="python">&#34;&#34;&#34;Defines the possible routines for computing the gravitational forces in the
simulation.


All the methods in these file require a position (n, 3) vector, a mass (n, )
vector and an optional softening scale float.&#34;&#34;&#34;

import ctypes
import numpy.ctypeslib as ctl
import numpy as np
from numba import jit


def bruteForce(r_vec, mass, soft=0.):
    &#34;&#34;&#34;Calculates the acceleration generated by a set of masses on themselves.
       Complexity O(n*m) where n is the total number of masses and m is the
       number of massive particles.

    Parameters:
        r_vec (array): list of particles positions.
            Shape (n, 3) where n is the number of particles
        mass (array): list of particles masses.
            Shape (n,)
        soft (float): characteristic plummer softening length scale
    Returns:
        forces (array): list of forces acting on each particle.
            Shape (n, 3)
    &#34;&#34;&#34;
    # Only calculate forces from massive particles
    mask = mass!=0
    massMassive = mass[mask]
    rMassive_vec = r_vec[mask]
    #  x m x 1 matrix (m = number of massive particles) for broadcasting
    mass_mat = massMassive.reshape(1, -1, 1)
    # Calculate displacements
    # r_ten is the direction of the pairwise displacements. Shape (n, m, 3)
    # r_mat is the absolute distance of the pairwise displacements. (n, m, 1)
    r_ten = rMassive_vec.reshape(1, -1, 3) - r_vec.reshape(-1, 1, 3)
    r_mat = np.linalg.norm(r_ten, axis=-1, keepdims=True)
    # Avoid division by zeros
    # $a = M / (r + \epsilon)^2$, where $\epsilon$ is the softening scale
    # r_ten/r_mat gives the direction unit vector
    accel = np.divide(r_ten * mass_mat/(r_mat+soft)**2, r_mat, 
        where=r_ten.astype(bool), out=r_ten) # Reuse memory from r_ten
    return accel.sum(axis=1) # Add all forces on each particle

@jit(nopython=True) # Numba annotation
def bruteForceNumba(r_vec, mass, soft=0.):
    &#34;&#34;&#34;Calculates the acceleration generated by a set of masses on themselves.
    It is done in the same way as in bruteForce, but this
    method is ran through Numba&#34;&#34;&#34;
    mask = mass!=0
    massMassive = mass[mask]
    rMassive_vec = r_vec[mask]
    mass_mat = massMassive.reshape(1, -1, 1)
    r_ten = rMassive_vec.reshape(1, -1, 3) - r_vec.reshape(-1, 1, 3)
    # Avoid np.linalg.norm to allow Numba optimizations
    r_mat = np.sqrt(r_ten[:,:,0:1]**2 + r_ten[:,:,1:2]**2 + r_ten[:,:,2:3]**2)
    r_mat = np.where(r_mat == 0, np.ones_like(r_mat), r_mat)
    accel = r_ten/r_mat * mass_mat/(r_mat+soft)**2
    return accel.sum(axis=1) # Add all forces in each particle

@jit(nopython=True) # Numba annotation
def bruteForceNumbaOptimized(r_vec, mass, soft=0.):
    &#34;&#34;&#34;Calculates the acceleration generated by a set of masses on themselves.
    This is optimized for high performance with Numba. All massive particles
    must appear first.&#34;&#34;&#34;
    accel = np.zeros_like(r_vec) 
    # Use superposition to add all the contributions
    n = r_vec.shape[0] # Number of particles
    delta = np.zeros((3,)) # Only allocate this once
    for i in range(n):
        # Only consider pairs with at least one massive particle i
        if mass[i] == 0: break
        for j in range(i+1, n):
            # Explicitely separate components for high performance
            # i.e. do not do delta = r_vec[j] - r_vec[i]
            # (The effect of this is VERY relevant (x10) and has to do with
            # memory reallocation) Numba will vectorize the loops.
            for k in range(3): delta[k] = r_vec[j,k] - r_vec[i,k]
            r = np.sqrt(delta[0]*delta[0] + delta[1]*delta[1] + delta[2]*delta[2])
            tripler = (r+soft)**2 * r
            
            # Compute acceleration on first particle
            mr3inv = mass[i]/(tripler)
            # Again, do NOT do accel[j] -= mr3inv * delta
            for k in range(3): accel[j,k] -= mr3inv * delta[k]
            
            # Compute acceleration on second particle
            # For pairs with one massless particle, no reaction force
            if mass[j] == 0: break
            # Otherwise, opposite direction (+)
            mr3inv = mass[j]/(tripler)
            for k in range(3): accel[i,k] += mr3inv * delta[k]
    return accel

# C++ interface, load library
ACCLIB = None
def loadCPPLib():
    &#34;&#34;&#34;Loads the C++ shared library to the global variable ACCLIB. Must be
    called before using the library.&#34;&#34;&#34;
    global ACCLIB
    ACCLIB = ctypes.CDLL(&#39;cpp/acclib.so&#39;)
    # Define appropiate types for library functions
    doublepp = np.ctypeslib.ndpointer(dtype=np.uintp) # double**
    doublep = ctl.ndpointer(np.float64, flags=&#39;aligned, c_contiguous&#39;)#double*
    # Check cpp/acclib.cpp for function signatures
    ACCLIB.bruteForceCPP.argtypes = [doublepp, doublep, 
        ctypes.c_int, ctypes.c_double]
    ACCLIB.barnesHutCPP.argtypes = [doublepp, doublep, 
        ctypes.c_int, ctypes.c_double, ctypes.c_double, 
        ctypes.c_double, ctypes.c_double, ctypes.c_double]

def bruteForceCPP(r_vec, m_vec, soft=0.):
    &#34;&#34;&#34;Calculates the acceleration generated by a set of masses on themselves.
    This is ran in a shared C++ library through Brute Force (pairwise sums)
    Massive particles must appear first.&#34;&#34;&#34;
    # Convert array to data required by C++ library
    if ACCLIB is None: loadCPPLib() # Singleton pattern
    # Change type to be appropiate for calling library
    r_vec_c = (r_vec.ctypes.data + np.arange(r_vec.shape[0]) 
        * r_vec.strides[0]).astype(np.uintp)
    # Set return type as double*
    ACCLIB.bruteForceCPP.restype = np.ctypeslib.ndpointer(dtype=np.float64, 
        shape=(r_vec.shape[0]*3,))
    # Call the C++ function: double* bruteForceCPP
    accel =  ACCLIB.bruteForceCPP(r_vec_c, m_vec, r_vec.shape[0], soft)
    # Change shape to get the expected Numpy array (n, 3)
    accel.shape = (-1, 3)
    return accel

def barnesHutCPP(r_vec, m_vec, soft=0.):
    &#34;&#34;&#34;Calculates the acceleration generated by a set of masses on themselves.
    This is ran in a shared C++ library using a BarnesHut tree&#34;&#34;&#34;
    # Convert array to data required by C++ library
    if ACCLIB is None: loadCPPLib() # Singleton pattern
    # Change type to be appropiate for calling library
    r_vec_c = (r_vec.ctypes.data + np.arange(r_vec.shape[0]) 
        * r_vec.strides[0]).astype(np.uintp)
    # Set return type as double*
    ACCLIB.barnesHutCPP.restype = np.ctypeslib.ndpointer(dtype=np.float64, 
        shape=(r_vec.shape[0]*3,))
    # Explicitely pass the corner and size of the box for the top node
    px, py, pz = np.min(r_vec, axis=0)
    size = np.max(np.max(r_vec, axis=0) - np.min(r_vec, axis=0))
    # Call the C++ function: double* barnesHutCPP
    accel =  ACCLIB.barnesHutCPP(r_vec_c, m_vec, r_vec.shape[0], 
        size, px, py, pz, soft)
    # Change shape to get the expected Numpy array (n, 3)
    accel.shape = (-1, 3)
    return accel</code></pre>
</details>
</section>
<section>
</section>
<section>
</section>
<section>
<h2 class="section-title" id="header-functions">Functions</h2>
<dl>
<dt id="acceleration.barnesHutCPP"><code class="name flex">
<span>def <span class="ident">barnesHutCPP</span></span>(<span>r_vec, m_vec, soft=0.0)</span>
</code></dt>
<dd>
<section class="desc"><p>Calculates the acceleration generated by a set of masses on themselves.
This is ran in a shared C++ library using a BarnesHut tree</p></section>
<details class="source">
<summary>Source code</summary>
<pre><code class="python">def barnesHutCPP(r_vec, m_vec, soft=0.):
    &#34;&#34;&#34;Calculates the acceleration generated by a set of masses on themselves.
    This is ran in a shared C++ library using a BarnesHut tree&#34;&#34;&#34;
    # Convert array to data required by C++ library
    if ACCLIB is None: loadCPPLib() # Singleton pattern
    # Change type to be appropiate for calling library
    r_vec_c = (r_vec.ctypes.data + np.arange(r_vec.shape[0]) 
        * r_vec.strides[0]).astype(np.uintp)
    # Set return type as double*
    ACCLIB.barnesHutCPP.restype = np.ctypeslib.ndpointer(dtype=np.float64, 
        shape=(r_vec.shape[0]*3,))
    # Explicitely pass the corner and size of the box for the top node
    px, py, pz = np.min(r_vec, axis=0)
    size = np.max(np.max(r_vec, axis=0) - np.min(r_vec, axis=0))
    # Call the C++ function: double* barnesHutCPP
    accel =  ACCLIB.barnesHutCPP(r_vec_c, m_vec, r_vec.shape[0], 
        size, px, py, pz, soft)
    # Change shape to get the expected Numpy array (n, 3)
    accel.shape = (-1, 3)
    return accel</code></pre>
</details>
</dd>
<dt id="acceleration.bruteForce"><code class="name flex">
<span>def <span class="ident">bruteForce</span></span>(<span>r_vec, mass, soft=0.0)</span>
</code></dt>
<dd>
<section class="desc"><p>Calculates the acceleration generated by a set of masses on themselves.
Complexity O(n*m) where n is the total number of masses and m is the
number of massive particles.</p>
<h2 id="parameters">Parameters</h2>
<dl>
<dt><strong><code>r_vec</code></strong> :&ensp;<code>array</code></dt>
<dd>list of particles positions.
Shape (n, 3) where n is the number of particles</dd>
<dt><strong><code>mass</code></strong> :&ensp;<code>array</code></dt>
<dd>list of particles masses.
Shape (n,)</dd>
<dt><strong><code>soft</code></strong> :&ensp;<code>float</code></dt>
<dd>characteristic plummer softening length scale</dd>
</dl>
<h2 id="returns">Returns</h2>
<dl>
<dt><strong><code>forces</code></strong> :&ensp;<code>array</code></dt>
<dd>list of forces acting on each particle.
Shape (n, 3)</dd>
</dl></section>
<details class="source">
<summary>Source code</summary>
<pre><code class="python">def bruteForce(r_vec, mass, soft=0.):
    &#34;&#34;&#34;Calculates the acceleration generated by a set of masses on themselves.
       Complexity O(n*m) where n is the total number of masses and m is the
       number of massive particles.

    Parameters:
        r_vec (array): list of particles positions.
            Shape (n, 3) where n is the number of particles
        mass (array): list of particles masses.
            Shape (n,)
        soft (float): characteristic plummer softening length scale
    Returns:
        forces (array): list of forces acting on each particle.
            Shape (n, 3)
    &#34;&#34;&#34;
    # Only calculate forces from massive particles
    mask = mass!=0
    massMassive = mass[mask]
    rMassive_vec = r_vec[mask]
    #  x m x 1 matrix (m = number of massive particles) for broadcasting
    mass_mat = massMassive.reshape(1, -1, 1)
    # Calculate displacements
    # r_ten is the direction of the pairwise displacements. Shape (n, m, 3)
    # r_mat is the absolute distance of the pairwise displacements. (n, m, 1)
    r_ten = rMassive_vec.reshape(1, -1, 3) - r_vec.reshape(-1, 1, 3)
    r_mat = np.linalg.norm(r_ten, axis=-1, keepdims=True)
    # Avoid division by zeros
    # $a = M / (r + \epsilon)^2$, where $\epsilon$ is the softening scale
    # r_ten/r_mat gives the direction unit vector
    accel = np.divide(r_ten * mass_mat/(r_mat+soft)**2, r_mat, 
        where=r_ten.astype(bool), out=r_ten) # Reuse memory from r_ten
    return accel.sum(axis=1) # Add all forces on each particle</code></pre>
</details>
</dd>
<dt id="acceleration.bruteForceCPP"><code class="name flex">
<span>def <span class="ident">bruteForceCPP</span></span>(<span>r_vec, m_vec, soft=0.0)</span>
</code></dt>
<dd>
<section class="desc"><p>Calculates the acceleration generated by a set of masses on themselves.
This is ran in a shared C++ library through Brute Force (pairwise sums)
Massive particles must appear first.</p></section>
<details class="source">
<summary>Source code</summary>
<pre><code class="python">def bruteForceCPP(r_vec, m_vec, soft=0.):
    &#34;&#34;&#34;Calculates the acceleration generated by a set of masses on themselves.
    This is ran in a shared C++ library through Brute Force (pairwise sums)
    Massive particles must appear first.&#34;&#34;&#34;
    # Convert array to data required by C++ library
    if ACCLIB is None: loadCPPLib() # Singleton pattern
    # Change type to be appropiate for calling library
    r_vec_c = (r_vec.ctypes.data + np.arange(r_vec.shape[0]) 
        * r_vec.strides[0]).astype(np.uintp)
    # Set return type as double*
    ACCLIB.bruteForceCPP.restype = np.ctypeslib.ndpointer(dtype=np.float64, 
        shape=(r_vec.shape[0]*3,))
    # Call the C++ function: double* bruteForceCPP
    accel =  ACCLIB.bruteForceCPP(r_vec_c, m_vec, r_vec.shape[0], soft)
    # Change shape to get the expected Numpy array (n, 3)
    accel.shape = (-1, 3)
    return accel</code></pre>
</details>
</dd>
<dt id="acceleration.bruteForceNumba"><code class="name flex">
<span>def <span class="ident">bruteForceNumba</span></span>(<span>r_vec, mass, soft=0.0)</span>
</code></dt>
<dd>
<section class="desc"><p>Calculates the acceleration generated by a set of masses on themselves.
It is done in the same way as in bruteForce, but this
method is ran through Numba</p></section>
<details class="source">
<summary>Source code</summary>
<pre><code class="python">@jit(nopython=True) # Numba annotation
def bruteForceNumba(r_vec, mass, soft=0.):
    &#34;&#34;&#34;Calculates the acceleration generated by a set of masses on themselves.
    It is done in the same way as in bruteForce, but this
    method is ran through Numba&#34;&#34;&#34;
    mask = mass!=0
    massMassive = mass[mask]
    rMassive_vec = r_vec[mask]
    mass_mat = massMassive.reshape(1, -1, 1)
    r_ten = rMassive_vec.reshape(1, -1, 3) - r_vec.reshape(-1, 1, 3)
    # Avoid np.linalg.norm to allow Numba optimizations
    r_mat = np.sqrt(r_ten[:,:,0:1]**2 + r_ten[:,:,1:2]**2 + r_ten[:,:,2:3]**2)
    r_mat = np.where(r_mat == 0, np.ones_like(r_mat), r_mat)
    accel = r_ten/r_mat * mass_mat/(r_mat+soft)**2
    return accel.sum(axis=1) # Add all forces in each particle</code></pre>
</details>
</dd>
<dt id="acceleration.bruteForceNumbaOptimized"><code class="name flex">
<span>def <span class="ident">bruteForceNumbaOptimized</span></span>(<span>r_vec, mass, soft=0.0)</span>
</code></dt>
<dd>
<section class="desc"><p>Calculates the acceleration generated by a set of masses on themselves.
This is optimized for high performance with Numba. All massive particles
must appear first.</p></section>
<details class="source">
<summary>Source code</summary>
<pre><code class="python">@jit(nopython=True) # Numba annotation
def bruteForceNumbaOptimized(r_vec, mass, soft=0.):
    &#34;&#34;&#34;Calculates the acceleration generated by a set of masses on themselves.
    This is optimized for high performance with Numba. All massive particles
    must appear first.&#34;&#34;&#34;
    accel = np.zeros_like(r_vec) 
    # Use superposition to add all the contributions
    n = r_vec.shape[0] # Number of particles
    delta = np.zeros((3,)) # Only allocate this once
    for i in range(n):
        # Only consider pairs with at least one massive particle i
        if mass[i] == 0: break
        for j in range(i+1, n):
            # Explicitely separate components for high performance
            # i.e. do not do delta = r_vec[j] - r_vec[i]
            # (The effect of this is VERY relevant (x10) and has to do with
            # memory reallocation) Numba will vectorize the loops.
            for k in range(3): delta[k] = r_vec[j,k] - r_vec[i,k]
            r = np.sqrt(delta[0]*delta[0] + delta[1]*delta[1] + delta[2]*delta[2])
            tripler = (r+soft)**2 * r
            
            # Compute acceleration on first particle
            mr3inv = mass[i]/(tripler)
            # Again, do NOT do accel[j] -= mr3inv * delta
            for k in range(3): accel[j,k] -= mr3inv * delta[k]
            
            # Compute acceleration on second particle
            # For pairs with one massless particle, no reaction force
            if mass[j] == 0: break
            # Otherwise, opposite direction (+)
            mr3inv = mass[j]/(tripler)
            for k in range(3): accel[i,k] += mr3inv * delta[k]
    return accel</code></pre>
</details>
</dd>
<dt id="acceleration.loadCPPLib"><code class="name flex">
<span>def <span class="ident">loadCPPLib</span></span>(<span>)</span>
</code></dt>
<dd>
<section class="desc"><p>Loads the C++ shared library to the global variable ACCLIB. Must be
called before using the library.</p></section>
<details class="source">
<summary>Source code</summary>
<pre><code class="python">def loadCPPLib():
    &#34;&#34;&#34;Loads the C++ shared library to the global variable ACCLIB. Must be
    called before using the library.&#34;&#34;&#34;
    global ACCLIB
    ACCLIB = ctypes.CDLL(&#39;cpp/acclib.so&#39;)
    # Define appropiate types for library functions
    doublepp = np.ctypeslib.ndpointer(dtype=np.uintp) # double**
    doublep = ctl.ndpointer(np.float64, flags=&#39;aligned, c_contiguous&#39;)#double*
    # Check cpp/acclib.cpp for function signatures
    ACCLIB.bruteForceCPP.argtypes = [doublepp, doublep, 
        ctypes.c_int, ctypes.c_double]
    ACCLIB.barnesHutCPP.argtypes = [doublepp, doublep, 
        ctypes.c_int, ctypes.c_double, ctypes.c_double, 
        ctypes.c_double, ctypes.c_double, ctypes.c_double]</code></pre>
</details>
</dd>
</dl>
</section>
<section>
</section>
</article>
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<h1>Index</h1>
<div class="toc">
<ul></ul>
</div>
<ul id="index">
<li><h3><a href="#header-functions">Functions</a></h3>
<ul class="">
<li><code><a title="acceleration.barnesHutCPP" href="#acceleration.barnesHutCPP">barnesHutCPP</a></code></li>
<li><code><a title="acceleration.bruteForce" href="#acceleration.bruteForce">bruteForce</a></code></li>
<li><code><a title="acceleration.bruteForceCPP" href="#acceleration.bruteForceCPP">bruteForceCPP</a></code></li>
<li><code><a title="acceleration.bruteForceNumba" href="#acceleration.bruteForceNumba">bruteForceNumba</a></code></li>
<li><code><a title="acceleration.bruteForceNumbaOptimized" href="#acceleration.bruteForceNumbaOptimized">bruteForceNumbaOptimized</a></code></li>
<li><code><a title="acceleration.loadCPPLib" href="#acceleration.loadCPPLib">loadCPPLib</a></code></li>
</ul>
</li>
</ul>
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