SimClass/BlogEulerCromerTimeStep.html at master · blackencino/SimClass · GitHub

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<!DOCTYPE html>
<html>
<head>
  <title>Encino Sim Class : The Euler-Cromer Time Step</title>

  <style type="text/css" title="currentStyle">
    @import "css/SimClass.css";
  </style>

<script src='http://processingjs.org/js/processing.min.js' type='text/javascript'/></script>
<!Install the MathJax stuff so we can show LaTeX>
<script type="text/javascript"
  src="https://c328740.ssl.cf1.rackcdn.com/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML">
</script>
</head>
<body>
<h2>Forward Euler Problems</h2>
In the previous class, we observed that a Forward Euler time step quickly
produced an unusable solution to our simple spring system. This illustration
from the <a href="http://en.wikipedia.org/wiki/Euler_method">
Wikipedia entry on Forward Euler</a> illustrates how the method works,
and also why its solutions drift away from the correct result.
<img src="http://upload.wikimedia.org/wikipedia/commons/thumb/a/ae/Euler_method.png/613px-Euler_method.png">
<p>
In this illustration, you can see that at each point, the slope of the function
is used to construct a line segment that estimates the position of the next point.
The error in each estimate is accumulated with each subsequent estimate, and the solution
veers off course.  In order to study these systems better, we first need some debug tools.

<h2>Adding Debug Tools to our Spring Simulation</h2>
The Forward Euler Time Step that we added to the Spring Simulation in the
previous class produced an oscillatory motion, but did so in a way that
produced an unstable result, with the mass oscillating wildly off screen
in a short period of time.
<p>
The instability shown by our simple spring simulation is a constant problem
for most simulations. It's easy to see the problem in this simple example
(which is why we're using it!), but the problem affects large-scale fluid
and cloth sims, rigid body sims, and so on.  Strangely, despite the general
knowledge that Forward Euler is not a good time integration tool, it is still
used in many commercial tools, including Maya and Houdini.
<p>
In order to analyze our improvements, we first need a measuring stick. To quote
my good friend Adam Shand, "You can't improve what you can't measure.". We
chose the simple spring because it did have an exact answer. Therefore, we'll
add some code to our drawing routine that draws the right answer just above
the simulated spring. Here's that code - note that this uses the expression
for the explicit position of the spring that we established in the previous
class. We'll also add some text to the top of the drawn image to remind
us which time integration we're using.
<pre><code>
void DrawState()
{
    // Compute end of arm.
    float SpringEndX = PixelsPerMeter * State[StatePositionX];

    // Compute the CORRECT position.
    float sqrtKoverM = sqrt( Stiffness / BobMass );
    float x0 = InitState[StatePositionX];
    float v0 = InitState[StateVelocityX];
    float t = State[StateCurrentTime];
    float CorrectPositionX = ( x0 * cos( sqrtKoverM * t ) ) +
        ( ( v0 / sqrtKoverM ) * sin( sqrtKoverM + t ) );

    // Compute draw pos for "correct"
    float CorrectEndX = PixelsPerMeter * State[CorrectPositionX];

    // Draw the spring.
    strokeWeight( 1.0 );
    line( 0.0, 0.0, SpringEndX, 0.0 );

    // Draw the spring pivot
    fill( 0.0 );
    ellipse( 0.0, 0.0,
             PixelsPerMeter * 0.03,
             PixelsPerMeter * 0.03 );

    // Draw the spring bob
    fill( 1.0, 0.0, 0.0 );
    ellipse( SpringEndX, 0.0,
             PixelsPerMeter * 0.1,
             PixelsPerMeter * 0.1 );

    // Draw the correct bob in blue
    fill( 0.0, 0.0, 1.0 );
    ellips( CorrectEndX, -PixelsPerMeter * 0.25,
            PixelsPerMeter * 0.1,
            PixelsPerMeter * 0.1 );
}
</code></pre>

<p>

And here that is:

<p>
<script type="application/processing" data-processing-target="pjsSpringSimp3">
float Stiffness = 5.0;
float BobMass = 0.5;
int StateSize = 3;
float[] InitState = new float[StateSize];
float[] State = new float[StateSize];
int StateCurrentTime = 0;
int StatePositionX = 1;
int StateVelocityX = 2;

int WindowWidthHeight = 300;
float WorldSize = 2.0;
float PixelsPerMeter;
float OriginPixelsX;
float OriginPixelsY;

void setup()
{
    // Create initial state.
    InitState[StateCurrentTime] = 0.0;
    InitState[StatePositionX] = 0.65;
    InitState[StateVelocityX] = 0.0;

    // Copy initial state to current state.
    // notice that this does not need to know what the meaning of the
    // state elements is, and would work regardless of the state's size.
    for ( int i = 0; i < StateSize; ++i )
    {
        State[i] = InitState[i];
    }

    // Set up normalized colors.
    colorMode( RGB, 1.0 );

    // Set up the stroke color and width.
    stroke( 0.0 );
    //strokeWeight( 0.01 );

    // Create the window size, set up the transformation variables.
    size( WindowWidthHeight, WindowWidthHeight );
    PixelsPerMeter = (( float )WindowWidthHeight ) / WorldSize;
    OriginPixelsX = 0.5 * ( float )WindowWidthHeight;
    OriginPixelsY = 0.5 * ( float )WindowWidthHeight;

    textSize( 24 );
}

// Draw our State, with the unfortunate units conversion.
void DrawState()
{
    // Compute end of arm.
    float SpringEndX = PixelsPerMeter * State[StatePositionX];

    // Compute the CORRECT position.
    float sqrtKoverM = sqrt( Stiffness / BobMass );
    float x0 = InitState[StatePositionX];
    float v0 = InitState[StateVelocityX];
    float t = State[StateCurrentTime];
    float CorrectPositionX = ( x0 * cos( sqrtKoverM * t ) ) +
        ( ( v0 / sqrtKoverM ) * sin( sqrtKoverM + t ) );

    // Compute draw pos for "correct"
    float CorrectEndX = PixelsPerMeter * CorrectPositionX;

    // Draw the spring.
    strokeWeight( 1.0 );
    line( 0.0, 0.0, SpringEndX, 0.0 );

    // Draw the spring pivot
    fill( 0.0 );
    ellipse( 0.0, 0.0,
             PixelsPerMeter * 0.03,
             PixelsPerMeter * 0.03 );

    // Draw the spring bob
    fill( 1.0, 0.0, 0.0 );
    ellipse( SpringEndX, 0.0,
             PixelsPerMeter * 0.1,
             PixelsPerMeter * 0.1 );

    // Draw the correct bob in blue
    fill( 0.0, 0.0, 1.0 );
    ellipse( CorrectEndX, -PixelsPerMeter * 0.25,
             PixelsPerMeter * 0.1,
             PixelsPerMeter * 0.1 );
}

// Time Step function.
void TimeStep( float i_dt )
{
    // Compute acceleration from current position.
    float A = ( -Stiffness / BobMass ) * State[StatePositionX];

    // Update position based on current velocity.
    State[StatePositionX] += i_dt * State[StateVelocityX];

    // Update velocity based on acceleration.
    State[StateVelocityX] += i_dt * A;

    // Update current time.
    State[StateCurrentTime] += i_dt;
}

// Processing Draw function, called every time the screen refreshes.
void draw()
{
    // Time Step.
    TimeStep( 1.0/24.0 );

    // Clear the display to a constant color.
    background( 0.75 );

    // Label.
    fill( 1.0 );
    text( "Forward Euler", 10, 30 );

    pushMatrix();

    // Translate to the origin.
    translate( OriginPixelsX, OriginPixelsY );

    // Draw the simulation
    DrawState();
}

// Reset function. If the key 'r' is released in the display,
// copy the initial state to the state.
void keyReleased()
{
    if ( key == 114 )
    {
        for ( int i = 0; i < StateSize; ++i )
        {
            State[i] = InitState[i];
        }
    }
}

</script>
<canvas id="pjsSpringSimp3"> </canvas>
<p>
I also added a "reset" capability to the simple spring code by using
Processing's keyRelease function. In response to the key 'r' being released
in the view (which has ascii-code 114), we copy the initial state onto the
state. (See how handy state vectors are turning out to be!)  Because we
included the Current Time in our state vector, this is easy:

<pre><code>
// Reset function. If the key 'r' is released in the display,
// copy the initial state to the state.
void keyReleased()
{
    if ( key == 114 )
    {
        for ( int i = 0; i < StateSize; ++i )
        {
            State[i] = InitState[i];
        }
    }
}
</code></pre>

<h2>The Euler-Cromer Time Step</h2>
When we created the forward finite difference approximation to our
<b>ODE</b> equation of motion in the previous class, we had to create two
forward difference equations - one for velocity from acceleration, and then
one for position from velocity. As you remember, we computed acceleration
from the current position, then updated position based on the current
velocity, and then finally updated velocity based on the acceleration.
What if we updated velocity first, and then updated the position from the
current velocity?  That would look like this:

<div class="LaTexEquation">
    \(
    a( t ) = f( t, x( t ) ) = \frac{-k}{m} x( t ) \\
    v( t + \Delta t ) \approx v( t ) + \Delta t a( t ) \\
    x( t + \Delta t ) \approx x( t ) + \Delta t v( t + \Delta t )
    \)
</div>

This seems almost trivially different from our Forward-Euler time step. This
method is called many different things:
<a href="http://en.wikipedia.org/wiki/Semi-implicit_Euler_method">
Semi-Implicit Euler Method</a>, Symplectic Euler, Semi-Explicit Euler,
Euler-Cromer, and Newton–Størmer–Verlet (NSV) (at least). One of the things
I've found in years of writing simulations is that just learning the names
of all these different techniques, and learning that the different names
often mean exactly the same thing, is a huge part of simplifying the giant
mess of understanding.
<p>
The code changes to implement Euler Cromer are trivial, we just reorder the
time step - literally just swap the velocity update line and the position
update line:
<p>
<pre><code>
// Euler-Cromer Time Step function.
void TimeStep( float i_dt )
{
    // Compute acceleration from current position.
    float A = ( -Stiffness / BobMass ) * State[StatePositionX];

    // Update velocity based on acceleration.
    State[StateVelocityX] += i_dt * A;

    // Update position based on current velocity.
    State[StatePositionX] += i_dt * State[StateVelocityX];

    // Update current time.
    State[StateCurrentTime] += i_dt;
}
</code></pre>
<p>
And let's see how this ONE LINE (okay, TWO-LINE) change affects the sim:

<p>
<script type="application/processing" data-processing-target="pjsSpringSimp4">
float Stiffness = 5.0;
float BobMass = 0.5;
int StateSize = 3;
float[] InitState = new float[StateSize];
float[] State = new float[StateSize];
int StateCurrentTime = 0;
int StatePositionX = 1;
int StateVelocityX = 2;

int WindowWidthHeight = 300;
float WorldSize = 2.0;
float PixelsPerMeter;
float OriginPixelsX;
float OriginPixelsY;

void setup()
{
    // Create initial state.
    InitState[StateCurrentTime] = 0.0;
    InitState[StatePositionX] = 0.65;
    InitState[StateVelocityX] = 0.0;

    // Copy initial state to current state.
    // notice that this does not need to know what the meaning of the
    // state elements is, and would work regardless of the state's size.
    for ( int i = 0; i < StateSize; ++i )
    {
        State[i] = InitState[i];
    }

    // Set up normalized colors.
    colorMode( RGB, 1.0 );

    // Set up the stroke color and width.
    stroke( 0.0 );
    //strokeWeight( 0.01 );

    // Create the window size, set up the transformation variables.
    size( WindowWidthHeight, WindowWidthHeight );
    PixelsPerMeter = (( float )WindowWidthHeight ) / WorldSize;
    OriginPixelsX = 0.5 * ( float )WindowWidthHeight;
    OriginPixelsY = 0.5 * ( float )WindowWidthHeight;

    textSize( 24 );
}

// Draw our State, with the unfortunate units conversion.
void DrawState()
{
    // Compute end of arm.
    float SpringEndX = PixelsPerMeter * State[StatePositionX];

    // Compute the CORRECT position.
    float sqrtKoverM = sqrt( Stiffness / BobMass );
    float x0 = InitState[StatePositionX];
    float v0 = InitState[StateVelocityX];
    float t = State[StateCurrentTime];
    float CorrectPositionX = ( x0 * cos( sqrtKoverM * t ) ) +
        ( ( v0 / sqrtKoverM ) * sin( sqrtKoverM + t ) );

    // Compute draw pos for "correct"
    float CorrectEndX = PixelsPerMeter * CorrectPositionX;

    // Draw the spring.
    strokeWeight( 1.0 );
    line( 0.0, 0.0, SpringEndX, 0.0 );

    // Draw the spring pivot
    fill( 0.0 );
    ellipse( 0.0, 0.0,
             PixelsPerMeter * 0.03,
             PixelsPerMeter * 0.03 );

    // Draw the spring bob
    fill( 1.0, 0.0, 0.0 );
    ellipse( SpringEndX, 0.0,
             PixelsPerMeter * 0.1,
             PixelsPerMeter * 0.1 );

    // Draw the correct bob in blue
    fill( 0.0, 0.0, 1.0 );
    ellipse( CorrectEndX, -PixelsPerMeter * 0.25,
             PixelsPerMeter * 0.1,
             PixelsPerMeter * 0.1 );
}

// Euler-Cromer Time Step function.
void TimeStep( float i_dt )
{
    // Compute acceleration from current position.
    float A = ( -Stiffness / BobMass ) * State[StatePositionX];

    // Update velocity based on acceleration.
    State[StateVelocityX] += i_dt * A;

    // Update position based on current velocity.
    State[StatePositionX] += i_dt * State[StateVelocityX];

    // Update current time.
    State[StateCurrentTime] += i_dt;
}

// Processing Draw function, called every time the screen refreshes.
void draw()
{
    // Time Step.
    TimeStep( 1.0/24.0 );

    // Clear the display to a constant color.
    background( 0.75 );

    // Label.
    fill( 1.0 );
    text( "Euler-Cromer", 10, 30 );

    pushMatrix();

    // Translate to the origin.
    translate( OriginPixelsX, OriginPixelsY );

    // Draw the simulation
    DrawState();
}

// Reset function. If the key 'r' is released in the display,
// copy the initial state to the state.
void keyReleased()
{
    if ( key == 114 )
    {
        for ( int i = 0; i < StateSize; ++i )
        {
            State[i] = InitState[i];
        }
    }
}

</script>
<canvas id="pjsSpringSimp4"> </canvas>
<p>
This is not bad! Our simulation seems to have become stable, which is good
news. However, if we watch the simulation for a while, it becomes clear that
the simulated positions diverge more and more over time from the correct
positions.  In order to fix this, we're going to need to explore even
higher-order time integration techniques, which is the topic of the next
class!
</body>