xna - Why do objects interpenetrate in this simple collision solver?

The code below is from a Microsoft XNA sample here. This is quite a simple rigid body simulation that ignores many physical effects (such as angular momentum), but it does try to push objects (spheres) apart so that they are not penetrating one-another.

However, the simulation allows spheres not only to penetrate, but when many spheres are stacked on top of each other, small spheres can be almost completely inside of larger spheres. If I make all spheres have the same radius and mass, then the simulation performs reasonably well (with minimal interpenetration).

Can someone explain why there is any interpenetration at all? Since it moves the positions of the spheres, it seems like interpenetration should be impossible.

For each sphere in the simulation, this method is called on every other sphere.

/// 

// Given 2 spheres with velocity, mass and size, evaluate whether
// a collision occured, and if so, excatly where, and move sphere 2
// at the contact point with sphere 1, and generate new velocities.
/// 
private void SphereCollisionImplicit(Sphere sphere1, Sphere sphere2)
{
    const float K_ELASTIC = 0.75f;

    Vector3 relativepos = sphere2.Position - sphere1.Position;

    float distance = relativepos.Length();
    float radii = sphere1.Radius + sphere2.Radius;
    if (distance >= radii)
    {
        return; // No collision
    }

    // Add epsilon to avoid NaN.
    distance += 0.000001f;


    Vector3 relativeUnit = relativepos * (1.0f / distance);
    Vector3 penetration = relativeUnit * (radii - distance);

    // Adjust the spheres' relative positions
    float mass1 = sphere1.Mass;
    float mass2 = sphere2.Mass;

    float m_inv = 1.0f / (mass1 + mass2);
    float weight1 = mass1 * m_inv; // relative weight of sphere 1
    float weight2 = mass2 * m_inv; // relative weight of sphere 2. w1+w2==1.0


    sphere1.Position -= weight2 * penetration;
    sphere2.Position += weight1 * penetration;

    // Adjust the objects’ relative velocities, if they are
    // moving toward each other.
    //
    // Note that we're assuming no friction, or equivalently, no angular momentum.
    //
    // velocityTotal = velocity of v2 in v1 stationary ref. frame

    // get reference frame of common center of mass
    Vector3 velocity1 = sphere1.Velocity;
    Vector3 velocity2 = sphere2.Velocity;

    Vector3 velocityTotal = velocity1 * weight1 + velocity2 * weight2;
    Vector3 i2 = (velocity2 - velocityTotal) * mass2;
    if (Vector3.Dot(i2, relativeUnit) < 0)
    {
        // i1+i2 == 0, approx
        Vector3 di = Vector3.Dot(i2, relativeUnit) * relativeUnit;

        i2 -= di * (K_ELASTIC + 1);
        sphere1.Velocity = (-i2) / mass1 + velocityTotal;
        sphere2.Velocity = i2 / mass2 + velocityTotal;
    }
}

In particular, I though that this:

sphere1.Position -= weight2 * penetration;
sphere2.Position += weight1 * penetration;

Should completely disallow any interpenetration, why doesn't it?

Answer

I think your problem comes from multiple collisions.

Consider three spheres A, B & C.

A sits there.

B hits A. Fine.

C hits B. B is nudged into A. Oops.