SpringPart.h

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// Jolt Physics Library (https://github.com/jrouwe/JoltPhysics)
// SPDX-FileCopyrightText: 2021 Jorrit Rouwe
// SPDX-License-Identifier: MIT
 
#pragma once
 
JPH_NAMESPACE_BEGIN
#ifndef JPH_PLATFORM_DOXYGEN // Somehow Doxygen gets confused and thinks the parameters to CalculateSpringProperties belong to this macro
JPH_MSVC_SUPPRESS_WARNING(4723) // potential divide by 0 - caused by line: outEffectiveMass = 1.0f / inInvEffectiveMass, note that JPH_NAMESPACE_BEGIN already pushes the warning state
#endif // !JPH_PLATFORM_DOXYGEN
 
class SpringPart
{
private:
    JPH_INLINE void             CalculateSpringPropertiesHelper(float inDeltaTime, float inInvEffectiveMass, float inBias, float inC, float inStiffness, float inDamping, float &outEffectiveMass)
    {
        // Soft constraints as per: Soft Constraints: Reinventing The Spring - Erin Catto - GDC 2011
 
        // Note that the calculation of beta and gamma below are based on the solution of an implicit Euler integration scheme
        // This scheme is unconditionally stable but has built in damping, so even when you set the damping ratio to 0 there will still
        // be damping. See page 16 and 32.
 
        // Calculate softness (gamma in the slides)
        // See page 34 and note that the gamma needs to be divided by delta time since we're working with impulses rather than forces:
        // softness = 1 / (dt * (c + dt * k))
        // Note that the spring stiffness is k and the spring damping is c
        mSoftness = 1.0f / (inDeltaTime * (inDamping + inDeltaTime * inStiffness));
 
        // Calculate bias factor (baumgarte stabilization):
        // beta = dt * k / (c + dt * k) = dt * k^2 * softness
        // b = beta / dt * C = dt * k * softness * C
        mBias = inBias + inDeltaTime * inStiffness * mSoftness * inC;
 
        // Update the effective mass, see post by Erin Catto: http://www.bulletphysics.org/Bullet/phpBB3/viewtopic.php?f=4&t=1354
        //
        // Newton's Law:
        // M * (v2 - v1) = J^T * lambda
        //
        // Velocity constraint with softness and Baumgarte:
        // J * v2 + softness * lambda + b = 0
        //
        // where b = beta * C / dt
        //
        // We know everything except v2 and lambda.
        //
        // First solve Newton's law for v2 in terms of lambda:
        //
        // v2 = v1 + M^-1 * J^T * lambda
        //
        // Substitute this expression into the velocity constraint:
        //
        // J * (v1 + M^-1 * J^T * lambda) + softness * lambda + b = 0
        //
        // Now collect coefficients of lambda:
        //
        // (J * M^-1 * J^T + softness) * lambda = - J * v1 - b
        //
        // Now we define:
        //
        // K = J * M^-1 * J^T + softness
        //
        // So our new effective mass is K^-1
        outEffectiveMass = 1.0f / (inInvEffectiveMass + mSoftness);
    }
 
public:
    inline void                 CalculateSpringPropertiesWithBias(float inBias)
    {
        mSoftness = 0.0f;
        mBias = inBias;
    }
 
    inline void                 CalculateSpringPropertiesWithFrequencyAndDamping(float inDeltaTime, float inInvEffectiveMass, float inBias, float inC, float inFrequency, float inDamping, float &outEffectiveMass)
    {
        outEffectiveMass = 1.0f / inInvEffectiveMass;
 
        if (inFrequency > 0.0f)
        {
            // Calculate angular frequency
            float omega = 2.0f * JPH_PI * inFrequency;
 
            // Calculate spring stiffness k and damping constant c (page 45)
            float k = outEffectiveMass * Square(omega);
            float c = 2.0f * outEffectiveMass * inDamping * omega;
 
            CalculateSpringPropertiesHelper(inDeltaTime, inInvEffectiveMass, inBias, inC, k, c, outEffectiveMass);
        }
        else
        {
            CalculateSpringPropertiesWithBias(inBias);
        }
    }
 
    inline void                 CalculateSpringPropertiesWithStiffnessAndDamping(float inDeltaTime, float inInvEffectiveMass, float inBias, float inC, float inStiffness, float inDamping, float &outEffectiveMass)
    {
        if (inStiffness > 0.0f)
        {
            CalculateSpringPropertiesHelper(inDeltaTime, inInvEffectiveMass, inBias, inC, inStiffness, inDamping, outEffectiveMass);
        }
        else
        {
            outEffectiveMass = 1.0f / inInvEffectiveMass;
 
            CalculateSpringPropertiesWithBias(inBias);
        }
    }
 
    inline bool                 IsActive() const
    {
        return mSoftness != 0.0f;
    }
 
    inline float                GetBias(float inTotalLambda) const
    {
        // Remainder of post by Erin Catto: http://www.bulletphysics.org/Bullet/phpBB3/viewtopic.php?f=4&t=1354
        //
        // Each iteration we are not computing the whole impulse, we are computing an increment to the impulse and we are updating the velocity.
        // Also, as we solve each constraint we get a perfect v2, but then some other constraint will come along and mess it up.
        // So we want to patch up the constraint while acknowledging the accumulated impulse and the damaged velocity.
        // To help with that we use P for the accumulated impulse and lambda as the update. Mathematically we have:
        //
        // M * (v2new - v2damaged) = J^T * lambda
        // J * v2new + softness * (total_lambda + lambda) + b = 0
        //
        // If we solve this we get:
        //
        // v2new = v2damaged + M^-1 * J^T * lambda
        // J * (v2damaged + M^-1 * J^T * lambda) + softness * total_lambda + softness * lambda + b = 0
        //
        // (J * M^-1 * J^T + softness) * lambda = -(J * v2damaged + softness * total_lambda + b)
        //
        // So our lagrange multiplier becomes:
        //
        // lambda = -K^-1 (J v + softness * total_lambda + b)
        //
        // So we return the bias: softness * total_lambda + b
        return mSoftness * inTotalLambda + mBias;
    }
 
private:
    float                       mBias  = 0.0f;
    float                       mSoftness  = 0.0f;
};
 
JPH_NAMESPACE_END