Class PositionBasedDynamics

Class Documentation

class PositionBasedDynamics

Public Static Functions

static bool solve_DistanceConstraint(const Vector3r &p0, Real invMass0, const Vector3r &p1, Real invMass1, const Real restLength, const Real stiffness, Vector3r &corr0, Vector3r &corr1)

Determine the position corrections for a distance constraint between two particles:\(C(\mathbf{p}_0, \mathbf{p}_1) = \| \mathbf{p}_0 - \mathbf{p}_1\| - l_0 = 0\) More information can be found in the following papers: [12], [2], [5], [6],

Parameters

  • p0 – position of first particle

  • invMass0 – inverse mass of first particle

  • p1 – position of second particle

  • invMass1 – inverse mass of second particle

  • restLength – rest length of distance constraint

  • stiffness – stiffness coefficient

  • corr0 – position correction of first particle

  • corr1 – position correction of second particle

static bool solve_DihedralConstraint(const Vector3r &p0, Real invMass0, const Vector3r &p1, Real invMass1, const Vector3r &p2, Real invMass2, const Vector3r &p3, Real invMass3, const Real restAngle, const Real stiffness, Vector3r &corr0, Vector3r &corr1, Vector3r &corr2, Vector3r &corr3)

Determine the position corrections for a dihedral bending constraint. For a pair of adjacent triangles \((\mathbf{p}_1, \mathbf{p}_3, \mathbf{p}_2)\) and \((\mathbf{p}_1, \mathbf{p}_2, \mathbf{p}_4)\) with the common edge \((\mathbf{p}_3, \mathbf{p}_4)\) a bilateral bending constraint is added by the constraint function

\[\begin{equation*} C_{bend}(\mathbf{p}_1, \mathbf{p}_2,\mathbf{p}_3, \mathbf{p}_4) = \text{acos}\left( \frac{\mathbf{p}_{2,1} \times \mathbf{p}_{3,1}}{|\mathbf{p}_{2,1} \times \mathbf{p}_{3,1}|} \cdot \frac{\mathbf{p}_{2,1} \times \mathbf{p}_{4,1}}{|\mathbf{p}_{2,1} \times \mathbf{p}_{4,1}|}\right)-\varphi_0 \end{equation*}\]
and stiffness \(k_{bend}\). The scalar \(\varphi_0\) is the initial dihedral angle between the two triangles and \(k_{bend}\)

is a global user parameter defining the bending stiffness.

More information can be found in the following papers:

[12], [2], [5], [6],

Parameters

  • p0 – position of first particle

  • invMass0 – inverse mass of first particle

  • p1 – position of second particle

  • invMass1 – inverse mass of second particle

  • p2 – position of third particle

  • invMass2 – inverse mass of third particle

  • p3 – position of fourth particle

  • invMass3 – inverse mass of fourth particle

  • restAngle – rest angle \(\varphi_0\)

  • stiffness – stiffness coefficient

  • corr0 – position correction of first particle

  • corr1 – position correction of second particle

  • corr2 – position correction of third particle

  • corr3 – position correction of fourth particle

static bool solve_VolumeConstraint(const Vector3r &p0, Real invMass0, const Vector3r &p1, Real invMass1, const Vector3r &p2, Real invMass2, const Vector3r &p3, Real invMass3, const Real restVolume, const Real stiffness, Vector3r &corr0, Vector3r &corr1, Vector3r &corr2, Vector3r &corr3)

Determine the position corrections for a constraint that conserves the volume of single tetrahedron. Such a constraint has the form

\[\begin{equation*} C(\mathbf{p}_1, \mathbf{p}_2, \mathbf{p}_3, \mathbf{p}_4) = \frac{1}{6} \left(\mathbf{p}_{2,1} \times \mathbf{p}_{3,1}\right) \cdot \mathbf{p}_{4,1} - V_0, \end{equation*}\]
where \(\mathbf{p}_1\), \(\mathbf{p}_2\), \(\mathbf{p}_3\) and \(\mathbf{p}_4\) are the four corners of the tetrahedron and \(V_0\)

is its rest volume.

More information can be found in the following papers:

[12], [2], [5], [6],

Parameters

  • p0 – position of first particle

  • invMass0 – inverse mass of first particle

  • p1 – position of second particle

  • invMass1 – inverse mass of second particle

  • p2 – position of third particle

  • invMass2 – inverse mass of third particle

  • p3 – position of fourth particle

  • invMass3 – inverse mass of fourth particle

  • restVolume – rest angle \(V_0\)

  • stiffness – stiffness coefficient

  • corr0 – position correction of first particle

  • corr1 – position correction of second particle

  • corr2 – position correction of third particle

  • corr3 – position correction of fourth particle

static bool solve_EdgePointDistanceConstraint(const Vector3r &p, Real invMass, const Vector3r &p0, Real invMass0, const Vector3r &p1, Real invMass1, const Real restDist, const Real compressionStiffness, const Real stretchStiffness, Vector3r &corr, Vector3r &corr0, Vector3r &corr1)

Determine the position corrections for a constraint that preserves a rest distance between a point and an edge.

Parameters

  • p – position of point particle

  • invMass – inverse mass of point particle

  • p0 – position of first edge particle

  • invMass0 – inverse mass of first edge particle

  • p1 – position of second edge particle

  • invMass1 – inverse mass of second edge particle

  • restDist – rest distance of point and edge

  • compressionStiffness – stiffness coefficient for compression

  • stretchStiffness – stiffness coefficient for stretching

  • corr – position correction of point particle

  • corr0 – position correction of first edge particle

  • corr1 – position correction of second edge particle

static bool solve_TrianglePointDistanceConstraint(const Vector3r &p, Real invMass, const Vector3r &p0, Real invMass0, const Vector3r &p1, Real invMass1, const Vector3r &p2, Real invMass2, const Real restDist, const Real compressionStiffness, const Real stretchStiffness, Vector3r &corr, Vector3r &corr0, Vector3r &corr1, Vector3r &corr2)

Determine the position corrections for a constraint that preserves a rest distance between a point and a triangle.

Parameters

  • p – position of point particle

  • invMass – inverse mass of point particle

  • p0 – position of first triangle particle

  • invMass0 – inverse mass of first triangle particle

  • p1 – position of second triangle particle

  • invMass1 – inverse mass of second triangle particle

  • p2 – position of third triangle particle

  • invMass2 – inverse mass of third triangle particle

  • restDist – rest distance of point and triangle

  • compressionStiffness – stiffness coefficient for compression

  • stretchStiffness – stiffness coefficient for stretching

  • corr – position correction of point particle

  • corr0 – position correction of first triangle particle

  • corr1 – position correction of second triangle particle

  • corr2 – position correction of third triangle particle

static bool solve_EdgeEdgeDistanceConstraint(const Vector3r &p0, Real invMass0, const Vector3r &p1, Real invMass1, const Vector3r &p2, Real invMass2, const Vector3r &p3, Real invMass3, const Real restDist, const Real compressionStiffness, const Real stretchStiffness, Vector3r &corr0, Vector3r &corr1, Vector3r &corr2, Vector3r &corr3)

Determine the position corrections for a constraint that preserves a rest distance between two edges.

Parameters

  • p0 – position of first particle of edge 0

  • invMass0 – inverse mass of first particle of edge 0

  • p1 – position of second particle of edge 0

  • invMass1 – inverse mass of second particle of edge 0

  • p2 – position of first particle of edge 1

  • invMass2 – inverse mass of first particle of edge 1

  • p3 – position of second particle of edge 1

  • invMass3 – inverse mass of second particle of edge 1

  • restDist – rest distance between both edges

  • compressionStiffness – stiffness coefficient for compression

  • stretchStiffness – stiffness coefficient for stretching

  • corr0 – position correction of first particle of edge 0

  • corr1 – position correction of second particle of edge 0

  • corr2 – position correction of first particle of edge 1

  • corr3 – position correction of second particle of edge 1

static bool init_IsometricBendingConstraint(const Vector3r &p0, const Vector3r &p1, const Vector3r &p2, const Vector3r &p3, Matrix4r &Q)

Initialize the local stiffness matrix Q. The matrix is required by the solver step. It must only be recomputed if the rest shape changes.

Bending is simulated for the angle on (p2, p3) between the triangles (p0, p2, p3) and (p1, p3, p2).

Parameters

  • p0 – point 0 of stencil

  • p1 – point 1 of stencil

  • p2 – point 2 of stencil

  • p3 – point 3 of stencil

  • Q – returns the local stiffness matrix which is required by the solver

static bool solve_IsometricBendingConstraint(const Vector3r &p0, Real invMass0, const Vector3r &p1, Real invMass1, const Vector3r &p2, Real invMass2, const Vector3r &p3, Real invMass3, const Matrix4r &Q, const Real stiffness, Vector3r &corr0, Vector3r &corr1, Vector3r &corr2, Vector3r &corr3)

Determine the position corrections for the isometric bending constraint. This constraint can be used for almost inextensible surface models.

More information can be found in:

[6], [4]

Parameters

  • p0 – position of first particle

  • invMass0 – inverse mass of first particle

  • p1 – position of second particle

  • invMass1 – inverse mass of second particle

  • p2 – position of third particle

  • invMass2 – inverse mass of third particle

  • p3 – position of fourth particle

  • invMass3 – inverse mass of fourth particle

  • Q – local stiffness matrix which must be initialized by calling init_IsometricBendingConstraint()

  • stiffness – stiffness coefficient for bending

  • corr0 – position correction of first particle

  • corr1 – position correction of second particle

  • corr2 – position correction of third particle

  • corr3 – position correction of fourth particle

static bool init_ShapeMatchingConstraint(const Vector3r x0[], const Real invMasses[], const int numPoints, Vector3r &restCm)

Initialize rest configuration infos for one shape matching cluster which are required by the solver step. It must only be reinitialized if the rest shape changes.

Parameters

  • x0 – rest configuration of all particles in the cluster

  • invMasses – inverse masses of all particles in the cluster

  • numPoints – number of particles in the cluster

  • restCm – returns the center of mass of the rest configuration

static bool solve_ShapeMatchingConstraint(const Vector3r x0[], const Vector3r x[], const Real invMasses[], const int numPoints, const Vector3r &restCm, const Real stiffness, const bool allowStretch, Vector3r corr[], Matrix3r *rot = NULL)

Determine the position corrections for a shape matching constraint.

More information can be found in:

[6], [2], [5], [11], [3], [8]

Parameters

  • x0 – rest configuration of all particles in the cluster

  • x – current configuration of all particles in the cluster

  • invMasses – invMasses inverse masses of all particles in the cluster

  • numPoints – number of particles in the cluster

  • restCm – center of mass of the rest configuration

  • stiffness – stiffness coefficient

  • allowStretch – allow stretching

  • corr – position corrections for all particles in the cluster

  • rot – returns determined rotation matrix

static bool init_StrainTriangleConstraint(const Vector3r &p0, const Vector3r &p1, const Vector3r &p2, Matrix2r &invRestMat)

Initialize rest configuration infos which are required by the solver step. Recomputation is only necessary when rest shape changes.

The triangle is defined in the xy plane.

Parameters

  • p0 – point 0 of triangle

  • p1 – point 1 of triangle

  • p2 – point 2 of triangle

  • invRestMat – returns a matrix required by the solver

static bool solve_StrainTriangleConstraint(const Vector3r &p0, Real invMass0, const Vector3r &p1, Real invMass1, const Vector3r &p2, Real invMass2, const Matrix2r &invRestMat, const Real xxStiffness, const Real yyStiffness, const Real xyStiffness, const bool normalizeStretch, const bool normalizeShear, Vector3r &corr0, Vector3r &corr1, Vector3r &corr2)

Solve triangle constraint with strain based dynamics and return position corrections.

More information can be found in:

[6], [13]

Parameters

  • p0 – position of first particle

  • invMass0 – inverse mass of first particle

  • p1 – position of second particle

  • invMass1 – inverse mass of second particle

  • p2 – position of third particle

  • invMass2 – inverse mass of third particle

  • invRestMat – precomputed matrix determined by init_StrainTriangleConstraint()

  • xxStiffness – stiffness coefficient for xx stretching

  • yyStiffness – stiffness coefficient for yy stretching

  • xyStiffness – stiffness coefficient for xy shearing

  • normalizeStretch – should stretching be normalized

  • normalizeShear – should shearing be normalized

  • corr0 – position correction for point 0

  • corr1 – position correction for point 1

  • corr2 – position correction for point 2

static bool init_StrainTetraConstraint(const Vector3r &p0, const Vector3r &p1, const Vector3r &p2, const Vector3r &p3, Matrix3r &invRestMat)

Initialize rest configuration infos which are required by the solver step. Recomputation is only necessary when rest shape changes.

Parameters

  • p0 – point 0 of tet

  • p1 – point 1 of tet

  • p2 – point 2 of tet

  • p3 – point 3 of tet

  • invRestMat – returns a matrix required by the solver

static bool solve_StrainTetraConstraint(const Vector3r &p0, Real invMass0, const Vector3r &p1, Real invMass1, const Vector3r &p2, Real invMass2, const Vector3r &p3, Real invMass3, const Matrix3r &invRestMat, const Vector3r &stretchStiffness, const Vector3r &shearStiffness, const bool normalizeStretch, const bool normalizeShear, Vector3r &corr0, Vector3r &corr1, Vector3r &corr2, Vector3r &corr3)
static void computeGradCGreen(Real restVolume, const Matrix3r &invRestMat, const Matrix3r &sigma, Vector3r *J)
static void computeGreenStrainAndPiolaStress(const Vector3r &x1, const Vector3r &x2, const Vector3r &x3, const Vector3r &x4, const Matrix3r &invRestMat, const Real restVolume, const Real mu, const Real lambda, Matrix3r &epsilon, Matrix3r &sigma, Real &energy)
static void computeGreenStrainAndPiolaStressInversion(const Vector3r &x1, const Vector3r &x2, const Vector3r &x3, const Vector3r &x4, const Matrix3r &invRestMat, const Real restVolume, const Real mu, const Real lambda, Matrix3r &epsilon, Matrix3r &sigma, Real &energy)
static bool init_FEMTriangleConstraint(const Vector3r &p0, const Vector3r &p1, const Vector3r &p2, Real &area, Matrix2r &invRestMat)

Implementation of the finite element method described in

Jan Bender, Dan Koschier, Patrick Charrier and Daniel Weber,

”Position-Based Simulation of Continuous Materials”,

Computers & Graphics 44, 2014

Initialize rest configuration infos which are required by the solver step. Recomputation is only necessary when rest shape changes.

static bool solve_FEMTriangleConstraint(const Vector3r &p0, Real invMass0, const Vector3r &p1, Real invMass1, const Vector3r &p2, Real invMass2, const Real &area, const Matrix2r &invRestMat, const Real youngsModulusX, const Real youngsModulusY, const Real youngsModulusShear, const Real poissonRatioXY, const Real poissonRatioYX, Vector3r &corr0, Vector3r &corr1, Vector3r &corr2)

Implementation of the finite element method described in

Jan Bender, Dan Koschier, Patrick Charrier and Daniel Weber,

”Position-Based Simulation of Continuous Materials”,

Computers & Graphics 44, 2014

Solve the continuum mechanical constraint defined for a triangle.

static bool init_FEMTetraConstraint(const Vector3r &p0, const Vector3r &p1, const Vector3r &p2, const Vector3r &p3, Real &volume, Matrix3r &invRestMat)

Implementation of the finite element method described in

Jan Bender, Dan Koschier, Patrick Charrier and Daniel Weber,

”Position-Based Simulation of Continuous Materials”,

Computers & Graphics 44, 2014

Initialize rest configuration infos which are required by the solver step. Recomputation is only necessary when rest shape changes.

static bool solve_FEMTetraConstraint(const Vector3r &p0, Real invMass0, const Vector3r &p1, Real invMass1, const Vector3r &p2, Real invMass2, const Vector3r &p3, Real invMass3, const Real restVolume, const Matrix3r &invRestMat, const Real youngsModulus, const Real poissonRatio, const bool handleInversion, Vector3r &corr0, Vector3r &corr1, Vector3r &corr2, Vector3r &corr3)

Implementation of the finite element method described in

Jan Bender, Dan Koschier, Patrick Charrier and Daniel Weber,

”Position-Based Simulation of Continuous Materials”,

Computers & Graphics 44, 2014

Solve the continuum mechanical constraint defined for a tetrahedron.

static bool init_ParticleTetContactConstraint(const Real invMass0, const Vector3r &x0, const Vector3r &v0, const Real invMass[], const Vector3r x[], const Vector3r v[], const Vector3r &bary, const Vector3r &normal, Eigen::Matrix<Real, 3, 3, Eigen::DontAlign> &constraintInfo)

Initialize contact between a particle and a tetrahedron and return info which is required by the solver step.

Parameters

  • invMass0 – inverse mass of particle which collides with tet

  • x0 – particle position

  • v0 – particle velocity

  • invMass – inverse masses of tet particles

  • x – positions of tet particles

  • v – velocities of tet particles

  • bary – barycentric coordinates of contact point in tet

  • normal – contact normal in body 1

  • constraintInfo –

    Stores the local and global position of the contact points and the contact normal.

    The joint info contains the following columns:

    0: contact normal in body 1 (global)

    1: contact tangent (global)

    0,2: 1.0 / normal^T * K * normal

    1,2: maximal impulse in tangent direction

static bool solve_ParticleTetContactConstraint(const Real invMass0, const Vector3r &x0, const Real invMass[], const Vector3r x[], const Vector3r &bary, Eigen::Matrix<Real, 3, 3, Eigen::DontAlign> &constraintInfo, Real &lambda, Vector3r &corr0, Vector3r corr[])

Perform a solver step for a contact constraint between a particle and a tetrahedron. A contact constraint handles collisions and resting contacts between the bodies. The contact info must be generated in each time step.

Parameters

  • invMass0 – inverse mass of particle which collides with tet

  • x0 – particle position

  • invMass – inverse masses of tet particles

  • x – positions of tet particles

  • bary – barycentric coordinates of contact point in tet

  • constraintInfo – information which is required by the solver. This information must be generated in the beginning by calling init_RigidBodyContactConstraint().

  • corr0 – position correction of particle

  • corr – position corrections of tet particles

static bool velocitySolve_ParticleTetContactConstraint(const Real invMass0, const Vector3r &x0, const Vector3r &v0, const Real invMass[], const Vector3r x[], const Vector3r v[], const Vector3r &bary, const Real lambda, const Real frictionCoeff, Eigen::Matrix<Real, 3, 3, Eigen::DontAlign> &constraintInfo, Vector3r &corr_v0, Vector3r corr_v[])

Perform a solver step for a contact constraint between a particle and a tetrahedron. A contact constraint handles collisions and resting contacts between the bodies. The contact info must be generated in each time step.

Parameters

  • invMass0 – inverse mass of particle which collides with tet

  • x0 – particle position

  • v0 – particle velocity

  • invMass – inverse masses of tet particles

  • x – positions of tet particles

  • v – velocities of tet particles

  • bary – barycentric coordinates of contact point in tet

  • frictionCoeff – friction coefficient

  • constraintInfo – information which is required by the solver. This information must be generated in the beginning by calling init_RigidBodyContactConstraint().

  • corr_v0 – velocity correction of particle

  • corr_v – velocity corrections of tet particles