Class XPBD

Class Documentation

class XPBD

Public Static Functions

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

Determine the position corrections for a distance constraint between two particles using XPBD:\(C(\mathbf{p}_0, \mathbf{p}_1) = \| \mathbf{p}_0 - \mathbf{p}_1\| - l_0 = 0\) More information can be found in the following papers: [12], [2], [5], [6], [10]

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

  • dt – Time step size

  • lambda – Lagrange multiplier (XPBD)

  • corr0 – position correction of first particle

  • corr1 – position correction of second 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, const Real dt, Real &lambda, 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

  • dt – Time step size

  • lambda – Lagrange multiplier (XPBD)

  • 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_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, const Real dt, Real &lambda, Vector3r &corr0, Vector3r &corr1, Vector3r &corr2, Vector3r &corr3)

Determine the position corrections for the isometric bending constraint using XPBD

. 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

  • dt – Time step size

  • lambda – Lagrange multiplier (XPBD)

  • 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_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, const Real dt, Real &lambda, Vector3r &corr0, Vector3r &corr1, Vector3r &corr2, Vector3r &corr3)

XPBD

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.