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Projective constraint

Different types of constraint exist in SOFA. The projective constraint are method allowing to project the velocity of the constraint points of an object to a desired value.

Matrix approach

A projection matrix noted \(\mathbf{P}\) multiplies the matrix \(\mathbf{A}\) of the linear system \(\mathbf{A}x=b\) (where our unknown \(x\) is actually \(\Delta{v}\)) to enforce the so-called project constraint. The system thus becomes: \(\mathbf{P}^T\mathbf{A}\mathbf{P} \Delta v=\mathbf{P}^Tb\). Implicit integration has the advantage of being more stable for stiff forces or large time steps. The solution of these equation systems requires linear solvers. Due to the superlinear time complexity of equation solvers, it is generally more efficient to process independent interaction groups using separated solvers rather than a unique solver.

Another type of constraints is available in SOFA focusing on constraint-based interactions which requires the computation of Lagrange multipliers based on interaction Jacobians. This will be discussed in the next article about Lagrange constraint.

API of projective constraint

In SOFA, you can find several of these projective constraints in the SofaBoundaryConditions module, among them:

  • the FixedProjectiveConstraint: projecting a constant velocity, if the vertex is initially fixed, then it is attached to its initial position
  • the PartialFixedProjectiveConstraint: inheriting from FixedProjectiveConstraint, this constraint is projected only along certain degrees of freedom (e.g. only in x direction)

Classes considering on single object inherit from the class ProjectiveConstraintSet. The usual API functions associated to projective constraints are:

/// Project dx to constrained space (dx models an acceleration):
void projectResponse(const core::MechanicalParams* mparams, DataVecDeriv& resData);

/// Project v to constrained space (v models a velocity):
void projectVelocity(const core::MechanicalParams* mparams, DataVecDeriv& vData);

/// Project x to constrained space (x models a position):
void projectPosition(const core::MechanicalParams* mparams, DataVecCoord& xData);

/// Project c to constrained space (c models a constraint):
/// this method must be implemented by the component to handle Lagrange Multiplier based constraint
void projectJacobianMatrix(const core::MechanicalParams* mparams, DataMatrixDeriv& cData);

/// Project the global Mechanical Matrix to constrained space using offset parameter
void applyConstraint(const MechanicalParams*, const sofa::core::behavior::MultiMatrixAccessor*);

/// Project the global Mechanical Vector to constrained space using offset parameter
void applyConstraint(const MechanicalParams* , defaulttype::BaseVector*, const sofa::core::behavior::MultiMatrixAccessor*);