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PolynomialSpringsForceField

This component belongs to the category of ForceField. This component allows to simulate springs with Polynomial stress strain behavior. If we note:

  • \(F\) the spring force
  • \(S\) the cross section (always 1.0)
  • \(\sigma\) the stress-strain non-linear function
  • \(l_0\) the original length and \(l\) the current length of the spring
  • \(\Delta u %3D \left\{\begin{matrix} \Delta x\\ \Delta y\\ \Delta z\\ \end{matrix}\right.\) the point displacement

the generic non-linear force can thus be written:

\[ F=S%7E\sigma \left( \frac{l-l_0}{l_0} \right%29 \frac{\Delta u}{l} \]

where \(\sigma\) is polynom as follows:

\[ \sigma \left( \frac{l-l_0}{l_0} \right) =\sigma(L)=a_1L+a_2L^2+\cdots+a_nL^n \]

and

\[ \frac{\partial \sigma}{\partial L}=a_1+a_2L+\cdots+a_nL^{n-1} \]

The dedication of Jacobian matrix for PolynomialSpringForceField is given below:

\[ J_F(u)=\left(S\frac{\partial&space;\sigma}{\partial&space;L}\cdot\frac{1}{l_0}-S\sigma\cdot\frac{1}{|l|}&space;\right)\begin{bmatrix}\frac{\Delta&space;x^2}{l^2}&\frac{\Delta&space;x\Delta&space;y}{l^2}&\frac{\Delta&space;x\Delta&space;z}{l^2}\\\frac{\Delta&space;y\Delta&space;x}{l^2}&\frac{\Delta&space;y^2}{l^2}&\frac{\Delta&space;y\Delta&space;z}{l^2}\\\frac{\Delta&space;z\Delta&space;x}{l^2}&\frac{\Delta&space;z\Delta&space;y}{l^2}&&space;\frac{\Delta&space;z^2}{l^2}\end{bmatrix}+S\sigma\cdot\frac{1}{|l|}\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix} \]

Note that a RestShapePolynomialSpringsForceField does exist. It will compute the same non-linear force with regards to the rest shape of one single object. To avoid Nan problems when a spring has a zero length, an exponential addition to the denominator has been added. As a result, the stress simulation is shifted compared with polynomial values, but it keeps its nonlinearity:

\[ J_F(u)=\left(S\frac{\partial&space;\sigma}{\partial&space;L}\cdot\frac{(1-sc\cdot&space;e^{sh-sc(\Delta&space;x^2+\Delta&space;y^2+\Delta&space;z^2)})}{l_0}-S\sigma\cdot\frac{(1-sc\cdot&space;e^{sh-sc(\Delta&space;x^2+\Delta&space;y^2+\Delta&space;z^2)})}{\sqrt{\Delta&space;x^2+\Delta&space;y^2+\Delta&space;z^2+e^{sh-sc(\Delta&space;x^2+\Delta&space;y^2+\Delta&space;z^2)}}}&space;\right)\cdot\frac{1}{\sqrt{\Delta&space;x^2+\Delta&space;y^2+\Delta&space;z^2+e^{sh-sc(\Delta&space;x^2+\Delta&space;y^2+\Delta&space;z^2)}}}\cdot\begin{bmatrix}\Delta&space;x^2&\Delta&space;x\Delta&space;y&\Delta&space;x\Delta&space;z\\\Delta&space;y\Delta&space;x&\Delta&space;y^2&\Delta&space;y\Delta&space;z\\\Delta&space;z\Delta&space;x&\Delta&space;z\Delta&space;y&\Delta&space;z^2\end{bmatrix}+S\sigma\cdot\frac{1}{\sqrt{\Delta&space;x^2+\Delta&space;y^2+\Delta&space;z^2+e^{sh-sc(\Delta&space;x^2+\Delta&space;y^2+\Delta&space;z^2)}}}\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix} \]

More details were given in the pull-request #1342.

Usage

The PolynomialSpringsForceField requires two different objects to link, which means two MechanicalObjects on which the non-linear spring will act. On the other hand, RestShapePolynomialSpringsForceField will act on one single body, i.e. one MechanicalObject.

Simple elastic springs applied to given degrees of freedom between their current and rest shape position.

Vec3d

Templates:

  • Vec3d

Target: Sofa.Component.SolidMechanics.Spring

namespace: sofa::component::solidmechanics::spring

parents:

  • PairInteractionForceField

Data

Name Description Default value
name object name unnamed
printLog if true, emits extra messages at runtime. 0
tags list of the subsets the objet belongs to
bbox this object bounding box
componentState The state of the component among (Dirty, Valid, Undefined, Loading, Invalid). Undefined
listening if true, handle the events, otherwise ignore the events 0
rayleighStiffness Rayleigh damping - stiffness matrix coefficient 0
firstObjectPoints points related to the first object
secondObjectPoints points related to the second object
polynomialStiffness coefficients for all spring polynomials
polynomialDegree vector of values that show polynomials degrees
computeZeroLength flag to compute initial length for springs 1
zeroLength initial length for springs
recompute_indices Recompute indices (should be false for BBOX) 0
compressible Indicates if object compresses without any reaction force 0
springColor spring color 0 1 0 1
Visualization
drawMode The way springs will be drawn: - 0: Line - 1:Cylinder - 2: Arrow 0
showArrowSize size of the axis 0.01
showIndicesScale Scale for indices display (default=0.02) 0.02
Name Description Destination type name
context Graph Node containing this object (or BaseContext::getDefault() if no graph is used) BaseContext
slaves Sub-objects used internally by this object BaseObject
master nullptr for regular objects, or master object for which this object is one sub-objects BaseObject
mechanicalStates List of mechanical states to which this component is associated BaseMechanicalState
object1 First object associated to this component MechanicalState<Vec3d>
object2 Second object associated to this component MechanicalState<Vec3d>

Examples

PolynomialSpringsForceField.scn

<?xml version="1.0" ?>
<Node name="lroot" gravity="0 0 0" dt="0.02">
    <RequiredPlugin name="Sofa.Component.Constraint.Projective"/> <!-- Needed to use components [FixedProjectiveConstraint] -->
    <RequiredPlugin name="Sofa.Component.IO.Mesh"/> <!-- Needed to use components [MeshGmshLoader MeshOBJLoader] -->
    <RequiredPlugin name="Sofa.Component.LinearSolver.Iterative"/> <!-- Needed to use components [CGLinearSolver] -->
    <RequiredPlugin name="Sofa.Component.Mapping.Linear"/> <!-- Needed to use components [BarycentricMapping] -->
    <RequiredPlugin name="Sofa.Component.Mass"/> <!-- Needed to use components [DiagonalMass UniformMass] -->
    <RequiredPlugin name="Sofa.Component.ODESolver.Backward"/> <!-- Needed to use components [EulerImplicitSolver] -->
    <RequiredPlugin name="Sofa.Component.SolidMechanics.FEM.Elastic"/> <!-- Needed to use components [TetrahedralCorotationalFEMForceField] -->
    <RequiredPlugin name="Sofa.Component.SolidMechanics.Spring"/> <!-- Needed to use components [PolynomialSpringsForceField] -->
    <RequiredPlugin name="Sofa.Component.StateContainer"/> <!-- Needed to use components [MechanicalObject] -->
    <RequiredPlugin name="Sofa.Component.Topology.Container.Dynamic"/> <!-- Needed to use components [TetrahedronSetGeometryAlgorithms TetrahedronSetTopologyContainer] -->
    <RequiredPlugin name="Sofa.Component.Visual"/> <!-- Needed to use components [VisualStyle] -->
    <RequiredPlugin name="Sofa.GL.Component.Rendering3D"/> <!-- Needed to use components [OglModel] -->

    <VisualStyle displayFlags="showInteractionForceFields"/>
    <DefaultAnimationLoop/>
    <MeshOBJLoader name="LiverSurface" filename="mesh/liver-smooth.obj" />

    <Node name="Liver" >
        <EulerImplicitSolver name="cg_odesolver"   rayleighStiffness="0.1" rayleighMass="0.1" />
        <CGLinearSolver name="linear solver" iterations="25" tolerance="1e-09" threshold="1e-09" />
        <MeshGmshLoader name="meshLoader" filename="mesh/liver.msh" />
        <TetrahedronSetTopologyContainer name="topo" src="@meshLoader" />
        <MechanicalObject name="dofs" src="@meshLoader" />
        <TetrahedronSetGeometryAlgorithms template="Vec3" name="GeomAlgo" />
        <DiagonalMass  name="computed using mass density" massDensity="1" />
        <TetrahedralCorotationalFEMForceField template="Vec3" name="FEM" method="large" poissonRatio="0.3" youngModulus="3000" computeGlobalMatrix="0" />
        <FixedProjectiveConstraint  name="FixedProjectiveConstraint" indices="3 39 64" />
        <Node name="Visu" tags="Visual" gravity="0 -9.81 0">
            <OglModel  name="VisualModel" src="@../../LiverSurface" />
            <BarycentricMapping name="visual mapping" input="@../dofs" output="@VisualModel" />
        </Node>

        <Node name="Weight" >
            <MechanicalObject template="Vec3" name="myParticle" rest_position="0 0 0" position="0 0 0" />
            <UniformMass totalMass="30" />
            <PolynomialSpringsForceField polynomialDegree="3" polynomialStiffness="20 10 50" object1='@.' firstObjectPoints='0' object2='@../dofs' secondObjectPoints='15' drawMode='0' showIndicesScale="1"/>
        </Node>
    </Node>

</Node>

def createScene(root_node):

   lroot = root_node.addChild('lroot', gravity="0 0 0", dt="0.02")

   lroot.addObject('RequiredPlugin', name="Sofa.Component.Constraint.Projective")
   lroot.addObject('RequiredPlugin', name="Sofa.Component.IO.Mesh")
   lroot.addObject('RequiredPlugin', name="Sofa.Component.LinearSolver.Iterative")
   lroot.addObject('RequiredPlugin', name="Sofa.Component.Mapping.Linear")
   lroot.addObject('RequiredPlugin', name="Sofa.Component.Mass")
   lroot.addObject('RequiredPlugin', name="Sofa.Component.ODESolver.Backward")
   lroot.addObject('RequiredPlugin', name="Sofa.Component.SolidMechanics.FEM.Elastic")
   lroot.addObject('RequiredPlugin', name="Sofa.Component.SolidMechanics.Spring")
   lroot.addObject('RequiredPlugin', name="Sofa.Component.StateContainer")
   lroot.addObject('RequiredPlugin', name="Sofa.Component.Topology.Container.Dynamic")
   lroot.addObject('RequiredPlugin', name="Sofa.Component.Visual")
   lroot.addObject('RequiredPlugin', name="Sofa.GL.Component.Rendering3D")
   lroot.addObject('VisualStyle', displayFlags="showInteractionForceFields")
   lroot.addObject('DefaultAnimationLoop', )
   lroot.addObject('MeshOBJLoader', name="LiverSurface", filename="mesh/liver-smooth.obj")

   liver = lroot.addChild('Liver')

   liver.addObject('EulerImplicitSolver', name="cg_odesolver", rayleighStiffness="0.1", rayleighMass="0.1")
   liver.addObject('CGLinearSolver', name="linear solver", iterations="25", tolerance="1e-09", threshold="1e-09")
   liver.addObject('MeshGmshLoader', name="meshLoader", filename="mesh/liver.msh")
   liver.addObject('TetrahedronSetTopologyContainer', name="topo", src="@meshLoader")
   liver.addObject('MechanicalObject', name="dofs", src="@meshLoader")
   liver.addObject('TetrahedronSetGeometryAlgorithms', template="Vec3", name="GeomAlgo")
   liver.addObject('DiagonalMass', name="computed using mass density", massDensity="1")
   liver.addObject('TetrahedralCorotationalFEMForceField', template="Vec3", name="FEM", method="large", poissonRatio="0.3", youngModulus="3000", computeGlobalMatrix="0")
   liver.addObject('FixedProjectiveConstraint', name="FixedProjectiveConstraint", indices="3 39 64")

   visu = Liver.addChild('Visu', tags="Visual", gravity="0 -9.81 0")

   visu.addObject('OglModel', name="VisualModel", src="@../../LiverSurface")
   visu.addObject('BarycentricMapping', name="visual mapping", input="@../dofs", output="@VisualModel")

   weight = Liver.addChild('Weight')

   weight.addObject('MechanicalObject', template="Vec3", name="myParticle", rest_position="0 0 0", position="0 0 0")
   weight.addObject('UniformMass', totalMass="30")
   weight.addObject('PolynomialSpringsForceField', polynomialDegree="3", polynomialStiffness="20 10 50", object1="@.", firstObjectPoints="0", object2="@../dofs", secondObjectPoints="15", drawMode="0", showIndicesScale="1")