BDFOdeSolver

This component belongs to the category of integration schemes or ODE Solver. It is an implicit method for the numerical integration of the ODE resulting from Newton's second law of motion.

The method relies on Backward Differentiation Formula (BDF). To integrate the ODE in time, it uses information from the previous time steps to compute the next step. It establishes a linear combination of the unknown states, the previous states and the values of the ODE function when applied on those states. This equation leads to a nonlinear function to solve. That is why this component requires a NewtonRaphsonSolver, which the purpose is to solve nonlinear functions.

The coefficients of the linear combination come from the approximation of the function by a Lagrange interpolation polynomial. The order of the BDF is the number of previous time steps required to approximate the interpolation polynomial. The first-order BDF requires a single time step in the past to compute the next. It corresponds to the backward Euler method. The coefficients are unique for a given order, but can be influenced by a change of time step size. The SOFA component supports any order, and any change of time step size.

Details

The ODE resulting from Newton's second law of motion is:

\[ \begin{bmatrix} \frac{d q}{d t}\\ M \frac{d \dot{q}}{d t} \end{bmatrix} = \begin{bmatrix} \dot{q}\\ F(q, \dot{q}) \end{bmatrix} \]

where \(q\) and \(\dot{q}\) are respectively the position and the velocity, \(M\) is the mass matrix, and \(F\) is the sum of forces.

We define \(y(t)=\begin{bmatrix} q \\ \dot{q} \end{bmatrix}\), and \(f(t,y)=\begin{bmatrix} \dot{q} \\ M^{-1} F(q,\dot{q}) \end{bmatrix}\), such that the ODE is \(y'=f(t,y)\). We are interested in computing \(y(t_{n+s})\) and we know the values of \(y(t_{n+j})\) for \(j \lt s\). The Lagrange interpolation polynomial is the linear combination \(L(t) = \sum_{j=0}^s y(t_n+j) l_j(t)\), where \(l_j\) is the basis polynomials defined as \(l_j(t)=\prod_{0 \leq m \leq s, m \neq j} \frac{t-t_{n+m}}{t_{n+j}-t{n+m}}\). Then, we approximate \(y'\) by \(L'\), leading to the equation \(\sum_{j=0}^s y_{n+j} l'_j(t_{n+s}) = f(t_{n+s},y_{n+s})\). We can now define a nonlinear function \(r(q, \dot{q}) = \left[\sum_{j=0}^s y_{n+j} l'_j(t_{n+s})\right] - f(t_{n+s},y_{n+s})\). The roots of this function corresponds to \(y_{tn+s}\), i.e. the next values of the state.

To find the root of this function, we use a Newton-Raphson algorithm. It means the derivative of the function is necessary. At each iteration \(i\) of the algorithm, we solve the following linear system:

\[ \begin{bmatrix} l'_s(t_{n+s}) I & -dt I \\ -dt \frac{\partial F}{\partial q} & l'_s(t_{n+s}) M - dt \frac{\partial F}{\partial \dot{q}} \end{bmatrix} \begin{bmatrix} q^{i+1} - q^i \\ \dot{q}^{i+1} - \dot{q}^{i} \end{bmatrix} = -r(q^i, \dot{q}^i) \]

This system is solved by a block Gaussian elimination leading to the velocity-based linear system:

\[ \left(l'_s(t_{n+s}) M - dt \frac{\partial F}{\partial \dot{q}} - \frac{dt^2}{l'_s(t_{n+s})} \frac{\partial F}{\partial q} \right) (\dot{q}^{i+1} - \dot{q}^i) = -r_{\dot{q}}(q^i, \dot{q}^i) - \frac{dt}{l'_s(t_{n+s})} \frac{\partial F}{\partial q} r_{q}(q^i, \dot{q}^i) \]

Velocity-based ODE solver using Backward Differentiation Formula (BDF), at any order, supporting variable time step size.

Target: Sofa.Component.ODESolver.Backward

namespace: sofa::component::odesolver::backward

parents:

• BaseLinearMultiStepMethod

Data

Name	Description	Default value
name	object name	unnamed
printLog	if true, emits extra messages at runtime.	0
tags	list of the subsets the object 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
order	Order of the numerical method	1
rayleighStiffness	Rayleigh damping coefficient related to stiffness, > 0	0
rayleighMass	Rayleigh damping coefficient related to mass, > 0	0

Links

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
linearSolver	Linear solver used by this component	LinearSolver
newtonSolver	Link to a Newton-Raphson solver to solve the nonlinear equation produced by this numerical method	NewtonRaphsonSolver

Examples

BDFOdeSolver_spring3d.scn

XMLPython

<Node name="root" gravity="0 -9.81 0" dt="0.01">

    <Node name="plugins">
        <RequiredPlugin name="Sofa.Component.Constraint.Projective"/> <!-- Needed to use components [FixedProjectiveConstraint] -->
        <RequiredPlugin name="Sofa.Component.LinearSolver.Direct"/> <!-- Needed to use components [EigenSimplicialLDLT] -->
        <RequiredPlugin name="Sofa.Component.Mass"/> <!-- Needed to use components [UniformMass] -->
        <RequiredPlugin name="Sofa.Component.ODESolver.Backward"/> <!-- Needed to use components [BDFOdeSolver,NewtonRaphsonSolver] -->
        <RequiredPlugin name="Sofa.Component.SolidMechanics.Spring"/> <!-- Needed to use components [SpringForceField] -->
        <RequiredPlugin name="Sofa.Component.StateContainer"/> <!-- Needed to use components [MechanicalObject] -->
        <RequiredPlugin name="Sofa.Component.Topology.Container.Dynamic"/> <!-- Needed to use components [EdgeSetTopologyContainer] -->
        <RequiredPlugin name="Sofa.Component.Visual"/> <!-- Needed to use components [LineAxis,VisualGrid,VisualStyle] -->
    </Node>

    <VisualStyle displayFlags="showWireframe showVisual showBehaviorModels showForceFields" />
    <DefaultAnimationLoop/>
    <VisualGrid size="2"/>
    <LineAxis size="2"/>

    <Node name="BDF-1">

        <BDFOdeSolver order="1" printLog="false" rayleighMass="0.01" rayleighStiffness="0.01"/>
        <NewtonRaphsonSolver name="newton" printLog="false" maxNbIterationsNewton="4" maxNbIterationsLineSearch="2" absoluteResidualStoppingThreshold="1e-6"/>
        <EigenSimplicialLDLT template="CompressedRowSparseMatrix"/>

        <MechanicalObject name="dofs" template="Vec3" position="0 0 0  1 1 0 " showObject="true" showObjectScale="20"/>
        <EdgeSetTopologyContainer name="topologyContainer" edges="0 1" position="@dofs.position"/>
        <UniformMass totalMass="15" topology="@topologyContainer"/>
        <SpringForceField spring="0 1 1000 0 1"/>
        <FixedProjectiveConstraint indices="0" />

    </Node>

    <Node name="BDF-2">

        <BDFOdeSolver order="2" printLog="false" rayleighMass="0.01" rayleighStiffness="0.01"/>
        <NewtonRaphsonSolver name="newton" printLog="false" maxNbIterationsNewton="4" maxNbIterationsLineSearch="2" absoluteResidualStoppingThreshold="1e-6"/>
        <EigenSimplicialLDLT template="CompressedRowSparseMatrix"/>

        <MechanicalObject name="dofs" template="Vec3" position="0 0 0  1 1 0 " showObject="true" showObjectScale="20"/>
        <EdgeSetTopologyContainer name="topologyContainer" edges="0 1" position="@dofs.position"/>
        <UniformMass totalMass="15" topology="@topologyContainer"/>
        <SpringForceField spring="0 1 1000 0 1"/>
        <FixedProjectiveConstraint indices="0" />

    </Node>

    <Node name="BDF-3">

        <BDFOdeSolver order="3" printLog="false" rayleighMass="0.01" rayleighStiffness="0.01"/>
        <NewtonRaphsonSolver name="newton" printLog="false" maxNbIterationsNewton="4" maxNbIterationsLineSearch="2" absoluteResidualStoppingThreshold="1e-6"/>
        <EigenSimplicialLDLT template="CompressedRowSparseMatrix"/>

        <MechanicalObject name="dofs" template="Vec3" position="0 0 0  1 1 0 " showObject="true" showObjectScale="20"/>
        <EdgeSetTopologyContainer name="topologyContainer" edges="0 1" position="@dofs.position"/>
        <UniformMass totalMass="15" topology="@topologyContainer"/>
        <SpringForceField spring="0 1 1000 0 1"/>
        <FixedProjectiveConstraint indices="0" />

    </Node>
</Node>

def createScene(root_node):

   root = root_node.addChild('root', gravity="0 -9.81 0", dt="0.01")

   plugins = root.addChild('plugins')

   plugins.addObject('RequiredPlugin', name="Sofa.Component.Constraint.Projective")
   plugins.addObject('RequiredPlugin', name="Sofa.Component.LinearSolver.Direct")
   plugins.addObject('RequiredPlugin', name="Sofa.Component.Mass")
   plugins.addObject('RequiredPlugin', name="Sofa.Component.ODESolver.Backward")
   plugins.addObject('RequiredPlugin', name="Sofa.Component.SolidMechanics.Spring")
   plugins.addObject('RequiredPlugin', name="Sofa.Component.StateContainer")
   plugins.addObject('RequiredPlugin', name="Sofa.Component.Topology.Container.Dynamic")
   plugins.addObject('RequiredPlugin', name="Sofa.Component.Visual")

   root.addObject('VisualStyle', displayFlags="showWireframe showVisual showBehaviorModels showForceFields")
   root.addObject('DefaultAnimationLoop', )
   root.addObject('VisualGrid', size="2")
   root.addObject('LineAxis', size="2")

   bdf_1 = root.addChild('BDF-1')

   bdf_1.addObject('BDFOdeSolver', order="1", printLog="false", rayleighMass="0.01", rayleighStiffness="0.01")
   bdf_1.addObject('NewtonRaphsonSolver', name="newton", printLog="false", maxNbIterationsNewton="4", maxNbIterationsLineSearch="2", absoluteResidualStoppingThreshold="1e-6")
   bdf_1.addObject('EigenSimplicialLDLT', template="CompressedRowSparseMatrix")
   bdf_1.addObject('MechanicalObject', name="dofs", template="Vec3", position="0 0 0  1 1 0 ", showObject="true", showObjectScale="20")
   bdf_1.addObject('EdgeSetTopologyContainer', name="topologyContainer", edges="0 1", position="@dofs.position")
   bdf_1.addObject('UniformMass', totalMass="15", topology="@topologyContainer")
   bdf_1.addObject('SpringForceField', spring="0 1 1000 0 1")
   bdf_1.addObject('FixedProjectiveConstraint', indices="0")

   bdf_2 = root.addChild('BDF-2')

   bdf_2.addObject('BDFOdeSolver', order="2", printLog="false", rayleighMass="0.01", rayleighStiffness="0.01")
   bdf_2.addObject('NewtonRaphsonSolver', name="newton", printLog="false", maxNbIterationsNewton="4", maxNbIterationsLineSearch="2", absoluteResidualStoppingThreshold="1e-6")
   bdf_2.addObject('EigenSimplicialLDLT', template="CompressedRowSparseMatrix")
   bdf_2.addObject('MechanicalObject', name="dofs", template="Vec3", position="0 0 0  1 1 0 ", showObject="true", showObjectScale="20")
   bdf_2.addObject('EdgeSetTopologyContainer', name="topologyContainer", edges="0 1", position="@dofs.position")
   bdf_2.addObject('UniformMass', totalMass="15", topology="@topologyContainer")
   bdf_2.addObject('SpringForceField', spring="0 1 1000 0 1")
   bdf_2.addObject('FixedProjectiveConstraint', indices="0")

   bdf_3 = root.addChild('BDF-3')

   bdf_3.addObject('BDFOdeSolver', order="3", printLog="false", rayleighMass="0.01", rayleighStiffness="0.01")
   bdf_3.addObject('NewtonRaphsonSolver', name="newton", printLog="false", maxNbIterationsNewton="4", maxNbIterationsLineSearch="2", absoluteResidualStoppingThreshold="1e-6")
   bdf_3.addObject('EigenSimplicialLDLT', template="CompressedRowSparseMatrix")
   bdf_3.addObject('MechanicalObject', name="dofs", template="Vec3", position="0 0 0  1 1 0 ", showObject="true", showObjectScale="20")
   bdf_3.addObject('EdgeSetTopologyContainer', name="topologyContainer", edges="0 1", position="@dofs.position")
   bdf_3.addObject('UniformMass', totalMass="15", topology="@topologyContainer")
   bdf_3.addObject('SpringForceField', spring="0 1 1000 0 1")
   bdf_3.addObject('FixedProjectiveConstraint', indices="0")

BDFOdeSolver.scn

XMLPython

<Node name="root" gravity="-1.8 0 100" dt="0.01">

    <Node name="plugins">
        <RequiredPlugin name="Sofa.Component.Constraint.Projective"/> <!-- Needed to use components [FixedProjectiveConstraint] -->
        <RequiredPlugin name="Sofa.Component.Engine.Select"/> <!-- Needed to use components [BoxROI] -->
        <RequiredPlugin name="Sofa.Component.IO.Mesh"/> <!-- Needed to use components [MeshGmshLoader,MeshOBJLoader] -->
        <RequiredPlugin name="Sofa.Component.LinearSolver.Direct"/> <!-- Needed to use components [EigenSimplicialLDLT] -->
        <RequiredPlugin name="Sofa.Component.LinearSystem"/> <!-- Needed to use components [ConstantSparsityPatternSystem] -->
        <RequiredPlugin name="Sofa.Component.Mapping.Linear"/> <!-- Needed to use components [BarycentricMapping] -->
        <RequiredPlugin name="Sofa.Component.Mass"/> <!-- Needed to use components [MeshMatrixMass] -->
        <RequiredPlugin name="Sofa.Component.ODESolver.Backward"/> <!-- Needed to use components [BDFOdeSolver,NewtonRaphsonSolver] -->
        <RequiredPlugin name="Sofa.Component.SolidMechanics.FEM.HyperElastic"/> <!-- Needed to use components [TetrahedronHyperelasticityFEMForceField] -->
        <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] -->
    </Node>

    <VisualStyle displayFlags="showWireframe showVisual showBehaviorModels hideForceFields" />
    <DefaultAnimationLoop parallelODESolving="false"/>

    <Node name="DeformableObject">

        <BDFOdeSolver order="2" printLog="false" rayleighMass="0.01" rayleighStiffness="0.01"/>
        <NewtonRaphsonSolver name="newton" printLog="false"
                             maxNbIterationsNewton="10" absoluteResidualStoppingThreshold="1e-5"
                             maxNbIterationsLineSearch="5" lineSearchCoefficient="0.5"
                             relativeInitialStoppingThreshold="1e-3"
                             absoluteEstimateDifferenceThreshold="1e-5"
                             relativeEstimateDifferenceThreshold="1e-5"/>
        <ConstantSparsityPatternSystem template="CompressedRowSparseMatrix" name="A" checkIndices="false"/>
        <PCGLinearSolver name="PCG" iterations="1000" preconditioner="@preconditioner"/>
        <AsyncSparseLDLSolver name="preconditioner"/>

        <MeshGmshLoader name="loader" filename="mesh/truthcylinder1.msh" />
        <TetrahedronSetTopologyContainer src="@loader" name="topologyContainer"/>
        <TetrahedronSetGeometryAlgorithms name="geomAlgo"/>
        <MechanicalObject src="@loader" />
        <MeshMatrixMass totalMass="15" topology="@topologyContainer"/>

        <BoxROI name="box" box="-10 -20 -10  10 -17.5 10" drawBoxes="true"/>
        <FixedProjectiveConstraint indices="@box.indices" />
        <TetrahedronHyperelasticityFEMForceField name="FEM" ParameterSet="3448.2759 31034.483" materialName="StVenantKirchhoff"/>

        <Node name="visual">
            <MeshOBJLoader name="meshLoader_0" filename="mesh/truthcylinder1.obj" handleSeams="1" />
            <OglModel name="Visual" src="@meshLoader_0" color="red" />
            <BarycentricMapping input="@.." output="@Visual" />
        </Node>
    </Node>

</Node>

def createScene(root_node):

   root = root_node.addChild('root', gravity="-1.8 0 100", dt="0.01")

   plugins = root.addChild('plugins')

   plugins.addObject('RequiredPlugin', name="Sofa.Component.Constraint.Projective")
   plugins.addObject('RequiredPlugin', name="Sofa.Component.Engine.Select")
   plugins.addObject('RequiredPlugin', name="Sofa.Component.IO.Mesh")
   plugins.addObject('RequiredPlugin', name="Sofa.Component.LinearSolver.Direct")
   plugins.addObject('RequiredPlugin', name="Sofa.Component.LinearSystem")
   plugins.addObject('RequiredPlugin', name="Sofa.Component.Mapping.Linear")
   plugins.addObject('RequiredPlugin', name="Sofa.Component.Mass")
   plugins.addObject('RequiredPlugin', name="Sofa.Component.ODESolver.Backward")
   plugins.addObject('RequiredPlugin', name="Sofa.Component.SolidMechanics.FEM.HyperElastic")
   plugins.addObject('RequiredPlugin', name="Sofa.Component.StateContainer")
   plugins.addObject('RequiredPlugin', name="Sofa.Component.Topology.Container.Dynamic")
   plugins.addObject('RequiredPlugin', name="Sofa.Component.Visual")
   plugins.addObject('RequiredPlugin', name="Sofa.GL.Component.Rendering3D")

   root.addObject('VisualStyle', displayFlags="showWireframe showVisual showBehaviorModels hideForceFields")
   root.addObject('DefaultAnimationLoop', parallelODESolving="false")

   deformable_object = root.addChild('DeformableObject')

   deformable_object.addObject('BDFOdeSolver', order="2", printLog="false", rayleighMass="0.01", rayleighStiffness="0.01")
   deformable_object.addObject('NewtonRaphsonSolver', name="newton", printLog="false", maxNbIterationsNewton="10", absoluteResidualStoppingThreshold="1e-5", maxNbIterationsLineSearch="5", lineSearchCoefficient="0.5", relativeInitialStoppingThreshold="1e-3", absoluteEstimateDifferenceThreshold="1e-5", relativeEstimateDifferenceThreshold="1e-5")
   deformable_object.addObject('ConstantSparsityPatternSystem', template="CompressedRowSparseMatrix", name="A", checkIndices="false")
   deformable_object.addObject('PCGLinearSolver', name="PCG", iterations="1000", preconditioner="@preconditioner")
   deformable_object.addObject('AsyncSparseLDLSolver', name="preconditioner")
   deformable_object.addObject('MeshGmshLoader', name="loader", filename="mesh/truthcylinder1.msh")
   deformable_object.addObject('TetrahedronSetTopologyContainer', src="@loader", name="topologyContainer")
   deformable_object.addObject('TetrahedronSetGeometryAlgorithms', name="geomAlgo")
   deformable_object.addObject('MechanicalObject', src="@loader")
   deformable_object.addObject('MeshMatrixMass', totalMass="15", topology="@topologyContainer")
   deformable_object.addObject('BoxROI', name="box", box="-10 -20 -10  10 -17.5 10", drawBoxes="true")
   deformable_object.addObject('FixedProjectiveConstraint', indices="@box.indices")
   deformable_object.addObject('TetrahedronHyperelasticityFEMForceField', name="FEM", ParameterSet="3448.2759 31034.483", materialName="StVenantKirchhoff")

   visual = DeformableObject.addChild('visual')

   visual.addObject('MeshOBJLoader', name="meshLoader_0", filename="mesh/truthcylinder1.obj", handleSeams="1")
   visual.addObject('OglModel', name="Visual", src="@meshLoader_0", color="red")
   visual.addObject('BarycentricMapping', input="@..", output="@Visual")

BDFOdeSolver_spring1d.scn

XMLPython

<Node name="root" gravity="-9.81 0 0" dt="0.01">
    <RequiredPlugin name="Sofa.Component.Constraint.Projective"/> <!-- Needed to use components [FixedProjectiveConstraint] -->
    <RequiredPlugin name="Sofa.Component.LinearSolver.Direct"/> <!-- Needed to use components [EigenSimplicialLDLT] -->
    <RequiredPlugin name="Sofa.Component.Mass"/> <!-- Needed to use components [MeshMatrixMass] -->
    <RequiredPlugin name="Sofa.Component.ODESolver.Backward"/> <!-- Needed to use components [NewtonRaphsonSolver] -->
    <RequiredPlugin name="Sofa.Component.ODESolver.Integration"/> <!-- Needed to use components [BDF1] -->
    <RequiredPlugin name="Sofa.Component.SolidMechanics.Spring"/> <!-- Needed to use components [SpringForceField] -->
    <RequiredPlugin name="Sofa.Component.StateContainer"/> <!-- Needed to use components [MechanicalObject] -->
    <RequiredPlugin name="Sofa.Component.Visual"/> <!-- Needed to use components [VisualStyle] -->
    <RequiredPlugin name="Sofa.GL.Component.Rendering3D"/> <!-- Needed to use components [OglCylinderModel] -->
    <RequiredPlugin name="Sofa.Component.Topology.Container.Dynamic"/> <!-- Needed to use components [EdgeSetTopologyContainer] -->

    <VisualStyle displayFlags="showWireframe showVisual showBehaviorModels showForceFields" />
    <DefaultAnimationLoop/>
    <VisualGrid size="2"/>
    <LineAxis size="2"/>

    <Node name="BDF-1">

        <BDFOdeSolver order="1" printLog="false" rayleighMass="0.01" rayleighStiffness="0.01"/>
        <NewtonRaphsonSolver name="newton" printLog="false" maxNbIterationsNewton="4" maxNbIterationsLineSearch="2" absoluteResidualStoppingThreshold="1e-6"/>
        <EigenSimplicialLDLT template="CompressedRowSparseMatrix"/>

        <MechanicalObject name="dofs" template="Vec1" position="1 " rest_position="0" showObject="true" showObjectScale="20"/>
        <PointSetTopologyContainer name="topologyContainer" position="1"/>
        <UniformMass totalMass="1" topology="@topologyContainer"/>
        <RestShapeSpringsForceField points="0"/>
    </Node>

    <Node name="BDF-2">

        <BDFOdeSolver order="2" printLog="false" rayleighMass="0.01" rayleighStiffness="0.01"/>
        <NewtonRaphsonSolver name="newton" printLog="false" maxNbIterationsNewton="4" maxNbIterationsLineSearch="2" absoluteResidualStoppingThreshold="1e-6"/>
        <EigenSimplicialLDLT template="CompressedRowSparseMatrix"/>

        <MechanicalObject name="dofs" template="Vec1" position="1 " rest_position="0" showObject="true" showObjectScale="20"/>
        <PointSetTopologyContainer name="topologyContainer" position="1"/>
        <UniformMass totalMass="1" topology="@topologyContainer"/>
        <RestShapeSpringsForceField points="0"/>
    </Node>

    <Node name="BDF-3">

        <BDFOdeSolver order="3" printLog="false" rayleighMass="0.01" rayleighStiffness="0.01"/>
        <NewtonRaphsonSolver name="newton" printLog="false" maxNbIterationsNewton="4" maxNbIterationsLineSearch="2" absoluteResidualStoppingThreshold="1e-6"/>
        <EigenSimplicialLDLT template="CompressedRowSparseMatrix"/>

        <MechanicalObject name="dofs" template="Vec1" position="1 " rest_position="0" showObject="true" showObjectScale="20"/>
        <PointSetTopologyContainer name="topologyContainer" position="1"/>
        <UniformMass totalMass="1" topology="@topologyContainer"/>
        <RestShapeSpringsForceField points="0"/>
    </Node>

</Node>

def createScene(root_node):

   root = root_node.addChild('root', gravity="-9.81 0 0", dt="0.01")

   root.addObject('RequiredPlugin', name="Sofa.Component.Constraint.Projective")
   root.addObject('RequiredPlugin', name="Sofa.Component.LinearSolver.Direct")
   root.addObject('RequiredPlugin', name="Sofa.Component.Mass")
   root.addObject('RequiredPlugin', name="Sofa.Component.ODESolver.Backward")
   root.addObject('RequiredPlugin', name="Sofa.Component.ODESolver.Integration")
   root.addObject('RequiredPlugin', name="Sofa.Component.SolidMechanics.Spring")
   root.addObject('RequiredPlugin', name="Sofa.Component.StateContainer")
   root.addObject('RequiredPlugin', name="Sofa.Component.Visual")
   root.addObject('RequiredPlugin', name="Sofa.GL.Component.Rendering3D")
   root.addObject('RequiredPlugin', name="Sofa.Component.Topology.Container.Dynamic")
   root.addObject('VisualStyle', displayFlags="showWireframe showVisual showBehaviorModels showForceFields")
   root.addObject('DefaultAnimationLoop', )
   root.addObject('VisualGrid', size="2")
   root.addObject('LineAxis', size="2")

   bdf_1 = root.addChild('BDF-1')

   bdf_1.addObject('BDFOdeSolver', order="1", printLog="false", rayleighMass="0.01", rayleighStiffness="0.01")
   bdf_1.addObject('NewtonRaphsonSolver', name="newton", printLog="false", maxNbIterationsNewton="4", maxNbIterationsLineSearch="2", absoluteResidualStoppingThreshold="1e-6")
   bdf_1.addObject('EigenSimplicialLDLT', template="CompressedRowSparseMatrix")
   bdf_1.addObject('MechanicalObject', name="dofs", template="Vec1", position="1 ", rest_position="0", showObject="true", showObjectScale="20")
   bdf_1.addObject('PointSetTopologyContainer', name="topologyContainer", position="1")
   bdf_1.addObject('UniformMass', totalMass="1", topology="@topologyContainer")
   bdf_1.addObject('RestShapeSpringsForceField', points="0")

   bdf_2 = root.addChild('BDF-2')

   bdf_2.addObject('BDFOdeSolver', order="2", printLog="false", rayleighMass="0.01", rayleighStiffness="0.01")
   bdf_2.addObject('NewtonRaphsonSolver', name="newton", printLog="false", maxNbIterationsNewton="4", maxNbIterationsLineSearch="2", absoluteResidualStoppingThreshold="1e-6")
   bdf_2.addObject('EigenSimplicialLDLT', template="CompressedRowSparseMatrix")
   bdf_2.addObject('MechanicalObject', name="dofs", template="Vec1", position="1 ", rest_position="0", showObject="true", showObjectScale="20")
   bdf_2.addObject('PointSetTopologyContainer', name="topologyContainer", position="1")
   bdf_2.addObject('UniformMass', totalMass="1", topology="@topologyContainer")
   bdf_2.addObject('RestShapeSpringsForceField', points="0")

   bdf_3 = root.addChild('BDF-3')

   bdf_3.addObject('BDFOdeSolver', order="3", printLog="false", rayleighMass="0.01", rayleighStiffness="0.01")
   bdf_3.addObject('NewtonRaphsonSolver', name="newton", printLog="false", maxNbIterationsNewton="4", maxNbIterationsLineSearch="2", absoluteResidualStoppingThreshold="1e-6")
   bdf_3.addObject('EigenSimplicialLDLT', template="CompressedRowSparseMatrix")
   bdf_3.addObject('MechanicalObject', name="dofs", template="Vec1", position="1 ", rest_position="0", showObject="true", showObjectScale="20")
   bdf_3.addObject('PointSetTopologyContainer', name="topologyContainer", position="1")
   bdf_3.addObject('UniformMass', totalMass="1", topology="@topologyContainer")
   bdf_3.addObject('RestShapeSpringsForceField', points="0")