EulerExplicitSolver

The EulerExplicitSolver component belongs to the category of integration schemes or ODE Solver. This scheme allows to solve dynamic systems explicitly: all forces will be computed based on the state information at the current time step $Current position$ .

Looking at continuum mechanics, the linear system $Linear system$ arises from the dynamic equation. This dynamic is written as follows but other physics (like heat transfer) result in a similar equation:

$Dynamic system$

where $DOF at next time step system$ is the degrees of freedom, $Mass matrix$ the mass matrix and $Forces$ a function of $DOF$ (and possibly its derivatives) acting on our system. In the case of the EulerExplicitSolver, this equation can be written:

$Explicit dynamic system$

since forces only depend on known state (at our current time step). These forces are computed by the ForceField in the addForce() function. The system matrix $System matrix$ is only equal to the mass matrix $Mass matrix$ .

Depending on whether the mass matrix is diagonal or not, SOFA supports two cases:

1) The mass matrix is diagonal. It makes the resolution of the linear system trivial (best performances). In this case, the system matrix $System matrix$ equals a diagonal mass matrix $Mass matrix$ which is diagonal and it can be stored as a vector $Mass vector$ . Moreover, its inverse can directly be obtained as: $Inverse mass matrix$ . The solution $Explicit resolution$ finally corresponds to a division operation of $Explicit forces$ by the mass. This computation is actually performed by the Mass component in the accFromF() function. Therefore, no LinearSolver is needed to compute directly or iteratively a solution. 2) The mass matrix is not diagonal. Solving the system requires a linear solver.

Sequence diagram

Data

The data symplectic allows to modify the scheme to make it symplectic, i.e. velocities are updated before the positions. It allows to update the positions from the newly computed velocities, instead of velocities from the previous time step. This option makes the scheme more robust in time. EulerExplicitSolver is symplectic by default.

Usage

The EulerExplicitSolver requires a MechanicalObject to store the state vectors. However, as explained above, no LinearSolver is needed and the EulerExplicitSolver is only working using a UniformMass or DiagonalMass, which ensures to have a diagonal system matrix.

Example

This component is used as follows in XML format:

<EulerExplicitSolver name="odeExplicitSolver" />

or using SofaPython3:

node.addObject('EulerExplicitSolver', name='odeExplicitSolver')

Examples of scenes involving a EulerExplicitSolver are available in examples/Component/ODESolver/Forward/EulerExplicitSolver:

EulerExplicitSolver.scn: non-symplectic and non-diagonal mass matrix
EulerExplicitSolver_diagonal.scn: non-symplectic and diagonal mass matrix
EulerSymplecticSolver.scn: symplectic and non-diagonal mass matrix
EulerSymplecticSolver_diagonal.scn: symplectic and diagonal mass matrix

Target: Sofa.Component.ODESolver.Forward

namespace: sofa::component::odesolver::forward

parents:

OdeSolver

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
symplectic	If true, the velocities are updated before the positions and the method is symplectic (more robust). If false, the positions are updated before the velocities (standard Euler, less robust).	1
threadSafeVisitor	If true, do not use realloc and free visitors in fwdInteractionForceField.	0

Links:

Name	Description
context	Graph Node containing this object (or BaseContext::getDefault() if no graph is used)
slaves	Sub-objects used internally by this object
master	nullptr for regular objects, or master object for which this object is one sub-objects
linearSolver	Linear solver used by this component

Examples

Component/ODESolver/Forward/EulerExplicitSolver_diagonal.scn

XMLPython

<!--
This scene shows an example of a forward Euler integration scheme.
This is the variant of the component EulerExplicitSolver where the Data
'symplectic' is set to false (true by default).
In this example, the mass is diagonal. Since it can be inverted
trivially, it does not require a linear solver.
-->

<Node name="root" gravity="0 -9.81 0" dt="0.00001">
    <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 [MeshOBJLoader] -->
    <RequiredPlugin name="Sofa.Component.Mapping.Linear"/> <!-- Needed to use components [IdentityMapping] -->
    <RequiredPlugin name="Sofa.Component.Mass"/> <!-- Needed to use components [UniformMass] -->
    <RequiredPlugin name="Sofa.Component.ODESolver.Forward"/> <!-- Needed to use components [EulerExplicitSolver] -->
    <RequiredPlugin name="Sofa.Component.SolidMechanics.Spring"/> <!-- Needed to use components [MeshSpringForceField] -->
    <RequiredPlugin name="Sofa.Component.StateContainer"/> <!-- Needed to use components [MechanicalObject] -->
    <RequiredPlugin name="Sofa.Component.Topology.Container.Dynamic"/> <!-- Needed to use components [HexahedronSetGeometryAlgorithms QuadSetTopologyContainer QuadSetTopologyModifier] -->
    <RequiredPlugin name="Sofa.Component.Topology.Container.Grid"/> <!-- Needed to use components [RegularGridTopology] -->
    <RequiredPlugin name="Sofa.Component.Topology.Mapping"/> <!-- Needed to use components [Hexa2QuadTopologicalMapping] -->
    <RequiredPlugin name="Sofa.Component.Visual"/> <!-- Needed to use components [VisualStyle] -->
    <RequiredPlugin name="Sofa.GL.Component.Rendering3D"/> <!-- Needed to use components [OglModel] -->
    <DefaultAnimationLoop/>
    <VisualStyle displayFlags="showBehaviorModels showForceFields" />

    <Node name="DeformableObject">

        <!-- Matrix system is diagonal because integration is explicit and Mass is diagonal -->
        <!-- No need for a LinearSolver -->
        <EulerExplicitSolver name="odeExplicitSolver" symplectic="false"/>

        <MechanicalObject name="dofs"/>

        <RegularGridTopology name="topology" nx="4" ny="4" nz="11" xmin="-1.5" xmax="1.5" ymin="-1.5" ymax="1.5" zmin="0" zmax="10" />
        <HexahedronSetGeometryAlgorithms/>
        <UniformMass totalMass="15"/>

        <BoxROI box="-1.5 -1.5 0 1.5 1.5 0.0001" name="box"/>
        <FixedProjectiveConstraint indices="@box.indices" />
        <MeshSpringForceField stiffness="3E2"/>

        <Node name="visual">
            <QuadSetTopologyContainer  name="Container" />
            <QuadSetTopologyModifier/>
            <Hexa2QuadTopologicalMapping input="@../topology" output="@Container" />
            <OglModel name="Visual" color="yellow" quads="@Container.quads" />
            <IdentityMapping input="@../dofs" output="@Visual" />
        </Node>

    </Node>

    <Node name="floor-visual">
        <MeshOBJLoader name="meshLoader" filename="mesh/floorFlat.obj" scale3d="0.5 0.5 0.5"/>
        <OglModel src="@meshLoader" dy="-8" dz="10"/>
        <OglModel src="@meshLoader" rx="90" dy="2"/>
    </Node>
</Node>

def createScene(rootNode):

    root = rootNode.addChild('root', gravity="0 -9.81 0", dt="0.00001")
    root.addObject('RequiredPlugin', name="Sofa.Component.Constraint.Projective")
    root.addObject('RequiredPlugin', name="Sofa.Component.Engine.Select")
    root.addObject('RequiredPlugin', name="Sofa.Component.IO.Mesh")
    root.addObject('RequiredPlugin', name="Sofa.Component.Mapping.Linear")
    root.addObject('RequiredPlugin', name="Sofa.Component.Mass")
    root.addObject('RequiredPlugin', name="Sofa.Component.ODESolver.Forward")
    root.addObject('RequiredPlugin', name="Sofa.Component.SolidMechanics.Spring")
    root.addObject('RequiredPlugin', name="Sofa.Component.StateContainer")
    root.addObject('RequiredPlugin', name="Sofa.Component.Topology.Container.Dynamic")
    root.addObject('RequiredPlugin', name="Sofa.Component.Topology.Container.Grid")
    root.addObject('RequiredPlugin', name="Sofa.Component.Topology.Mapping")
    root.addObject('RequiredPlugin', name="Sofa.Component.Visual")
    root.addObject('RequiredPlugin', name="Sofa.GL.Component.Rendering3D")
    root.addObject('DefaultAnimationLoop')
    root.addObject('VisualStyle', displayFlags="showBehaviorModels showForceFields")

    DeformableObject = root.addChild('DeformableObject')
    DeformableObject.addObject('EulerExplicitSolver', name="odeExplicitSolver", symplectic="false")
    DeformableObject.addObject('MechanicalObject', name="dofs")
    DeformableObject.addObject('RegularGridTopology', name="topology", nx="4", ny="4", nz="11", xmin="-1.5", xmax="1.5", ymin="-1.5", ymax="1.5", zmin="0", zmax="10")
    DeformableObject.addObject('HexahedronSetGeometryAlgorithms')
    DeformableObject.addObject('UniformMass', totalMass="15")
    DeformableObject.addObject('BoxROI', box="-1.5 -1.5 0 1.5 1.5 0.0001", name="box")
    DeformableObject.addObject('FixedProjectiveConstraint', indices="@box.indices")
    DeformableObject.addObject('MeshSpringForceField', stiffness="3E2")

    visual = DeformableObject.addChild('visual')
    visual.addObject('QuadSetTopologyContainer', name="Container")
    visual.addObject('QuadSetTopologyModifier')
    visual.addObject('Hexa2QuadTopologicalMapping', input="@../topology", output="@Container")
    visual.addObject('OglModel', name="Visual", color="yellow", quads="@Container.quads")
    visual.addObject('IdentityMapping', input="@../dofs", output="@Visual")

    floor-visual = root.addChild('floor-visual')
    floor-visual.addObject('MeshOBJLoader', name="meshLoader", filename="mesh/floorFlat.obj", scale3d="0.5 0.5 0.5")
    floor-visual.addObject('OglModel', src="@meshLoader", dy="-8", dz="10")
    floor-visual.addObject('OglModel', src="@meshLoader", rx="90", dy="2")

Component/ODESolver/Forward/EulerExplicitSolver.scn

XMLPython

<!--
This scene shows an example of a forward Euler integration scheme.
This is the variant of the component EulerExplicitSolver where the Data
'symplectic' is set to false (true by default).
In this example, the mass is not diagonal. Since it cannot be inverted
trivially, it requires a linear solver, here SparseLDLSolver.
-->

<Node name="root" gravity="0 -9.81 0" dt="0.00001">
    <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 [MeshOBJLoader] -->
    <RequiredPlugin name="Sofa.Component.LinearSolver.Direct"/> <!-- Needed to use components [SparseLDLSolver] -->
    <RequiredPlugin name="Sofa.Component.Mapping.Linear"/> <!-- Needed to use components [IdentityMapping] -->
    <RequiredPlugin name="Sofa.Component.Mass"/> <!-- Needed to use components [MeshMatrixMass] -->
    <RequiredPlugin name="Sofa.Component.ODESolver.Forward"/> <!-- Needed to use components [EulerExplicitSolver] -->
    <RequiredPlugin name="Sofa.Component.SolidMechanics.Spring"/> <!-- Needed to use components [MeshSpringForceField] -->
    <RequiredPlugin name="Sofa.Component.StateContainer"/> <!-- Needed to use components [MechanicalObject] -->
    <RequiredPlugin name="Sofa.Component.Topology.Container.Dynamic"/> <!-- Needed to use components [HexahedronSetGeometryAlgorithms QuadSetTopologyContainer QuadSetTopologyModifier] -->
    <RequiredPlugin name="Sofa.Component.Topology.Container.Grid"/> <!-- Needed to use components [RegularGridTopology] -->
    <RequiredPlugin name="Sofa.Component.Topology.Mapping"/> <!-- Needed to use components [Hexa2QuadTopologicalMapping] -->
    <RequiredPlugin name="Sofa.Component.Visual"/> <!-- Needed to use components [VisualStyle] -->
    <RequiredPlugin name="Sofa.GL.Component.Rendering3D"/> <!-- Needed to use components [OglModel] -->
    <DefaultAnimationLoop/>

    <VisualStyle displayFlags="showBehaviorModels showForceFields" />

    <Node name="DeformableObject">

        <EulerExplicitSolver name="odeExplicitSolver" symplectic="false"/>
        <SparseLDLSolver />

        <MechanicalObject name="dofs"/>

        <RegularGridTopology name="topology" nx="4" ny="4" nz="11" xmin="-1.5" xmax="1.5" ymin="-1.5" ymax="1.5" zmin="0" zmax="10" />
        <HexahedronSetGeometryAlgorithms/>
        <MeshMatrixMass totalMass="15"/>

        <BoxROI box="-1.5 -1.5 0 1.5 1.5 0.0001" name="box"/>
        <FixedProjectiveConstraint indices="@box.indices" />
        <MeshSpringForceField stiffness="3E2"/>

        <Node name="visual">
            <QuadSetTopologyContainer  name="Container" />
            <QuadSetTopologyModifier/>
            <Hexa2QuadTopologicalMapping input="@../topology" output="@Container" />
            <OglModel name="Visual" color="yellow" quads="@Container.quads" />
            <IdentityMapping input="@../dofs" output="@Visual" />
        </Node>

    </Node>

    <Node name="floor-visual">
        <MeshOBJLoader name="meshLoader" filename="mesh/floorFlat.obj" scale3d="0.5 0.5 0.5"/>
        <OglModel src="@meshLoader" dy="-8" dz="10"/>
        <OglModel src="@meshLoader" rx="90" dy="2"/>
    </Node>
</Node>

def createScene(rootNode):

    root = rootNode.addChild('root', gravity="0 -9.81 0", dt="0.00001")
    root.addObject('RequiredPlugin', name="Sofa.Component.Constraint.Projective")
    root.addObject('RequiredPlugin', name="Sofa.Component.Engine.Select")
    root.addObject('RequiredPlugin', name="Sofa.Component.IO.Mesh")
    root.addObject('RequiredPlugin', name="Sofa.Component.LinearSolver.Direct")
    root.addObject('RequiredPlugin', name="Sofa.Component.Mapping.Linear")
    root.addObject('RequiredPlugin', name="Sofa.Component.Mass")
    root.addObject('RequiredPlugin', name="Sofa.Component.ODESolver.Forward")
    root.addObject('RequiredPlugin', name="Sofa.Component.SolidMechanics.Spring")
    root.addObject('RequiredPlugin', name="Sofa.Component.StateContainer")
    root.addObject('RequiredPlugin', name="Sofa.Component.Topology.Container.Dynamic")
    root.addObject('RequiredPlugin', name="Sofa.Component.Topology.Container.Grid")
    root.addObject('RequiredPlugin', name="Sofa.Component.Topology.Mapping")
    root.addObject('RequiredPlugin', name="Sofa.Component.Visual")
    root.addObject('RequiredPlugin', name="Sofa.GL.Component.Rendering3D")
    root.addObject('DefaultAnimationLoop')
    root.addObject('VisualStyle', displayFlags="showBehaviorModels showForceFields")

    DeformableObject = root.addChild('DeformableObject')
    DeformableObject.addObject('EulerExplicitSolver', name="odeExplicitSolver", symplectic="false")
    DeformableObject.addObject('SparseLDLSolver')
    DeformableObject.addObject('MechanicalObject', name="dofs")
    DeformableObject.addObject('RegularGridTopology', name="topology", nx="4", ny="4", nz="11", xmin="-1.5", xmax="1.5", ymin="-1.5", ymax="1.5", zmin="0", zmax="10")
    DeformableObject.addObject('HexahedronSetGeometryAlgorithms')
    DeformableObject.addObject('MeshMatrixMass', totalMass="15")
    DeformableObject.addObject('BoxROI', box="-1.5 -1.5 0 1.5 1.5 0.0001", name="box")
    DeformableObject.addObject('FixedProjectiveConstraint', indices="@box.indices")
    DeformableObject.addObject('MeshSpringForceField', stiffness="3E2")

    visual = DeformableObject.addChild('visual')
    visual.addObject('QuadSetTopologyContainer', name="Container")
    visual.addObject('QuadSetTopologyModifier')
    visual.addObject('Hexa2QuadTopologicalMapping', input="@../topology", output="@Container")
    visual.addObject('OglModel', name="Visual", color="yellow", quads="@Container.quads")
    visual.addObject('IdentityMapping', input="@../dofs", output="@Visual")

    floor-visual = root.addChild('floor-visual')
    floor-visual.addObject('MeshOBJLoader', name="meshLoader", filename="mesh/floorFlat.obj", scale3d="0.5 0.5 0.5")
    floor-visual.addObject('OglModel', src="@meshLoader", dy="-8", dz="10")
    floor-visual.addObject('OglModel', src="@meshLoader", rx="90", dy="2")