NewmarkImplicitSolver

This component belongs to the category of integration schemes or ODE Solver.

This scheme is an implicit time integrator for dynamic system using the Newmark scheme. To compute the new position or new velocity, the NewmarkImplicitSolver is based on the following equations:

\[ x_{t+h}=x_t+h v_t+\frac{h^2}{2}((1-2\beta)a_t+2\beta a_{t+h}) \]

\[ v_{t+h}=v_t+h((1-\gamma)a_t+\gamma a_{t+h}) \]

Applied to a mechanical system where \(\small Ma_t+(r_MM+r_KK)v_t+Kx_t=f_{ext}\), we need to solve the following system:

\[ \tiny Ma_{t+h}+(r_MM+r_KK)v_{t+h}+Kx_{t+h}=f_{ext} \]

\[ \tiny Ma_{t+h}+(r_MM+r_KK)(v_t+h((1-\gamma)a_t+\gamma a_{t+h}))+K(x_t+hv_t+\frac{h^2}{2}((1-2\beta)a_t+2\beta a_{t+h}))=f_{ext} \]

\[ \tiny (M+h\gamma(r_MM+r_KK)+h^2\beta K)a_{t+h}=f_{ext}-(r_MM+r_KK)(v_t+h(1-\gamma)a_t)-K(x_t+hv_t+\frac{h^2(1-2\beta)}{2}a_t) \]

\[ \tiny ((1+h\gamma r_M)M+(h^2\beta +h\gamma r_K)K)a_{t+h}=f_{ext}-(r_MM+r_KK)v_t-Kx_t-(r_MM+r_KK)(h(1-\gamma)a_t)-K(hv_t+\frac{h^2(1-2\beta)}{2}a_t) \]

\[ \tiny ((1+h\gamma r_M)M+(h^2\beta+h\gamma r_K)K)a_{t+h}=a_t-(r_MM+r_KK)(h(1-\gamma)a_t)-K(hv_t+\frac{h^2(1-2\beta)}{2}a_t) \]

Sequence diagram

Usage

At each simulation step and each Newton Raphson iteration, the NewmarkImplicitSolver requires:

a LinearSolver to solve the linear system
and a MechanicalObject to store the state vectors.

Implicit time integrator using Newmark scheme.

Target: Sofa.Component.ODESolver.Backward

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

parents:

OdeSolver
LinearSolverAccessor

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
rayleighStiffness	Rayleigh damping coefficient related to stiffness	0
rayleighMass	Rayleigh damping coefficient related to mass	0
vdamping	Velocity decay coefficient (no decay if null)	0
gamma	Newmark scheme gamma coefficient	0.5
beta	Newmark scheme beta coefficient	0.25
threadSafeVisitor	If true, do not use realloc and free visitors in fwdInteractionForceField.	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

Examples

NewmarkImplicitSolver.scn

XMLPython

<Node name="root" gravity="-1.8 0 100" 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.Direct"/> <!-- Needed to use components [SparseLDLSolver] -->
    <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 [UniformMass] -->
    <RequiredPlugin name="Sofa.Component.ODESolver.Backward"/> <!-- Needed to use components [NewmarkImplicitSolver] -->
    <RequiredPlugin name="Sofa.Component.SolidMechanics.FEM.Elastic"/> <!-- Needed to use components [TetrahedronFEMForceField] -->
    <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.Constant"/> <!-- Needed to use components [MeshTopology] -->
    <RequiredPlugin name="Sofa.GL.Component.Rendering3D"/> <!-- Needed to use components [OglModel] -->
    <DefaultAnimationLoop/>

    <Node name="Reference">
        <MeshOBJLoader name="meshLoader_0" filename="mesh/truthcylinder1-bent.obj" scale="0.95" handleSeams="1" />
        <OglModel src="@meshLoader_0" dx="0" dy="-1" dz="0" color="green" />
    </Node>
    <Node name="Springs">
        <NewmarkImplicitSolver rayleighMass="0" rayleighStiffness="0.1" />
        <SparseLDLSolver/>
        <MeshGmshLoader name="loader" filename="mesh/truthcylinder1.msh" />
        <MeshTopology src="@loader" />
        <MechanicalObject src="@loader" dx="15" />
        <UniformMass totalMass="15" />
        <FixedProjectiveConstraint indices="0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 268 269 270 271 343 345" />
        <MeshSpringForceField name="Spring" tetrasStiffness="1870" tetrasDamping="0" />

        <Node>
            <MeshOBJLoader name="meshLoader_3" filename="mesh/truthcylinder1.obj" handleSeams="1" />
            <OglModel name="Visual" src="@meshLoader_3" color="yellow" dx="15" />
            <BarycentricMapping input="@.." output="@Visual" />
        </Node>
    </Node>

    <Node name="CoFEM">
        <NewmarkImplicitSolver rayleighMass="0" rayleighStiffness="0.1" />
        <CGLinearSolver iterations="100" tolerance="1e-5" threshold="1e-5"/>
        <MeshGmshLoader name="loader" filename="mesh/truthcylinder1.msh"/>
        <MeshTopology src="@loader" />
        <MechanicalObject src="@loader" dx="30" />
        <UniformMass totalMass="15" />
        <FixedProjectiveConstraint indices="0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 268 269 270 271 343 345" />
        <TetrahedronFEMForceField name="FEM" youngModulus="1116" poissonRatio="0.49" method="polar" />

        <Node>
            <MeshOBJLoader name="meshLoader_2" filename="mesh/truthcylinder1.obj" handleSeams="1" />
            <OglModel name="Visual" src="@meshLoader_2" color="cyan" dx="30" />
            <BarycentricMapping input="@.." output="@Visual" />
        </Node>
    </Node>
    <Node name="LinearFEM">
        <NewmarkImplicitSolver rayleighMass="0" rayleighStiffness="0.1" />
        <CGLinearSolver iterations="100" tolerance="1e-5" threshold="1e-5"/>
        <MeshGmshLoader name="loader" filename="mesh/truthcylinder1.msh"/>
        <MeshTopology src="@loader" />
        <MechanicalObject src="@loader" dx="45" />
        <UniformMass totalMass="15" />
        <FixedProjectiveConstraint indices="0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 268 269 270 271 343 345" />
        <TetrahedronFEMForceField name="FEM" youngModulus="1116" poissonRatio="0.49" method="small" />

        <Node>
            <MeshOBJLoader name="meshLoader_1" filename="mesh/truthcylinder1.obj" handleSeams="1" />
            <OglModel name="Visual" src="@meshLoader_1" color="red" dx="45" />
            <BarycentricMapping input="@.." output="@Visual" />
        </Node>
    </Node>
</Node>

def createScene(root_node):

   root = root_node.addChild('root', gravity="-1.8 0 100", dt="0.02")

   root.addObject('RequiredPlugin', name="Sofa.Component.Constraint.Projective")
   root.addObject('RequiredPlugin', name="Sofa.Component.IO.Mesh")
   root.addObject('RequiredPlugin', name="Sofa.Component.LinearSolver.Direct")
   root.addObject('RequiredPlugin', name="Sofa.Component.LinearSolver.Iterative")
   root.addObject('RequiredPlugin', name="Sofa.Component.Mapping.Linear")
   root.addObject('RequiredPlugin', name="Sofa.Component.Mass")
   root.addObject('RequiredPlugin', name="Sofa.Component.ODESolver.Backward")
   root.addObject('RequiredPlugin', name="Sofa.Component.SolidMechanics.FEM.Elastic")
   root.addObject('RequiredPlugin', name="Sofa.Component.SolidMechanics.Spring")
   root.addObject('RequiredPlugin', name="Sofa.Component.StateContainer")
   root.addObject('RequiredPlugin', name="Sofa.Component.Topology.Container.Constant")
   root.addObject('RequiredPlugin', name="Sofa.GL.Component.Rendering3D")
   root.addObject('DefaultAnimationLoop', )

   reference = root.addChild('Reference')

   reference.addObject('MeshOBJLoader', name="meshLoader_0", filename="mesh/truthcylinder1-bent.obj", scale="0.95", handleSeams="1")
   reference.addObject('OglModel', src="@meshLoader_0", dx="0", dy="-1", dz="0", color="green")

   springs = root.addChild('Springs')

   springs.addObject('NewmarkImplicitSolver', rayleighMass="0", rayleighStiffness="0.1")
   springs.addObject('SparseLDLSolver', )
   springs.addObject('MeshGmshLoader', name="loader", filename="mesh/truthcylinder1.msh")
   springs.addObject('MeshTopology', src="@loader")
   springs.addObject('MechanicalObject', src="@loader", dx="15")
   springs.addObject('UniformMass', totalMass="15")
   springs.addObject('FixedProjectiveConstraint', indices="0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 268 269 270 271 343 345")
   springs.addObject('MeshSpringForceField', name="Spring", tetrasStiffness="1870", tetrasDamping="0")

   node = Springs.addChild('node')

   node.addObject('MeshOBJLoader', name="meshLoader_3", filename="mesh/truthcylinder1.obj", handleSeams="1")
   node.addObject('OglModel', name="Visual", src="@meshLoader_3", color="yellow", dx="15")
   node.addObject('BarycentricMapping', input="@..", output="@Visual")

   co_fem = root.addChild('CoFEM')

   co_fem.addObject('NewmarkImplicitSolver', rayleighMass="0", rayleighStiffness="0.1")
   co_fem.addObject('CGLinearSolver', iterations="100", tolerance="1e-5", threshold="1e-5")
   co_fem.addObject('MeshGmshLoader', name="loader", filename="mesh/truthcylinder1.msh")
   co_fem.addObject('MeshTopology', src="@loader")
   co_fem.addObject('MechanicalObject', src="@loader", dx="30")
   co_fem.addObject('UniformMass', totalMass="15")
   co_fem.addObject('FixedProjectiveConstraint', indices="0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 268 269 270 271 343 345")
   co_fem.addObject('TetrahedronFEMForceField', name="FEM", youngModulus="1116", poissonRatio="0.49", method="polar")

   node = CoFEM.addChild('node')

   node.addObject('MeshOBJLoader', name="meshLoader_2", filename="mesh/truthcylinder1.obj", handleSeams="1")
   node.addObject('OglModel', name="Visual", src="@meshLoader_2", color="cyan", dx="30")
   node.addObject('BarycentricMapping', input="@..", output="@Visual")

   linear_fem = root.addChild('LinearFEM')

   linear_fem.addObject('NewmarkImplicitSolver', rayleighMass="0", rayleighStiffness="0.1")
   linear_fem.addObject('CGLinearSolver', iterations="100", tolerance="1e-5", threshold="1e-5")
   linear_fem.addObject('MeshGmshLoader', name="loader", filename="mesh/truthcylinder1.msh")
   linear_fem.addObject('MeshTopology', src="@loader")
   linear_fem.addObject('MechanicalObject', src="@loader", dx="45")
   linear_fem.addObject('UniformMass', totalMass="15")
   linear_fem.addObject('FixedProjectiveConstraint', indices="0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 268 269 270 271 343 345")
   linear_fem.addObject('TetrahedronFEMForceField', name="FEM", youngModulus="1116", poissonRatio="0.49", method="small")

   node = LinearFEM.addChild('node')

   node.addObject('MeshOBJLoader', name="meshLoader_1", filename="mesh/truthcylinder1.obj", handleSeams="1")
   node.addObject('OglModel', name="Visual", src="@meshLoader_1", color="red", dx="45")
   node.addObject('BarycentricMapping', input="@..", output="@Visual")