SimTK::SemiExplicitEuler2Integrator Class Reference
This is an implementation of a variable-step, first-order semi-explicit Euler method, also known as semi-implicit Euler or symplectic Euler. More...
Inheritance diagram for SimTK::SemiExplicitEuler2Integrator:
Public Member Functions | |
| SemiExplicitEuler2Integrator (const System &sys) | |
| Create a SemiExplicitEuler2Integrator for integrating a System with variable size steps. More... | |
| Public Member Functions inherited from SimTK::Integrator | |
| Integrator () | |
| ~Integrator () | |
| const char * | getMethodName () const |
| Get the name of this integration method. More... | |
| int | getMethodMinOrder () const |
| Get the minimum order this Integrator may use. More... | |
| int | getMethodMaxOrder () const |
| Get the maximum order this Integrator may use. More... | |
| bool | methodHasErrorControl () const |
| Get whether this Integrator provides error control. More... | |
| void | initialize (const State &state) |
| Supply the integrator with a starting state. More... | |
| void | reinitialize (Stage stage, bool shouldTerminate) |
| After an event handler has made a discontinuous change to the Integrator's "advanced state", this method must be called to reinitialize the Integrator. More... | |
| const State & | getState () const |
| Return a State corresponding to the "current" time at the end of the last call to stepTo() or stepBy(). More... | |
| Real | getTime () const |
| Get the time of the current State. This is equivalent to calling getState().getTime(). More... | |
| bool | isStateInterpolated () const |
| Get whether getState() will return an interpolated state or just the same thing as getAdvancedState() does. More... | |
| const State & | getAdvancedState () const |
| Return the state representing the trajectory point to which the integrator has irreversibly advanced. More... | |
| Real | getAdvancedTime () const |
| Get the time of the advanced State. This is equivalent to calling getAdvancedState().getTime(). More... | |
| State & | updAdvancedState () |
| Get a non-const reference to the advanced state. More... | |
| Real | getAccuracyInUse () const |
| Get the accuracy which is being used for error control. More... | |
| Real | getConstraintToleranceInUse () const |
| Get the constraint tolerance which is being used for error control. More... | |
| SuccessfulStepStatus | stepTo (Real reportTime, Real scheduledEventTime=Infinity) |
| Integrate the System until something happens which requires outside processing, and return a status code describing what happened. More... | |
| SuccessfulStepStatus | stepBy (Real interval, Real scheduledEventTime=Infinity) |
| Integrate the System until something happens which requires outside processing, and return a status code describing what happened. More... | |
| Vec2 | getEventWindow () const |
| Get the window (tLow, tHigh] within which one or more events have been localized. More... | |
| const Array_< EventId > & | getTriggeredEvents () const |
| Get the IDs of all events which have been localized within the event window. More... | |
| const Array_< Real > & | getEstimatedEventTimes () const |
| Get the estimated times of all events which have been localized within the event window. More... | |
| const Array_< Event::Trigger > & | getEventTransitionsSeen () const |
| Get EventTriggers describing the events which have been localized within the event window. More... | |
| bool | isSimulationOver () const |
| Get whether the simulation has terminated. More... | |
| TerminationReason | getTerminationReason () const |
| Get the reason the simulation terminated. More... | |
| void | resetAllStatistics () |
| Reset all statistics to zero. More... | |
| Real | getActualInitialStepSizeTaken () const |
| Get the size of the first successful step after the last initialize() call. More... | |
| Real | getPreviousStepSizeTaken () const |
| Get the size of the most recent successful step. More... | |
| Real | getPredictedNextStepSize () const |
| Get the step size that will be attempted first on the next call to stepTo() or stepBy(). More... | |
| int | getNumStepsAttempted () const |
| Get the total number of steps that have been attempted (successfully or unsuccessfully) since the last call to resetAllStatistics(). More... | |
| int | getNumStepsTaken () const |
| Get the total number of steps that have been successfully taken since the last call to resetAllStatistics(). More... | |
| int | getNumRealizations () const |
| Get the total number of state realizations that have been performed since the last call to resetAllStatistics(). More... | |
| int | getNumQProjections () const |
| Get the total number of times a state positions Q have been projected since the last call to resetAllStatistics(). More... | |
| int | getNumUProjections () const |
| Get the total number of times a state velocities U have been projected since the last call to resetAllStatistics(). More... | |
| int | getNumProjections () const |
| Get the total number of times a state has been projected (counting both Q and U projections) since the last call to resetAllStatistics(). More... | |
| int | getNumErrorTestFailures () const |
| Get the number of attempted steps that have failed due to the error being unacceptably high since the last call to resetAllStatistics(). More... | |
| int | getNumConvergenceTestFailures () const |
| Get the number of attempted steps that failed due to non-convergence of internal step iterations. More... | |
| int | getNumRealizationFailures () const |
| Get the number of attempted steps that have failed due to an error when realizing the state since the last call to resetAllStatistics(). More... | |
| int | getNumQProjectionFailures () const |
| Get the number of attempted steps that have failed due to an error when projecting the state positions (Q) since the last call to resetAllStatistics(). More... | |
| int | getNumUProjectionFailures () const |
| Get the number of attempted steps that have failed due to an error when projecting the state velocities (U) since the last call to resetAllStatistics(). More... | |
| int | getNumProjectionFailures () const |
| Get the number of attempted steps that have failed due to an error when projecting the state (either a Q- or U-projection) since the last call to resetAllStatistics(). More... | |
| int | getNumConvergentIterations () const |
| For iterative methods, get the number of internal step iterations in steps that led to convergence (not necessarily successful steps). More... | |
| int | getNumDivergentIterations () const |
| For iterative methods, get the number of internal step iterations in steps that did not lead to convergence. More... | |
| int | getNumIterations () const |
| For iterative methods, this is the total number of internal step iterations taken regardless of whether those iterations led to convergence or to successful steps. More... | |
| void | setFinalTime (Real tFinal) |
| Set the time at which the simulation should end. More... | |
| void | setInitialStepSize (Real hinit) |
| Set the initial step size that should be attempted. More... | |
| void | setMinimumStepSize (Real hmin) |
| Set the minimum step size that should ever be used. More... | |
| void | setMaximumStepSize (Real hmax) |
| Set the maximum step size that should ever be used. More... | |
| void | setFixedStepSize (Real stepSize) |
| Set the integrator to use a single fixed step size for all steps. More... | |
| void | setAccuracy (Real accuracy) |
| Set the overall accuracy that should be used for integration. More... | |
| void | setConstraintTolerance (Real consTol) |
| Set the tolerance within which constraints must be satisfied. More... | |
| void | setUseInfinityNorm (bool useInfinityNorm) |
| (Advanced) Use infinity norm (maximum absolute value) instead of default RMS norm to evaluate whether accuracy has been achieved for states and for constraint tolerance. More... | |
| bool | isInfinityNormInUse () const |
| (Advanced) Are we currently using the infinity norm? More... | |
| void | setInternalStepLimit (int nSteps) |
| Set the maximum number of steps that may be taken within a single call to stepTo() or stepBy(). More... | |
| void | setReturnEveryInternalStep (bool shouldReturn) |
| Set whether the Integrator should return from stepTo() or stepBy() after every internal step, even if no event has occurred and the report time has not been reached. More... | |
| void | setProjectEveryStep (bool forceProject) |
| Set whether the system should be projected back to the constraint manifold after every step. More... | |
| void | setAllowInterpolation (bool shouldInterpolate) |
| Set whether the Integrator is permitted to return interpolated states for reporting purposes which may be less accurate than the "real" states that form the trajectory. More... | |
| void | setProjectInterpolatedStates (bool shouldProject) |
| Set whether interpolated states should be projected back to the constraint manifold after interpolation is performed. More... | |
| void | setForceFullNewton (bool forceFullNewton) |
| (Advanced) Constraint projection may use an out-of-date iteration matrix for efficiency. More... | |
Additional Inherited Members | |
| Public Types inherited from SimTK::Integrator | |
| enum | SuccessfulStepStatus { ReachedReportTime =1, ReachedEventTrigger =2, ReachedScheduledEvent =3, TimeHasAdvanced =4, ReachedStepLimit =5, EndOfSimulation =6, StartOfContinuousInterval =7, InvalidSuccessfulStepStatus = -1 } |
| When a step is successful, it will return an indication of what caused it to stop where it did. More... | |
| enum | TerminationReason { ReachedFinalTime = 1, AnUnrecoverableErrorOccurred = 2, EventHandlerRequestedTermination = 3, InvalidTerminationReason = -1 } |
| Once the simulation has ended, getTerminationReason() may be called to find out what caused it to end. More... | |
| Static Public Member Functions inherited from SimTK::Integrator | |
| static String | getSuccessfulStepStatusString (SuccessfulStepStatus) |
| Get a human readable description of the reason a step returned. More... | |
| static String | getTerminationReasonString (TerminationReason) |
| Get a human readable description of the termination reason. More... | |
| static String | successfulStepStatusString (SuccessfulStepStatus stat) |
| OBSOLETE: use getSuccessfulStepStatusString(). More... | |
| Protected Member Functions inherited from SimTK::Integrator | |
| const IntegratorRep & | getRep () const |
| IntegratorRep & | updRep () |
| Protected Attributes inherited from SimTK::Integrator | |
| IntegratorRep * | rep |
Detailed Description
This is an implementation of a variable-step, first-order semi-explicit Euler method, also known as semi-implicit Euler or symplectic Euler.
The fixed-step, first-order semi-explicit Euler method is very popular for game engines such as ODE, and Simbody offers that method as SemiExplicitEulerIntegrator. See the documentation there for a discussion and theory. This integrator is an attempt to extend that method to give some error control. For equivalent performance to the fixed-step method, this one needs to run at twice the step size (because it takes two substeps). Be sure to set its maximum allowable step large enough, and use a very loose accuracy setting.
Theory
See SemiExplicitEulerIntegrator for the theory behind the underlying first order method. Here we use Richardson Extrapolation (step doubling) to get an error estimate. See Solving Ordinary Differential Equations I: Nonstiff Problems, 2nd rev. ed., Hairer, Norsett, & Wanner, section II.4, pp 164-165. Note that we are not using the local extrapolation trick to get a higher-order result; see below for why.
The idea is to take two steps of length h, and a single big step of length 2h, and combine the results. Starting at t=t0 we can get to t2=t0+2h two different ways:
u1 = u0 + h udot(q0,u0) ub = u0 + 2h udot(q0,u0)
q1 = q0 + h N(q0)u1 and qb = q0 + 2h N(q0)ub
u2 = u1 + h udot(q1,u1) (b = big step)
q2 = q1 + h N(q1)u2
Note that we can use udot(q0,u0) in both methods so it doesn't cost much to calculate both solutions y2(t2)=(q2,u2) and yb(t2)=(qb,ub).
Now assume the true solution at t2 is y(t2)=(q,u). Then
y = y2 + 2C h^2 + O(h^3)
y = yb + C(2h)^2 + O(h^3)
where C is some unknown constant of the underlying method. Ignoring the 3rd order terms, subtract these two equations to get:
y2-yb = 2Ch^2
which is the 2nd order error term for y2. We can use that for error control.
Hairer suggests using that error term to improve on y2 to get a better estimate for y (this is called "local extrapolation") :
y2' = y2 + (y2-yb) = 2 y2 - yb
y = y2' + O(h^3) <-- we don't actually do this
y2' is a 2nd order accurate estimate for y(t2). But ... we don't have an error estimate for the revised value and we don't actually know that it is more accurate. In fact, although this sounds lovely in theory, in practice we will often be running at close to a stability boundary and using the "big step" value to update the result causes stability problems. So we'll skip the local extrapolation and stick with the first order result from the two half-steps.
Constructor & Destructor Documentation
◆ SemiExplicitEuler2Integrator()
| SimTK::SemiExplicitEuler2Integrator::SemiExplicitEuler2Integrator | ( | const System & | sys | ) |
Create a SemiExplicitEuler2Integrator for integrating a System with variable size steps.
The documentation for this class was generated from the following file:
