SECTION 8 INTEGRATOR SETTINGS

COURSE OBJECTIVES What’s in this section: INTEGRATOR Settings: ERROR HMAX PATTERN HINIT, HMIN, MAXIT MODIFIED CORRECTOR ADAPTIVITY INTERPOLATE KMAX

INTEGRATOR CONTROL

INTEGRATOR CONTROL: ERROR Specifies the relative and absolute local error tolerance that the integrator must satisfy at each step, based on comparisons of predicted and corrected values For I3 BDF integrators, Adams/Solver monitors the integration errors in the displacement and state variables that the other differential equations (DIFFs, LSEs, GSEs, and TFSISOs) define SI2 also monitors errors in velocities The value of ERROR is units sensitive (for example, if length unit is mm, specifying ERROR = .001 implies monitoring changes on the order of 1 micron) Default: ERROR=1e-3 except for HHT and Newmark (1e-5) Range: ERROR>=0

INTEGRATOR CONTROL: HMAX Defines the maximum time step that the integrator is allowed to take (can influence accuracy). Default: If INTERPOLATE = ON, the integration step size is limited to HMAX; if HMAX is not defined, no limit is placed on integration step size If INTERPOLATE = OFF, the maximum step size is limited to the output step Use small HMAX with large ERROR value for the integrator to behave like a constant step-size integrator Range: 0 < HMIN < HINIT < HMAX

INTEGRATOR CONTROL: PATTERN Controls how often the Jacobian matrix is to be re-evaluated during the correction phase A value of true (T) indicates that Adams/Solver is evaluating a new Jacobian matrix for that iteration. A value of false (F) indicates that Adams/Solver is using a previously calculated Jacobian matrix as an approximation of the current one. PATTERN=F will adaptively* evaluate the Jacobian. For legacy Fortran Solver, will only evaluate when the corrector has not converged. Useful for almost linear problems. Default (GSTIFF, WSTIFF): T:F:F:F:T:F:F:F:T:F Default (HHT, NEWMARK): F * This algorithm determines a corrector convergence speed and based on this value, extrapolates the system configuration after MAXIT iterations. The Jacobian is updated if the algorithm indicates the convergence rate is too slow for the corrector to meet the convergence criteria.

INTEGRATOR CONTROL: HINIT Defines the initial time step that the integrator attempts Default: 1/20 of the output step Specify small value if high initial transients are present in simulation Can remove spikes in accelerations by giving the integrator chance to get to higher order (3 and above) before first output step Range: 0 < HMIN < HINIT < HMAX

INTEGRATOR CONTROL: HMIN, MAXIT Defines the minimum time step that the integrator is allowed to take. Default: 1.0E-6*HMAX for GSTIFF, WSTIFF, and I3 Range: 0 < HMIN < HINIT < HMAX MAXIT The maximum number of iterations allowed for corrector Default: 10 Range: MAXIT > 0 Tip: if anything, try less iterations rather than more, as too many corrections probably means the prediction is too far from the correction and the step will be rejected. The corrector should converge quadratically so a solution should typically be found within 4 iterations.

INTEGRATOR CONTROL: MODIFIED CORRECTOR Unlike the original corrector, the modified corrector does not require that the error on all solution variables be less than the corrector tolerance The modified corrector is helpful for models containing discontinuities in the forcing functions (problems with contacts fall in this category) Real-world benefits Contact 200% faster Fewer corrector convergence failures Also recommended for Friction and Durability (flexible bodies)

INTEGRATOR CONTROL: CORRECTOR Original Corrector Modified Corrector GSTIFF I3, WSTIFF I3 Error control on displacements, flex modes, applied forces and state variables from DIFFs; i.e., all states except Lagrange multipliers and contact forces. Error control on displacements, flex modes and state variables from differential equations (DIFFs). GSTIFF SI2, WSTIFF SI2, HASTIFF SI1/SI2 Error control on displacements, velocities of body equations, flex modes, flex mode velocities, applied forces, and state variables from DIFFs; i.e., all states except Lagrange multipliers, Lagrange multiplier velocities, and contact forces. Error control on displacements and velocities of body equations, flex modes, flex mode velocities, and state variables from DIFFs. HHT, NEWMARK Error control on accelerations of body equations, flex mode accelerations, and state variables from differential equations (DIFFs). GSTIFF SI1, WSTIFF SI1, CONSTANT_BDF Not supported in the C++ solver

INTEGRATOR CONTROL: ADAPTIVITY Makes the corrector stopping criteria easier to satisfy (legacy Fortran Solver only) The original stopping criteria: becomes: Start small, for example ADAPTIVITY=1e-09. The corrector tolerance should be 10 to 1000 times stricter than the integration tolerance.

INTEGRATOR CONTROL: INTERPOLATE Specifies that the integrator not control its step size to hit an output point. Therefore, when the integrator crosses an output point, it computes a preliminary solution by interpolating to the output point. It then refines or reconciles the solution to satisfy the equations of motion and constraint. Sometimes helps eliminating spikes in the results at the price of slightly larger CPU times INTERPOLATE=OFF turns off interpolation for the chosen integrator Default: OFF for GSTIFF and WSTIFF, ON for ABAM and RKF_45

INTEGRATOR CONTROL: KMAX Indicates the maximum order of the predicting polynomial that the integrator can use Accuracy primarily achieved on integrator order, so any modification can adversely affect the integrator’s accuracy and robustness Useful for models with contact and/or very stiff elements: Reduce to KMAX=2 Irrelevant (ignored) for HHT and Newmark since these are constant order integrators (2 and 1, respectively). Default: KMAX =6 for GSTIFF and WSTIFF. KMAX =12 for ABAM (Fortran solver only).

Available Integrators Stiffly stable, multi-step, variable-order, variable-step integrators: GSTIFF WSTIFF HASTIFF Stiffly stable, single-step, fixed-order, variable-step integrators: Newmark HHT Non-stiff, variable step integrators (Fortran solver only) ABAM (multi-step, variable-order) RKF45 (single-step, constant-order) Stiffly stable, multi-step, variable-order, predominantly fixed-step integrators (Fortran solver only) CONSTANT_BDF

Available Integrators GSTIFF (Gear) Most widely used and tested Adams/SimXpert integrator Implicit, stiffly stable, variable-order, variable-step, multi-step integrator with maximum order of 6. Available in I3 and SI2 formulations. BDF coefficients assume the step size is mostly constant (when step size changes, a small error is introduced) Benefits: Limitations: High speed High accuracy of system displacements Robust in handling a variety of analysis problems Velocities and especially accelerations can have errors. Easy way to minimize this is to control HMAX so that integrator runs at a constant step size and consistently at high order (3 or more) Can encounter corrector failures at small step sizes (using index-3 formulation), which occur because Jacobian is a function of the inverse of the step size and becomes ill-conditioned at small steps.

Available Integrators WSTIFF (Wielenga) Implicit, stiffly stable, variable-order, variable-step, multi-step integrator BDF coefficients are a function of the step size Can change step size without any loss of accuracy, which can reduce artificial acceleration spikes as seen with GSTIFF. Benefits and limitations are similar to those for GSTIFF Available in I3 and SI2 formulations

Available Integrators HASTIFF (Hiller-Anantharaman) SI1 or SI2 variable-order, variable-step, multi-step integrator with maximum order of 6. BDF coefficients assume mostly constant step size. Solve for instead of y (y is discretized as a function of ) Lagrange multipliers appear in differential form, thus allowing an SI1 formulation An index-1 formulation is preferable because it’s more stable numerically, potentially allowing for bigger steps Some characteristics of the SI1 formulation (default): Ensures that the solution satisfies both position and velocity constraints Ensures that velocities and accelerations satisfy all first and second time derivatives Monitors integration error only in system displacements and velocities Is as fast as GSTIFF SI2 Jacobian matrix generally cannot be come ill-conditioned even at small time steps Application Any model where force, acceleration, or velocity is of importance

Available Integrators Newmark and HHT Priority of HHT and Newmark is speed rather than accuracy The default ERROR for HHT and Newmark is 1e-5. This was calibrated so that HHT will give roughly the same accuracy as GSTIFF with its default ERROR (1e-3). Benefits: Limitations: More efficient than GSTIFF – fewer Jacobian evaluations and corrector iterations; no regions of instability in left half plane; no divisions by step size Unlike BDF formulas, it behaves like a low-pass filter and cuts high frequency spurious oscillations Can control the cutoff frequency by adjusting a; the smaller the value, the lower the cutoff frequency Stable at small integration time steps Because of the reduced order, the acceleration and reaction forces obtained with this I3 formulation will be more spiky. Use HMAX to reduce acceleration spikes. The numerical damping associated with HHT is smaller than that produced by BDF integrators used by the GSTIFF family If a BDF integrator manages to run at order 4 or higher, it will take step sizes significantly larger than HHT. This is because HHT is a low-order integrator and needs to limit the step size based on accuracy characteristics.

Available Integrators Newmark Implicit I3 integrator of the form: Thus it is required to specify g and b parameters or accept the defaults A-stable (stability in the whole left plane) is guaranteed when First order method unless g = ¼ and b = ½, which makes Newmark the same as the trapezoidal rule (second order). Trapezoidal rule has poor damping properties, which is unacceptable for problems having either High oscillations of no interest Parasitic oscillations that are a byproduct of the FE discretization process

Available Integrators HHT (Hilbert-Hughes-Taylor) Implicit I3 integrator similar to Newmark Requires you to specify a parameter or accept the default. The following parameters are then calculated: Rather than discretizing the EOM according to like Newmark, HHT discretizes according to: The idea is that by scaling the current value of the force by 1 + a and the previous value of force by a, the equations of motion become more efficient to solve with this type of integrator (Newmark effectively sets a =0) where