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Adams Solver Training notes

Solver phases

In short, the Adams solver is of a predictor-corrector type. Time-steps are taken by estimating the

next step based on a number of previous step, and subsequently correcting the guess based on an

extended set of equations.

The Adams Solver works in four phases for each solver step during dynamic simulation. The first

phase is a predictor phase, where the system state at next time-step is estimated. Second step is a

corrector phase, where the estimated state is corrected. Third phase is a error checking phase, that

forces the solver to find the solution of the set of equations closest to the previous solution. The

fourth phase is a preparation phase, that prepares the solver to estimate the next time step.

Phase 1 - predictor

As mentioned above, the predictor

bases the guess q for the state-vector

q for the next step as a function of

previous values.

The number of previous states included in

the extrapolation, and the order of

extrapolation is determined in phase four.

In order to start the simulation, the solver

uses some special routines and small step

sizes in order to obtain a state history to for

the extrapolation.

Phase 2 corrector phase

The corrector phase evaluates the predicted value. The purpose of this evaluation is to obtain the

correct state vector for the next time-step ( tn1 ). The cost function Gq, q , t evaluates the

set of equations describing constraints (reaction forces), applied forces, and user written differential

equations for positions (formulation i3) or position and velocity (formulation si2). The correct

value is found by using a modified Newton-Raphson algorithm to minimize the output of

Gq, q , t .

If the corrector failes in obtaining a corrected value within HMAX attempts, the solver returns to

the predictor phase to attempt with a new time-step.

Time

q

q

tn1tntn1tn2tn3

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Phase 3 - error checking phase

The basis of the error checking is the difference q between the predicted state vector and the

corrected. This is normed into a scalar value and then compared to the ERROR value in the solver

settings dialog box. The comparison is based on:

qERROR

1000ADAPTIVITY

t

If this requirement is met, the corrected state-vector is accepted.

The ERROR value has a very direct connection to the precision of the output, as for some setup and

simple models, the value of the error refers directly to the unit system used for modelling.

The ADAPTIVITY was previously used to relax the error-checking, but is with new improvements

of the solver less used. It is worth noticing, that the ADAPTIVITY can have severe influence when

the solver is forced to use very small time-steps!

If the new state vector failes to comply with the ERROR requirement, the solver returns to phase 1

to attempt once again.

Phase 4 Preparation phase

This phase prepares the solver for the next time step. It determines the size of the time-step, the

number of previous states to be used for predicting the next state (predictor order), it saves data to

the output file.

Solver setting dialog box

KMAX

Extrapolation polynomium order for the predictor phase.

Note: for GSTIFF, KMAX may not be set higher than 6!!

HINIT

Initial time-step.

Can improve solver performance at simulation start-up.

HMIN

Minimum allowed time-step

HMAX

Maximum allowed time-step

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Default setting is based on solver command inputs -> 5sec/500 step = HMAX_default =

0.01s

ERROR

Accuracy of solution

Se phase 3

MAXIT

Max number of iteration in phase 2.

General comment: Never set higher than 10 if the NR have not converged after 10

iterations it is better to reduce step-size.

ADAPTIVITY

Relaxing error-checking. See phase 3.

FORMULATION

Different ways of formulating the set of equations. i3 (default) has only error checking on

positions, whereas si2 includes error-checking on velocities.

i3 is more compact = faster solution, for small time-steps the jacobian become ill-

conditioned!

si2 better accuracy on velocities. Stable for all time-step!

Generally a switch from i3 to si2 can be followed by a larger error tolerance (test it!)

INTERPOLATE ON/OFF

Interpolate is used in phase four, when the solver writes to the output file. If OFF, the solver

MUST match a time-step with the output write time. When ON, the solver will interpolate to

find values for output steps when not matched by a solver step.

Advantage: Some solver behave very bad with large variation in time-step (especially

WSTIFF) noticable as peaks in the accelaration output.

SCALE

PATTERN FOR JACOBIAN

The Jacobian used by the NR is not generated for every iteration in phase 2. By pattern for

Jacobian the update of the jacobian can be altered.

Error tuning

1. Run a simulation with fx ERROR = 1E-3 (mm)

2. Plot a displacement of interest

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3. Run a new simulation with 10 times finer ERROR (1E-4) and plot the displacement from 2.

4. Repeat step 2 and 3 until there no longer are any significant difference in the results.

5. Select the larger ERROR setting of the two latest simulations.

Hmax tuning

1. Run a simulation with fx HMAX = 0.01s

2. Plot acceleration of interest

3. Run a new simulation with 2-3 times finer HMAX

4. Repeat until there are no longer a significant difference in simulation output5. Select the larger HMAX of the last two simulations.