ACOS numerical errors in Adams/Solver

Question

I'm using vector operations to extract an angle during a simulation. The measurement consists of the dot product of two unit vectors which I then take the arc-cosine to extract the angle.

data_element create variable &

variable_name=.model_1.VAR_MdotX &

function= "UVX(MARKER_1)*UVX(MARKER_2)"

data_element create variable &

variable_name=.model_1.VAR_Angle &

function = "ACOS(VARVAL(VAR_MdotX))"

Occasionally in large (5000-10000) case Monte Carlo simulations I will get a handful of cases that fail in the middle of the simulation with errors like the following:

---- START: ERROR ----

an invalid argument value occurred for function ACOS. Argument

of ACOS must have an absolute value less than or equal to 1.

The absolute value of the argument is 20.11355. The function is not

defined for this value. If this is a numerical problem, consider

using an expression involving ASIN, ATAN or ATAN2 instead.

I'm confused on why VAR_MdotX would ever evaluate to greater than 1 given that its the dot product of two unit vectors. Is storing MdotX in a separate state variable messing up the ACOS operation? I did this because MdotX is used elsewhere in the model for some other calculations.

Anyone have any thoughts on how to resolve/understand this issue?

Jesper Slattengren · Accepted Answer

To further spin on Henrik's suggestion:

By storing the intermediate results in a state variable, you have created exactly that; a state in the equation system. That means that this will be part of Adams solver routines, not only be subjected to error control but also to predictor behavior.

When Adams attempts to take a step forward, it starts with predicting all states for the new time point t+dt based on the current and previous values of the states {t,t-dt,t-2dt...} (it also uses the state derivative at the current point). Then it feeds those predicted states into the equation system to see how far from zero the solution is (Adams uses implicit equation formulation; F-ma=0 basically). In this step it is easy for the predictor to "guess" an impossible value as it is a simple extrapolation at this stage. This is where you might see the impossible solution that might crash your run.

By doing what Henrik suggests, putting the argument directly inside the ACOS function, you remove one state from the equation system and this intermediate result will never be extrapolated.

PS. After the prediction phase, Adams solver uses a Newton-Raphson method to refine the solution and bring it close enough to zero to satisfy the error criteria.