The global equation of motion in the time domain is given by:

(13–86)

where is the assembled structural and added mass matrix described in Equation 13–3 in the fixed reference axes, is the unknown acceleration vector, and

is the total applied force vector.

In order to integrate the estimated acceleration for the structure velocity and position, Aqwa uses a 2-stage predictor-corrector algorithm.

Stage 1 - Predictor Stage

First, the total applied force is calculated, typically a function of the known time, position, and velocity:

(13–87)

If a user-defined external force routine is used, the external routine is called at this point with stage = 1.

The acceleration is solved by substituting Equation 13–87 into Equation 13–86. The results are output at this point (during the first stage of the calculation).

The predicted velocity and position at time t + dt are given by:

(13–88)

where the superscript * indicates intermediate results in the predictor-corrector algorithm.

Stage 2 - Corrector Stage

To begin stage 2, the total applied force is estimated at time t + dt:

(13–89)

If a user-defined external force routine is used, the external routine is called at this point with stage = 2.

The acceleration at time t + dt is solved by substituting Equation 13–89 into Equation 13–86.

The corrected velocity and position is calculated at time t + dt, using Taylor’s theorem:

(13–90)

The structure is then moved to the new position, the time is incremented, and the process is repeated from stage 1.