 # Overview

This document presents a discussion on the -omega option that is part of the &EQUI command. The &EQUI command in MOSES computes a static equilibrium position with a modified Newton method.  Also presented is an example when the -omega option in the &EQUI command is used to reach equilibrium.

# Basic Theory

You can find a good description of the Newton method on Wikipedia.  The article in Wikipedia shows a sample of a single curve with one x-axis intercept.  The x-axis intercept is found after various tangent lines are calculated.  In our engineering world, the curve represents the force and moments at various positions and the tangents represent the stiffness of the system at each position.  The method presented works very efficiently for a single curve.  In the offshore world, bodies have six degrees of freedom.  There are at least six curves f(x), f(y), . . . f(rz) needed.  The axis we are interested in intercepting is the one represented by all forces and moments equal to zero.

In an ideal world, one would find the position where the forces and the moments equal zero.  The tangent lines would find a position where all six degrees of freedom intercept the axis.  Since this is an iterative method, it can be very time-consuming and thus inefficient, but there are options to help with efficiency.  The tolerance option is unique, and it controls how close to zero the forces and moments must get for the iterative process to stop.   The other options control the size, or distances, between each iteration.  This work discusses the omega option which controls the calculation of the tangent lines in the Newton method.

A modified Newton method is used on the force and moment curves vs. position.  These curves will include any non-linearities due to hydrostatic stiffness and connectors.  Non-linearities in hydrostatic stiffness can be due to changes in shape.  Connectors such as catenary mooring lines are non-linear.  When there are several degrees of freedom to consider one cannot calculate the curves at all possible positions before starting the iterative process.  One method to get around this is to compute the tangent at each iteration from the change in forces and moments from the previous position.  For a change in position that does not result in a change in forces and moments, the tangent is flat.  For a change in position that changes forces and moments drastically, the tangent is near vertical. Depending on the offshore procedure being analyzed, there can be one or more curves that result in tangent lines that do not get the solution near the axis intercept. In these cases, the other &EQUI options can help.

Heave, roll and pitch motions are normally well behaved for an analysis of a single floating body without mooring lines.  Since there are no restraints in the surge, sway, and yaw motions the tangents for those curves could be flat.  Another example would be a barge moored to a quay with fenders and mooring lines.  For analysis with several fenders where not all fenders are engaged at each position, the non-linearities can cause the tangent to be almost vertical.  To avoid possible singularities in the stiffness matrix, MOSES augments the stiffness with a fraction of the inertia.  The equation used for stiffness is: In the above examples, for an analysis of a single floating body, omega defaults to 0.2236.  For the example where flexible connectors are used, omega defaults to 0.0236. In essence, the stiffness is augmented with a fraction of the inertia.

# Sample Problem

Here an example where a change in omega is needed is reviewed.  For this example, the up_damage sample files are used as found in ..\hdesk\runs\samples\how_to directory. A training exercise is also based on this set of files; please contact support@bentley.com for a copy.

The files model three compartments in a three-legged jacket.  The analysis files use the time domain to flood compartment Three and find an equilibrium position with the flooded leg.  In this example, the &EQUI command is used to find the new equilibrium position.  Reviewing the geometry, one would expect the flooding of compartment Three to rotate the jacket such that compartment Three is lowest in the water, or nearer to the seabed.  One also expects a pitch angle since the jacket buoyancy properties are not evenly distributed.  The jacket undamaged free-floating position is shown in Figure 1 and Figure 2.  The jacket in the equilibrium position after damaging compartment Three is shown in Figure 3 and Figure 4. Figure 1: View from jacket top elevation in undamaged floating position (Click picture for larger view) Figure 2:  View showing face floating on water plane in undamaged floating position (Click picture for larger view) Figure 3: Side view of damaged floating position – Compartment Three is flooded (Click picture for larger view) Figure 4:  View from jacket top elevation for damaged floating position – Compartment Three is flooded (Click picture for larger view)

The calculations presented here find equilibrium with the &EQUI command and three different values of omega: 0.1, 0.2236 and 0.6.  The default value of 0.2236 will be referred to as “default.”   The intent is to show a value smaller and a value larger in comparison to the default value. The files monitor the heave, roll, and pitch position at each iteration.  Plots showing the position vs. iteration are shown below in Figure 5 to Figure 7.  These plots also show the time domain results.  Plots showing the change in position vs. iteration are shown in Figure 8 to Figure 10.  A total of 150 iterations are shown.  The plots show the equilibrium position for omega = 0.6 and the time domain results agree.

In Figure 5 to Figure 7, one can see that for the default value and omega = 0.1 value plots, the trace oscillates.  This oscillation will be called chatter.  For roll, Figure 6, and pitch, Figure 7, the results for omega = 0.1 are farther from the solution in comparison to the default omega.  Reducing the omega value caused the solution to be farther from the solution.  The chatter is amplified in Figure 8 to Figure 10.

Chatter usually implies the slope of the tangent is overestimating the next step.  The overestimate results in a reverse-overestimate to compensate the prior step.  This constant overestimate results in the chatter.  To exit the chatter a different step between the iterations is needed; however, we do not know how the step size.  Not knowing the step size, makes finding a different step size difficult.

The other &EQUI options let the user specify the step size, whereas the omega option lets the program change the step size depending on the stiffness calculations.  The omega option eliminates the need to know the step size. An increase in omega will decrease the step size.  Taking the recommendation, we should increase the value of omega.  It is evident that increasing omega to 0.6 results in a smooth curve without any chattering.  This is shown in figure 5 to 7, position vs. iteration.

Instead of chattering, oscillations, the motion could show values that move without a definite convergence.  Motion without a convergence will be referred to as wandering. Wandering is common in systems with mooring systems or other connectors. For a system that is wandering a decrease in omega is the recommended action.  A decrease in omega will decrease the dependence on the mass properties and increase the dependence on the connector stiffness.  The default omega for systems with connectors is 0.02236.  The default for systems with connectors is a factor of ten smaller in comparison to systems without connectors.  The default for systems with connectors is ten-squared times less dependent on the mass properties in comparison to systems without connectors.

Lastly, it is worth comparing the results to those obtained in the time domain.  Another method of finding equilibrium is via the time domain.  We see in Figure 5 to Figure7 that the time domain results and the omega = 0.6 are converging on the same values.  The convergence to the same values lends confidence in our results. Figure 5: Heave position vs iteration (Click picture for larger view) Figure 6:  Roll position vs iteration (Click picture for larger view) Figure 7: Pitch position vs iteration Figure 8:  Change in heave position vs iterations (Click picture for larger view) Figure 9:  Change in roll position vs iterations (Click picture for larger view) Figure 10:  Change in Pitch position vs iterations (Click picture for larger view)

# Summary

In summary, for multi-body, procedures with connectors, or procedures with an unrestrained degree of freedom the static equilibrium can be difficult to find.  MOSES offers the omega option to increase efficiency.  It would be difficult to recommend specific values since stiffness is problem-specific and relative to the mass of the bodies and the stiffness of the connectors in the analysis.  We hope the example used here is helpful.