Solution Convergence |
 
|
Solution Convergence |
|
Introduction
This article provides some background theory regarding the evaluation of solution convergence within Flexcom.
If you experience convergence issues with a numerical simulation, you may find the Troubleshooting section helpful. It presents some general advice regarding model building, provides recommendations regarding the selection of appropriate solution parameters, and includes some practical advice on what to do if an analysis fails to complete successfully.
Standard Convergence Measure
The standard convergence measure is based on evaluating the change in solution (considering all the degrees of freedom of the hybrid beam column element) from iteration to iteration, and this is the default method for static and time domain dynamic analyses. Specifically, Flexcom determines if convergence has been achieved at each analysis time step by computing a convergence measure t as follows:
t = maximum (t1, t2, t3, t4, t5, t6, t7, t8) = maximum (ti)
where:
(1)
and:
•
is the global degree of freedom. 
varies from 1 to 8, where values 1 to 6 correspond to the six spatial DOFs (motions and rotations), and values 7 and 8 correspond to axial force and torque, respectively
•
is the number of nodes for 
= 1 to 6, and the number of elements for 
= 7 and 8
•
is the solution variable at node or element 
, degree of freedom 
, iteration 
(present iteration)
•
is the solution variable at node or element 
, degree of freedom 
, iteration 
(previous iteration)
Note that the axial force and torque terms are included in the convergence calculations because they are solution variables in the Flexcom hybrid beam element formulation.
If the computed value of t at a particular time step is less than the analysis tolerance measure, then convergence has been achieved and the solution advances to the next time step. If not, a further iteration begins. A maximum number of such iterations is set for each analysis, to prevent indefinite looping. The convergence tolerance measure and the maximum number of iterations are optional inputs, with suitable defaults provided.
The inclusion of torque in the convergence calculations can in some analyses increase the number of iterations required for convergence. This is particularly true of flexible riser systems subjected to waves and current approaching at an angle oblique to the initial plane of the riser. In reality, the actual torque values may be small, and the influence of torque on the solution (as opposed to the rate of achieving this solution) minimal. The Small Torque Value input can be used in such cases to effectively instruct Flexcom to ignore torque in determining convergence.
Specifically, when Flexcom computes t8 it ignores those elements where the torque is below the user-specified or default Small Torque Value. If convergence is slow in a 3D riser analysis, you may wish to increase from the default value to 100, 500 or possibly 1000. It is strongly recommended that if you exercise this option for a series of analyses that you do at least one verification analysis to ensure the assumption about the relative unimportance of torque is valid.
The small torque value is not dimensionless and so the default value is typically 10 Nm when using the SI system of units and 10 ft.lb for the Imperial system.
Further information on this topic is contained in the following sections:
•*TOLERANCE is used to define the analysis convergence tolerance measure and related data.