Trans RINA, Vol 153, Part B2, Intl J Small Craft Tech, 2011 Jul-Dec and 212   H H32 H H33


H H31 H H32  23


 22


 


 


 21  22


  11 


structure is described by a dampened vibrating system with virtual masses on the nodes and link forces to describe the elements. The solution then is based on a time strepping scheme.


Under the observations above we now can state that the formulation for 12 tension, while 22


tension. As well we can state that 11 ε T ε   from the above formulae with  11 11  


of  11 


 22 22 


  .   


11  


  , 11 is positive  


 22w  


 holds true for general uniaxial  is restricted to natural uniaxial  is the same


regardless of natural or general uniaxial tension. Therefore we can calculate  from element rotation


or ε  for the state of natural uniaxial tension   .


Given the two methods to determine , now we have to numerically find the angle  where, under the condition


is given by 


definite,  12 w   22


  


12 22 22      and and is


always positive finite. At this angle the above condition for 


σm will be fulfilled. This is done by bracketing of the values of 11


roots [14] as recommended in [6] and using Ridders’s method to find it to a predefined degree (e.g. 0.01*pi). Assuming the case of a wrinkled element two roots are returned, one gives the angle of  the other the angle of   and 22


 from the material axes


.  is determined by checking the  at these angles. This approach is


quite stable as long as the structural properties of axes of the sail’s material are basically balanced (meaning not uni-directional).


For the special case of an isotropic material can be calculated directly from the directional strains  by


tan 2 


   


 2 12 11 22


In the current (debug) implementation, the analysis of anisotropic wrinkling increases runtime four to five times compared to a purely isotropic wrinkling analysis. This is due to the computational effort for finding the wrinkling angle numerically as opposed to direct calculation and depends a lot on the fraction of wrinkled elements in the total structure.


5. FE – SOLVER


The solver used so far for the structural part of FlexSail was based based on the minimization of total potential energy using a modified Newton approach [2]. However it was not able to treat the strong structural nonlinearities associated with wrinkling. Thus a new solver has been implemented.


Promising the necessary stability, a kinetically damped Dynamic Relaxation approach was chosen to solve the finite-element case [8].


In this approach separate equations for equilibrium and compatibility are used. The B-74 t  t 2:


Basically the motion of any node i at time t can be described by Newton’s 2nd law of motion as


i m V R


t with


i t


i


t Ri being the vectorial sum of all forces (internal


and external) acting on node i at time t.


In centred difference form this acceleration term can be approximated as:


V V i  t


i


tt 2 Vi t


 


tt 2 .


This yields the following term for nodal velocities at time 


Vi tt 2  Vi tt 2   


t m R


i t


i


The updated geometry projected to time t therefore given by:


t  x


t i


t  x   Vi


t i


t tt 2


The isotropic virtual masses used above are calculated by i


m t S 2


2 i


With Si being the largest direct stiffness that may occur during analysis.


To get the dynamic relaxation solver to converge some kind of damping method is necessary. Typically used is either viscous or kinetic damping. For FlexSail kinetic damping was chosen as it gives robust performance with little computational overhead. In a kinetically damped system kinetic energy peaks of the whole vibrating system are detected and all nodal velocities set to zero before releasing the nodes again.


As described in [8], due to “the separation of equilibrium and compatibility” giving a vectorial formulation of the problem, no global stiffness matrix has to be constructed, keeping computational


is


overhead low. The vectorial


formulation lends itself to parallelising using a SPMD paradigm on a multi-core machine. An initial, un- optimised OpenMP implementation has shown a significant reduction of runtime.


©2011: The Royal Institution of Naval Architects


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