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     Jerry Brown (jerrybrown11743) Member Username: jerrybrown11743 Post Number: 57 Registered: 03-2004 |
I’m having trouble getting a model of a thin cantilevered beam to converge. This is thin steel strip, 10 inches long, 1 inch wide and 1/16 inch thick. My original intention was to demonstrate the benefit of using nonlinear elasticity theory for large deflections by testing a real sample and comparing the results with both a linear and nonlinear model. I expected meshing to be a problem with a long, thin 3D model. So I tried both mesh controls and coordinate scaling. Neither seems to help. The non-scaled model starts out looking promising. But, slows exponentially and is still 56% away from the load boundary condition after 2 days of running. I’ve only run the scaled model for a day. But, it was no further along at that point than the non-scaled model. I’ve tinkered with a number of things. But, nothing seems to help. I’ve learned that if the end surface (normal to the long dimension of the beam), where the load is applied, is not finely meshed, things are really hopeless. In the FEA literature I’ve read about something called shear locking. Could this be my problem. See the attached file. Any suggestions would be appreciated? Thank you, Jerry Brown
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| Jerry Brown (jerrybrown11743) Member Username: jerrybrown11743 Post Number: 58 Registered: 03-2004 |
I see that the pdf on shear locking didn't upload. I'll send that by email. | ||||
| Jerry Brown (jerrybrown11743) Member Username: jerrybrown11743 Post Number: 59 Registered: 03-2004 |
I should also mention that I'm using version 5.1.2 on an Intel 2.4 GHz quad core CPU running Windows XP. | ||||
| Robert G. Nelson (rgnelson) Moderator Username: rgnelson Post Number: 1304 Registered: 06-2003 |
1. I don't believe the shear locking discussion is related to your problems. In the first place, he talks primarily about accuracy, and mostly with linear interpolation models. FlexPDE defaults to a quadratic model. We don't have a problem with accuracy of deflection in the standard model, as shown in the Bentbar.pde example. 2. The trouble you are having is that the solver is unable to find a solution to the coupling equations. This implies that the matrix is nearly singular, so that the real solution is lost in the roundoff error of the computation. Successive attempts to modify the trial solution merely superimpose new noise on the previous noise. The nonlinear equations are extremely nasty, and when they are differentiated to form the Jacobian matrix, they will be even nastier. Many (all?) of the nonlinear terms are products of derivatives, which is a tricky situation. The product is positive as long as the two factors have the same sign, but it can't tell the difference between both positive and both negative. This makes the dependency matrix indeterminate and allows oscillations. I would be happier if this could be recast into a form that has the linear terms in front and a secondary nonlinear term with something simple like ux multiplied by a horrible coefficient. The horrible coefficient could then be hidden in a SAVE() so that it is not differentiated. SAVE() is updated at the beginning of each Newton step. So, ultimately, the value will be consistent with the solution, unless the SAVED quantity is so dominant that no convergence is achieved. It hides its contents from differentiation in computing the slope for Newton's method, and if used properly it can avoid expensive computations and avoid erratic Jacobian terms. 3. A pseudo-time model might allow relaxation from the linear to the nonlinear solution. If you start with a solution of the linear system and add the nonlinear terms in a pseudo-time model, the timestep controls should allow the solution to creep toward the nearest nonlinear solution, rather than trying to jump instantly to the (perhaps multi-valued) steady-state solution. 4. If this is simply a matter of modeling large deflections, why not use a moving mesh model? I have attached a moving mesh version of the Bentbar example problem.
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Unscaled model