22.8. Exercises

22.8.1. Reproduce the simulation

of the cover page of: Nonlinear Solid Mechanics for Finite Element Analysis: Statics, J. Bonet, A.J. Gil, A.D. Wood

https://www.amazon.com/Nonlinear-Mechanics-Finite-Element-Analysis/dp/1107115795

Increase the force using load-stepping. To rotate the direction of the force use the deformation gradient \(F\):

def F(u):
    return Id(u.dim) + Grad(u)
a += (F(u.Trace())*force)*v*ds("top")

22.8.2. Dynamics of beams

• start from the notebook dynamics, and replace the Dirichlet boundary condition by mean value constraints for the displacement (but not for the rotation).

• simpulate a double pendulum by coupling two beams via mean value constraints

22.8.3. Snap-through problem

Apply mean value displacement constraints left and right, and increase the vertical force. Plot force vs mean vertical displacement in the middle (matplotlib).

from netgen.occ import *
from ngsolve.webgui import Draw

shape = MoveTo(0,0).Rotate(20).Arc(2,-40).Rotate(90).Line(0.05,name="right") \
    .Rotate(90).Arc(2.05,40).Close(name="left").Face()
cx,cy,cz = shape.center
edge = Segment((cx,0,0), (cx,1,0)) * shape
edge.name = "mid"
shape = Glue([ shape, edge] )
Draw (shape);