Lecture 17--> Flashcards
What are superconvergent points
Points where the order of convergence of displacement and stresses is at leas one order higher than the convergence rate anticipated from the polynomial order of the shape functions
Best accuracy for the gradient or stresses is obtained at
Gauss point corresponding to a quadrature rule one order lower than full integration
What are the coordinates r and s used for?
Used when interpolating Gauss point stresses.
r = square(3) xi
s = square (3) eta
Element smoothing is done to
To obtain improved stresses as to have accurate stresses within each element
Nodal averaging
Method to obtain improved stresses. Average stress at a node is the sum of the FE stress components computed at node “a” in element “e”, divided by number of elements
What is global smoothing
Nodal values of the smoothed stress field is determined by a least-squares fit, minimising the square of the difference between the smoothed stress field and the discontinuous FE stress field
What is superconvergent patch recovery?
Local smoothing based kn stresses sampled at superconvergent points. The smoothed stress field is expressed as a polynomial.
Why is the unsmoothed FE stress field sometimes better when assessing the accuracy
The amount of discontinuity gives a measure of the accuracy if the FE solution
How many displacement patterns does the Q4 element have?
8
3 rigid body
3 constant strain
2 bending
Bending deformation of the Q4 element always displays
spurious shear strains as well as the expected bending
Does the Q4 exhibit shear locking behaviour?
Yes
What is the parasitic shear effect?
The reason behind shear locking. Absorbs strain energy. If the Q4 element is exposed to bending deformation, the bending moment needed to produce it is larger than the correct value
What is the remedy for the overly stiff Q4 element when bent?
Adding two additional modes, referred to as incompatible displacement modes. Allow edges of the element to become curved.
—> Q6
How many shape functions do the Q6 element have?
6 shape functions
Why is Q6 called incompatible?
Because overlap or gaps may appear between adjacent elements
What is the QM6 element?
A modified version of Q6, that satisfies the strong form of the patch test and thus converges much faster
Can the QM6 represent pure bending exactly?
Yes, but only if the element is rectangular
What is reduced integration
Integration with a quadrature rule of less than full order
What can reduced integration lead to in regards to the rank?
It may lead to a rank-deficient k, which means that the rank of k is less than its order minus the possible rigid body modes. Will exist one or more spurious modes
What are spurious modes?
Deformations that may be activated without any increase in the strain energy. Also called zero-energy modes
For n Gauss point, rank (k) =
Min {3n, ndof-nr}
Can selective integration avoid shear locking behavior?
Yes, by applying reduced integration to the shear part of the strain only
What can you say about the CST element and bending?
Bending deformation of the CST element always displays spurious shear strains.
The CST element exhibit shear locking behaviour
What can you say about the LST element and bending?
The LST element can represent pure bending exactly
The ___ element cannot model the linearly varying normal stress, neither the quadratic varying sheer stress
CST
The ___ element can exactly model the normal stress, but not the quadratic varying shear stress
LST
The ___ element can model both stresses exactly
QST
CST and bending
The constant strain tetrahedron is poor at representing fields of bending or twisting strains
LST and bending
Unless the element edges become curved, the linear strain tetrahedron can exactly represent fields of pure bending exactly
Does the tri-linear hexahedron (H8) exhibit shear locking?
Yes
H11 element is what?
The H8 with added incompatible displacement modes. Passes patch test, and may represent bending fairly well
