Higher Order Shape Functions

Question 1

Why use a complete quadratic

Answer 1

gives epsilon_x and epsilon_y to be functions of both x and y
- possible to use higher order + different ones for both rectangle and triangles

Question 2

Lagrangian

Answer 2

• A perfect grid with 9 nodes
• Value of 1 at the specific node and zero at all others is a C0 requirement

Question 3

Serendipity

Answer 3

-Complete quadratic but no internal node
-Shape function is quadratic along the edge of the node under question and goes linearly to zero at the other end for mid-side nodes
-For corner nodes there is bilinear variation but there is zero enforced at the adjacent nodes

Question 4

Element mixing

Answer 4

can be bad if C0 requirements are violated @ the displacement of edges linking quadratic and linear elements

Question 5

Error types

Answer 5

Modelling: from constructing the mathematical model of the physical problem incorrectly

Round-off: from machine precision so very small numbers shouldnt be added to very big ones

Discretisation: a finite number of DoFs will never complete the exact solution but tends to it with more elements and and more flexible response/accuracy
-geometric domain and material characteristics are correct
-continuity of shape functions
-accurate stiffness matrix and nodal F
-essential BC applied

Question 6

Order of errors

Answer 6

Displacement: O(h to the p+1)
Stress/strain: O(h to the p+1-m)

-p = order of complete polynomial
-m = order of diff. relating disp to strains
-used to estimate exact solutions but not @ positions of concentrated loading

Ue - U_h/2 dived by Ue - U_h = (h/2) squared/h squared

Question 7

p-refinement

Answer 7

-higher order elements of the same shape used
-imporved accuracy not guaanteed by swithing from triangular to rectangular since the shape function wont be subsumed

Question 8

h-refinement

Answer 8

each element replaced by >=2 of the same type so that the shape functions are subsumed