Hermite main-interpolation functions for two-dimensional surface (update:24-07-07)

Quadratic shape functions

DESCRIPTION:

Under the bending or twisting effect a thin-plate element's conforming shape functions be derivative basis displacement functions on Pascal Triangle. This solution technique is non-conforming [1] high-order-degree plate element's shape functions and plate's bending and twisting motion.

for example [1];
(9node-27Dof plate element)
Rank 27 (non-conforming)
w=[ 1 ...
x y ...
x^2 x*y y^2 ...
x^3 x^2*y x*y^2 y^3 ...
x^4+x^3*y x^2*y^2 x*y^3+y^4 ... Pascal tree
x^5 x^4*y x^3*y^2 x^2*y^3 x*y^4 y^5 ...
x^5*y+x^4*y^2 x^3*y^3 x^2*y^4+x*y^5 ...
x^5*y^2+x^4*y^3 x^3*y^4+x^2*y^5 ...
x^5*y^3 x^4*y^4 x^3*y^5]

SYNTAX:

[Nshape,Shapefunctions]=Hermiten(totalx,totaly,nodevalue)

Syntax.inputs
totalx : two-dimensional interpolation area's "x" axis length
totaly : two-dimensional interpolation area's "y" axis length
nodevalue: hermite basis functions nodevalues (interpolation node-degree)

Syntax.outputs
Nshape : n. order-degree Hermite basis interpolation functions
Shapefunctions :two-dimensional area plate shape functions for bending
motion

EXAMPLE:
Example.inputs

[A,B,C]=hermiten(1,1,3,1) <---|

A =

1-92*x^2+528*x^3-1088*x^4+768*x^5
x-12*x^2+52*x^3-96*x^4+64*x^5
64*x^2-256*x^3+256*x^4
-16*x^2+128*x^3-320*x^4+256*x^5
28*x^2-272*x^3+832*x^4-768*x^5
-2*x^2+20*x^3-64*x^4+64*x^5

B =
Node(1) ---->Node(n)
w [ (1-23*x^2+66*x^3-68*x^4+24*x^5)*(1-23*y^2+66*y^3-68*y^4+24*y^5), ..
Qxx [ (-46*x+198*x^2-272*x^3+120*x^4)*(1-23*y^2+66*y^3-68*y^4+24*y^5), ..
Qyy [ (1-23*x^2+66*x^3-68*x^4+24*x^5)*(-46*y+198*y^2-272*y^3+120*y^4), ..

C =
Node(1)------->Node(n)
w [ (1-23*x^2+66*x^3-68*x^4+24*x^5)*(1-23*y^2+66*y^3-68*y^4+24*y^5), ..
Qxx [ (-46*x+198*x^2-272*x^3+120*x^4)*(1-23*y^2+66*y^3-68*y^4+24*y^5), ..
Qyy [ (1-23*x^2+66*x^3-68*x^4+24*x^5)*(-46*y+198*y^2-272*y^3+120*y^4), ..
Qxy [ (-46*x+198*x^2-272*x^3+120*x^4)*(-46*y+198*y^2-272*y^3+120*y^4), ..

Example.inputs

[A,B,C]=hermiten(1,1,3,0) <---| or
[A,B,C]=hermiten(1,1,3) <---|
Example.outputs

A =

1-23*x^2/a^2+66*x^3/a^3-68*x^4/a^4+24*x^5/a^5
x-6*x^2/a+13*x^3/a^2-12*x^4/a^3+4*x^5/a^4
16*x^2/a^2-32*x^3/a^3+16*x^4/a^4
-8*x^2/a+32*x^3/a^2-40*x^4/a^3+16*x^5/a^4
7*x^2/a^2-34*x^3/a^3+52*x^4/a^4-24*x^5/a^5
-x^2/a+5*x^3/a^2-8*x^4/a^3+4*x^5/a^4

B =
Node(1)----->Node(n)
[ (1-23*x^2+66*x^3-68*x^4+24*x^5)*(1-23*y^2+66*y^3-68*y^4+24*y^5), ..
[ (-46*x+198*x^2-272*x^3+120*x^4)*(1-23*y^2+66*y^3-68*y^4+24*y^5), ..
[ (1-23*x^2+66*x^3-68*x^4+24*x^5)*(-46*y+198*y^2-272*y^3+120*y^4), ..

C = Node(1)----->Node(n)
[(1-23*x^2+66*x^3-68*x^4+24*x^5)*(1-23*y^2+66*y^3-68*y^4+24*y^5), .. [(-46*x+198*x^2-272*x^3+120*x^4)*(1-23*y^2+66*y^3-68*y^4+24*y^5), ..
[(1-23*x^2+66*x^3-68*x^4+24*x^5)*(-46*y+198*y^2-272*y^3+120*y^4), .. [ (-46*x/a^2+198*x^2/a^3-272*x^3/a^4+120*x^4/a^5)*(-46*y+198*y^2- ..