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1

Finite element method - basis functions

Finite Elements: Basis functions

1-D elements

coordinate transformation

1-D elements

linear basis functions quadratic basis functions cubic basis functions

2-D elements

coordinate transformation triangular elements linear basis functions quadratic basis functions rectangular elements linear basis functions quadratic basis functions Scope : Understand the origin and shape of basis functions used in classical finite element techniques. 2

Finite element method - basis functions

1-D elements: coordinate transformation

We wish to approximate a function u(x) defined in an interval [a,b] by some set of basis functions n i ii cxu 1 where i is the number of grid points (the edges of our elements) defined at locations x i . As the basis functions look the same in all elements (apart from some constant) we make life easier by moving to a local coordinate system iii xxxx 1 so that the element is defined for x=[0,1]. 3

Finite element method - basis functions

1-D elements - linear basis functions

There is not much choice for the shape of a

(straight) 1-D element! Notably the length can vary across the domain.

We require that our function u() be approximated

locally by the linear function 21
)(ccu

Our node points are defined at

1,2 =0,1 and we require that 21211
21211
uucuc ccucu Auc

11-01A

4

Finite element method - basis functions

1-D elements - linear basis functions

As we have expressed the coefficients c

i as a function of the function values at node points 1,2 we can now express the approximate function using the node values )()()1()()(

21121211

NNuuuuuuu

.. and N 1,2 (x) are the linear basis functions for 1-D elements. 5

Finite element method - basis functions

1-D quadratic elements

Now we require that our function u(x) be

approximated locally by the quadratic function 2 321
cccu

Our node points are defined at

1,2,3 =0,1/2,1 and we require that

3213321211

25.05.0

cccucccucu Auc

242143001

A 6

Finite element method - basis functions

1-D quadratic basis functions

... again we can now express our approximated function as a sum over our basis functions weighted by the values at three node points ... note that now we re using three grid points per element ...

Can we approximate a

constant function? 3 12 32
22
12 321
)()2()44()231()( iii

Nuuuucccu

7

Finite element method - basis functions

1-D cubic basis functions

... using similar arguments the cubic basis functions can be derived as 32
432
332
232
13 42
321
)(23)(2)(231)()( N

NNNccccu

... note that here we need derivative information at the boundaries ...

How can we

approximate a constant function? 8

Finite element method - basis functions

2-D elements: coordinate transformation

Let us now discuss the geometry and basis

functions of 2-D elements, again we want to consider the problems in a local coordinate system, first we look at trianglesP 3 P 2 P 1 xy P 3 P 2 P 1 beforeafter 9

Finite element method - basis functions

2-D elements: coordinate transformation

Any triangle with corners P

i (x i ,y i ), i=1,2,3 can be transformed into a rectangular, equilateral triangle with P 3 P 2 P 1 P 1 (0,0) P 3 (0,1)P 2 (1,0)

1312113121

yyyyyyxxxxxx using counterclockwise numbering. Note that if =0, then these equations are equivalent to the 1-

D tranformations. We seek to approximate a

function by the linear form 321
),(cccu we proceed in the same way as in the 1-D case 10

Finite element method - basis functions

2-D elements: coefficients

... and we obtain P 3 P 2 P 1 P 1 (0,0) P 3 (0,1)P 2 (1,0) ... and we obtain the coefficients as a function of the values at the grid nodes by matrix inversion

31321211

)1,0()0,1()0,0( ccuu ccuucuu Auc

101011001

A containing the

1-D case

11-01A

11

Finite element method - basis functions

triangles: linear basis functions from matrix A we can calculate the linear basis functions for triangles P 3 P 2 P 1 P 1 (0,0) P 3 (0,1)P 2 (1,0) ),(),(1),( 321
NNN 12

Finite element method - basis functions

triangles: quadratic elements Any function defined on a triangle can be approximated by the quadratic function 2 652
4321
),(yxyxyxyxu and in the transformed system we obtain 2 652
4321
ccccccu as in the 1-D case we need additional points on the element. P 3 P 2 P 1 P 1 (0,0) P 3 (0,1)P 2 (1,0) P 5 (1/2,1/2)P 4 (1/2,0) P 6 (0,1/2) P 5 P 4 P 6 13

Finite element method - basis functions

triangles: quadratic elements

To determine the coefficients we calculate the

function u at each grid point to obtain

6316654321542146313421211

6/12/14/14/14/12/12/14/12/1

cccu ccccccucccucccucccucu P 3 P 2 P 1 P 1 (0,0) P 3 (0,1)P 2 (1,0) P 5 (1/2,1/2)P 4 (1/2,0) P 6 (0,1/2) P 5 P 4 P 6 ... and by matrix inversion we can calculate the coefficients as a funct ion of the values at P i Aucquotesdbs_dbs19.pdfusesText_25