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(ξ) be
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[PDF] Finite Elements: 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(ξ) be
[PDF] PE281 Finite Element Method Course Notes
finite elements are the subregions of the domain over which each basis function is defined Hence each basis function has compact support over an element Each element has length h The lengths of the elements do NOT need to be the same (but generally we will assume that they are )
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It is with the construction of such basis functions for elements with relatively the solution over n by a function from the nodal finite element space, i e from the
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May 23 2006 · any set of linearly independent functions will work to solve the ODE Now we are ?nally going to talk about what kind of functions we will want to use as basis functions The ?nite element method is a general and systematic technique for constructing basis functions for Galerkin approximations In 5
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Finite element method - basis functions
Finite Elements: Basis functions
1-D elements
coordinate transformation1-D elements
linear basis functions quadratic basis functions cubic basis functions2-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. 2Finite 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]. 3Finite 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 2121121211
uucuc ccucu Auc
11-01A
4Finite 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. 5Finite element method - basis functions
1-D quadratic elements
Now we require that our function u(x) be
approximated locally by the quadratic function 2 321cccu
Our node points are defined at
1,2,3 =0,1/2,1 and we require that3213321211
25.05.0
cccucccucu Auc242143001
A 6Finite 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 3222
12 321
)()2()44()231()( iii
Nuuuucccu
7Finite element method - basis functions
1-D cubic basis functions
... using similar arguments the cubic basis functions can be derived as 32432
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? 8Finite 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 9Finite 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 inversion31321211
)1,0()0,1()0,0( ccuu ccuucuu Auc101011001
A containing the1-D case
11-01A
11Finite 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),( 321NNN 12
Finite element method - basis functions
triangles: quadratic elements Any function defined on a triangle can be approximated by the quadratic function 2 6524321
),(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