1-D and 2-D Elements. 1-D and 2-D Elements

Transcription

1 merial Methods in Geophysis -D and -D Elements -D and -D Elements -D elements -D elements - oordinate transformation - linear elements linear basis fntions qadrati basis fntions bi basis fntions - oordinate transformation - trianglar elements linear basis fntions qadrati basis fntions - retanglar elements linear basis fntions qadrati basis fntions

2 merial Methods in Geophysis -D and -D Elements -D elements: oordinate tranformation We wish to approximate a fntion x defined in an interval [ab] by some set of basis fntions n x ϕ i i i where i is the nmber of grid points the edges of or elements defined at loations x i. As the basis fntions look the same in all elements apart from some onstant we make life easier by moving to a loal oordinate system x xi x x i i so that the element is defined for [].

3 merial Methods in Geophysis -D and -D Elements -D linear elements There is not mh hoie for the shape of a straight -D element! otably the length an vary aross the domain. We reqire that or fntion be approximated loally by the linear fntion Or node points are defined at and we reqire that A A -

4 merial Methods in Geophysis -D and -D Elements -D elements linear basis fntions As we have expressed the oeffiients i as a fntion of the fntion vales at node points we an now express the approximate fntion sing the node vales.. and x are the linear basis fntions for -D elements.

5 merial Methods in Geophysis -D and -D Elements -D qadrati elements -D qadrati elements ow we reqire that or fntion x be approximated loally by the qadrati fntion Or node points are defined at / and we reqire that.5.5 A A

6 merial Methods in Geophysis -D and -D Elements -D qadrati basis fntions... again we an now express or approximated fntion as a sm over or basis fntions weighted by the vales at three node points i i i... note that now we re sing three grid points per element... Can we approximate a onstant fntion?

7 merial Methods in Geophysis -D and -D Elements -D bi basis fntions -D bi basis fntions... sing similar argments the bi basis fntions an be derived as ζ... note that here we need derivative information at the bondaries... How an we approximate a onstant fntion?

8 merial Methods in Geophysis -D and -D Elements -D elements: oordinate transformation Let s now disss the geometry and basis fntions of -D elements again we want to onsider the problems in a loal oordinate system first we look at triangles y P P P P x P P before after

9 merial Methods in Geophysis -D and -D Elements -D elements: oordinate transformation Any triangle with orners P i x i y i i an be transformed into a retanglar eqilateral triangle with x y x y x y x y x y x y P P P P P P sing onterlokwise nmbering. ote that if then these eqations are eqivalent to the -D tranformations. We seek to approximate a fntion by the linear form we proeed in the same way as in the -D ase

10 merial Methods in Geophysis -D and -D Elements -D elements: oeffiients... and we obtain P P P P P P... and we obtain the oeffiients as a fntion of the vales at the grid nodes by matrix inversion A A ontaining the -D ase A -

11 merial Methods in Geophysis -D and -D Elements triangles: linear basis fntions from matrix A we an allate the linear basis fntions for triangles P P P P P P

12 merial Methods in Geophysis -D and -D Elements triangles: qadrati elements Any fntion defined on a triangle an be approximated by the qadrati fntion x y α α x α y α x α xy α y and in the transformed system we obtain P P P P P / P 5 // as in the -D ase we need additional points on the element. P 6 P 5 P 6 / P P P

13 merial Methods in Geophysis -D and -D Elements triangles: qadrati elements To determine the oeffiients we allate the fntion at eah grid point to obtain 5 6 / / / 6 / / / 6 6 / / 5 / 6 P 6 P P P P 5 P P P P / P 5 // P 6 / P... and by matrix inversion we an allate the oeffiients as a fntion of the vales at P i A

14 merial Methods in Geophysis -D and -D Elements triangles: basis fntions triangles: basis fntions P P P P P P P 5 // P / P 6 / P 5 P P 6 A A... to obtain the basis fntions 5... and they look like...

15 merial Methods in Geophysis -D and -D Elements triangles: qadrati basis fntions The first three qadrati basis fntions... P P P P P / P 5 // P 5 P 6 / P 6 P P P

16 merial Methods in Geophysis -D and -D Elements triangles: qadrati basis fntions.. and the rest... P P P P P / P 5 // P 5 P 6 / P 6 P P P

17 merial Methods in Geophysis -D and -D Elements retangles: transformation Let s onsider retanglar elements and transform them into a loal oordinate system y P P P P P P x P P before after

18 merial Methods in Geophysis -D and -D Elements retangles: linear elements retangles: linear elements With the linear Ansatz we obtain matrix A as A and the basis fntions

19 merial Methods in Geophysis -D and -D Elements retangles: qadrati elements With the qadrati Ansatz we obtain an 8x8 matrix A... and a basis fntion look e.g. like 5 P P 8 P P 7 P 5 P P 6 P

20 merial Methods in Geophysis -D and -D Elements -D and -D elements: smmary The basis fntions for finite element problems an be obtained by:. Transforming the system in to a loal to the element system. Making a linear qadrati bi Ansatz for a fntion defined aross the element.. Using the interpolation ondition whih states that the partilar basis fntions shold be one at the orresponding grid node to obtain the oeffiients as a fntion of the fntion vales at the grid nodes.. Using these oeffiients to derive the n basis fntions for the n node points or onditions.