Convergence of the FEM. by Hui Zhang Jorida Kushova Ruwen Jung

Transcription

1 Covergece o te FEM by Hi Zg Jorid Ksov Rwe Jg

2 I order to proo FEM soltios to be coverget, mesremet or teir qlity is reqired. A simple pproc i ect soltio is ccessible is to qtiy te error betwee FEMd te ect soltio. At irst orm o ctio, wic mesres te ctio s size, s to be developed. Te orm o vector is commoly sed orm d tke s etry poit: It is deied by wit it yields ϖ i ϖ y z i [] + 3 ϖ i y + i z

3 d Δ i i,, 0 Δ d i i i i ϖ I logy to te orm o vector te orm o ctio c be writte s d is clled te ebesqe orm. ike te orm o vector it will lwys be positive vle. Te similrity betwee te orms c be see i te orm o vector [] is ormlized d te ollowig sbstittios re mde [] Wit tese cges [] trsorms to

4 Usig [] leds to deiitio or te error i FEM soltio were e is te ect d te FEM soltio. Wit [3.] te distce betwee y poit o bot ctios is mesred d dded wic gives te totl error i displcemet. To get more sel vle d te possibility to compre dieret FEM soltios te error [3.] is ormlized, wic becomes [3.] d llows esy iterprettio: vle o 0. mes verge error i te displcemet o 0%. e e d e e e e e d d e [3.]

5 Aoter, more reqetly sed, pproc o qtiyig te error is to evlte te error i eergy. Tis ist doe by d c lso be ormlized wic gives [4.] e e e e d E e ε ε e e e e e e e d E d E e ε ε ε [4.]

6 Covergece by Nmericl Eperimets Sice it is ow possible to mesre te error, simple mericl emple will be cosidered to sow te FEM s covergece. b c A, E l t -cl²/a Te igre bove sows br o legt l wit Yog s Modls E 0 4 Nm d crosssectiol re A m². Te ect soltios or tis problem re 3 c e d c + l AE 6 d ε + l d AE e

7 Te log o te error c ow be clclted d sow s ctio o te log o elemet legt : log log e wic leds to te ollowig grps or lier d qdrtic elemet lier elemet qdrtic elemet

8 As c be see, te log o error vries lierly wit te log o elemet legt d te slope depeds o te order o te elemet. Tis c be epressed wit te eqtio or strigts s y m + b log C + log e [5] wit te slope epressed by. Tkig te power o bot sides o [5] gives e C For lier elemets d or qdrtic elemets 3, wic leds to te coclsio p+ wit p s te order o te elemet. Usig tis eqtio te bove epressio becomes e C p+ Ad te orm o eergy e e C p Now it c be esily see, tt te error decreses wit elemet legt. For te ebesqeorm tis reds s: to lve elemet s size leds to error tt is oly ort o te previos error or lier elemets d eigt or qdrtic elemets.

9 Colclsio: Te FEM is coveget. It s covergece rte icreses wit te order o te elemet d o corse it s size. So does it s compleity. Qdrtic elemets oer good blce betwee ccrcy d compleity d re tereore recommeded. A esy wy to evlte te qlity o soltio, i o ect soltio is preset or te FEM sotwre does ot provide estimte o te error o elemet by elemet bsis, is to reie te mes d compre te ew soltio wit te previos oe. I tere re lrge cges te origil mes ws ideqte d te reied migt be lso rter reiemet is ecessry.

10 Ricrdso Etrpoltio Te FEM s covergece bevior c be sed to obti more ect soltio e i two soltios wit dieret discretistios m, o te sme problem re preset. Tis metod ist clled Ricrdso Etrpoltio. It is ssmed tt bot soltios oly dier i teir mese s step size m, d te irst member o te tylor series epsios is decisive or te totl vles. I Additio te order o covergece p mst be kow. It is sed to deie te order o error. Te better soltio c be clclted by e m + K m [6.] d e + K [6.] Bot eqtios [6.], [6.] combied d K elimited give e e m m Reordered it becomes e m + m wic represets more ccrte soltio.

11 Tks or yor ttetio!