GENG2140 Modelling and Computer Analysis for Engineers

Transcription

1 GENG4 Moelling n Computer Anlysis or Engineers Letures 9 & : Gussin qurture Crete y Grn Romn Joles, PhD Shool o Mehnil Engineering, UWA

2 GENG4 Content Deinition o Gussin qurture Computtion o weights n points or -point Gussin qurture Chnge o intervl or Gussin qurture Wys o inresing integrtion ury Multiimensionl integrls Rel lie emple o usge Improper integrls Crete y Grn Romn Joles, PhD Shool o Mehnil Engineering, UWA

3 GENG4 Deinition: A qurture rule is n pproimtion o the einite integrl o untion, usully stte s weighte sum o untion vlues t speiie points within the omin o integrtion The numeril integrtion lgorithms presente so r Trpezoil rule, Simpson s rule worke on evenly spe points Trpezoil rule: n i wi i w i weight tors, i evlution points Crete y Grn Romn Joles, PhD Shool o Mehnil Engineering, UWA

4 GENG4 Crl Frierih Guss Germn mthemtiin n sientist notie tht y suitle hoosing oth the weights n the evlution points the ury o the integrl n e improve He propose to hoose the weights n points so tht the proeure shoul e et or polynomils o egree s high s possile I the proeure requires the evlution o the untion in n points, it hs n prmeters to e etermine w i n i, i = n see eq. Crete y Grn Romn Joles, PhD Shool o Mehnil Engineering, UWA 4

5 Beuse generl polynomil o egree N hs N+ oeiients, the Gussin qurture with n points is require to e et or ny polynomil o egree N = n- whih lso hve n prmeters or less The Gussin qurture lgorithms re onventionlly stte or the integrtion omin [-, ], with symmetril weights n points: w k i w i k i w i [ i [ i i ] i ] or o n, k or even n, k n GENG4 n- Crete y Grn Romn Joles, PhD Shool o Mehnil Engineering, UWA 5

6 Computtion o weights n points or -point Gussin qurture n=, the prmeters re w n oring to eq., n the proeure shoul e et or polynomils o egree up to N = n- = Any polynomil o egree up to N = n e written s liner omintion o the ollowing monomils:,,, GENG4 p,,, - onstnts I the proeure is et or ny polynomil o egree up to, it hs to e et or the ove monomils s well Crete y Grn Romn Joles, PhD Shool o Mehnil Engineering, UWA 6

7 By sustituting the 4 monomils in eq. : Reltions 5 n 7 re utomtilly stisie euse o the symmetry o weights n points From eq. 4 n 6: w = n = / 7 GENG4 Crete y Grn Romn Joles, PhD Shool o Mehnil Engineering, UWA 7 ] [ 6 ] [ / 5 ] [ 4 ] [ w w w w

8 The -point Gussin qurture: 8 GENG4 Although very simple, eq. 8 is et or polynomils up to r egree! I the untion tht is integrte is not polynomil o egree or less, eq. 8 will give n pproimtion only the ury epens on how muh the untion resemles polynomil o r egree Crete y Grn Romn Joles, PhD Shool o Mehnil Engineering, UWA 8

9 Emple: Et integrl: GENG4 os sin sin sin Gussin qurture: os os os The sme proeure n e pplie to in the weights n points or higher egree Gussin qurtures n =,4, very teious The weights n points re usully given in tulr orm Crete y Grn Romn Joles, PhD Shool o Mehnil Engineering, UWA 9

10 Chnge o intervl or Gussin qurture I the integrtion intervl is not [-, ], hnge o vrile n e use to moiy it: GENG4 Crete y Grn Romn Joles, PhD Shool o Mehnil Engineering, UWA, 9 n m n m n m z z z m n z m I

11 From eq. 9 - : I z z GENG4 Emple: ; ; m m z n.5z.5 z z / ln.5z ; n 5. ;.5;. 5z; /.5 Crete y Grn Romn Joles, PhD Shool o Mehnil Engineering, UWA et vlue. 5z.5.5 z z Gussin qurture

12 Wys o inresing integrtion ury Inrese the numer o Guss points Composite Gussin qurture ivie the integrtion omin into su-omins the integrl vlue is the sum o Gussin qurtures or eh su-omin A omintion o the ove GENG4 Crete y Grn Romn Joles, PhD Shool o Mehnil Engineering, UWA

13 GENG4 Emple o Mtl implementtion: untion g = guss_hnle,, %GAUSS_hnle,, will integrte the untion _hnle % over the intervl << using point Guss qurture % pproimtion. Emple o use: % + sin; g = guss,,. %====================================================== = [ ;.49594; ; ; ]; w = [ ; ;.98665; ; ]; t =.5*++.5*-*[-; ]; W = [w; w]; g = sumw.*_hnlet*-/; en Crete y Grn Romn Joles, PhD Shool o Mehnil Engineering, UWA

14 GENG4 Multiimensionl integrls The multiimensionl integrl is ompute s repete one-imensionl integrls This is vli or ll numeril integrtion methos, not only or Gussin qurture Numeril integrtion over more thn one imension is sometimes esrie s uture Dierent integrtion methos n e pplie in ierent imensions e.g. points Gussin integrtion in one imension n point Gussin integrtion in nother imension Crete y Grn Romn Joles, PhD Shool o Mehnil Engineering, UWA 4

15 5 GENG4 Crete y Grn Romn Joles, PhD Shool o Mehnil Engineering, UWA Emple: 4-point Gussin qurture in D,,,, t,, y,, y, y, y, t t y y z y z y y I

16 Emple o use The Finite GENG4 Element Metho: The Finite Element metho is numeril metho or solving prtil ierentil equtions or integrl equtions When pplie to soli mehnis, it requires mny vriles to e integrte over the sptil omin eining the system. For emple, the stiness mtri o system is eine s: K V B T EBV Crete y Grn Romn Joles, PhD Shool o Mehnil Engineering, UWA 6

17 GENG4 The integrtion omin is ivie in smller suomins the elements K V B T EBV e K The integrl over eh element is ompute using Gussin qurture usully or points in eh imension e e V e B T EBV e Crete y Grn Romn Joles, PhD Shool o Mehnil Engineering, UWA 7

18 GENG4 Improper integrls Ininite integrtion limits Integrls o untions with vertil symptotes the untion eomes unoune within the integrtion omin Crete y Grn Romn Joles, PhD Shool o Mehnil Engineering, UWA 8

19 Improper integrl on inite omin: Improper integrl on n ininite omin: Osilltory improper integrl: Improper integrls tht o not onverge: Crete y Grn Romn Joles, PhD Shool o Mehnil Engineering, UWA sin Si GENG4 9

20 GENG4 Improper integrls re iiult to integrte. For the se where there is n ininite intervl o integrtion, one my mke hnge o vriles tht trnsorms the ininite rnge o integrtion into inite one. e e - ; t - t - Crete y Grn Romn Joles, PhD Shool o Mehnil Engineering, UWA e - t t e - - t t; - t e t t proper integrl t

21 GENG4 Methos suh s Gussin qurture o not use the vlue o the integrn t en points, n hene integrns tht re uneine t en points n e integrte using suh methos. I I pointsgussin qurture I 4 pointsgussin qurture I I I ln Crete y Grn Romn Joles, PhD Shool o Mehnil Engineering, UWA

22 GENG4 For onvergent improper integrl we n get etter pproimtions y using more n more Guss points this is wsteul In generl, n improper integrl is esy to lulte wy rom its singulrity For emple, or I we get similr vlue using the points Gussin qurture We wnt to use lots o Guss points ner the singulrity ut not so mny elsewhere Crete y Grn Romn Joles, PhD Shool o Mehnil Engineering, UWA

23 GENG4 An ptive metho o numeril integrtion: Compute the ollowing integrls using one o the numeril integrtion methos tht uses only interior points: We epet tht I I +I, n i the eqution hols with resonle ury we ept I +I s the vlue o the integrl I the vlues re not lose enough, we lulte I +I seprtely using the sme metho Crete y Grn Romn Joles, PhD Shool o Mehnil Engineering, UWA I I I

24 Thnk you! Crete y Grn Romn Joles, PhD Shool o Mehnil Engineering, UWA 4